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The proposed solution does **not** answer the exercise that was stated.
A plane intersects a tetrahedron along a triangle and a sphere along a circle.
We are asked to simplify the interpolation formula (45) when the nodes are equally spaced, that is, when $x_k = x_0 + kh, \qquad 0 \le k \le n.$ Formula (45) in Section 4.
Let S=\sum_{\epsilon_1,\ldots,\epsilon_n} (-1)^{\epsilon_1+\cdots+\epsilon_n} \prod_{1\le i\le n} \sum_{1\le j\le n}\epsilon_jx_{ij},
The position of the game is completely determined by the current number of matches in the pile and the set of numbers already written on the sheet.
Newton's interpolation polynomial (42) has the form u_n(x) = a_0 +a_1(x-x_0)
Let f[x_0]=f(x_0), and for $n>0$ define the divided differences recursively by
Let $N=2^n,\qquad \omega=e^{2\pi i/N},$ and consider the discrete Fourier transform \qquad 0\le s<N.
I do not yet have the full textual version of Kvant problem M1209.
Yes.
I need the textual statement of Kvant problem M1208 in order to produce a rigorous solution according to your strict rules.
Let the general discrete Fourier transform (37) be f(s_1,\ldots,s_n) = \sum_{t_1=0}^{m_1-1} \cdots
I see the problem statement itself is not fully provided yet.
Let P(x)=u_nx^{\underline n}+u_{n-1}x^{\underline{n-1}}+\cdots+u_1x^{\underline1}+u_0, where
I see that the problem statement for Kvant M1206 is not yet provided.
I cannot write a solution to Kvant problem M1205 because the actual problem statement is not present in the conversation.
Let $X = (x_{ij})$ be an $n \times n$ matrix.
Consider three circles centered at points $A$, $B$, and $C$, each expanding at the same constant rate.
We wish to compute $\beta_0, \ldots, \beta_r$ so that the polynomial $u(x_0 + kh) = \beta_0 + \beta_1 k + \cdots + \beta_r k^r \eqno(6)$ for all integers $k$.
Let the side of the large square be $1000$ m.
Let u(z) = u_n z^n + u_{n-1} z^{n-1} + \cdots + u_1 z + u_0, \qquad z, u_k \in \mathbb{C}, be a polynomial of degree $n$ with **complex coefficients**, evaluated at a complex number $z = x + iy$.
Let $u(x) = \sum_{k=0}^n u_k x^k$ be a polynomial of degree $n$, and let $x_0$ be the point at which we wish to evaluate $u(x_0)$ and its derivatives.
Let u(x)=u_nx^n+u_{n-1}x^{n-1}+\cdots+u_1x+u_0 .
Let u(x,y)=\sum_{i+j\le n}u_{ij}x^iy^j be a polynomial of total degree $n$ in two variables.
We are asked to design an exponentiation procedure analogous to Algorithm A, but based on radix $m = 2^\nu$, such that the algorithm performs approximately \lg n + \nu + m multiplications, where $\nu$...
Let $u(x) = u_n x^n + u_{n-1} x^{n-1} + \cdots + u_1 x + u_0$ be a polynomial over a ring $R$, and suppose we wish to compute $u(x)$ where $x$ itself is a polynomial, or more generally an element of a...
The statement is false.
Let the two rays from $A$ be $r_1$ and $r_2$.
Write the polynomial in the form u(x)=x\bigl(u_{2n+1}x^{2n}+u_{2n-1}x^{2n-2}+\cdots+u_1\bigr).
**Exercise 4.
Let the numbers of voters for parties $A,B,C$ in a district be $a,b,c$, expressed as fractions of the district electorate.
Consider a small number of arcs on a circle.
Consider the quartic polynomial $P(x) = ax^4 + bx^3 + cx^2 + dx + e$ and the quadratic polynomial $Q(x) = ax^2 + (c-b)x + (e-d)$.
**Exercise 4.
The problem asks for the largest set of 10-digit binary words such that no two words can be obtained from each other by reversing a contiguous subsequence of even sum.
The reviewer's criticism is directed at the proposed response as a _solution_ to Exercise 33.
Working
Let
**Exercise 4.
Let $n = 31$.
Denote the transformation applied to a chosen pair $(a,b)$ by
The answer is **yes**.
For (a), the smallest $z=x\nabla y$ is obtained by binary addition with carries propagated maximally: at each digit position, $z_j=1$ whenever a $1$ can occur there in some sum $x'+y'$ with $x'\le x$...
Let 1 = a_0 < a_1 < \cdots < a_r = n be an addition chain.
Consider the given inequality
We wish to compute the $n$th Fibonacci number $F_n$ modulo $m$, where $n$ and $m$ are large integers.
Let y=(.
Let the rectangle have vertices
Let R_n(B) = \frac{B^n-1}{B-1} = 1 + B + B^2 + \cdots + B^{\,n-1}.
Let n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}, \quad p_1 < p_2 < \cdots < p_k be the prime factorization of the positive integer $n$.
Let
Let $l(n)$ denote the minimum length of an addition chain for $n$, and let $l^F(n)$ denote the length obtained by the factor method.
**Solution to Exercise 4.
The polyhedron has all edges of equal length and every edge is tangent to a sphere.
We follow Hansen's **structural decomposition of star chains** as defined in Section 4.
Let $S_A$ denote the central symmetry with center $A$.
Let the multiplicity of an element $x$ in a multiset $A$ be denoted by $m_A(x)$.
For the first part, reinterpret the table as a bipartite graph.
For $n=1$, a single line divides the plane into two regions.
I cannot write a solution to Kvant problem M1188 from the information provided because the actual problem statement is missing.
Let the required order be $a_1,a_2,\dots,a_{m-1}$.
Solution to Kvant math problem 1186.
I do not yet have the full text of Kvant problem M1185.
I do not have access to the published graphical version of Kvant M1184, and you mentioned the textual version is being prepared.
I cannot write a solution to Kvant problem M1183 because the actual problem statement is not present in your message.
Before I begin, could you provide the exact text of Kvant problem M1182? I need the precise statement to produce a complete, rigorous solution in the requested format.
Consider small instances of the chessboard and examine the parity of black squares occupied by pieces when each row and each column contains exactly one piece.
Consider two spheres intersecting along a circle.
Consider the first recurrence, $a_{n+1} = \frac{n}{n+1}(a_n+1)$ with $a_1 = 0$.
The inequality involves the three classical quantities
Consider the inequality for small values of $n$ to understand its structure.
The expression to be proved resembles the area formula for a quadrilateral written in terms of the lengths of two diagonals of some auxiliary figure.
Consider the problem for small values of $n$ first.
Compute the first few values of the sequence to look for patterns.
Let the three lines through the interior point $P$ meet the sides of triangle $ABC$ in such a way that they cut off three corner triangles of areas $S_1,S_2,S_3$.
Let
Let
A triangulation of a convex $n$-gon contains exactly $n-3$ diagonals.
Consider a rectangle $ABCD$ with sides $AB = CD = a$ and $BC = AD = b$.
Model the country by a graph with $1989$ vertices and $4000$ edges.
Let $p(i)$ denote the position of $i$ in the permutation.
The inequality involves the side lengths $a$, $b$, $c$ of a triangle and three numbers $p$, $q$, $r$ summing to zero.
Consider a square of side length $n$ on a standard graph paper with $1\times1$ cells.
Let $\sigma(n)$ denote the sum of all positive divisors of $n$.
Let the position of the first turtle at time $t$ be $P(t)$ and the position of the second turtle be $Q(t)$.
Consider the Diophantine equation
Consider first the configuration of ten identical billiard balls arranged snugly in a triangular container.
Consider the situation with only two kangaroos first.
Before I can write the full six-section solution you requested, I need the **textual statement of Kvant problem M1159**, since I do not have the content of the problem from memory.
We are asked to minimize $(x+y)(x+z)$ under the constraint $xyz(x+y+z)=1$, with $x$, $y$, $z$ positive.
Let the three triangles be $T_W,T_R,T_G$, and let $M$ be a point lying in the interior of each of them.
There are eight teams, each playing once against every other team, so each team plays $7$ games.
A complete solution cannot be written from the information provided.
I cannot write a solution to Kvant problem M1154 because the actual problem statement is not present in your message.
I can do that.
I do not have the statement of Kvant problem M1152, and the prompt says that only the graphical version is currently available.
I cannot write a solution to Kvant problem M1151 from the information provided, because the actual problem statement is missing.
For $n=3$ the inequality becomes
Consider two rays $p$ and $q$ with vertices $P$ and $Q$, respectively.
In the proof of Lemma J, the intervals $J_1,\ldots,J_k$ are introduced in order to partition a finite set of admissible values into maximal consecutive blocks.
Consider small values of $a$ and $n$ to understand the pattern.
Let $l^{(0)}(n)$ denote the length of an addition chain for $n$ produced by the binary S-and-X method described in Section 4.
The reviewer's objections are correct.
Let 1=a_0<a_1<\cdots<a_{r-1}<a_r=n be an addition chain of length
The condition says that every closed path contains an even number of red edges.
We are asked to construct a _star chain_ of length $A+2$ for each of the four cases in Theorem C, thereby showing that Theorem C remains valid when $l$ is replaced by $l^*$.
Place the equilateral triangle $ABC$ in the plane with convenient coordinates.
No, it is not possible to extend the tree of Fig.
Let 1=a_0<a_1<\cdots<a_r=n be an addition chain.
Consider a circle with a point $P$ outside it and two tangents $PB$ and $PC$, forming an angle $\angle BPC > 90^\circ$.
Let
Each node in the tree represents an exponent $n$, and the tree specifies a parent for each $n$ corresponding to the immediately preceding exponent used to compute $x^n$.
Let $b(n)$ denote the number of multiplications used by the left-to-right binary method to compute $x^n$.
Consider first small circular arrangements of weights with integer masses and total mass divisible into parts.
Let $B(n)$, $F(n)$, and $P(n)$ denote the numbers of multiplications used by the binary method, factor method, and power-tree method, respectively.
We are asked to show that the **decreasing-order power tree** produces a method of computing $x^n$ that requires **exactly the same number of multiplications as the binary method**.
Consider small tables first.
Maintain, for each integer $j$ already present in the tree, two links: LINK0[$j$], pointing to the predecessor of $j$ on the unique path from the root; and LINK1[$j$], pointing to the next node on the...
Consider a trapezoid $ABCD$ with $AB$ and $CD$ as the bases, $AB \parallel CD$, and a circle inscribed within it.
Consider the binary and octal ($m = 8$) methods for evaluating $x^n$.
**Problem.
Let $M(n)$ denote the number of multiplications.
**Corrected Solution for Exercise 4.
**Solution to Exercise 4.
Each intersection point is a crossing of two branches.
Let u(x) = u_n x^n + u_{n-1} x^{n-1} + \cdots + u_0 \in \mathbb{Z}[x], and assume that $u(x)$ is reducible in $\mathbb{Z}[x]$:
For a convex polyhedron whose faces are all squares, every face angle equals $90^\circ$.
Let u(x)=x^n+u_{n-1}x^{n-1}+\cdots+u_1x+u_0, \qquad u_0\ne0, with integer coefficients.
Consider a random monic polynomial $u(x)$ of degree $n$ with coefficients chosen uniformly from ${0,1,\ldots,p-1}$.
Let u(x)=\prod_{i\ge1} u_i(x)^i be the squarefree representation of the given polynomial modulo $p$.
**Exercise 4.
Let u(x)=u_1(x)u_2(x)^2u_3(x)^3\cdots u_e(x)^e, where the $u_i(x)$ are squarefree and pairwise relatively prime, and where $u_e(x)\ne1$.
The expression $n^2+n+3\sqrt n$ is not always an integer.
Let $u(x)$ be a primitive polynomial over the integers, and suppose that the unique factorization of $u(x)$ in a UFD is $u(x) = \prod_{i \ge 1} \prod_{j=1}^{r_i} p_{i,j}(x)^i,$ where $p_{i,j}(x)$ are...
**Exercise 4.
Consider first small polygons.
Let $T(t)=t(x)+t(x)^p+t(x)^{p^2}+\cdots+t(x)^{p^{d-1}}.$ Since $q(x)$ is irreducible of degree $d$, the residue class ring modulo $q(x)$ is the finite field with $p^d$ elements.
Testing small integer values for $A$, $M$, and $S$ helps to gain intuition about the inequality.
Let g(x)=p_1(x)p_2(x)\cdots p_r(x), where the $p_j(x)$ are distinct irreducible polynomials of degree $d$, and let $q=p^d$.
Let
Fix a degree $n \ge 1$.
Let $N_d$ denote the number of monic irreducible polynomials of degree $d$ over $\mathbb{F}_p$.
Consider a right triangle $ABC$ with right angle at $A$ and altitude $AD$.
Error in message stream
Let $u(x)=p_1(x)^{e_1}\cdots p_r(x)^{e_r}$ be the factorization of $u(x)$ modulo $p$, and let $d_j=\deg p_j(x)$ for $1\le j\le r$, with multiplicities included.
Consider the sum
**Solution.
Let
Let u(x)=u_nx^n+\cdots+u_0 =u_n\prod_{r=1}^{n}(x-\alpha_r), and define
Assume that for some $r \ge 1$ we are given polynomials u(x),\ v_s(x),\ w_s(x),\ \alpha(x),\ \beta(x)\in\mathbb Z[x] such that
Consider the case $n=1$ first.
Let $u(x) = u_n x^n + u_{n-1} x^{n-1} + \cdots + u_0$ be a primitive polynomial with integer coefficients, so that $\gcd(u_0, u_1, \ldots, u_n) = 1$, and define the monic polynomial $v(x) = u_n^{-1} \...
Assume that $u(x)$ is reducible over the integers.
Let $p$ be a prime, and let $f(x)$ be an irreducible polynomial modulo $p$ of degree $n$.
Let $F$ be a finite field with $13^2 = 169$ elements.
We are asked to give an explicit factorization of x^8 + 1 modulo an odd prime $p$, using the quantities $\sqrt{-1}, \sqrt{2}, \sqrt{-2}$ whenever these exist in $\mathbb{F}_p$.
Given a prime $p$ and an integer $u$ for which a square root modulo $p$ is known to exist, we seek an efficient algorithm for finding an integer $v$ satisfying v^2 \equiv u \pmod p.
The exercise asks for the complete factorization of the polynomial $u(x)$ in equation (22) by Berlekamp's method.
Consider first a simple convex polygon, such as a triangle or a square.
Since $2$ is a primitive root modulo $101$, every nonzero residue modulo $101$ is of the form 2^k \pmod{101}, \qquad 0\le k\le 99.
I cannot write a solution to Kvant problem M1129 because the actual problem statement is not present in the conversation.
Let A_{n,p}=\sum_{m=1}^n \frac{a_{m,p}}{p^m}, where $a_{m,p}$ denotes the number of monic irreducible polynomials of degree $m$ over $\mathbf F_p$.
We are asked to prove the congruence x^p - x \equiv (x-0)(x-1)\cdots(x-(p-1)) \pmod{p}, \eqno(9) where $p$ is a prime and arithmetic is in the field $\mathbb{F}_p$ of $p$ elements.
Algorithm N triangularizes the matrix $Q-I$ by means of elementary column operations.
Consider first the case of a $2 \times 2$ chessboard with two pieces.
**Solution.
Let U(x)=u_1(x)u_2(x)\cdots u_r(x), where the polynomials $u_1,\ldots,u_r$ are pairwise relatively prime over the field $S$.
Let $p$ be a prime number, and let $a_{n,p}$ denote the number of monic irreducible polynomials of degree $n$ over the finite field $\mathbb{F}_p$.
(a) Let $R$ be a unique factorization domain (UFD), and let $u(x) \in R[x]$ be monic.
A fully corrected solution cannot be written from the information provided.
The proposed solution does not answer the exercise that was asked.
Let u_1(x),u_2(x),u_3(x),\ldots be the subresultant polynomial remainder sequence defined by equation (16) of Algorithm C.
I notice that the problem statement itself is not yet provided.
Let $u(x),v(x)\in\mathbf{R}[x]$ be polynomials whose coefficients are represented in floating-point arithmetic.
Let L(A,B)=\{\,XA+YB : X,Y\in M_n(\mathbb Z)\,\}.
Let the input polynomials be u_0(x),\qquad u_1(x), with
Let $f_0(x),f_1(x),\ldots,f_m(x)$ be the Sturm sequence associated with a squarefree polynomial $f_0(x)$, defined by $f_{i-1}(x)=q_i(x)f_i(x)-f_{i+1}(x)\qquad(1\le i<m),$ where $f_m(x)$ is a nonzero c...
The exercise asks for an investigation rather than a theorem with a single conclusion.
**Solution.
Let $S$ be a unique factorization domain, and let $u(x),v(x)\in S[x]$.
The statement resembles a converse of a familiar fact about equal angles subtending the same segment.
Let N(S_1,\ldots,S_n) = |S_1|\cdots |S_n| -
Let $\mathcal F=\mathbb Q\langle A\rangle$ denote the free associative algebra generated by the alphabet $A$ over the rationals.
Let u(x)=u_mx^m+\cdots,\qquad v(x)=ax^n+\cdots, where $a=l(v)$, and let $m=\deg(u)$, $n=\deg(v)$.
Let $u(x)$ and $v(x)$ be polynomials over a field $S$, with $\deg(u) = m$ and $\deg(v) = n$, and let $u_1(x), u_2(x), \ldots$ be the sequence of polynomials obtained during a run of Algorithm C (the E...
We are asked to prove Hadamard's inequality, equation (25) in Section 4.
Let $S$ be a unique factorization domain, and let $u(z),v(z)\in S[z]$ be primitive polynomials.
Table 1 in Section 4.
I cannot write a solution to Kvant problem M1125 from the information provided, because the actual problem statement is missing.
Let $S$ be a field, and let $u(x),v(x)\in S[x]$.
**Exercise 4.
Consider a trapezoid $ABCD$ with bases $AB$ and $CD$, where $AB$ is the shorter base.
Let $S$ be a unique factorization domain.
Let $S$ be a unique factorization domain, and let $f(x)\in S[x]$ be a unit.
Let $f(x)$ be a polynomial with integer coefficients.
Let $P_n=\Pr\bigl(\gcd(u(x),v(x))=1\bigr),$ where $u(x)$ and $v(x)$ are independently and uniformly distributed monic polynomials of degree $n$ over the field $\mathbf F_p$.
Let f(x) = 3x^6 + x^5 + 4x^4 + 3x^3 + 4x + 3 and let its reverse be
Let u(x)=x^6+x^5-x^4+2x^3+3x^2-x+2, and
Label the cells by coordinates $(i,j)$, where $i,j\in\mathbb N$ and $i,j\ge 1$.
Let $d_0=m,\qquad d_1=n,\qquad d_2=n_1,\qquad \ldots,\qquad d_{t+1}=n_t,\qquad d_{t+2}=-\infty,$ be the sequence of degrees occurring in Euclid's algorithm modulo $p$.
Let f(X) = a_0 + a_1 X + a_2 X^2 + a_3 X^3 and suppose we are given integers $m_1, \dots, m_7 > 10^{72}$ that are pairwise coprime, with
Let $T(n)$ denote the number of elementary operations needed to multiply two polynomials of degree $\le n$ modulo $2$.
Yes, polynomial multiplication modulo $2$ can be facilitated by packing coefficients into machine words, but **ordinary integer multiplication cannot be used directly**, because its carries do not cor...
(a) Let $u(x)$ and $v(x)$ be monic polynomials, with leading coefficients $\ell(u) = \ell(v) = 1$.
Let c_1 \equiv x_1^2 \pmod N,\qquad c_2 \equiv x_2^2 \pmod N, where the hexadecimal values of $c_1,c_2$, and $N$ are given in the statement.
By equation (4), w_k=u_0v_k+u_1v_{k-1}+\cdots+u_kv_0, where coefficients with indices exceeding $s$ are taken to be $0$.
Let
In polynomial arithmetic modulo $10$, coefficients are reduced modulo $10$ after each operation.
Let $p$ be a prime, let $a$ be a primitive root modulo $p$, and let b \equiv a^n \pmod p, \qquad 0\le n\le p-2.
Given odd $n$ with $\gcd(ab,n)=1$, we seek an efficient algorithm that finds integers $x,y$ satisfying x^2-ay^2\equiv b \pmod n, without knowing the factorization of $n$.
**Problem restated.
Let m=pq, where $p\equiv q\equiv3\pmod4$, and let $Q_m$ denote the set of quadratic residues modulo $m$ whose unique square roots also lie in $Q_m$.
The machine performs addition, subtraction, multiplication, integer division, and equality tests on arbitrarily large integers in unit time.
Let $N$ be a positive integer.
Let D=\{\,d:d\mid N,\ d\equiv r \pmod s\,\}, where
The statement of the problem is incomplete:
The exercise is inherently computational.
Let N = pq, \qquad p \equiv 3 \pmod 8, \qquad q \equiv 7 \pmod 8 be the product of two distinct odd primes.
The reviewer's criticism is misplaced in one essential respect: it evaluates the submission as though the missing information were available.
Let \sqrt D=[R;\overline{a_1,a_2,\ldots,a_m}], \qquad R=\lfloor \sqrt D\rfloor , where $D$ is not a perfect square.
Consider the sequence defined by $a_0 = 0$ and $a_n = P(a_{n-1})$ for $n \ge 1$, where $P(x)$ is a polynomial with integer coefficients and $P(x) > 0$ for $x \ge 0$.
Consider first small values of $k$.
**Solution to Exercise 4.
Let N=pq, where $p$ and $q$ are the RSA primes.
In textbook RSA, a message $M$ is encoded as C \equiv M^3 \pmod N, where $N=pq$ and the public exponent is $3$.
Let $P_k$ denote the probability that the first $k$ parity vectors produced by Dixon's algorithm are linearly independent in $GF(2)^m$, under the usual heuristic assumption that the vectors are indepe...
Let $N = pq$ be a product of two distinct primes $p$ and $q$ such that $p \equiv q \equiv 2 \pmod{3}.$ Suppose we are given an integer $y \equiv x^3 \pmod{N}$ and assume the existence of a "reasonably...
Expanding the left-hand side gives
**Exercise 4.
**Solution.
Let S=\{(e_1,\ldots,e_m): e_1+\cdots+e_m\le r/2\}.
**Solution.
Let $v_p(n)$ denote the exponent of $p$ in $n$.
**Solution.
The previous argument fails because it interprets $m(p)$ as the number of repeated divisions by $p$ after the algorithm has already reached the trial divisor $p$.
Let the sides of the given triangle $ABC$ be
For odd integers $q>1$, the Jacobi symbol is defined by \left(\frac{p}{q}\right)\in\{-1,0,1\}, with
**Exercise 4.
Let $n\ge 3$ be an odd integer.
Let D=\prod_{q\le B} q^{e_q}, where $q^{e_q}$ is the largest power of the prime $q$ not exceeding the prescribed bound (later $B=10^5$).
Let $N$ be a positive integer with prime factorization N = p_1 p_2 \cdots p_t, \qquad p_1 \le p_2 \le \cdots \le p_t.
Consider a rectangle drawn on a square grid where the unit squares are the cells.
A _Mersenne number_ is a number of the form $M_p = 2^p - 1,$ where $p$ is a positive integer.
We prove simultaneously, by induction on the height of the tree, that for every node $(q,x)$: 1.
Let $T$ be the integer formed in step E3 of Algorithm E.
Let N=p_1p_2\cdots p_d have $d$ distinct prime factors.
Let $P$ and $Q$ be integers with $\gcd(P, Q) = 1$, and define the Lucas sequence $(U_n)$ by $U_0 = 0, \quad U_1 = 1, \quad U_{n+1} = P U_n - Q U_{n-1} \text{ for } n \ge 1.$ Let $N$ be a positive inte...
Assume that for every prime divisor $p$ of $n-1$ there exists an integer $x_p$ such that x_p^{(n-1)/p}\equiv 1 \pmod n, \qquad x_p^{\,n-1}\not\equiv 1 \pmod n.
Algorithm E is the continued-fraction factoring method applied to $\sqrt{kN}$.
**Problem:** In the notation of Exercise 3.
Consider the first problem.
Let $n$ be an odd integer with $n \ge 3$, and let $\lambda(n)$ be the Carmichael function of $n$, as defined in Theorem 3.
We wish to transform the classical sieve of Eratosthenes into a form suitable for efficient computation, avoiding multiplications.
Let $p$ be an odd prime, and let $N$ be an integer such that $p \nmid N$.
Algorithm D uses, for each modulus $m_i$, a table that indicates whether a residue class modulo $m_i$ can occur as a quadratic residue.
Let N=11111.
Let P=\prod_{p\le 1000} p, where the product extends over all prime numbers not exceeding $1000$.
Consider a tetrahedron with vertices $A$, $B$, $C$, $D$ and let $a = AB$ and $b = CD$ be two skew edges.
Model the situation as a graph on $21$ vertices, the cities.
No.
Let X = \sum_{n\ge 0} \frac{1}{2^{2^n}} = \frac{1}{2} + \frac{1}{4} + \frac{1}{16} + \frac{1}{256} + \cdots.
If $d_k$ is not prime, then $d_k$ has a prime factor $p < d_k$.
Let $T(u,v)$ denote the number of iterations performed by Harris's binary Euclidean algorithm after the common factor $2^{\min(\nu_2(u),\nu_2(v))}$ has been removed.
Let $X$ be a real number with regular continued fraction expansion X = //A_1, A_2, \ldots// and let $q_n = K_n(A_1, \ldots, A_n)$ denote the $n$-th continuant (denominator of the $n$-th convergent) as...
**Statement:** Let L(n) = \max_{m \ge 0} T(m,n), where $T(m,n)$ denotes the number of division steps performed by Algorithm 4.
Let $h$ be the number of hits and $n$ the number of times at bat.
Let us consider the infinite binary tree described in the exercise, commonly known as the _Stern–Brocot tree_.
Starting with the numbers $1$ and $2$ on the board, the rule allows us to produce $ab + a + b$ whenever $a$ and $b$ are present.
Let $a_1, \ldots, a_n$ be positive integers, and let $K_s(x_1, \ldots, x_n)$ denote the continuant of $n$ variables as defined in equation (4) of Section 4.
**Problem.
Algorithm B is the binary gcd algorithm.
We are asked: > What is the smallest value of $u_n$ such that the calculation of $\gcd(u_1, \dots, u_n)$ by Algorithm 4.
Consider triangle $ABC$ with acute angles and its circumcircle $\Gamma$.
Consider the first few natural numbers and compute the greatest common divisors of all distinct pairs.
Let the vertices of an inscribed equilateral triangle be
Let S(n)=\sum_{m=1}^{n-1}\sum_{j\ge1}A_j(m/n), where
Consider small cases first.
Let h(n)=\#\{(x,x',y,y'):\ n=xx'+yy',\ x>y>0,\ x'>y'>0,\ x\perp y\}, as in Exercise 33.
The inequality is homogeneous in the ratios of the sides.
**Exercise 4.
Consider a convex hexagon $ABCDEF$.
Associate the positions $1,2,\ldots,n$ with the variables $x_1,x_2,\ldots,x_n$.
The problem concerns unfolding a convex polyhedron along straight-line cuts so that its surface lies flat as a single polygon, with specified identifications of points on the boundary.
Let the modified algorithm of Exercise 30 be written in the form $u=qv+r,\qquad 0\le r<v,$ and replace the pair $(u,v)$ by $\bigl(v,\min(r,v-r)\bigr).$ Thus each division step uses the remainder of le...
Let
Let u=qv+r,\qquad 0\le r<v .
Begin with the first part of the problem, which concerns tiling an infinite plane with $1\times 2$ dominoes after some non-overlapping dominoes are already placed.
A rigorous proof cannot be supplied from the information given.
For $n=3$ it is natural to search among classical identities involving sums of three cubes.
Let $\angle A=\alpha$.
Let $1(n)$ denote the constant function $1$, and let $\delta_{n1}$ be $1$ when $n=1$ and $0$ otherwise.
Let S(M,N)=\sum_{m\le M}\sum_{n\le N}T(m,n).
**Exercise 4.
Consider a finite set of logs lying on a straight riverbank, each forming an angle less than $45^\circ$ with the bank.
We are asked to find a set $\mathcal{I} = I_1 \cup I_2 \cup I_3 \cup \cdots \subseteq [0,1]$, with the $I_k$ disjoint intervals, such that equation (45) does not hold.
Let $A_n$ denote the $n$th partial quotient in the regular continued fraction expansion of a random real number $x\in(0,1)$, x=[0;A_1,A_2,\ldots].
**Exercise 4.
Consider small examples to gain intuition.
Consider the game for small values of $n$.
We are asked to **develop efficient means to calculate accurate approximations** to the quantities $\lambda_1$ and $\Psi_2(x)$ defined in equation (44) of TAOCP §4.
Consider small examples of isosceles triangles whose vertices have integer coordinates.
Let the circle have radius $R=\dfrac d2$.
Let $\varphi_c$ be the comparison function obtained by taking T_g(x)=\frac1{x+c}, \qquad c>0.
The problem involves constructing a chord $MN$ of a circle with center $O$ seen from $A$ under a given angle $\alpha$, with additional geometric constraints.
The two inequalities are
We are asked to deduce equation (38) from equation (37) in _TAOCP_ Volume 2.
Equation (24) is F(x)=\sum_{k\ge1}\left(F\!
Represent the configuration by numbers $a_1,\dots,a_n\in{0,1,2}$ arranged cyclically.
Consider a single fold of a convex polygon and a subsequent straight cut.
Testing small values helps build intuition about the inequality.
Let the inradius of triangle $AOB$ be $r_1$, of $BOC$ be $r_2$, of $COD$ be $r_3$, and of $DOA$ be $r_4$.
The condition is
Let $h_a,h_b,h_c$ be the altitudes of triangle $ABC$.
Consider the problem of reaching a target number from $0$ using only two operations: doubling the current number or adding $1$.
Consider the problem geometrically.
We adopt Knuth's notation: //a_1,a_2,\dots,a_n// = \cfrac1{a_1 + \cfrac1{a_2 + \cdots + \cfrac1{a_n}}}.
Let the two given circles be $\omega_1$ and $\omega_2$, intersecting at $A$ and $B$.
Consider small values of $n$ to understand the inequality.
The given equality resembles the identity for the sum of squares of the sides of a quadrilateral.
Compute a few values:
We define recursively f_0(z) = \tanh z = \frac{e^z - e^{-z}}{e^z + e^{-z}}, \qquad f_{n+1}(z) = \frac{1}{f_n(z)} - \frac{2n+1}{z}.
Let
For $n=3$ the problem asks for a single triangle whose three side lengths are irrational and whose area is a nonzero rational number.
Assume that a function $f:\mathbb N_0\to\mathbb N_0$ satisfies
Let $X(\sigma)$ denote the number of fixed points of a permutation $\sigma$ of an $n$ element set.
We are asked to design an online algorithm that computes the continued fraction y = \frac{ax+b}{cx+d} given the continued fraction
Let $\angle A=2\alpha$.
Consider the problem in terms of digit patterns.
Let $m=2n+1$.
Consider a hexagon $A_1A_2A_3A_4A_5A_6$ with a point $O$ from which all sides are seen under an angle of $60^\circ$.
Let X=//a_0,a_1,a_2,\ldots// denote a regular continued fraction.
The expression $989 \cdot 1001 \cdot 1007 + 320$ appears to involve three numbers spaced by six units: $989$, $1001$, $1007$.
Consider smaller versions of the game to understand the parity dynamics.
Let the tetrahedron have vertices $A,B,C,D$.
Let f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \ne 0 be a polynomial with exactly one real root $\xi > 1$, where $\xi$ is irrational and $f'(\xi) \ne 0$.
Consider a small number of families, say three or four, each in a distinct apartment.
Consider an angle $AOB$ with points $A$ on one side and $B$ on the other.
Let
**Exercise 4.
Consider six points in the plane with all pairwise distances at most $1$.
We are asked to study vectors $(x;y)$ with non-negative integer coordinates and to decide when they can be written as sums of _generating vectors_, i.
Let the closed broken line have vertices $V_1,\dots,V_n$ and segments $e_i=V_iV_{i+1}$, where indices are taken modulo $n$.
Let X=A_0+//\!
Let the digits of the $n$-digit number $a$ be
The first part of the problem deals with a triangle $ABC$ with points $D$ on $AC$ and $E$ on $AB$, forming the intersecting lines $BD$ and $CE$ at $M$.
Let X=A_0+//\!
Interpret the cities and roads as a graph.
Consider two closed polygonal chains in the plane, each with an odd number of sides.
Let
For (a), write T=//x_{k+1},\ldots,x_n//.
Consider a finite subset of $\mathbb{Z}^2$ as a candidate for the marked points and examine what happens when we translate each by all vectors from the given finite set.
Let X = [A_0;A_1,A_2,\ldots] be the regular continued fraction generated by the process of §4.
A move consists of writing a number that is not a divisor of any previously written number.
Consider small cases first.
By Euler's description of continuants, $K_n(x_1,\ldots,x_n)$ is the sum of all monomials obtained by deleting disjoint adjacent pairs $x_jx_{j+1}$.
Consider a circle with a small number of points to understand the behavior of arcs subtending at most $120^\circ$.
Consider four spheres in three-dimensional space.
Let X=//B_1,B_2,\ldots//.
The first few Fibonacci numbers with at least four digits are
Let q_n=K_n(x_1,\ldots,x_n).
Consider a convex $n$-gon with vertices labeled cyclically as $A_1, A_2, \dots, A_n$.
Let D_n= K_n(x_1,\ldots,x_n)\,K_n(x_2,\ldots,x_{n+1}) - K_{n+1}(x_1,\ldots,x_{n+1})\,K_{n-1}(x_2,\ldots,x_n).
Consider a $3\times3$ cluster of pieces on an $8\times8$ chessboard.
Let the chosen points be
Consider a cylinder $\text{Ц}_1$ with radius $R_1$ and height $H_1$, and define its diameter-to-height ratio $k = \frac{2R_1}{H_1}$.
Consider the simplest nontrivial cases of the knight’s tour game.
Consider a small round-robin tournament with $n$ players.
Consider an acute-angled triangle $ABC$ with $\angle A = 60^\circ$.
Let D_n= \det \begin{pmatrix} x_1 & 1 & 0 & \cdots & 0 \\
Consider the geometry of the kingdom, which is a square of side $2$ km.
Let us consider the product $M_n = \begin{pmatrix}1 & 1 \\ 1 & 0\end{pmatrix} \begin{pmatrix}x_1 & 1 \\ 1 & 0\end{pmatrix} \begin{pmatrix}x_2 & 1 \\ 1 & 0\end{pmatrix} \cdots \begin{pmatrix}x_n & 1 \\...
Let
Let $T(u,v)$ denote the number of MIX division instructions executed by Program 4.
Let the three subsets be colored by the colors $A,B,C$.
We are asked to evaluate the determinant D_n = \begin{vmatrix} \gcd(1,1) & \gcd(1,2) & \cdots & \gcd(1,n) \\ \gcd(2,1) & \gcd(2,2) & \cdots & \gcd(2,n) \\ \vdots & & & \vdots \\
Let the class contain $n$ students.
Let G(m,n)=\gcd(10^m-1,\;10^n-1), where $m,n\ge 0$.
A regular pentagon is determined up to congruence by any three consecutive vertices.
Let M=\max(|u|,|v|), where $u$ is initially odd.
For $n=1$ the three groups are ${1},{2},{3}$, and $3=1+2$.
Label the tetrahedron vertices as $A$, $B$, $C$, $D$.
The rectangle contains $mn$ cells.
Algorithm B is the binary gcd algorithm.
Consider the equation $x^y - y^x = x + y$ with $x, y \in \mathbb{N}$.
Let B(u,v) denote the number of subtraction steps performed by Algorithm B.
Consider a pentagon and imagine cutting it into two smaller pentagons of equal area and shape.
No.
**Exercise 4.
The exercise asks for a rigorous justification of Brent's probabilistic model.
**Exercise 4.
**Corrected Solution for Exercise 4.
After the first subtract-and-shift cycle of Algorithm B, the ratio $x = \min(u, v)/\max(u, v)$ is transformed according to the rule $x \mapsto x/(1 + 2^k x)$, where $k$ is the number of trailing zeros...
We are asked to prove that, for a function $G(x)$ satisfying (36)–(40), 2G(x) - 5G(2x) + 2G(4x) = G(1+2x) - 2G(1+4x) + 2G(1 + 1/x) - G(1 + 1/(2x)).
**Exercise 4.
Equation (58) expresses $\psi_n$ as a constant multiple of the Bernoulli number $B_{2n}$.
The proposed solution is mathematically correct as far as it goes, but it does not answer the exercise that Knuth intended.
Let $\psi_n$ be defined by the logarithmic generating function that precedes (58): \log\frac{x}{e^x-1}=\sum_{n\ge1}\psi_n x^n .
Let F_n(x)=\Pr\!
We are asked to analyze the asymptotic behavior of the average number of subtraction steps $C_{mn}$ and shift steps $D_{mn}$ in **Algorithm B**, when $u$ and $v$ are odd integers with \lfloor \lg u \r...
**Exercise 4.
**Exercise 4.
Let $M=2^{n'-1}.$ The odd integers in the interval $2^{n'}\le u,v<2^{n'+1}$ are $u=2^{n'}+(2a+1),\qquad v=2^{n'}+(2b+1),$ with $0\le a,b<M$.
For part (a), consider the system 3x + 7y + 11z = 1, \quad 5x - 7y - 3z = 3.
We are asked to determine the terminal values of $v_1$ and $v_2$ in **Algorithm X** from Section 4.
Algorithm L already performs several Euclidean divisions at once by using the leading digits of $u$ and $v$.
Since $v \perp m$, the extended Euclidean algorithm yields integers $a$ and $b$ such that $av+bm=1.$ Reducing modulo $m$ gives $av\equiv 1 \pmod m,$ so $a$ is the multiplicative inverse of $v$ modulo...
Let G=\gcd(u,v).
Let P_d=\Pr(\gcd(u,v)=d).
If $u$ and $v$ are random positive integers, let $d$ be a positive integer.
Let $q_n$ denote the number of ordered pairs $(u,v)$ with $1 \le u,v \le n$ such that $\gcd(u,v) = 1$.
Let the notation be that of Knuth's analysis of Program B.
By Theorem 4.
We are asked to compute $\gcd(31408, 2718)$ using **Algorithm B**, and then to find integers $m$ and $n$ such that $31408 \, m + 2718 \, n = \gcd(31408, 2718)$ using **Algorithm X**.
Let $u$ and $v$ be random positive integers.
In Program B of §4.
Exercise 4.
Let $u$ and $v$ be positive integers.
**Exercise 4.
We derive equations (8) through (12) from the prime-factor definitions (6) and (7).
Let n=\prod_{p}p^{a_p} be the canonical factorization of $n$, as in (5), where only finitely many exponents $a_p$ are nonzero.
Let u=\prod_{p} p^{u_p}, \qquad v_i=\prod_{p} p^{v_{i,p}} \quad (1\le i\le n)
The fractions $(1/0)$ and $(-1/0)$ arise naturally in the fraction representation scheme, because ordinary fractions are represented by pairs $(u,v)$ with $v>0$.
Consider marking points on $[0,1]$ sequentially.
Suppose $1 \le u' < 2^k$ and $1 \le v' < 2^k$.
Let \frac{u}{u'},\qquad \frac{v}{v'} be the input fractions, with
Consider small chocolate bars first.
Let u = (u_7 u_6 u_5 u_4 u_3 u_2 u_1 u_0)_{10} = \sum_{j=0}^{7} u_j 10^j, \qquad 0 \le u_j \le 9 be an 8-digit decimal number, and let
Let \frac{u}{u'},\qquad \frac{v}{v'} be fractions in canonical form, with
**Corrected Solution for Exercise 4.
Let $d = \gcd(u, v)$.
Let A=\gcd(u,v'), \qquad B=\gcd(u',v).
Let \mathcal D(p)=\{\pm (0.
A satisfactory method is based on continued reduction rather than cross multiplication, because the products $uv'$ and $vu'$ may overflow even when all four quantities fit in a single word.
Let the square have vertices $A,B,C,D$ in cyclic order.
Let $C(n)$ denote the time required to convert an $n$-digit decimal integer to binary notation, or conversely.
Let $b$ and $B$ be two integer bases, and let $u$ be a $p$-digit floating point number in base $b$, with representation $u = (d_0.d_1 d_2 \ldots d_{p-1})_b \times b^e,$ where $0 \le d_i < b$ for $0 \l...
Let $(d_0 d_1 \dots d_{n-1})_{10}$ be an $n$-digit decimal integer.
Section 4.
Let x=(.
We are asked to find, for each nonnegative integer $k$, the smallest nonnegative integer $u$ such that \left\lfloor \frac{v_k(u)}{16} \right\rfloor \ne \left\lfloor \frac{u}{10} \right\rfloor, where
Begin with small values of $n$ to detect a pattern.
We are asked to devise a **rapid pencil-and-paper method** for converting integers from ternary $(0,1,2)_3$ to decimal $(0,1,\dots,9)_{10}$ and to illustrate the method on the number $(12120112120210)...
Reflecting on the problem, the point $M$ is chosen on the line $\ell$ to minimize the sum $MA + MB$.
**5.
We are asked to convert $(5772)_{10}$ to decimal.
Let a binary-coded decimal (BCD) number be represented as u = \cdots u_3\,u_2\,u_1\,u_0, where each 4-bit group $u_3 u_2 u_1 u_0$ encodes a decimal digit $d$ with $0 \le d \le 9$.
We are asked to write a MIX program analogous to program (1) that computes the decimal digits of a nonnegative integer $u$ using **formula (5)** and contains **no division instructions**.
We are given a real number $r$ and integers $u$ and $w$ such that 0 < \alpha \le r \le \alpha + \frac{1}{w}, \quad 0 \le u \le w.
Methods 1a, 1b, 2a, and 2b are defined for conversion between positional systems of radix $b$ and radix $B$.
Let $x$ be a real number with a terminating binary representation.
Consider simple convex polyhedra such as nested cubes, tetrahedra, or pyramids.
Let $u$ and $v$ be $N$-digit integers stored in $N$ consecutive memory locations.
We are asked to compute uv \bmod m using a **bounded number of operations** allowed in Exercise 3.
We generalize Method 2a by introducing a stopping criterion based on the desired precision $\epsilon$.
Let the arithmetic progression be
Let u=a_m b_{m-1}\cdots b_1 b_0+\cdots+a_1 b_0+a_0, where $0\le a_j<b_j$, and suppose we wish to express the same quantity in the mixed-radix system
Method 1a extends to mixed radices by repeated division with changing divisors.
We are asked to solve the Karatsuba recurrence K_1 = 1, \quad K_{2n} = 3 K_n, \quad K_{2n+1} = 2 K_{n+1} + K_n \quad (n \ge 1), and to find an explicit formula for $K_n$ when
**Exercise 4.
Begin by considering the configuration of two intersecting lines and points $D$ and $E$ on them.
Algorithm T is Knuth's Toom-Cook multiplication algorithm.
We are asked to determine the fastest possible online multiplication algorithms on various species of automata.
The machine of the exercise has only node creation, pointer manipulation, equality tests, input, and output.
**Exercise 4.
Let $T(n)$ denote the time needed to multiply two $n$-bit numbers by one of the fast multiplication methods discussed in this section.
Let the linear iterative array be the family of automata M_0,M_1,M_2,\ldots defined by equations (37) and (38).
Let $k$ be the length of the discrete Fourier transform and let $\omega = e^{2\pi i / k}$ be a primitive $k$th root of unity.
Let $n$ be the bit length of the inputs to the Schönhage–Strassen multiplication algorithm, and let $\tilde{u}_s$, $\tilde{v}_s$, $\tilde{w}_s$ denote the discrete Fourier transforms (DFTs) used in th...
The number $1987$ is prime, since it is not divisible by any prime not exceeding $\sqrt{1987}<45$.
The assertion that $u_{j+n} = 0$ at the beginning of step D3 of Algorithm D is false.
Let $x$ be an $n$-digit decimal number.
We are asked to prove that the six numbers in equation (24) of Section 4.
Let the common measure of each arc be $x$.
We are asked to show that if in step T1 of Algorithm T we replace the original initialization $R \leftarrow \lfloor \sqrt{Q} \rfloor$ by $R \leftarrow \lceil \sqrt{2Q} \rceil + 1,$ then the bound on t...
We are asked to compute 1234 \cdot 2341 using the decimal analogue of the method in (2) (Karatsuba-type divide-and-conquer multiplication).
For $k>0$, the desired inequality is 2^{q_k+1}(2r_k)^{r_k}\le 2^{q_{k-1}+q_k}.
Consider a convex quadrilateral $ABCD$ with extensions of opposite sides $AB$ and $CD$, and $AD$ and $BC$, intersecting at points $P$ and $Q$ respectively.
Consider two triangles with angles $\alpha, \beta, \gamma$ and $\alpha_1, \beta_1, \gamma_1$.
We are asked to compute the representation of $w = (uv) \bmod (2^q - 1)$ for $q$-bit integers $u$ and $v$ that are expressed in a nonuniform radix representation $u = \sum_{k=0}^{n-1} u_k 2^{\lfloor k...
Let the moduli $m_1,\ldots,m_r$ be pairwise relatively prime, and let x=v_1+m_1\bigl(v_2+m_2(\cdots+m_{r-1}v_r)\cdots\bigr) \tag{25} be the mixed-radix representation.
Suppose $0 \le u, v < m$, and let $w = u + v$ computed in a modular representation with moduli $m_1, \ldots, m_r$.
We claim that 2^e \equiv 2^f \pmod{2^g - 1} \quad \Longleftrightarrow \quad e \equiv f \pmod{g}.
Let u \equiv u_j \pmod{m_j}, \qquad 0\le u_j<m_j \qquad (1\le j\le r), where the moduli $m_j$ are odd and $u$ is known to be even.
Suppose an integer $u$ satisfies $-m/2 < u < m/2$, where $m = m_1 m_2 \cdots m_r$, and consider a modular representation of $u$ using symmetric residues $u_1, \ldots, u_r$ such that $-m_j/2 < u_j < m_...
Let $b$ be the radix, $w = (w_{n-1} \ldots w_1 w_0)_b$ an $n$-place integer with $\gcd(w,b) = 1$, and $w'$ a one-place integer satisfying $u_0 w' \equiv 1 \pmod b.$ We are asked to: a) Given $u = \pm(...
For small numbers of triangles the statement is false.
Let $(u_1,\ldots,u_r)$ be a modular representation with pairwise relatively prime moduli $m_1,\ldots,m_r$, and let $u$ be reconstructed by the procedure defined in equations (24) and (25).
Let M_i=m_1m_2\cdots m_i , and recall that in Knuth's derivation of (24) the constants $c_{ij}$ ($i<j$) satisfy
Let M=m_1m_2\cdots m_r, where the $m_j$ are odd positive integers $<100$ and are pairwise relatively prime.
Let $u$ be a $2n$-digit number and $v$ an $n$-digit number in radix $b$, such that $u \bmod v = 0$.
Let m=\operatorname{lcm}(m_1,m_2,\ldots,m_r).
Equation (13) is obtained by choosing successively the largest odd integer below the preceding modulus that is relatively prime to every modulus already chosen.
For numbers $1,2,\dots,2n$, suppose they are arranged in two rows and $n$ columns.
Consider how the mountaineer’s progress depends on the day’s starting point.
No.
Let S=\sum_{k\ge0}\frac1{16^k}\left(\frac4{8k+1}-\frac2{8k+4}-\frac1{8k+5}-\frac1{8k+6}\right).
Find all integers $u$ such that u \bmod 7 = 1,\qquad u \bmod 11 = 0,\qquad u \bmod 13 = 5,
Consider a sphere of radius $1$ with a curve drawn on it, either open of length less than $\pi$ or closed of length less than $2\pi$.
Consider first a small example on a $3 \times 3$ portion of the grid.
Consider a regular $n$-gon $A_1 A_2 \dots A_n$ with center $O$.
Consider assigning integers to the vertices of a regular pentagon and performing the prescribed operation whenever a vertex carries a negative number.
For a polygon circumscribed about a circle of radius $r$, let the sides be $s_1,\dots,s_n$, with corresponding side lengths $\ell_1,\dots,\ell_n$.
The polynomial is
Consider small examples of pairwise coprime numbers, such as $a_1=2$, $a_2=3$, $a_3=5$.
Consider triangle $ABC$ with points $M$ on $AB$ and $N$ on $BC$.
Consider arrangements of circles in the plane where each circle touches several others.
For the first inequality,
Consider the sequence defined by $r_1=2$ and $r_{n+1}=r_1 r_2 \cdots r_n + 1$.
Let the parallelogram be represented by vectors.
Number the steps from $1$ at the bottom to $2n+1$ at the top.
The equality
Let the triangle be $ABC$.
Consider a $3 \times 3$ table first.
Let a line through $A$ be fixed.
The quantities involve segments cut off by the feet of the altitudes.
The door opens as soon as some block of three consecutive pressed digits coincides with the code.
Let $S(n)$ denote the total sum of all recorded products when a pile of $n$ stones is repeatedly split until all piles contain one stone.
Let $O$ be the center of the circle containing the arc $AB$, and let $\angle AOB=2\alpha$.
Let
Consider a tetrahedron $AXBY$ circumscribed about a sphere with fixed points $A$ and $B$.
Let
The octagon is the intersection of two congruent squares.
Let
Let
Let $x$ be the smallest of $n$ consecutive natural numbers.
Consider small examples of social networks where each person has at least 10 friends.
Consider triangle $ABC$ with an altitude $CH$ and median $CK$.
Consider three lines in space, each pair of which is skew, and they are not all parallel to the same plane.
I begin by examining small natural numbers $a$ to see which of them satisfy the given conditions.
Consider small values of $n$ and $k$ to build intuition.
Consider small instances to gain intuition.
The inequality is
We are asked to count configurations of three lines through a point in space with prescribed pairwise angles, up to congruence.
Consider a square $ABCD$ and an arbitrary point $K$ inside it.
The tournament is a complete directed graph on $16$ vertices.
Construct triangle $ABC$ on paper and build the external squares $ABB_1A_2$, $BCB_1C_2$, $CAA_1C_2$.
Consider small repunit numbers of the form $R_n = 11\ldots1$ with $n$ ones.
Consider first a convex polygon in the plane with vertices $A_1, A_2, \dots, A_n$ and a point $O$ inside it.
Consider the definition of an exceptional set of $k$ numbers $a_1, a_2, \dots, a_k$, all strictly between 0 and 1.
The threshold $\sqrt{2/3}$ is suggestive because an equilateral triangle of side $a$ has altitude $\frac{\sqrt3}{2}a$, and when $a=\sqrt{2/3}$ the altitude equals $\frac1{\sqrt2}$.
The problem asks whether $x$ can be expressed using only addition, subtraction, and multiplication from given polynomials.
Place the square in coordinates:
Consider first a simplified scenario: a small $n\times n$ board, say $n=5$, with just a few hypothetical pieces each attacking a limited number of squares.
Suppose both players start with equal time and make alternating moves.
Let
The sequence $(x_n)$ begins with $x_1 = \frac12$ and satisfies the recurrence $x_{n+1} = x_n^2 + x_n$.
Consider a tournament of $8$ volleyball teams where each team plays every other team exactly once.
Unusual activity has been detected from your device.
For small values,
The statement asks for a dissection of an arbitrary triangle into four pieces such that the pieces can be rearranged into two triangles, each similar to the original triangle.
The sequence $(a_n)$ consists of distinct positive integers with the growth constraint $a_n < 100n$.
Let the side length of the square be $6$.
Consider first small examples.
Let
Consider first small numbers of participants.
Consider first the case of a rectangle inscribed in a triangle.
**Solution.
Write
The $25$ plots form the $5\times5$ grid graph.
Consider first small cases.
Let N=b^n, and let
**Solution (corrected)** Let $u$ and $v$ be integers with $0 \le u, v < 2^n$.
The statement is false.
If $d$ is a power of 2 on a binary computer, then multiplication or division by $d$ can be accomplished by simple bit shifts.
Position two parabolas in the plane with perpendicular axes.
Let \phi = 2^m\frac{1+x}{1-x}, where $m$ is an integer chosen so that $|x|<1$.
The essential requirement is that integers of arbitrary length be stored in linked memory, and that result digits be created directly in newly allocated nodes obtained from a free-storage list.
Let a decuple-precision floating-point number be represented by two consecutive machine words (X_1,X_0).
Let u=(u_{m+n-1}\cdots u_1 u_0)_3, \qquad v=(v_{n-1}\cdots v_1 v_0)_3, where the digits are balanced ternary,
For $n=1$, the partition is ${1}$ and ${2}$, hence
Something went wrong.
Let $\beta=2i$.
In Algorithms A and S, the computation of each output digit $w_j$ depends on the corresponding input digits $u_j$ and $v_j$ as well as on the carry or borrow from the previous step.
Consider first the case $k=2$, which corresponds to a regular decagon.
Step D8 in Algorithm D is the **add-back correction**.
We are asked to prove that at the beginning of step D8 in Algorithm D, the unnormalized remainder $(.u_{n-1} \ldots u_1 u_0)_b$ is always an exact multiple of the divisor $d$.
Step D1 of Algorithm D normalizes the divisor and dividend before quotient selection begins.
Let u=(u_nu_{n-1}\ldots u_0)_b,\qquad v=(v_{n-1}v_{n-2}\ldots v_0)_b, with
**Solution (Exercise 4.
We are asked to find a four-digit number $u$ divided by a three-digit number $v$ in base $b = 10$ for which step D6 of Algorithm D is necessary.
We adopt the notation of Exercises 19 and 20.
Let the exact quotient digit be $q$ in the classical long-division algorithm of Section 4.
For the planar statement, the natural idea is to look at one fixed side of the square, say the left side.
We are asked to multiply an $n$-place nonnegative integer U = (u_{n-1} u_{n-2} \dots u_1 u_0)_b by a single-digit integer $v$, $0 \le v < b$, producing an $(n+1)$-place result
Show that if $\hat{q}' = \left\lfloor \frac{u_n b + u_{n-1}}{v_{n-1} + 1} \right\rfloor,$ then $\hat{q}' \le q$, where $q$ is the quotient digit in the classical division algorithm.
Let U=(.
Let U=(.
Let U=(u_{m-1}\ldots u_1u_0)_b,\qquad V=(v_{n-1}\ldots v_1v_0)_b, and let
The problem has two parts.
We analyze the effect of interchanging instructions in Program S, which implements subtraction of nonnegative integers according to Algorithm S.
Something went wrong.
Let $u = (u_{n-1} \ldots u_1 u_0)_b$ and $v = (v_{n-1} \ldots v_1 v_0)_b$ be nonnegative $n$-place integers.
Let the angular speed be $\dfrac{360^\circ}{n}$ per second.
We are asked to generalize Algorithm A (addition of nonnegative $n$-place integers in a fixed radix $b$) to a _mixed-radix_ number system, where the digits have bases $b_0, b_1, \ldots, b_{n-1}$ from...
Consider the simplest nontrivial case $n=1$.
Let $u = (u_{n-1} \ldots u_0)_b$ and $v = (v_{n-1} \ldots v_0)_b$ be two independent, uniformly distributed $n$-place nonnegative integers in base $b$.
We are asked to **add two $n$-digit numbers in base $b$ from left to right**, writing each output digit only **once it is guaranteed that it cannot be affected by any carry from lower-order digits**.
Interpret the $2n$ points as vertices of a graph $G$ with $2n$ vertices and $n^2+1$ edges.
Exercise 2 generalizes Algorithm A to the addition of $m$ nonnegative $n$-place numbers (u^{(1)}_{n-1}\cdots u^{(1)}_0)_b,\; (u^{(2)}_{n-1}\cdots u^{(2)}_0)_b,\; \ldots,\; (u^{(m)}_{n-1}\cdots u^{(m)}...
**Exercise 4.
Let the clans be represented by labels.
The exercise is historical rather than theorem proving.
Let the $m$ given nonnegative $n$-place integers be (u^{(1)}_{n-1}\ldots u^{(1)}_0)_b,\; (u^{(2)}_{n-1}\ldots u^{(2)}_0)_b,\; \ldots,\; (u^{(m)}_{n-1}\ldots u^{(m)}_0)_b,
Let \phi=\frac{1+\sqrt5}{2},\qquad \psi=\frac{1-\sqrt5}{2}, so that Binet's formula gives
Consider triangle $ABC$ with an incircle touching sides $AB$, $BC$, and $CA$ at points $C_1$, $A_1$, and $B_1$ respectively.
Let A_k=\{n\ge 1:(\log_{10} n)\bmod 1<r\}, where $0\le r\le 1$.
Let ${P_1(n)}_{n \ge 1}$ be a sequence taking values 0 or 1.
Let $U$ and $V$ be independently distributed, normalized, positive floating point numbers in base $b = 10$, with exponents distributed according to probabilities $p_0, p_1, p_2, \ldots$, as in exercis...
Consider the equation $a^4 + b^4 + c^4 + d^4 = e^4$ modulo small primes to understand divisibility constraints.
Let $U$ and $V$ be independent, normalized, positive floating point numbers with fraction parts distributed according to the density functions $f(x)$ and $g(y)$, defined on the interval $[1/b, 1)$.
The exercise as quoted cannot be solved rigorously from the information provided.
Let $V=1/U$.
Let $f$ be the normalized fraction part of a positive radix-16 floating point number.
Consider small values of $N$ to understand the dynamics of the seat-shifting process.
Something went wrong.
Let $U$ be a random variable uniformly distributed on the interval $[0,1)$.
Let the averaging operator of Eq.
I do not have the statement of Kvant problem M926.
Consider a small blue region, for example, a disk of radius $r<1$.
Write the normalized decimal floating point numbers in Knuth's form u=f_u10^{e_u},\qquad v=f_v10^{e_v}, where
We are asked to show that no single distribution function $F(u)$ exists that satisfies equation (5) for **all integers $b \ge 2$** simultaneously and for all $r$ in the interval $1 \le r \le b$.
Consider a unit cube in three-dimensional space with edges parallel to the axes.
An antilogarithm table is indexed by values of $\log_{10} x$.
Let a normalized positive floating decimal number be written as $10^v \cdot f$, where $v$ is an integer exponent and $f \in [1,10)$ is the fraction part.
Program **A** implements a floating-point accumulator in MIX, with **8 bytes to the right of the radix point**.
Let $u$ and $v$ be normalized floating point numbers, and let the double-precision operations $\oplus$, $\ominus$, and $\otimes$ denote addition, subtraction, and multiplication carried out in the ext...
**Exercise 4.
The point of Kulisch's example is not merely that a subtraction of nearly equal quantities occurs.
Let the single-precision format of Section 4.
The problem involves a convex quadrilateral $ABCD$ with two given angles, $\angle A = \alpha$ and $\angle B = \beta$, and a special relation between its sides and area: the doubled area satisfies $2S…
We are asked to determine all pairs $(a,b)$ such that $\text{round}(b, \text{even}(x)) = \lfloor ax + b \rfloor + \lfloor ax - b \rfloor \quad \text{for all } x. \eqno(32.1)$ We interpret round$(b, \t...
Let u=(u_m+eu_l)b^{e_u}, \qquad v=(v_m+ev_l)b^{e_v}, where $u_m$ and $v_m$ are the most-significant halves of the fractions, $u_l$ and $v_l$ are the least-significant halves, and
Let the floating-point numbers in the interval $[x_0,x_1]$ be \xi_0<\xi_1<\cdots<\xi_m.
Inserting the instruction `ENTX 0` between lines 30 and 31 of Program **B** would not improve the accuracy of the computation.
**Statement:** Prove that 1 \ominus (1 \ominus (1 \ominus u)) = 1 \ominus u for all nonzero floating point numbers $u$, where $\ominus$ denotes floating point subtraction as defined in Section 4.
Let $f(x) = 1 + x + x^2 + \cdots + x^{106} = \frac{1 - x^{107}}{1 - x}, \quad x < 1,$ and define $g(y) = f\bigl((\tfrac{1}{3} - y^2)(3 + 3.45y^2)\bigr), \quad 0 < y < 1.$ We are asked to evaluate $g(y...
We want to show that the requirement $b^p \ge 3$ in Exercise 28 is necessary by giving an example in which repeated application of $h(x) = \tilde{g}(\tilde{f}(x))$ exhibits drift when $b^p = 2$.
For the first integral equality, the two integrals involve complementary functions: the tangent function on $[0,\pi/4]$ and the arctangent function on $[0,1]$.
We are asked to compute an exact decomposition of the product of two floating-point numbers $u$ and $v$ in the form uv = w + w', using only the floating-point operations $\oplus$, $\ominus$, and $\oti...
We are asked: > Can drift occur in floating point multiplication/division?
Let $u$, $u'$, $v$, and $v'$ be positive normalized floating point numbers, with $u \sim u' \ (\text{relative error } r), \qquad v \sim v' \ (\text{relative error } s).$ By definition of relative erro...
Let $\mathcal{F}$ denote the set of all normalized floating point numbers, together with the special symbols $+0$, $-0$, $+\infty$, and $-\infty$.
When people speak of "cancellation" in floating point subtraction, they refer not to an actual inaccuracy in the operation $u \ominus v$ itself, which is computed exactly according to equation (3), bu...
Let t_k=s_k-c_k .
Consider six-digit numbers from $000000$ to $999999$.
Let $u$ be a floating point number.
Let the floating point number in location ACC be u=s_u\,m_u\,b^{e_u}, and let the floating point number in register A be
Equation (17) asserts that for every real number $x$ in the range $b^{e-1}\le |x|<b^e,$ the rounded value satisfies $\operatorname{round}(x)=x(1+\delta(x)),$ with $\delta(x)$ defined by
Let P_1=x_1=1.
Let $b$ be the floating-point radix and $p$ the precision of the significand.
In unnormalized arithmetic, floating point addition and multiplication are still defined by but numbers need not be normalized.
Error in message stream
The inequality is cyclic rather than symmetric:
We are asked whether the computed midpoint of an interval always lies between the endpoints; that is, whether $(u \le v) \implies u \lesssim (u \oplus v) \oslash 2 \lesssim v,$ where $\oplus$ and $\os...
Assume, without loss of generality, that $e_u \ge e_v+p$.
Let $\eta=\frac12\,b^{\,1-p},$ so that every unnormalized floating point multiplication or division satisfies \qquad |\delta|\le \eta,
Let the floating point number $x$ have least significant digit in position $e$.
Let the approximation relation be defined as in §4.
We are asked to determine which of the relations $u < v, \qquad u \sim v, \qquad u > v, \qquad u \approx v$ hold for the given pairs of eight-digit, one's-complement floating point numbers, assuming $...
We seek floating binary numbers $u$ and $v$ for which (u\oplus v)^2>2(u^2\oplus v^2), where $\oplus$ denotes floating-point addition with rounding to the nearest floating binary number.
Consider triangle $ABC$ with circumcircle $\Gamma$.
We are asked whether the identity u \oslash v = u \otimes (1 \oslash v) holds for **all** floating-point numbers $u$ and $v \ne 0$, assuming no exponent overflow or underflow occurs.
By equation (6) of the floating point arithmetic defined in this section, 0 \oplus u = u for every floating point number $u$.
Yes.
We analyze the actual MIX **FADD** subroutine in **Program A**, which adds two floating-point numbers $u=(e_u,f_u)$ and $v=(e_v,f_v)$ in the MIX computer with **word size $w=5$ bytes** and fraction le...
Place quadrilateral $ABCD$ in the plane and select points $E$ on $AB$ and $F$ on $CD$.
We are asked to find eight-digit floating point numbers $u$, $v$, $w$ such that $u \oplus (v \oplus w) \ne (u \oplus v) \oplus w,$ where $\oplus$ denotes floating point addition in the sense of Sectio...
Since $y \ge 0$, property (8) implies that v \oplus 0 \le v \oplus y.
Let the regular hexagon be $P_1P_2P_3P_4P_5P_6$, and let the points of the problem be chosen on its sides so that $A_i\in P_iP_{i+1}$, indices modulo $6$.
We are asked to prove identity (7): $u \ominus v = -(v \ominus u)$ using only identities (2) through (6): \begin{aligned} &(2) && u \oplus v = v \oplus u, \\
A MIX floating-point number has the form x = s\,(0.
We are asked to define a rounding rule for a 36-bit binary machine using two's complement representation for negative floating point numbers, in which the combination of normalization and rounding can...
Let $A=\widehat A$, $B=\widehat B$, $C=\widehat C$.
We are asked to construct positive eight-digit floating decimal numbers $a,b,c,d,y$ with exponent range $-50 \le e < 50$ such that \frac{(a \otimes y) \oplus b}{(c \otimes y) \oplus d} \approx \frac{2...
A floating point number is represented by a single word containing a fraction and exponent.
We wish to compute the real and imaginary parts of $\frac{a + b i}{c + d i}$ without forming $c^2 + d^2$, which may overflow even if $|c|$ or $|d|$ is near the largest representable floating point num...
We are to design a MIX subroutine that computes u \pmod 1 = u-\lfloor u\rfloor, returning the result rounded to the nearest normalized floating point number.
Let u=(e_u,f_u), \qquad v=(e_v,f_v) be normalized floating point numbers in radix $b$, with $p$-digit fractions.
Balanced ternary uses radix $b=3$ and digits \bar 1=-1,\qquad 0,\qquad 1.
Let $\beta=10$, let the precision be $p$, and let the floating point operations be rounded to the nearest representable number, as in Algorithm N.
Consider the equation
We are asked to give an example of normalized, excess-50, eight-digit floating decimal numbers $u$ and $v$ such that multiplication produces a rounding overflow.
Let the floating point numbers have base $10$ and $p=8$ digits.
**Exercise 4.
Let the floating decimal numbers have $p=8$ digits in the fraction, and let normalized numbers satisfy $\frac1{10}\le |f|<1,$ unless the number is zero.
Let the quantity actually stored for exponent $e$ be denoted by $F_e$.
The question asks for the sign of register A when a floating point addition produces the value zero.
We apply Algorithm A with base $b=10$ and precision $p=8$.
In normalized binary floating-point arithmetic, every nonzero number is represented in the form x=\pm(0.
I cannot write a solution because the actual problem statement is missing.
A base $b$, excess $q$, $p$-digit floating point number has the form \pm f\,b^{\,e-q}, where the exponent satisfies
A plane section of a convex polyhedron changes combinatorially only when the plane passes through a vertex.
Let $D$ be any set of integers, let $b$ be any positive integer, and let $k_n$ be the number of distinct integers representable as $n$-digit numbers $(a_{n-1}\ldots a_0)_b$ with digits $a_i\in D$.
Let A=\left\{\sum_{i=0}^{m} a_i3^i \;:\; a_i\in\{0,1\},\ m\ge0\right\} be the set of nonnegative integers whose ternary representation uses only the digits $0$ and $1$.
Using the values given in §3, h \approx 6.
Consider triangle $ABC$ with bisectors $AK$ and $BM$ intersecting at $O$.
We are asked to represent an integer $n$ in a **balanced binary form**, that is, as $(\ldots a_2 a_1 a_0)_2 = \sum_{i=0}^{\infty} a_i 2^i, \quad a_i \in \{-1, 0, 1\},$ using the **fewest nonzero digit...
Consider the given odd natural numbers $a<b<c<d$ satisfying $ad=bc$, $a+d=2^k$, and $b+c=2^m$.
Let R(e_0,\ldots,e_r) =(1+i)^{e_0}+i(1+i)^{e_1}-(1+i)^{e_2}-i(1+i)^{e_3} +\cdots+i^r(1+i)^{e_r}, \qquad e_0<\cdots<e_r.
Let u=(\ldots u_3u_2u_1u_0.
**Exercise 4.
**Exercise 4.
The condition that the circle with diameter $AB$ is tangent to the line $CD$ has a simple metric interpretation.
Let S_n:=\sum_{k\le n}\epsilon_k\beta_k .
**Exercise 4.
Let \frac{u}{v}=a_0+\sum_{j\ge1}a_jb^{-j}, \qquad 0\le a_j\le b-1, be the standard radix-$b$ representation of $u/v$.
The complete graph on $n$ vertices is $K_n$.
Let D=\{d_0,d_1,\ldots,d_9\} be a set of ten nonnegative integers satisfying 1.
**Exercise 4.
We are asked to write a MIX subroutine, callable by JMP RAND1 with entry condition
Let the linear congruential generator be X_{n+1} \equiv a X_n + c \pmod m, \qquad m = 2^e, with full period $m$.
Let $Y = cX$, where $X$ is a continuous random variable with distribution function F(x) = \Pr\{X \le x\}.
Let T(x) = \int_x^\infty e^{-t^2/2}\, dt.
Write
The problem concerns connecting 51 cities in a square-shaped country of side 1000 km with 11,000 km of highways.
Consider a circle $\Gamma_1$ with tangents $CA$ and $CB$ meeting at $C$, so $A$ and $B$ are points of tangency.
Label the cells on the boundary of the $n\times n$ square cyclically by
Working
The sequence begins as $1, 0, 1, 0, 1, 0$ and each subsequent term is defined as the last digit of the sum of the preceding six terms.
Consider a pyramid with apex $A$ and base $B_1B_2\dots B_n$.
Consider a smaller version of the problem first.
We begin by testing small integer values to see whether the equation $(5+3\sqrt{2})^m = (3+5\sqrt{2})^n$ admits any obvious solutions.
Let $O_1,O_2,O_3$ be the centers of the circles $C_1,C_2,C_3$.
Let the occupied rooms be represented by the multiset of integer positions of all pianists.
Let the boys' heights be $b_1,\dots,b_{17}$ and the girls' heights be $g_1,\dots,g_{17}$.
Consider first a right triangle.
Consider a small board, $n=3$.
Consider small values of $n$ to understand the behavior of the sums modulo $1$.
Consider the triangle $ABC$ with circumcircle $(O)$ and incircle $(Z)$.
Let
Let
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Before I begin, could you provide the full text of Kvant problem M854? I need the problem statement in order to construct the rigorous solution according to your specifications.
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The function $f_0(x) = ||x-1|-2||x|-3||$ involves nested absolute values.
Let the regular polygon have $n$ sides and circumradius $R$.
We consider the problem of forming a centrally symmetric polygon using two types of tiles: a “corner” formed by four $1\times1$ squares arranged in an L-shape, and $4\times1$ rectangles.
For the first representation, the coefficients are constrained by $0\le a_k\le k$.
Consider the first identity $\sin\alpha + \sin\beta + \sin\gamma$ under the constraint $\alpha + \beta + \gamma = 0$.
Consider a right triangle $ABC$ with right angle at $C$.
For a set of integers with no three-term arithmetic progression, the classical example is obtained by writing numbers in base $3$ and allowing only digits $0$ and $1$.
Consider the simpler case where two of the numbers are coprime.
Represent the previous encounters by a bipartite graph.
Compute the first few terms to detect patterns.
Consider first the case of dividing a square into smaller squares.
Consider small values of $m$ to detect a pattern.
The figure consists of a triangle subdivided into smaller regions, four of which are blue triangles of equal area.
The operation described on the board replaces one number with the sum of the other two numbers minus one.
I need the diagram from Figure 3 to solve the problem rigorously.
Before I begin the full solution, I need clarification: does the photograph provide any perspective cues such as vanishing points, the relative positions of the corners, or angles of the visible wall?
A parallelogram has two pairs of parallel sides.
Consider small $n$-gons to understand what a uniform set of blue vertices looks like.
Let $s(n)$ denote the sum of the decimal digits of $n$.
A natural number is representable as a sum of two squares if and only if every prime congruent to $3 \pmod 4$ appears with even exponent in its prime factorization.
Consider three segments $OA$, $OB$, and $OC$ of equal length $R$, with $B$ inside the angle $AOC$.
The quantities involving medians and altitudes suggest rewriting everything in terms of the sides.
Compute the first few terms of the series to detect a pattern.
Begin by examining small values of $k$.
Consider the planar case first.
Place a right circular cylinder vertically with axis along the $z$-axis and center at the origin, so that $O=(0,0,0)$ is the midpoint of the axis.
The data involve two right triangles erected externally on sides $AB$ and $BC$.
Compute several small cases to see the pattern.
Consider first small values of $k$.
Consider a square $ABCD$ with a point $P$ inside it.
Consider small values of $n$ and their divisor sums $\sigma(n)$.
Let
Petya's calculator allows addition, subtraction, increment by one, and reciprocal operations.
Consider first a small number of points on a circle, for example, four points dividing the circle into four equal arcs.
Consider small values of $n$ and $k$ to identify a pattern.
The length of daylight at latitude $\varphi$ depends on the declination $\delta$ of the star.
Let $b > 1$ be an integer and $D$ a set of $b$ real numbers containing $0$, such that every positive real number $x$ has a representation $x = \sum_{k \le n} a_k b^k, \quad a_k \in D.$ Define the set...
Consider the first system of inequalities: $1 < x < 2$, $2 < x^2 < 3$, $3 < x^3 < 4$, and so on up to $n < x^n < n+1$.
Let $x$ be an arbitrary real number and $\epsilon > 0$ be given.
Let B=\left\{-\frac92,-\frac72,-\frac52,-\frac32,-\frac12, \frac12,\frac32,\frac52,\frac72,\frac92\right\}.
Consider triangle $ABC$ and a line parallel to $AC$ intersecting $AB$ at $D$ and $BC$ at $E$ such that $|AD| = |BE|$.
Let D=\{-1,0,1,0.
We are asked to study sequences of positive numbers $x_0, x_1, x_2, \dots$ with $x_0 = 1$ and $x_0 \ge x_1 \ge x_2 \ge \dots$.
Assume that every integer in the interval l\le m\le u, \qquad l=-\frac{\max D}{b-1}, \qquad
Let \beta=i-1.
**Exercise 4.
**Exercise 4.
**Exercise 4.
Let the triangle be isosceles with $A_2A_3=a_1$ as the base and $A_1A_2=A_1A_3$.
x=(0.
**Exercise 4.
**Exercise 4.
Let s_i=a_i+b_i+t_i, where $t_i$ is the carry entering position $i$.
Each octal digit corresponds to three binary digits, and each hexadecimal digit corresponds to four binary digits.
The given equation is cubic in two variables, $x$ and $y$:
Represent each real number by a decimal expansion extending infinitely in both directions, x=\cdots d_2d_1d_0.
Equation (5) asserts that a distribution function $F(x)$ is monotonically nondecreasing, with $F(x_1) \le F(x_2) \quad \text{if } x_1 \le x_2; \qquad F(-\infty) = 0, \quad F(+\infty) = 1.$ By definiti...
In signed magnitude notation, one bit is reserved for the sign, leaving $p-1$ bits for magnitude.
In MIX fixed-point arithmetic, the radix point is understood to be fixed relative to the register positions.
Consider a regular hexagon $ABCDEF$ with vertices labeled consecutively.
A negative integer $-N$ has a nines' complement representation obtained by replacing each decimal digit $d$ of $N$ with $9 - d$.
We are asked to represent the numbers -49, \quad -3\frac12, \quad \pi in four number systems: (a) binary (signed magnitude),
We are asked to express the integers -10, -9, \ldots, 9, 10 in the number system whose radix is $-2$.
The phenomenon occurs because the sequence $\langle X_n \rangle$ defined by $X_n = (X_{n-37} + X_{n-100}) \bmod 2$ is a linear recurrence modulo 2, which generates a purely periodic sequence of period...
The generator must advance through _all_ $1009$ values in each block, returning the first $100$ and discarding the remaining $909$.
Consider small values of $n$ to detect patterns.
**Exercise 3.
We consider the random-number generator `run_array` after initialization by `run_start(s)`.
We are asked to convert the C routines `run_array` and `run_start` to FORTRAN 77 subroutines that generate exactly the same sequences of numbers.
A lagged Fibonacci generator of the type recommended in Section 3.
Let the modulus be $MM=2^{52},$ and replace all arithmetic modulo $2^{30}$ by arithmetic modulo $2^{52}$.
Y_n=\left\lfloor \frac{X_n}{2^{10}}\right\rfloor,\qquad X_n=2^{10}Y_n+Z_n,\qquad 0\le Z_n<2^{10}.
A good test is to choose parameters $a$, $c$, $m$, and a seed $X_0$, then compute a short sequence of values independently from the defining recurrence X_{n+1}=(aX_n+c)\bmod m.
Let the original table be an infinite sequence of digits x_1,x_2,x_3,\ldots that is valid in the usual sense: Every block of $k$ digits occurs with limiting frequency $10^{-k}$.
This exercise is not a mathematical problem but a practical programming assignment.
Condition (2) resembles a midpoint convexity-type inequality, but in the reversed direction: usually convexity gives $f\left(\frac{x+y}{2}\right)\le \frac{f(x)+f(y)}{2}$, whereas here we have $f\left(…
Let us formalize the problem in terms of computational probability and sequence generation.
We are asked to simulate dice rolls, shuffle a deck of cards, play a common game to a winning outcome, and display results in a way consistent with a particular random number generator.
We simulate a game of craps as described, using the standard linear congruential generator from Section 3.
Lady Lovelace's statement concerns the relation between a machine and the instructions that govern its behavior.
Let $A$ be an algorithm that, given a cyclic binary sequence B_1,B_2,\ldots,B_N, attempts to predict each bit from the preceding bits.
Let Theorem P be the result asserting that a prediction algorithm for the output of the Blum-Blum-Shub generator can be converted into a factoring algorithm for a **random** $R$-bit Blum integer $M$.
Let $X_1, \ldots, X_n$ be random variables with mean $\mu = \mathrm{E},X_j$ and variance $\sigma^2 = \mathrm{E},X_j^2 - (\mathrm{E},X_j)^2$, for $1 \le j \le n$.
Complete the proof of Lemma P1.
Let z_n(u)=\#\{\,j\le n:0\le U_j<u\,\}, where $U_0,U_1,\ldots\in[0,1)$.
Let $A$ be a set of algorithms in the sense of Definition R6, and let $|A|=m$.
Let s_n=\frac{n(n+1)}2 .
Let $(X_n)$ be an R5-random binary sequence, and let $(a_n)$ be any computable increasing sequence of integers as in Definition R4.
We are asked to construct a sequence that satisfies **Definition R4** but fails **Definition R5**.
**Exercise 3.
Let $k$ be a positive integer.
Let the side lengths be
Let \nu_n=\#\{\,0\le j<n : U_j<\tfrac12\,\}.
**Solution.
**Exercise 3.
**Exercise 3.
Let A=G=\{(x,y)\in[0,1)^2:x-\tfrac12\le y\le x \text{ or } x+\tfrac12\le y\}.
Let $(X_n)$ be the periodic binary sequence (11), $0001\,0001\,1101\,1101\,0001\,0001\,1101\,1101\cdots,$ which is 3-distributed.
Let $(U_n)$ be a sequence in $[0,1)$.
Let $(U_n)$ be a $[0,1)$ sequence.
Let l_n^{(1)}\ge l_n^{(2)}\ge \cdots \ge l_n^{(n)} denote the lengths of the $n$ intervals determined by the first $n$ points $U_0,\ldots,U_{n-1}$, and define
Let $C_k(n)=\frac1n\sum_{0\le j<n}\left(U_j-\frac12\right)\left(U_{j+k}-\frac12\right).$ Since $(U_n)$ is a ${0,1}$ sequence, $U_j^2=U_j$, and =U_jU_{j+k}-\frac12(U_j+U_{j+k})+\frac14.
We are asked to prove: > If $(U_n)$ is $k$-distributed, so is the sequence > > > >
We are asked to train 8 workers on 5 machines so that any 3 absences still allow operation of all machines.
Suppose $(U_n)$ is $k$-distributed.
(a) No.
Definition R4 asserts that for every positive integer $s$, every residue class $t$ with $0 \le t < s$, and every fixed choice of preceding terms, the subsequence U_t,\ U_{t+s},\ U_{t+2s},\ \ldots is $...
Let $r = p/q$ be a rational number with $\gcd(p,q) = 1$, $q \ge 1$, and consider the sequence $U_n = r^n \bmod 1, \quad n = 0, 1, 2, \ldots,$ viewed as a $[0,1)$ sequence.
Let $\{U_n\}_{n\ge0}$ be an $\infty$-distributed sequence and define the stopping times $f(n)$ recursively by $f(0)=0$ and, for $n\ge1$, f(n) = \min\{ m > f(n-1) : U_{m-1} > U_m \}.
Let $L_n=f(n)-f(n-1).$ The definition of $f(n)$ implies that, beginning immediately after position $f(n-1)$, we inspect successive bits of the binary sequence until both symbols $0$ and $1$ have appea...
Let $b \ge 2$ be an integer representing the number of kinds of coupons.
Let I=[\alpha,\beta), \qquad p=\beta-\alpha .
**Exercise 3.
In TAOCP §3.
Let M_n=\max(U_n,U_{n+1},\ldots,U_{n+k-1}).
Let $S_1(n), S_2(n), \dots$ be an infinite sequence of statements about mutually disjoint events.
Lemma E states that if \lim_{n\to\infty}\frac1n\sum_{j=1}^{n}y_{jn}=a, \qquad \lim_{n\to\infty}\frac1n\sum_{j=1}^{n}y_{jn}^{\,2}=a^2,
Let $O$ be the common point which is simultaneously the circumcenter of $\triangle ABC$ and the incenter of $\triangle ABK$.
**Exercise 3.
Assume that the sequence $\langle U_n\rangle$ is $(m,k)$-distributed, and let $d$ be a divisor of $m$.
For each pair $(i,j)$, let $A_{ij}=\{\,n\ge 0 : S_{ij}(n)\text{ is true}\,\}.$ The hypothesis states that for every $n>0$ there is exactly one pair $(i,j)$ such that $S_{ij}(n)$ is true.
We are asked to determine $\Pr{U_n < \tfrac{1}{2}}$ for the sequence $U_n = \bigl(2^{9(n+1)/3}\bigr) \bmod 1.$ First, observe that $2^{9(n+1)/3} = 2^{3(n+1)} = 2^{3n+3} = 8 \cdot 2^{3n}.$ Hence we may...
Let A(n)=S(n)\text{ and }T(n),\qquad B(n)=S(n)\text{ or }T(n).
We are asked to construct a periodic ternary sequence that is 3-distributed.
The sequence has period $4$: $0,0,1,1,0,0,1,1,\ldots$ To test 2-distribution, examine the successive pairs: $00,\ 01,\ 11,\ 10,\ 00,\ 01,\ 11,\ 10,\ldots$ Each of the four binary numbers $00$, $01$, $...
Let q_k = \frac{U_k}{w_k}, \qquad 1 \le k \le N, where $U_1, \ldots, U_N$ are independent and uniformly distributed on $(0,1)$, and define, for any real $r$,
We consider $n$ items $(X_1, X_2, \ldots, X_n)$ initially in order $X_j = j$ for $1 \le j \le n$, and a sequence of exchanges X_j \leftrightarrow X_{k_j}, \quad 1 \le j \le n, where $k_1, \dots, k_n$...
Let $I=L$ be the incenter.
No.
We are asked to compute a random sample of size $n$ from $\{1,2,\ldots,N\}$, using hashing ideas, $O(n)$ storage, and expected $O(n)$ time.
Let the original deck be denoted in cyclic order as 2,3,4,5,6,7,8,9,10,J,Q,K,A \; \spadesuit, \dots, A\clubsuit and let $c^+$ denote the successor of card $c$ in this cyclic order (wrapping around fro...
Let m=N-n+1.
**Exercise 3.
Consider small values of $n$ first.
Algorithm P begins with X_k=k,\qquad 1\le k\le t, and executes steps P2, P4 only until $j=t-n$.
Let the positions of the cards be numbered modulo $2n-1$.
**Exercise 3.
Let $M$ be the number of elements that are placed into the reservoir during the first pass of Algorithm R, when a reservoir of size $n$ is used on a file of $N$ records.
Let $S = \{s_1 < s_2 < \dots < s_n\}$ be any fixed $n$-subset of $\{1,2,\dots,N\}$.
Let $T$ denote the value of $t$ when Algorithm S terminates.
Let - $N$ be the number of records still available, - $n$ be the number of selections still to be made, - $t$ be the number already selected.
For the first statement, choose coordinates so that the bisecting line $l$ is the vertical line $x=0$.
Let $p(m,t)$ denote the probability that exactly $m$ items have been selected from the first $t$ items processed by Algorithm S.
There is no contradiction because the two probabilities refer to different events.
Let $T$ denote the value of $t$ at termination of Algorithm S, i.
Let Y=a_1X_1+\cdots+a_nX_n, where $X_1,\ldots,X_n$ are independent $N(0,1)$ random variables and
The reviewer's principal objection is incorrect.
The task is purely existential.
Let $N$ be the total number of records in the input file, and let $n$ be the number of records to be selected by Algorithm S.
Equation (1) in Section 3.
Let $X$ and $Y$ be independent exponential random variables with mean $1$, so that their joint probability density function (pdf) is $f_{X,Y}(x,y) = e^{-x} e^{-y} = e^{-(x+y)}, \qquad x > 0, \ y > 0.$...
Let f(x)=4x(1-x), \qquad 0\le x\le 1, and suppose that every computed value is rounded to a fixed binary precision of $b$ fraction bits.
The two inequalities are
**Exercise 3.
The statement is affine in nature.
Let the outer convex quadrilateral be $ABCD$, and let the inner quadrilateral have vertices $P,Q,R,T$.
Let an arithmetic progression of reciprocals be
Consider a tetrahedron with vertices $A$, $B$, $C$, and $D$, and a point $M$ inside it.
The problem gives three numbers $a$, $b$, $c$ in the interval $(0, \frac{\pi}{2})$ satisfying
Begin by examining small examples.
The first statement is a classical rectangle theorem.
Consider first the planar problem with parabolas.
For the first part, write the numbers as $x_1,\dots,x_n$, let $M=\max x_i$ and $m=\min x_i$.
For the first problem, let
Part (1) is the classical two-color complete graph statement.
Let the points be represented by vectors $x_1,\dots,x_n$ from the center of the circle or sphere.
Consider a right circular cylindrical pot with radius $R$ and height $H$.
Consider a polygon in the plane, labeled $A_1 A_2 \dots A_n$.
Let the houses have populations $h_1,h_2,\ldots,h_n$, arranged in nonincreasing order.
Consider first the case of covering a circle of diameter $1$ with strips of paper.
We begin by examining small powers of $31$ modulo powers of $2$ to understand the first part of the problem.
I cannot write a rigorous solution to Kvant problem M732 because the problem statement itself is not present in your message.
The sequence $(a_n)$ is defined recursively by $a_1=0$ and $a_{2n}=a_{2n+1}=n-a_n$.
Consider a parallelepiped with vertex $P$ at the origin, edges along vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ leading to adjacent vertices $A = P + \vec{a}$, $B = P + \vec{b}$, $C = P + \vec{c}$.
Since the perimeter is $2$, we have
The sequence $r_n$ sums the $n$-th powers of the cosines of the angles $\pi/7$, $3\pi/7$, and $5\pi/7$.
Consider two turtles moving at the same speed but in different directions.
Consider the simplest nontrivial cases first.
The function $f$ is defined recursively on nonnegative integers.
Consider small values of $n$ to understand the property.
Consider small values of $n$ and $r$ to detect a pattern.
Let
Consider small values of $N$ to build intuition.
Let the finite set be $M$.
Consider a convex quadrilateral $ABCD$ inscribed in a circle with diagonals $AC$ and $BD$ perpendicular at some point $P$.
The rhombus tiles are the unit lozenges of the triangular lattice.
Let the clubs be represented by sets of students.
Consider two circles with centers $O_1$ and $O_2$ and radii $R_1$ and $R_2$.
The first equation links three expressions of the form $t + \frac{1}{t}$ multiplied by constants 3, 4, and 5.
Let the sides of the acute triangle $LMN$ be
Consider the set of all terminating decimal fractions.
Let the cyclic quadrilateral be $ABCD$, with side lengths
Consider the problem for small $k \times k$ squares.
Let the table have $m$ rows and $n$ columns.
Consider small-scale analogues of the village communication problem.
Consider triangle $ABC$ with arbitrary side lengths $AB=c$, $BC=a$, $CA=b$.
For a convex polygon, the quantity $\dfrac{2S}{P}$ has a geometric meaning.
Let
Let
Let the partition be
Represent each circle by a vertex, and join two vertices when the corresponding circles touch.
Let
The game is equivalent to building a connected graph on $n$ vertices by adding edges one at a time.
Consider small examples first.
We begin by computing small powers of $1981$ and observing the sums of their digits.
Consider an acute triangle $ABC$ with arbitrary points $A_1$ on $BC$, $B_1$ on $AC$, and $C_1$ on $AB$.
Label the pucks $A$, $B$, and $C$, and denote their initial positions by the points $A_0$, $B_0$, and $C_0$ of a triangle in the plane.
Let the cyclic quadrilateral be $ABCD$ with diagonals $AC$ and $BD$.
Consider a cyclic quadrilateral $ABCD$ with circumcircle $\Gamma$.
Consider a triangle $ABC$ with the smallest angle $\widehat A$ and suppose the differences $d = |AB| - |BC|$ and $e = |AC| - |BC|$ are given.
I cannot write a rigorous solution to Kvant problem M665 because the actual problem statement is missing.
Consider small prime numbers to understand the behavior of the expression $2^p + p^2$.
Let the speeds of the motorboat and rowboat be constant, equal to $v_M$ and $v_R$.
Consider the dynamics of the allowed operations on the circle.
Consider a square of side length $1$ with a collection of horizontal and vertical segments inside it.
Consider small cases by simulating the procedure described.
Consider small examples of six natural numbers and examine the divisibility patterns.
We are asked about sequences of numbers (natural numbers or integers) such that every element in a certain target set (all naturals, all integers, or subsets thereof) can be represented uniquely as a…
Consider the problem for small values of $n$ to understand the geometric constraints.
A convex equiangular $n$-gon has exterior angle $2\pi/n$ at every vertex.
The coefficients are restricted to the set ${-1,0,1}$, and two neighboring coefficients cannot both be nonzero.
Let the decimal expansion of $x_k$ be
Consider an equilateral triangle $ABC$ with side length normalized to $1$ for convenience.
Consider small examples to understand how the set $A$ might grow.
Let $\Pi$ be a Poisson random variable with mean $\mu$.
We want $X_1 \le X_2 \le \cdots \le X_n$ such that each $X_i$ lies in $[0,1]$ and the joint distribution is uniform over the simplex $0 \le X_1 \le X_2 \le \cdots \le X_n \le 1.$ Equivalently, we want...
Let \qquad a_1\ge \cdots \ge a_n>0.
Consider a spherical triangle with one side of length $120^\circ$.
Let a subroutine $\operatorname{Bin}(m,\tfrac12)$ be available; it returns a random variable having the binomial distribution $(m,\tfrac12)$.
Let $N_1$ and $N_2$ be independent Poisson random variables with means $\mu_1$ and $\mu_2$, respectively, where $\mu_1 > \mu_2 \ge 0$.
For part 1, suppose every natural number appears exactly once.
Let E_t=X_1\mid\bigl(X_2\mathbin{\&}(X_3\mid(X_4\mathbin{\&}X_5)\cdots)\bigr) denote the nested expression.
We are asked to determine whether the two methods described produce a random quantity $X$ with the same distribution.
Let the density to be sampled be proportional to e^{-x^{2}/2}, \qquad x\ge 0.
Can the exact Poisson distribution for large $\mu$ be obtained by generating an appropriate normal deviate, converting it to an integer in some convenient way, and applying a (possibly complicated) co...
I can proceed, but I need the text of problem M598 first.
Let $N$ be the number of executions of step R1 before the algorithm terminates.
**Exercise 3.
We are asked to generate a random integer $N$ such that \Pr\{N=n\} = n p^2 (1-p)^{\,n-1}, \qquad n \ge 0, with particular interest in the case where $p$ is small.
The sequence $x_n=1+\frac12+\dots+\frac1n$ is the $n$-th harmonic number.
Let f(x)=\frac{x^{a-1}e^{-x}}{\Gamma(a)}, \qquad x>0,\qquad 0<a\le1, be the gamma density of order $a$.
Let $X$ be a random variable representing the number of trials until the first success in a sequence of independent Bernoulli trials with success probability $p$, $0 < p \le 1$.
Label the vertices of the regular octagon cyclically by
Let $S=X_1+X_2.$ Since $X_1$ and $X_2$ are independent, the distribution function of $S$ is Condition on the value of $X_2$.
Algorithm M is designed to generate a discrete random variable X \in \{x_0, x_1, \dots, x_{n-1}\}, \quad \Pr\{X = x_j\} = p_j \ge 0, \quad \sum_{j=0}^{n-1} p_j = 1 using a uniform deviate $U \in [0,1)...
Let $X=(X_1,\ldots,X_n)^T,$ where the $X_i$ are independent normal deviates with mean $0$ and variance $1$.
**Exercise 3.
Let
The proposed solution does **not** answer Exercise 3.
We are asked to show that the alias method, as described in equation (3) of Section 3.
The solution does not answer the exercise as stated.
We are asked to generate a random variable $X$ with distribution function F(x) = p x + q x^2 + r x^3, \qquad 0 \le x \le 1, where $p \ge 0$, $q \ge 0$, $r \ge 0$, and $p + q + r = 1$.
The algorithm generates two independent uniform deviates $U$ and $V$, each distributed on $[0,1]$, and rejects pairs for which $U^2 + V^2 \ge 1$.
Consider a triangle $ABC$ with circumcircle $\Gamma$.
Suppose we have a uniform random variable $U$ between 0 and 1, represented in a computer word with $m$ possible discrete values, $0, 1, \ldots, m-1$, as in Section 3.
Let $U$ be a random variable uniformly distributed on $[0,1)$, and suppose that $mU$ is interpreted as a random integer between $0$ and $m-1$, namely $U_m = \lfloor mU \rfloor.$ Thus $\Pr{U_m = j} = 1...
Let $(X_n)$ be a linear congruential sequence modulo $m$, with multiplier $a$ and full period $m$, as in Section 3.
Let $U$ be a random variable uniformly distributed between 0 and 1.
Let $m=(m_1m_2)$, and let generator (38) be X_{n+1}=aX_n\bmod m_1,\qquad Y_{n+1}=aY_n\bmod m_2, with
Let \mathcal D_A=\Bigl\{q\ge1:\ q \text{ occurs as the denominator of a convergent } [0;a_1,\ldots,a_s],\ a_i\le A\Bigr\}.
We are asked to prove that r_{\max} \le \frac{1}{\sqrt{8}\, \nu_t}, where $r_{\max}$ is the maximal value of
Consider first the expression $|\cos x| + |\cos 2x|$.
Let X_{n+1}\equiv aX_n \pmod m, where $m$ is prime, $c=0$, $a$ is a primitive root modulo $m$, and $X_0\not\equiv0\pmod m$.
Consider a linear congruential sequence $(X_n)$ defined by $X_{n+1} = (a X_n + c) \bmod m, \quad 0 \le X_n < m,$ with full period length $m$.
Let r(u_1,\ldots,u_t)=\prod_{j=1}^t r(u_j), where, as in equation (46),
Let $r(k)$ be defined by Eq.
Let the given vectors be $v_1,\dots,v_n$.
Let X_n \equiv aX_{n-1}+bX_{n-2}\pmod p, where $p$ is prime, and suppose that the recurrence has period $p^2-1$.
Let $\mu_2$ and $\mu_3$ denote the two- and three-dimensional spectral constants associated with a linear congruential sequence, as defined in Section 3.
Let G=(g_{ij})_{1\le i,j\le t}, \qquad g_{ij}=U_i\cdot U_j .
The operation replaces two numbers $x,y$ by
Let $X_{n+1} \equiv aX_n \pmod{2^e}, \qquad X_0 \text{ odd}, \qquad a \equiv 3 \text{ or } 5 \pmod 8.$ By exercise 3.
Yes.
**Corrected Solution for Exercise 3.
The solution correctly addresses the exact question.
For each dimension $t$, Algorithm S examines all integer vectors U=(u_1,\ldots,u_t) satisfying (15), and computes
Perform Algorithm S by hand for $m=100$, $a=41$, $T=3$.
The majority are chemists, and chemists are perfectly reliable.
Let f(x_1,\ldots,x_t)=\sum_{i,j}u_{ij}x_ix_j be a quadratic form.
Let the hyperplanes defined by $U=(u_1,\ldots,u_t)$ be u_1x_1+\cdots+u_tx_t=h, where $h$ ranges over the integers.
Let f(\mathbf{x})=\mathbf{x}^{T}A\mathbf{x}, where $A$ is a positive definite symmetric matrix of order $t$.
Let m=2^e, \qquad R=\sqrt{\frac43}\,m .
Let $(u_1,\ldots,u_t)$ be a solution to problem (b) following Eq.
Let \frac{m}{a} =[a_0,a_1,\ldots,a_t] be the continued-fraction expansion used in §3.
Let $b=a-1$.
Suppose such a family of lines exists.
Since $(y_1,y_2)=1$, there exist integers $u_1,u_2$ such that u_1y_2-u_2y_1=m.
Let the black sectors have angular lengths $\alpha_1,\dots,\alpha_k$, where each
Let $m = p_1 p_2 \cdots p_t$, where $p_1, \ldots, p_t$ are distinct primes.
An alternative to Program A that updates all 55 entries of the $Y$ table every 55th output can be implemented by performing a full sweep of the table instead of a single additive step.
Suppose, contrary to the statement, that none of the given numbers is prime.
Consider a triangle $ABC$ with sides $BC < AC < AB$.
Each of the 20 values of $K_{10}^+$ was itself computed from 10 observations.
Let $O$ be the center of the circumcircle of the isosceles triangle $ABC$, and let $M$ be the midpoint of $PQ$.
The proposed solution answers the question being asked.
We are asked to determine which values of $m = w \pm 1$ in Table 3.
Consider the numbers $1978^m - 1$ and $1000^m - 1$ for small values of $m$.
Compute the first few values of $f$ for small natural numbers greater than $1$.
We are asked to consider sequences of $n$ distinct natural numbers $a_1 < a_2 < \dots < a_n < 2n$ with $n \ge 6$, and to find bounds for the minimum of their least common multiples and the maximum of…
The condition
Consider what it means for a number to be balanced.
Consider triangle $ABC$ with arbitrary points $A_1$ on $BC$, $B_1$ on $CA$, and $C_1$ on $AB$.
Each satellite moves along a circular orbit centered at $O$ with constant angular velocity.
Consider triangle $ABC$ and points $A_1$, $B_1$, $C_1$ on sides $BC$, $CA$, and $AB$, respectively, with cevians $AA_1$, $BB_1$, and $CC_1$ concurrent at $P$.
Let three consecutive terms be $a,ar,ar^2$, where all terms are integers.
The recurrence
The interval is
Consider a right triangle with legs $a$ and $b$ and hypotenuse $c$, where $c^2 = a^2 + b^2$.
Let
Consider a set of distinct natural numbers ${a_1, a_2, \dots, a_n}$ with the property that for any two elements $a_i$ and $a_j$, the sum $a_i + a_j$ is divisible by their difference $a_i - a_j$.
For the planar statement, the condition that no lattice points lie on the boundary except the vertices means that every side joins two lattice points with relatively prime coordinate differences.
We begin by examining the properties of perfect numbers modulo small integers.
Consider a cube of side length $1$ for simplicity.
We begin by examining the two sums for small values of $n$ to detect patterns.
Consider four points $A$, $B$, $C$, $D$ in the plane, and the scalar products $\overrightarrow{MA} \cdot \overrightarrow{MB}$ and $\overrightarrow{MC} \cdot \overrightarrow{MD}$ for a variable point $…
Consider first a smaller version of the problem.
Consider small examples to understand the problem concretely.
Consider a small number of weights, for instance $n=2$ or $n=3$, each with distinct masses $w_1<w_2<w_3$.
Consider the polynomial $x^{10}+a_9x^9+\dots+a_1x+1$ with all coefficients initially unspecified except for the leading and constant terms, which are $1$.
Let the vertices of the simple closed polygonal line be $A_1,A_2,\dots,A_n$ in cyclic order, and let $e_i=A_iA_{i+1}$, with indices taken modulo $n$.
Let the dwarfs act in order $1,2,\dots,7$ around the table.
Consider first the simplest nontrivial configuration of points, namely three points not lying on a line.
I cannot write a rigorous solution to Kvant problem M449 without the actual problem statement or the diagram.
I cannot write a solution to Kvant problem M447 because the actual problem statement is not present in your message.
I cannot write a solution to Kvant problem M445 from the information provided, because the actual problem statement is missing.
Before I begin writing the complete solution, I need the **full textual statement of Kvant problem M443**.
We are working with rooted binary trees where each node carries a binary label, either zero or one. The tree structure is arbitrary as long as every node has at most two children. On top of that structure, we assign labels independently.
We are given a fixed permutation p of size n. Alongside it, we must construct another permutation q of the same size.
We are given a binary array where each position contains either 0 or 1. A subset of positions is marked as special, and all special positions initially share the same value, call it $x$.
We are given a binary array of length n and a single special index p (since k=1 in this easy version). The bit at this index is considered "correct," and the goal is to make the entire array equal to this bit by performing the fewest number of flip operations.
Let the vertices of the convex $2n$-gon be $A_1,A_2,\dots,A_{2n}$ in cyclic order.
For part 1, write
This problem comes from an April Fools contest where the statement intentionally hides the real task. We are given two non-negative integers. The required operation is: 1. Reverse the decimal representation of the second number. 2. Add the result to the first number. 3.
The input contains two numbers. The first number, a, determines the order of a recursively constructed polyline. The second number, b, is an index along that polyline. The picture in the statement is the key.
We are asked to partition all pairwise sums of two sets of ten numbers each into ten groups of ten, each with the same total.
We are given three small positive integers, a1, a2, and a3, each ranging from 1 to 20. The problem asks us to compute a single integer as output based on these three numbers.
We are asked to find a special number associated with a single integer input d, where d represents a divisor or parameter in a number-theoretic sequence.
This is one of Codeforces' classic "special" problems. Unlike ordinary algorithmic tasks, there is no meaningful input to process and no data structure or optimization challenge to solve. The contest provides access to a language called Secret through the custom test environment.
Consider the sum
This is one of the most unusual problems on Codeforces. The input is supposed to contain a single integer between 1 and 5. There are only five official test cases. The output must be a single integer between 1 and 3. The crucial detail is that there is no actual task to solve.
Consider the sum of the digits of perfect squares.
We are asked to construct sequences of integers of length $n$ such that the greatest common divisor (gcd) of consecutive elements is always distinct. Each test case provides a number $n$, and for each, we must output one valid sequence of length $n$.
We are asked to build a rooted tree with exactly $n = x + y$ vertices. The root is fixed to be vertex $1$. For every vertex, consider the size of its rooted subtree.
For the planar statement, the number $2$ strongly suggests a relation between the area of a convex figure and the area of a rectangle determined by two orthogonal widths.
The problem asks us to pick an integer $y$ for a given integer $x$ so that the minimum of $x$ and $y$ is as large as possible. We are given multiple test cases, each consisting of a single integer $x$ between -67 and 67.
The city has n real residents. Among them, exactly x are wizards, and all of those wizards will attend a demonstration. Nobody else will attend. The administration measures attendance as a percentage of the real city population, which remains n even if the wizards create clones.
Let
Suppose such an $N$ exists.
The left-hand side contains the three quantities
Let the cells of the infinite graph paper be indexed by integer coordinates $(x,y)$, where each cell corresponds to one pair of integers.
For small values of $n$,
Interpret the drawn segments as the edges of a graph whose vertices are the given points.
Consider the equation $f(x+a)-f(x)=0$ for a function $f$ continuous on $[0,1]$ with $f(0)=f(1)=0$.
Consider a triangle with sides $a$, $b$, $c$, and a point inside it through which three segments pass, each parallel to a side and all of equal length $x$.
Consider the sphere of radius $1$ centered at the origin in $\mathbb{R}^3$, and let the equatorial plane be the $xy$-plane.
Consider small examples to understand the structure of the problem.
The condition concerns preservation of adjacency.
A convex polyhedron has the property that every vertex has even degree.
Consider small values of $N$ to understand the structure of universal sequences.
I cannot write a rigorous solution to Kvant problem M398 from the information provided, because the problem statement itself is missing.
The side length of the equilateral triangle is much larger than the lower bound $1$ imposed on the sides of the desired triangles.
Let
I cannot write a solution to Kvant problem M391 from the information provided, because the actual problem statement is missing.
I cannot write a rigorous solution to Kvant problem M388 because the problem statement itself is not provided in the conversation.
I can proceed with that, but I need the text of Kvant Problem M386 to provide the complete solution.
Consider small examples to understand the problem.
I cannot write a solution to Kvant problem M381 because the actual problem statement is not present in your message.
I can proceed with this framework, but I need the text of Kvant problem M379 in order to produce a rigorous solution.
Let the black piece start at cell $1$ and the white piece at cell $N$.
I cannot write a solution to Kvant problem M375 from the information provided, because the actual problem statement is missing.
Let p=\beta-\alpha .
Consider the triangle $ABC$ and the inequality $|AP| + |BP| + |CP| \ge |AC| + |BC|$ for an arbitrary point $P$ in the plane.
Let p=\beta-\alpha .
Let $N$ be the total number of values $U_j$ examined by Algorithm G when $n$ gaps have been recorded.
The circle $\gamma$ is centered at the orthocenter $H$ and lies inside the acute triangle $ABC$.
For triples, the serial test should count the occurrences of each ordered triple $(Y_{3j},Y_{3j+1},Y_{3j+2}), \qquad 0\le j<n,$ for all $d^3$ possible values $(q,r,s)$ with $0\le q,r,s<d$.
If the serial test is applied to the overlapping pairs $(Y_0, Y_1)$, $(Y_1, Y_2)$, $\ldots$, $(Y_{n-1}, Y_n)$, then each number $Y_j$ contributes to two successive pairs, except for the first and last...
Choose coordinates so that the three cylinder axes are parallel to the coordinate axes.
For fixed $(p,q,r)$ with $p+q+r=1$, the exact distribution of $V$ is obtained by enumerating all triples $(Y_1,Y_2,Y_3)$ satisfying $Y_1+Y_2+Y_3=n$.
Let D_n(u)=\sqrt n\,(G_n(u)-u), \qquad 0\le u\le1, where $G_n$ is the empirical distribution function of the transformed variables
Let $U_i=F(X_i)$, and let the values be renumbered so that U_1\le U_2\le\cdots\le U_n.
**Exercise 3.
Consider first a simple case of two numbers summing to $1$.
Suppose that $F(x)$ has jumps at points $\xi_1,\xi_2,\ldots$, with jump sizes $\Delta_j=F(\xi_j)-F(\xi_j-0)>0.$ If the sample contains $m_j$ occurrences of $\xi_j$, then the empirical distribution fun...
Let X^{(1)},X^{(2)},\ldots,X^{(n)} be independent observations from an unknown distribution on $\mathbf R^s$, where
The statement is affine in nature.
The statistics $K_n^+$ and $K_n^-$ are defined as the maximum positive and maximum negative deviations, respectively, of the empirical distribution function from the theoretical distribution function.
**Exercise 3.
Please provide the full text of Kvant problem M361 so I can write the complete, rigorous solution in the requested six-section format.
Let J_n=\det\!
I can produce a complete, rigorous Kvant-style solution, but I need the text of problem M358 to proceed.
We write $\gamma(x+1, x + z\sqrt{2x} + p) = \int_0^{x + z\sqrt{2x} + p} e^{-t} t^x \, dt / x!.$ Setting $t = x + s\sqrt{2x}$, we have $dt = \sqrt{2x}, ds$, and the integral becomes $\frac{\sqrt{2x}}{x...
Let F_n(x)=\frac1n\#\{x_j\le x\}, and let
Let $Y_s=np_s+\sqrt n\,Z_s,\qquad s=1,\ldots,k,$ where the variables $Z_s$ are bounded as $n\to\infty$.
Let
Let the unknown triangle be $ABC$, and suppose that $H$ is the foot of the altitude from $A$ onto $BC$.
**Exercise 3.
Let F_n(x)=\frac{1}{n}\#\{j:X_j\le x\} be the empirical distribution function of the original sample
Let the given triangle have sides $a,b,c$ opposite angles $A,B,C$.
**Solution (corrected)** We are asked to discuss the merits of pooling the 20 values of $K_{10}^{+}$ with the 20 values of $K_{10}^{-}$ and then applying a Kolmogorov-Smirnov test to the resulting 40...
Let the original observations produce counts $Y_1,\ldots,Y_k$, with probabilities $p_1,\ldots,p_k$.
Consider first small cases to understand the tension between a European team dominating the European Championship yet performing worst in the World Championship.
Let $F_n(x)$ be the empirical distribution function based on $n$ independent observations $X_1, \dots, X_n$ from a **continuous** distribution $F(x)$.
Let the observations be arranged in increasing order: \begin{aligned} X_{(1)}&=0.
Consider the operation on a small set of digits.
For two polygons the statement is immediate.
Let $F_n(x)$ be the empirical distribution function defined by equation (10) of Section 3.
Let the tetrahedron have edge length $1$.
To test whether the dice are fair, we must use the probability distribution for the sum of two ordinary dice.
Let the first die be fair, with outcomes $1,2,3,4,5,6$ equally likely, each with probability $\frac{1}{6}$.
The statement asks for a classification.
Let the two dice be labeled die A and die B.
Equation (5) arises from the dice-throwing experiment with eleven categories, namely the possible sums $2,3,\ldots,12$.
Let $T(x)=a x^{-1}+c \pmod p,$ where the value $\infty$ is adjoined in the usual way: $T(0)=\infty,\qquad T(\infty)=c.$ Since the sequence has period $p+1$, the transformation $T$ acts as a single cyc...
**Problem:** Determine the number of pairs $(a,c)\in \mathbb{F}_p \times \mathbb{F}_p$ satisfying the conditions of Exercise 34, i.
We consider the inversive congruential sequence defined by $X_{n+1} \equiv a X_n^{-1} + c \pmod{2^e}, \qquad e \ge 3, \eqno(1)$ where $a \bmod 4 = 1$ and $c \bmod 4 = 2$.
Let M=\begin{pmatrix} 0&1\\ a&c \end{pmatrix},
The sequence $\langle X_n \rangle$ satisfies the lagged Fibonacci recurrence $X_n = (X_{n-24} + X_{n-55}) \bmod m, \qquad n \ge 55. \eqno(7)$ Define the generating function $g_n(z) = X_{n+30} + X_{n+2...
We are asked to determine the period length of the sequence $\langle 7^n \rangle$ modulo $m = 2^e > 8$, under the assumption that the initial state $X_0, \dots, X_{54}$ is not all congruent to $\pm 1...
Consider a small graph representing countries, where vertices are countries and edges connect neighboring countries.
Let X_n \equiv X_{n-2}+X_{n-55}\pmod m .
Let f(x) = x^k - a_1 x^{k-1} - \cdots - a_k be primitive modulo 2, and suppose $X_0, \ldots, X_{k-1}$ are integers not all even.
Consider a single vertex where three hexagonal walls meet.
**29.
Exercise 28 asks for an experimental investigation of linear congruential sequences when the modulus is much larger than the machine word size, while the multiplier and increment remain single-precisi...
**Exercise 3.
The conjecture is false in general.
A suitable choice for CONTENTS$(A)$ in method (10) is any $k$-bit binary number in which roughly half of the bits are $0$ and half are $1$.
Let $\langle Y_n \rangle$ be the sequence with period $2^k - 1$, satisfying $Y_n = (Y_{n-l} + Y_{n-k}) \bmod 2$.
Let m=p_1p_2\cdots p_t, where the primes $p_1,\ldots,p_t$ are distinct.
We are asked to show that the sequence of $8$-bit blocks U_n = (X_{8n}, X_{8n+1}, \dots, X_{8n+7}) fails the serial test on pairs, where the $X_n$ are generated by method (10) with $k=35$ and
Let $m=\prod p_i^{e_i}$ be the prime factorization of $m$.
Let $X_n=(a_1X_{n-1}+a_2X_{n-2})\bmod p,$ with $k=2$.
For $m=5$ the consecutive fractions
Let T_n=(X_{n+k},X_{n+k-1},\ldots,X_{n+1}) \qquad (n\ge0), where $X_1=\cdots=X_k=0$.
Let $(X_n)$ and $(Y_n)$ be sequences of integers modulo $m$, with least periods $\lambda_1$ and $\lambda_2$, respectively.
Let $(X_n)$ and $(Y_n)$ be integer sequences modulo $m$ with periods $\lambda_1$ and $\lambda_2$, respectively.
Let $A$ be a $k$-bit word with binary representation $(a_1 a_2 \ldots a_k)_2, \quad a_i \in \{0,1\},$ so that $\text{CONTENTS}(A) = (a_1 a_2 \ldots a_k)_2.$ Consider method (10), which generates a seq...
Algorithm M combines a sequence $(X_n)$ of period length $\lambda_1$ with a sequence of skips $(q_n)$ of period length $\lambda_2$.
Let f(z) = 1 - a_1 z - \cdots - a_k z^k, \qquad G(z) = \frac{1}{f(z)} = \sum_{n \ge 0} A_n z^n.
We seek integers $X_0, X_1, a, b, c$ such that the sequence X_{n+1} = (a X_n + b X_{n-1} + c) \bmod 2^e, \qquad n \ge 1, has the maximal possible period modulo $2^e$.
**Exercise 3.
The problem involves a convex polyhedron intersected by three parallel planes $p_0$, $p_1$, $p_2$, with $p_1$ equidistant between the outer planes.
**Solution.
In the binary method (10), the contents of $X$ form a shift-register sequence determined by a primitive polynomial modulo $2$.
Let $P_e$ be the period of the Fibonacci sequence X_{n+1}=(X_n+X_{n-1})\bmod 2^e.
Let $m = w \pm 1$ be one of the moduli listed in Table 3.
**Corrected Solution to Exercise 3.
Consider Algorithm M as defined in Section 3.
We first compute the sequences $(X_n)$ and $(Y_n)$ modulo 8, given the recurrences $X_{n+1} = (5X_n + 3) \bmod 8, \quad X_0 = 0,$ $Y_{n+1} = (5Y_n + 1) \bmod 8, \quad Y_0 = 0,$ and $k = 4$.
Let $m(N)$ denote the minimum possible number of distinct marked points.
Let $m = 2^e \ge 8$, and consider a linear congruential generator of the form X_{n+1} \equiv a X_n + c \pmod{m}.
The recurrence U_{n+1}=(aU_n+c/m)\bmod 1 is mathematically equivalent to the linear congruential generator
Since $a \equiv 1 + b \pmod{m}$, we have $aa' \equiv (1 + b)a' \equiv 1 \pmod{m}.$ Solving for $a'$, we obtain $(1 + b)a' \equiv 1 \pmod{m} \implies a' \equiv \frac{1}{1 + b} \pmod{m}.$ We can expand...
Let $m = p_1^{e_1} \cdots p_r^{e_r}$ and $a = 1 + k p_1^{f_1} \cdots p_r^{f_r}$, where $k$ is relatively prime to $m$, and $a$ satisfies the conditions of Theorem 3.
**Solution.
The potency is the least integer $s$ such that b^s \equiv 0 \pmod{m}, where $b=a-1$; equivalently, $m\mid b^s$.
In the MIX code (3) we have $a = B^2 + 1$, so $b = a - 1 = B^2$.
Let $a=B^2+1$, so that $b=a-1=B^2$.
Consider smaller analogues of the problem to understand its structure.
**Solution.
**Exercise 3.
Assume $a$ is a primitive element modulo $p$, so that $a^{p-1} \equiv 1 \pmod{p}$ and $a^k \not\equiv 1 \pmod{p}$ for $1 \le k < p-1$.
Let X_{n+1}\equiv aX_n+c \pmod m,\qquad m=\prod_{j=1}^t p_j^{e_j}, and let $X_n^{(j)}$ denote the corresponding sequence modulo $p_j^{e_j}$.
From (9), $\lambda(m)=\varphi(m)$ holds for odd prime powers $p^e$, since $\lambda(p^e)=p^{e-1}(p-1)=\varphi(p^e), \qquad p>2.$ For powers of $2$, $\lambda(2)=1=\varphi(2),\qquad \lambda(4)=2=\varphi(...
**Exercise 3.
The condition
Assume $m = 2^e \ge 16$ and $c = 0$.
From Table 3.
We are asked to find all multipliers $a$ that satisfy the conditions of Theorem A when $m = 2^{35} + 1$.
We are given a linear congruential sequence $X_{n+1} = (a X_n + c) \bmod 2^e$ with $X_0 = 0$, and where $a$ and $c$ satisfy the conditions of Theorem A.
Let $m = 10^e$ with $e \ge 2$, and let $c$ be odd and not a multiple of 5.
Let $m = 2^e$, where $e \ge 1$.
Exercises 12 and 13 exploit moduli of the form $m=b^e\mp1,$ where multiplication by suitable powers of the radix $b$ can be reduced to cyclic shifts of digits.
The modulus $m = 10^{10}$ factors as $2^{10} \cdot 5^{10}$.
Let m=aq+r,\qquad q=\left\lfloor \frac{m}{a}\right\rfloor,\qquad 0\le r<a.
Let $m=9999999001=10^{10}-999=10^{10}-31^2.$ The congruence is the fundamental relation.
Let T_a(x)=ax \bmod m , where $0<a<m$.
Let the rectangle have coordinates
We are asked to write a MIX program analogous to program (2) of Section 3.
Let M=9999999999=10^{10}-1.
Let q=\left\lfloor \frac{m}{a}\right\rfloor , \qquad r=m\bmod a,
Examination of Table 1 reveals several recurring structural patterns in the factorizations of $w \pm 1$.
**Exercise 3.
Let P=ax, with
Consider the recurrence X_{n+1} \equiv aX_n-c \pmod m.
We are asked to write a MIX subroutine that computes the next value of a linear congruential sequence.
Consider a hoop of radius $R$ placed over a fixed circle of radius $r < R$.
Since $0 \le x,y<m$, we have $-(m-1)\le x-y\le m-1.$ Hence $(x-y)\bmod m$ is either $x-y$ or $x-y+m$.
Consider a convex quadrilateral with vertices $A$, $B$, $C$, $D$ in order, and let $K$, $L$, $N$ be the midpoints of three of its sides.
Let $w$ denote the machine word size of MIX.
Equation (6) states that for $k \ge 0$, X_{n+k}\equiv a^kX_n+\frac{a^k-1}{a-1}\,c \pmod m.
**Corrected Solution for Exercise 3.
We prove Eq.
Let X_{n+1}\equiv aX_n+c \pmod m, and assume that
Solution to TAOCP 3.1 Exercise 20.
**Exercise 3.
Using the sequence $f(0), f(1), f(2), \ldots$ where $f$ is a random function is not a practical way to generate random numbers.
Let $S$ denote the area of the polygon $A_1A_2\cdots A_n$.
Take, for example, $m=10$, $a=2$, $c=0$, and $X_0=1$.
The proposed disproof is invalid.
**Exercise 3.
Suppose a sequence is generated by a function $f$ of the preceding $k$ values, as in exercise 17.
Let $Y_n=(X_n,X_{n-1},\ldots,X_{n-k+1}).$ Then $Y_n$ ranges over the $m^k$ possible $k$-tuples, and the recurrence $X_{n+1}=f(X_n,\ldots,X_{n-k+1})$ induces a function $Y_{n+1}=F(Y_n)$
In the generalized scheme, each term is determined by the preceding $k$ values: X_{n+1}=f(X_n,X_{n-1},\ldots,X_{n-k+1}), where $0\le X_i<m$.
Let $C(f)$ denote the number of distinct final cycles of a mapping $f$ on an $m$-element set $M=\{1,\ldots,m\}$.
Let $L_m$ denote the length of the longest cycle in a random mapping $f:\{1,\ldots,m\}\to\{1,\ldots,m\}$, chosen uniformly from the $m^m$ possible functions.
Consider an acute-angled triangle with vertices $A$, $B$, and $C$ and corresponding angles $\alpha$, $\beta$, and $\gamma$, and sides $a = BC$, $b = AC$, $c = AB$.
Suppose $f$ is a random function from a set of $m$ elements to itself, as in exercise 11.
**Exercise 3.
Let f(x)=\left\lfloor \frac{x^{2}}{10}\right\rfloor \bmod 100, so that $f(x)$ is the middle two digits of the four-digit representation of $x^{2}$.
**Exercise 3.
Let $S=\{0,1,\dots,m-1\}$.
Let the square have side length $a$.
Suppose we are generating a sequence of $2n$-digit numbers in base $b$ by the middle-square method.
Algorithm K, as described in Section 3.
In the middle-square method with a fixed length of $10$ digits, we begin with x=1010101010.
Let $X_0, X_1, X_2, \ldots$ be a sequence generated by $X_{n+1} = f(X_n)$, where $f$ maps a finite set ${0, 1, \dots, m-1}$ into itself.
Step K11 of Algorithm K is the normalization step.
Since each $X_n$ lies in the set ${0,1,\ldots,m-1}$, there are only $m$ possible values.
A suitable method for obtaining a decimal digit at random is one that produces each of the digits $0,1,\dots,9$ with equal probability and independently of prior choices.
Consider a trihedral angle, that is, three planes meeting at a common vertex, forming three plane angles $\alpha$, $\beta$, and $\gamma$ at the vertex.
Consider small convex polyhedra such as the tetrahedron, cube, and octahedron.
The problem asks for an orientation of all edges between $n$ points.
Let
Let the requested block sizes be independent random variables with distribution $\Pr{2^k}=p_k$, where $\sum_k p_k=1$, and suppose that deallocations occur after independent holding times with finite m...
The dynamic storage allocation methods of Section 2.
**Exercise 2.
Let $N(n,m)$ denote the minimum amount of storage that guarantees successful allocation when the total amount of live data is at most $n$, and when block sizes are restricted to the interval $[1,m]$.
Suppose a request for a block of size $r$ is the first request that cannot be satisfied.
The reviewer's principal objection is valid: without the precise algorithm of Exercise 33, no correct MIX program can be written.
A block of size $4$ contains addresses that differ only in the last two bits.
Yes.
We assume a **binary MIX machine** with the new operation code \texttt{XOR} \quad (C=5, F=5), which complements in register `A` exactly the bits that are 1 in memory location `M`.
**Exercise 2.
Let $R\le n$ be the total space currently reserved.
The argument is incorrect even if it is known in advance that no request larger than $2^n$ will ever occur.
Working
**Exercise 2.
**Solution.
We are asked to design a **garbage collection and compaction algorithm** for a memory pool of nodes with the following properties: - Memory locations $1,2,\dots,AVAIL-1$ contain nodes `NODE(P)` of var...
Let M=\left\lfloor\frac{3n-1}{2}\right\rfloor .
Write M=\sum_{j=1}^{t}2^{m_j}, \qquad m_1>m_2>\cdots>m_t\ge0,
No.
When the distribution function $F(x)$ is continuous, the _distributed-fit method_ assigns the first $N$ slots according to the quantiles of $F$.
Exercise 2.
No.
Modify Algorithm B by first adjoining the interval $[P_0,P_0+N-1]$ to every free block that overlaps it, then perform the usual coalescing operation.
A COBOL data description is a rooted ordered forest in which each group item is followed in the source listing by the complete specification of its immediate subordinate items before the next item of...
Let rule (c) be the COBOL requirement that whenever two consecutive entries have level numbers $L_i$ and $L_{i+1}$, and $L_{i+1}>L_i$, the new level number must be the smallest level number that has a...
With first-fit, Algorithm A always takes the first available block large enough to satisfy a request.
Let the Data Table entries be stored in preorder, and let the structure stack be represented by the chain of `PREV` links exactly as in Algorithm B.
When there are no available blocks, the list headed by `AVAIL` must be empty.
To perform the function of Algorithm C with the representation of exercise 12, insert the new item exactly as before by setting its `PREV` field to the preceding item in the same group and setting its...
Consider an `AVAIL` list whose first two free blocks have sizes $25$ and $12$, in that order.
(a) In Algorithm B, the node $S$ is a Data Table entry.
A simple modification is to remember the position at which the previous successful search ended, and begin the next search there instead of always starting at the front of the `AVAIL` list.
Algorithm A needs only one change in the step that determines the parent of a newly read item.
If reservations and liberations occur strictly in last-in-first-out order, the free storage always consists of a single contiguous region at the top of the stack of allocated blocks.
The step M(x,y)=\text{visiting} \;\Rightarrow\; E(x,y)=\text{true} is invalid.
The previous answer fails because it assumes information that is not available after the trailing `SIZE` fields have been removed.
The statement "`MOVE CORRESPONDING \(\alpha\) TO \(\beta\)`" is exactly the same as "`MOVE \(\alpha\) TO \(\beta\)`" when the structures $\alpha$ and $\beta$ are _identical in layout and names of fiel...
Let the usable payload in a node be $k-b$ words.
Let the `AVAIL` list contain $m$ free blocks, arranged in increasing address order.
The reviewer’s criticisms are correct.
Algorithm B spends much of its time locating all occurrences of a given name.
In the buddy system, a block is repeatedly split and recombined with its unique buddy.
Rule (c) is removed, so Algorithm B must no longer reject repeated names within a group.
Algorithm A is modified as follows to satisfy the boundary-tag conventions (7)-(9), to use the modified step A4', and to incorporate the next-fit improvement of exercise 6.
**Corrected Solution to Exercise 2.
Let the memory contain permanent boundary sentinels at both extremes.
The essential idea is to organize the free blocks by size instead of by location.
Let $k$ be the node length and $b$ the control-word overhead per node.
Exercise 2.
To obtain the best-fit method, modify Algorithm A so that it does not stop when the first block with `SIZE(P) \ge N` is found.
**Exercise 2.
The reviewer is correct that the previous submission did not solve the exercise.
Algorithm B is modified by replacing the indexed access to $P_k$ with sequential traversal of the linked structure beginning at $T$.
Replace the explicit stack by a linked stack whose links occupy the unused `SIB` fields of the nodes that are on the stack.
Please provide the statement of Algorithm C from Section 2.
No.
Let t_1,t_2,t_3,\ldots be the sequence
“MOVE CORRESPONDING SALES TO PURCHASES” means that each elementary data item occurring in both group structures SALES and PURCHASES, identified by the same name within the COBOL record hierarchy, is t...
The proposed idea does not fit the buddy system as ordinarily defined.
A garbage collector can be implemented incrementally so that only a small, bounded portion of the memory is processed during each individual List operation.
Let the available-space list of Algorithm C be the doubly linked circular list maintained by the fields LINK(X),\qquad BACKLINK(X), with list header $AVAIL$.
The term $c_2 M$ arises from the tracing phase of the garbage collection procedure, in which all nodes that are reachable from the list heads are marked by following their structural links.
Let $T$ be the extended binary tree in (11), and let each circular node $v$ be assigned the value equal to the sum of the weights $w_j$ of all external nodes in the subtree rooted at $v$.
We want to construct an extended binary tree minimizing M = \max_{i=1}^m (w_i + l_i), where $l_i$ is the depth of leaf $i$.
**Solution to Exercise 2.
Let U_1,U_2,\ldots,U_q,\qquad V_1,V_2,\ldots,V_r be vertices of a directed graph.
Let the external nodes be numbered $m,m+1,\ldots,2m-1$, and let the internal nodes be numbered $m-1,m-2,\ldots,1$, so that the last internal node created is node $1$, the root.
The original solution fails because it treats the internal path length as if it were a simple additive marking that factors independently over subtrees.
Let $w_1,\ldots,w_m\ge 0$.
By Exercise 1 there is a one-to-one correspondence between binary trees with $n$ nodes and triangulations of a convex $(n+2)$-gon.
Let $n$ be the number of circular (internal) nodes and $s$ the number of square (external) nodes in an extended $t$-ary tree.
**Exercise 2.
The previous argument fails because it treats the choices for the arcs $t_j \to \cdot$ as independent.
Yes, List structures can be threaded analogously to trees.
**Exercise 2.
Let the vertices be labeled $1,\dots,n$, and let every edge be oriented toward the root.
Let $x$ denote the current node.
Let a forest be a collection of extended binary trees, each node carrying a positive weight, and let the weight of a tree be the sum of weights of its leaves.
**Corrected Solution for Exercise 2.
The flaw in the previous solution is that it treated the semicolon-separated display as a syntactic artifact rather than what it represents in the $((3,2,4),(1,4,2))$-construction.
Critical correction starts from the only valid backbone: part (a) and (b) already give a majorization principle, but the previous solution incorrectly tried to replace that with an assumed Huffman str...
Let the root node be $P$.
The structure of an ordered tree with $n$ vertices is independent of labels; only the relative left-to-right ordering of subtrees at each node matters.
Label the vertices of the convex $n$-gon by $1,2,\dots,n$ in cyclic order.
Let $K \leftarrow 0$.
**Solution.
A List can be described as a directed graph in which each node corresponds to a List element or atom, and each pointer (`DLINK`, `RLINK`, or `LLINK`) corresponds to a directed edge from one node to an...
**Corrected Solution to Exercise 2.
**Corrected Solution to Exercise 2.
We prove that a multiset of nonnegative integers $l_1,\dots,l_m$ occurs as the set of root-to-external-node path lengths in an extended binary tree if and only if \sum_{j=1}^m 2^{-l_j} = 1.
We give a fully precise in-place DFS marking algorithm that works for variable-size nodes and uses no auxiliary stack beyond the existing link fields.
Let $k_i$ denote the number of vertices of in-degree $i$.
Let a binary tree have $n$ nodes and internal path length $I$.
Yes.
The internal path length of a binary tree is the sum of the depths of its $n$ internal nodes.
The Prüfer-code framework is correct, but the constraint “exactly $k$ leaves” must be enforced as a surjectivity condition, not as a simple restriction on the alphabet.
Start from the correct expression that survives all prior reductions: Let the prescribed indegree sequence be $x_1,\dots,x_n$, a permutation of a multiset with $k_i$ occurrences of $i$, and $\sum_v x_...
Let $\mathcal{T}_n$ be the set of ordered trees with $n$ vertices whose vertices are labeled by ${1,2,\ldots,n}$.
A correct solution must first remove the ambiguity in the previous attempt: there is only one active control mechanism at any time, and “E-mode” is not a second concurrent process.
**Corrected Solution to Exercise 2.
Let the sides of the convex $(n+2)$-gon be distinct and let one of them be designated as the root side $r$.
**Corrected Solution to Exercise 2.
Use the `REF` fields temporarily to record the correspondence between original nodes and copied nodes.
Let $x(z)$ be the unique formal power series solution of zx(z)^t=\ln x(z), \qquad x(0)=1.
We prove Eq.
The flaw in the previous solution is that it tries to prove lexicographic improvement after swapping two inorder blocks without controlling what the first differing entry actually is.
For (a), first suppose that $G$ is a free tree and $m=n$.
We start from equation (3) of the section, which states that 1 + A(z) + A(z)^2 + A(z)^3 + \cdots = \prod_{n\ge1} \frac{1}{(1-z^n)^{a_n}}.
**Corrected Solution for Exercise 2.
\boxed{\text{True}} Let $G$ be a directed graph that is rooted and contains neither cycles nor oriented cycles.
Let the positive integers be covered by $S_1,\dots,S_k$.
Let $c_n$ be the number of $t$-ary trees with $n$ nodes.
Let $G$ be a directed graph, possibly infinite, and let $R$ be a root of $G$.
By exercise 2.
Let $T$ be the free tree in Fig.
Let $G$ be the directed graph with vertices $V_1,\dots,V_{13}$, where $V_{13}$ is the center pile, and where for $j\neq 13$ there is an arc e_j:V_j\to V_{b(j)}, $b(j)$ being the value of the bottom ca...
We restart from the structure that is actually relevant to the sequence, and we avoid the incorrect passage through Eulerian-trail counting entirely.
Let $(e_1,\ldots,e_n)$ be an oriented walk from $V$ to $V'$.
The given canonical sequence is 3,1,4,1,5,9,2,6,5.
Let the four quarters of a tetrad be denoted \begin{matrix} \text{NW} & \text{NE}\\ \text{SW} & \text{SE} \end{matrix}
Let t_j=\sum_{T_j} w(T_j),\qquad w(T_j)=\prod_{e\in T_j}p(e), where the sum is over all oriented spanning trees of $G$ rooted at $V_j$.
Let $R$ be the original root, and let V_j,V_{a_1},V_{a_2},\ldots,V_{a_t},R be the unique oriented path from $V_j$ to $R$.
Assume for contradiction that no such number $N$ exists.
Let $g_n$ denote the number of distinct oriented binary trees with $n$ vertices, where each vertex has in-degree at most two.
Let the oriented tree be rooted, with every edge directed toward the root.
We are asked to design a computer representation of directed graphs using **one node per arc**, two link fields `ALINK`, `BLINK`, and two one-bit tag fields `ATAG`, `BTAG`, such that: 1.
**Exercise 2.
**Answer:** No.
Let $C_e$ be the fundamental cycle determined by a non-tree arc $e$.
**Corrected Solution to Exercise 2.
Let $G$ be a directed graph with vertices $V_0, V_1, \dots, V_n$.
The solution fails because it tries to classify the six trees using degree sequences, but degree sequences alone do not enumerate all non-isomorphic trees on six vertices.
Let $G$ be a finite or locally finite oriented tree in which every vertex $v\neq R$ has exactly one outgoing arc and the root $R$ has no outgoing arc.
Let A(z)=\sum_{n\ge1}a_n z^n,\qquad a_n\ge 0, and
Let $G$ be a balanced directed graph, so for every vertex $V_j$ the indegree equals the outdegree $\sigma_j$.
Let $A$ be the incidence matrix of exercise 18.
Define the embedding $T \subseteq T'$ for finite ordered trees as follows.
Assume that $G$ contains an oriented walk $(e_1,\ldots,e_n)$ with $n\ge1$ and \operatorname{fin}(e_n)=\operatorname{init}(e_1).
The failure occurs at a single decisive point: the sequence $A_n$ was not derived from the correct functional equation for rooted unlabeled trees.
Let $T$ be a free tree.
The root of $G'$ is $V_1$, since the arcs of the oriented tree are $e_{01}$ and $e_{21}$.
Let $T$ be a free tree with $n$ vertices and two centroids, say $X$ and $Y$.
A finite directed graph can be topologically sorted if and only if it contains no oriented cycles.
True.
The degree of a node in the tree terminology of Section 2.
The root of the oriented tree is the empty sequence $(,)$, which corresponds to the case $n=0$.
Let $C(z)$ be the generating function for the numbers $c_n$, where $c_n$ counts oriented trees with $n$ leaves and with every nonleaf vertex having at least two subtrees.
Take three vertices $A,B,C$ and two arcs, A\to B,\qquad C\to B.
Expand the determinant by choosing, from each row, exactly one entry whose column is assigned to that row’s position in a permutation of ${1,\ldots,m+n}$.
The construction described after (16) already yields an efficient method.
Let the directed graph be processed exactly as in the solution to exercise 2.
**Corrected Solution to Exercise 2.
Let $t$ be any tetrad type in the given nonempty set of tetrad types.
Construct a new directed graph $G'$ by replacing each arc $e_j$ of $G$ with $E_j$ parallel arcs from $\operatorname{init}(e_j)$ to $\operatorname{fin}(e_j)$.
Let $G(z)=\sum_{n\ge1}g_nz^n$, where $g_n$ is the number of distinct oriented binary trees with $n$ vertices.
From exercise 1, write A(z)=z\,e^{C(z)},\qquad C(z)=\sum_{k\ge1}\frac{A(z^k)}{k}.
The claim is **true**: any finite directed graph satisfying (a) and (b) and having no directed cycles must already be an oriented tree, because condition (c) is forced by the other assumptions.
Let the given finite set of tetrad types be $T$.
We are asked to design an algorithm analogous to Algorithm `F` for the _preorder with degrees_ representation of a forest, traversing from **right to left**.
A direct implementation is obtained by maintaining, for each terminal not yet in the growing tree, the cheapest known connection from that terminal to any terminal already in the tree.
Let the graph of the flow chart have vertices corresponding to the boxes, and let the free subtree be $G'$ with edges $e_1,e_2,e_3,e_4,e_9$.
Let the original forest contain $n$ nodes, of which $m$ are terminal.
The exercise asks for a program that performs algebraic simplification on the tree representation of formulas described in Section 2.
The ordinary Algorithm `E` maintains a forest of equivalence classes.
Since $G'$ is a finite free tree with at least one edge, the auxiliary result in Section 2.
Algorithm `2.
Let a reduced flow chart be obtained from an original flow chart by combining a set of vertices into a supervertex and summing the corresponding edge flows.
Let $(V_0,V_1,\ldots,V_n)$ be a walk from $V = V_0$ to $V' = V_n$.
Let $P_T$ and $Q_T$ denote the preorder and postorder orderings of the nodes of a tree $T$.
Let the forest be given in preorder sequential representation: - `INFO1[j]` contains the node information.
Exercise `12` specifies `DIFF[8]` for exponentiation, corresponding to rule `(19)`: D(u \uparrow v) = D(u) \times \bigl(v \times (u \uparrow (v - 1))\bigr) +
Let the nodes be numbered $1,2,\ldots,n$ in their location order.
Let $F$ be a forest and let $u, v$ be nodes in $F$.
Process the edges one at a time, maintaining a set of connected components of the vertices already joined by the selected edges.
The reviewer's objections are decisive.
Let the terminals $T_1,\ldots,T_n$ be the vertices of a graph $G$, and let each wire correspond to an edge.
We reconstruct the solution by systematically enumerating all simple cycles that start and end at $B$, using the adjacency information from Fig.
Let the tree (or forest) be stored in the postorder-with-degrees representation (9).
The splitting of edges $e_{13}$ and $e_{19}$ was introduced solely to satisfy the formal definition of a graph in which no two edges may join the same pair of vertices.
Let the nodes be linked initially by the arbitrary linear list \text{FIRST} \to x_1 \to x_2 \to \cdots \to x_n \to \Lambda, through their present `RLINK` fields.
**Problem:** Give a formal proof (or disproof) of the validity of Algorithm `A`.
Let $S$ be any nonempty proper subset of the terminals, and let $\bar S$ be its complement.
Exercise 14 asks for the running time of the `COPY` subroutine of Exercise 13.
Let the given forest be represented in postorder with degrees as in representation `(9)`.
Assume first that the graph is connected.
Let $R(X,Y)$ denote the statement X \succeq Y.
The routine `DIV` computes the derivative of a formula of the form $u / v$ with respect to the variable $x$, according to rule `(18)`: D(u/v) = D(u)/v - (u \times D(v))/(v \uparrow 2).
We are asked to give a table analogous to `(15)` and a diagram analogous to `(16)` showing the trees present after Algorithm `E` has processed all equivalences in `(11)`.
We are asked to **give specifications for the routine `DIFF[8]`**, which handles differentiation of expressions of the form $u \uparrow v$ (exponentiation) in the context of the differentiation algori...
Yes.
The previous solution does not address the exercise.
**Problem restatement.
Algorithm `E` constructs a forest whose nodes are the elements $1,\ldots,n$.
A triply linked tree contains, for each node $x$, three pointers: $PARENT(x)$ to the parent of $x$, $LCHILD(x)$ to the leftmost child of $x$, and $RLINK(x)$ to the next sibling of $x$.
We are asked to reason about **descendant number sequences** in preorder.
The reviewer's objections are decisive: the exercise cannot be solved from a generic description of polynomial addition.
Algorithm `A` adds two polynomials represented as trees.
We are asked to design an algorithm that answers the query "`Is $j \equiv k$?
**Corrected Solution to Exercise 2.
We are asked: > If we had only `LTAG`, `INFO`, and `RTAG` fields (not `LLINK`) in a level-order sequential representation like (8), would it be possible to reconstruct the `LLINK`s?
The relation $9 \equiv 3$ serves only to place the element $9$ into the equivalence class containing $3$.
We are asked to determine whether the statement > "The terminal nodes of a tree occur in the same relative position in preorder and postorder.
Exercise 19 asks for an algorithm analogous to Algorithm `S`, but for the calculation of the preorder successor.
Let the right-threaded binary trees use the conventions of the section: - `LLINK(P)` is either a left child or `\Lambda`.
We aim to construct an algorithm analogous to Algorithm `T` that traverses a binary tree in _preorder_, visiting each node exactly once, and then prove its correctness by induction on the number of no...
Let $F$ and $F'$ be forests whose nodes in preorder are $u_1, u_2, \dots, u_n$ and $u'_1, u'_2, \dots, u'_{n'}$, respectively.
Let $T$ be a nonempty binary tree in which every node has either $0$ or $2$ children.
Let `T` be an unthreaded binary tree, represented in the standard form of (2), and let `P` be a pointer to a node of `T`.
Let `T` be a pointer to a binary tree of $n$ nodes with the conventional representation `(2)`, that is, each node `P` has two link fields `LLINK(P)` and `RLINK(P)` pointing either to the left or right...
Let $F$ be a forest containing $t$ trees.
Let a forest $F = (T_1, T_2, \dots, T_n)$ be given, with nodes numbered in Dewey decimal notation as in Section 2.
Algorithm `C` is intended to construct a new binary tree whose nodes contain the same information as the original tree and whose link structure is identical, regardless of whether a field represents a...
We first interpret the definition of $\preceq$ as a recursive lexicographic comparison of trees: the empty tree precedes every tree; among nonempty trees, the roots are compared first; if the roots ag...
Let `T` be the pointer to the right-threaded binary tree, and let `AVAIL` be the head of the list of available nodes.
Algorithm `C` constructs a new tree by creating a new node corresponding to each node of the original tree, copying the information field, and preserving the left and right subtree relationships.
The List $L = (a,(L))$ consists of two elements: the first element is the atom $a$, and the second element is a List identical to $L$ itself.
Let the preorder of the binary tree be $u_1 u_2 \dots u_n$ and the inorder be $v_1 v_2 \dots v_n$.
Let $H_n$ denote the largest number of entries simultaneously present in stack $A$ during the execution of Algorithm $T$ on a binary tree with $n$ nodes.
Let a binary tree have $n$ nodes.
The stack grows only in step `T3`, where the current value of `P` is pushed onto `A` and then `P` is replaced by `LLINK(P)`.
Let the Dewey decimal notation of a node be d_1.
Let us define the new traversal order recursively, as in the exercise: 1.
Algorithm `T` uses an auxiliary stack `A` in consecutive memory locations.
Let the nodes of a binary tree be distinct.
Let `T` denote the root of the binary tree in the figure.
Let us reformulate the ordering of Exercise `2.
Let $T$ be a binary tree represented in memory with nodes containing three fields: `LTAG(P), LLINK(P), RLINK(P)`.
Let $D(T)$ denote the double-order sequence of a binary tree $T$, as defined in exercise 18.
Exercise 31 refers to Algorithm `I` for insertion into a right-threaded binary tree.
Let $(S, \prec)$ be a well-ordered set.
Let `P` point to a node of a binary tree, and consider `Q = P*`, the successor of `NODE(P)` in preorder.
In tree (2), the root is $A$, so $RLINK(T)$ points to $C$.
Let a binary tree have $n$ nodes, with preorder sequence u_1 u_2 \dots u_n and inorder sequence
The double-order traversal visits each node twice.
A _right-threaded_ binary tree contains ordinary left links and either ordinary right links or right threads.
Let a binary tree with `n` nodes be traversed using Algorithm `T`.
Let $I(T)$ denote the number of internal nodes, namely nodes having two children, and let $L(T)$ denote the number of leaves.
A _ternary tree_ is defined recursively as either empty, or consisting of a root together with three ordered ternary subtrees, called the first, second, and third subtrees.
We are asked to give an algorithm analogous to Algorithm `S` that determines the preorder successor `P*` of a node `P` in a threaded binary tree with a list head as in `(8), (9), (10)`.
Exercise 25 defines a linear ordering on binary trees recursively.
Let the partial order on the nodes of a forest be defined by u < v whenever $v$ is a descendant of $u$.
A postorder traversal must process a node only after both of its subtrees have been traversed.
Let $N$ be the total number of nodes and let $t$ be the number of terminal nodes.
In the representation (2), each node contains exactly two links, `LLINK` and `RLINK`.
Let $B$ be a binary tree.
We are asked to write a MIX program that implements the algorithm of Exercise 21, which traverses an unthreaded binary tree in inorder _without using any auxiliary stack_, modifying the `LLINK` and `R...
No.
The statement claims that "The terminal nodes of a binary tree occur in the same relative position in preorder, inorder, and postorder.
Let a node $P$ of a threaded binary tree be given.
We are asked to draw trees analogous to those in `(7)` corresponding to the formula y = e^{-x^2}.
Let the roots of the trees of the forest $F$ be regarded as siblings arranged from left to right.
Let the representation of a node be the binary string $\alpha$, where the root is represented by `"1"`, the left child of $\alpha$ is $\alpha0$, and the right child is $\alpha1$.
We represent binary trees graphically using standard European paper sizes `A0`, `A1`, `A2`, .
Let $V[1], \dots, V[n]$ be the variables of the system, and let a step of the simulation specify a small subset of these variables to be updated simultaneously.
The desired change is that a user waiting on floor `IN` should enter the elevator only if the elevator is accepting passengers whose desired direction agrees with the user's destination.
Activity `E9` in the elevator coroutine is a scheduled action that occurs after the completion of certain steps in the elevator's operation, specifically following step `E6` (door-closing and possible...
A partially ordered set $(P, \preceq)$ corresponds to an unordered tree if and only if $P$ has a least element $r$ (the root) and, for every $x \in P \setminus {r}$, the set of elements less than $x$...
Suppose we have a $k$-dimensional array `A[I_1,I_2,\ldots,I_k]` with $l_r \le I_r \le u_r$ for $1 \le r \le k$.
In a sequential representation of a sparse matrix, the nonzero entries of each row may be stored consecutively, together with their column indices.
In lexicographic order, the elements are stored as A[0,0],A[0,1],\ldots,A[0,n],A[1,1],A[1,2],\ldots,A[1,n],\ldots,A[n,n].
Let a _general deque_ be a data structure that allows insertion and deletion at both ends.
Let `RESULT` be initially empty.
Let the matrix (1) have indices $0 \le J \le m$ and $0 \le K \le n$, and suppose each node occupies $c=2$ words.
Let the matrix elements be stored in the order in which they are created.
For the expression $2(a - b/c)$, the corresponding tree has the multiplication $2 \cdot (\cdot)$ at the root.
By definition, the _degree_ of a node is the number of its children.
Let $A$ be the nearest common ancestor of $X$ and $Y$.
The concepts of level, degree, parent, and child extend naturally to infinite trees defined as collections of nested sets.
A tridiagonal matrix contains exactly the elements $A[I,J]$ for which $|I-J|\le 1$.
If the lower triangular matrix is indexed by $1 \le k \le j \le n$, the lexicographic order becomes A[1,1],\; A[2,1],A[2,2],\; \ldots,\; A[n,1],A[n,2],\ldots,A[n,n].
The path from node $X$ to the root consists of all nodes that are ancestors of $X$, including $X$ itself and the root.
Represent the circular list by storing in each node $x_i$ a single link field L(x_i)=\operatorname{addr}(x_{i-1})\oplus \operatorname{addr}(x_{i+1}), where $\oplus$ denotes bitwise exclusive-or of mac...
With three nodes $A$, $B$, and $C$, every tree has exactly two edges.
Let `J` and `K` be stored in index registers `I1` and `I2`, respectively.
In representation (1) of a doubly linked list, there are distinguished variables `LEFT` and `RIGHT` giving the locations of the leftmost and rightmost nodes, respectively.
For $k=2$, the classical Cantor polynomial p(i_1,i_2)=\frac{(i_1+i_2)(i_1+i_2+1)}2+i_2 has the required property.
We begin with the given upper-triangular matrix A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}.
A direct transcription of Algorithm $S$ into MIXAL is possible if the fields `ROW`, `COL`, `VAL`, `LEFT`, and `UP` are assigned fixed offsets within each node, and if the floating-point instructions `...
The `DECISION` subroutine is called whenever the elevator is in a dormant condition and a new request may require a change of state.
False.
The scenario given concerns the discrete simulation of the Caltech Mathematics building elevator, using the routines described in Section 2.
Let the matrices be represented by orthogonal circular lists as in Fig.
In both binary trees of (1), the root is `A`.
A binary tree is not necessarily a tree in the sense of graph theory.
Step `E8` is the action that occurs after the elevator has moved one floor in its current direction.
Each nonzero entry is represented by a node containing three words.
To demonstrate that the elevator system requires three independent binary variables per floor, we must exhibit sequences of button presses that show each variable can be set or cleared independently o...
The statement `JANZ CYCLE` at line 154 was intended to skip the "give up" activity `U4` for a user if the elevator had already arrived at the user's floor.
For the array `A[i,j,k]` with $0 \le k \le j \le i \le n$, storing the elements in lexicographic order of the indices means that `k` varies fastest, then `j`, then `i`.
We can generalize the two-dimensional trick of equation (10) to the case of six tetrahedral arrays `A[I,J,K]`, `B[I,J,K]`, `C[I,J,K]`, `D[I,J,K]`, `E[I,J,K]`, `F[I,J,K]`.
Let `A` be the given sparse matrix represented with circularly linked row and column lists, with head nodes `BASEROW[i]` and `BASECOL[j]`.
Let T_k(n)=\#\{(i_1,\ldots,i_k):0\le i_k\le\cdots\le i_1\le n\}.
There are three nodes labeled $A$, $B$, and $C$.
Let the elements of $S$ be regarded as nodes, and define the parent of a node $\alpha.m$ to be $\alpha$ when $m=1$, and define the nodes $\alpha.1,\alpha.2,\ldots$ to be the children of $\alpha$ whene...
The node `Z[1,2,2]` is the second child of the second child of the first tree in the forest `Z`.
In the elevator simulation, the program must frequently insert and delete users from various lists representing queues on floors (`QUEUE[IN]`) and passengers inside the elevator (`ELEVATOR`).
Maintain three auxiliary arrays: A[\,],\qquad B[\,],\qquad C[\,], together with a counter $t$, initially $0$.
Circular lists are used because they eliminate end-of-list cases.
We proceed by induction on the number of nodes in the tree.
The matrix should remain in the sparse orthogonal-list representation throughout.
A natural notation for nodes of a binary tree extends the Dewey decimal system by encoding each branching choice as a binary digit.
Not every collection of nested sets corresponds to a tree in the sense defined in Section 2.
In the notation of (13), let the pivot element be $a$, let the other nonzero elements in the pivot row be represented by the coefficients $b$, and let the other nonzero elements in the pivot column be...
Two nodes `X` and `Y` of a tree are said to be _$m$th cousins, $n$ times removed_ if the following holds.
A more efficient organization of the personnel table can be obtained by employing _orthogonal indexing_ rather than single-directional linked lists.
Since $n$ is variable, the allocation function must encode the size of the matrix together with the indices.
In List (3), the notation `L[i_1, i_2, \dots]` refers to the node reached by starting at the root of `L`, following the $i_1$-th child, then the $i_2$-th child of that node, and so on.
Let the building under consideration have $F$ floors, numbered $0,1,\dots,F-1$, and a single elevator with automatic controls.
Let the circular list be N_1 \to N_2 \to \cdots \to N_k \to N_1, and suppose both `PTR₁` and `PTR₂` point to nodes of this same list.
In representation (4), a circular linked list is maintained with a distinguished head node `HEAD`.
Assume that the relation "$\subset$" satisfies properties (i) and (ii) of a partial ordering, namely: (i) Transitivity: x \subset y,\ y \subset z \implies x \subset z.
Program `T` in Section 2.
In Algorithm `T`, the front element of the queue is examined in step `T5`, but it remains on the queue until step `T7`.
Let $L_1$ consist of nodes a_1, a_2, \dots, a_m, with pointer $\mathrm{PTR}_1 = a_m$ and
Let $G=(V,E)$ be the finite directed graph of game positions.
Let $S = {x_1, \dots, x_n}$ be a finite set with a partial ordering $\preceq$, and let $(a_{ij})$ be its incidence matrix defined by a_{ij} = \begin{cases} 1,& \text{if } x_i \preceq x_j,\\ 0,& \text{...
Program `T` should reject any input that violates the assumptions under which the algorithm was designed.
Let $N=L_\infty-L_0$ be the total memory size.
Assume first that $\preceq$ is a linear ordering on the finite set $S$.
Let $n$ denote the number of sequential stacks, and let $m$ denote the total number of insertions $a_1, a_2, \ldots, a_m$, where $a_j \in {1, 2, \ldots, n}$ specifies the stack chosen for the $j$th in...
Let $a_1,a_2,\ldots,a_m$ be a sequence of insertions as in exercise 9, with $n=2$.
Yes.
The decreasing order of the `ABC` fields makes it possible to combine two polynomials efficiently by a single systematic scan through their lists.
Let $n$ be the number of nonzero terms of `polynomial(P)`.
In operation (14), a new node is inserted at the rear of the queue.
Exercise 2.
Let $B_n$ denote the family of all $2^n$ subsets of an $n$-element set, partially ordered by inclusion.
No.
Let E_m=\frac{m!
Let the linked linear list be represented by the pointer `FIRST`, with each node containing the fields `INFO` and `LINK` as in (3).
Let $(k_1,k_2)$ denote the numbers of occupied locations in the two tables when the process terminates.
The exercise asks for an empirical evaluation of Algorithm $G$ under varying stochastic patterns of insertions and deletions.
Let the queue be represented by link variables `F` and `R`, pointing respectively to the front and rear nodes of the queue, with the convention that `F = \Lambda` if and only if the queue is empty.
In the representation used in §2.
With $0$-origin indexing, the available array positions are X[0], X[1], \ldots, X[M-1].
We are asked to write a MIX program that implements the subroutine allocation algorithm described in Exercise 26.
Let the empty circular list be represented by PTR = LOC(PTR), instead of
Let the directed graph of Fig.
Let the linked list be represented by the pointer `FIRST` to the first node, and assume each node has fields `INFO` and `LINK` as in (3).
Algorithm `A` does **not** work properly when `P = Q`.
Algorithm `T` in Section 2.
We are asked to design an algorithm to determine the relocation addresses for a set of subroutines to be loaded from a tape library, taking into account the dependencies among subroutines.
A relation $\preceq$ is a partial ordering if and only if it is reflexive, antisymmetric, and transitive.
The principal advantage is that the circular representation eliminates special cases at the ends of the list.
Let $S$ be a set of $n$ elements.
We are asked to write a general-purpose MIX subroutine to perform the linked-stack insertion operation corresponding to equation (10).
Let - $p$ denote the number of nonzero terms of `polynomial(P)`, - $q$ denote the number of nonzero terms of `polynomial(Q)` at entry, - $m$ denote the number of nonzero terms of `polynomial(M)`.
We are asked to write a general-purpose MIX subroutine to perform the deletion operation from a linked stack, corresponding to operation (9) in Section 2.
We are asked to design an efficient algorithm to "erase" an entire circular list by placing all its nodes onto the `AVAIL` stack.
The pointer `Q1` trails `Q` by one node to simplify insertions and deletions in `polynomial(Q)`.
Let a circular list be represented as in (1), with `PTR` pointing to the last node, so that `LINK(PTR)` points to the first node.
Operation (8) does not mention `OVERFLOW` because it assumes that a new node can always be obtained from the `AVAIL` stack or the storage pool.
Let $S$ be a finite partially ordered set whose ordering relation is given as a collection of pairs $(u,v)$, meaning $u<v$.
We are asked to write a subroutine `ERASE` that adds a polynomial, represented as a circular list with a sentinel node, to the `AVAIL` list.
Let C=L_\infty-L_0 be the total amount of available memory.
In program (10), overflow is detected by LD1 AVAIL J1Z OVERFLOW When `AVAIL = Λ`, register `rI1` contains zero and control passes to `OVERFLOW`.
Algorithm `T` performs a topological sort on a directed graph whose vertices are the objects being ordered and whose relations are the pairs $j \prec k$ represented in the successor lists.
We are asked to create a subroutine `COPY` that produces a complete duplicate of a given polynomial represented as a circularly linked list with a sentinel node, preserving the original list and retur...
Each polynomial is represented by a circular list whose nodes are arranged in decreasing order of the field `ABC`.
Algorithm `T` of Section 2.
We wish to adapt Algorithms `A` (addition) and `M` (multiplication) for polynomials in a single variable $x$, allowing exponents up to $b^3 - 1$.
Let $(S,\preceq)$ be a partially ordered set.
Let $n$ be the number of stacks and $m$ the total number of insertions performed, as in exercise 9.
Let two buffers for output be \texttt{OUTBUF1}, \quad \texttt{OUTBUF2}, each of length $100$ words.
Exercise 3 introduces address modifications of the form `I1:I2`, where modification 7 means: interpret the addressed location as another address specification and continue the address-modification pro...
With a queue, the first item inserted is always the first item removed.
The output tape is used in two alternating buffer areas, 1000–1099 and 2000–2099.
We examine the linked-list structure depicted in equation (3), where `TOP` points to the first element of a stack and each element contains a `NEXT` field referencing the next element in the stack.
Let M=\max(L,C).
A queue permits insertion at the rear and deletion at the front.
We are asked to find, for each query, a hotel room whose size meets a minimum requirement and whose room number is closest to a preferred room number.
We have $\operatorname{LOC}(\operatorname{CONTENTS}(V)) = V$ precisely when $V$ is itself the location of some variable.
Let $b_n$ denote the number of permutations of $1,2,\ldots,n$ obtainable with an input-restricted deque.
The flaw in the previous solution is the assumption that link traversal alone can accumulate multiple index register contributions during effective address formation.
Consider the scenario of Fig.
Let $p_1p_2\ldots p_n$ be a permutation of $1,2,\ldots,n$.
We are asked to modify Algorithms and Programs A, R, and B of Section 1.
We are asked to extend the sequential allocation scheme for multiple stacks, as described by equations (9) and (10) and the repacking algorithm (Algorithm G), to the case in which one or more of the l...
We are asked to define the concept of an _admissible_ sequence of the symbols $S$, $Q$, and $X$ for an output-restricted deque, in such a way that every permutation of $n$ elements obtainable with the...
Let the successive operations be represented by a sequence a_1,a_2,\ldots,a_m, where each $a_t$ denotes the stack on which the $t$th insertion is performed.
A jump trace routine should retain only the information necessary to reconstruct the control flow of the program.
We consider permutations of $1234$ obtained by using deques, either input-restricted or output-restricted, following the definitions in Section 2.
We are asked to repeat exercise 9 under the assumption that there is only _one_ buffer.
Let a sequence consist of the symbols `S` and `X`, where `S` means "move the next car from the input into the stack" and `X` means "move the top car from the stack into the output.
We wish to design an algorithm analogous to Algorithm A, but instead of placing a new card on top of the pile, we place it _face down_ at the _bottom_ of the pile.
The previous submission failed because it attempted a high-level interpreter sketch without constructing an actual MIXAL self-tracing program.
In the multiple-buffering scheme described in the text, the computational program and the CONTROL coroutine communicate by means of the operations ASSIGN and RELEASE.
Yes.
Algorithm A inserts a card at the top of the pile.
Steps B1-B3 insert a new card at the top of the linked pile.
We model the system as a pool of $N$ indistinguishable buffers.
The COMPUTE coroutine, as described in Section 1.
To modify the CONTROL coroutine for card input so that input terminates upon reading a card with a `".
Let trace routines $T_A$ and $T_B$ occupy disjoint regions of memory.
Let $p = p_1 p_2 \ldots p_n$ be a permutation of $12\ldots n$, and let $D$ be a deque that is neither input- nor output-restricted.
Let the three buffers of exercise 9 be denoted by $\mathrm{BUF1}$, $\mathrm{BUF2}$, and $\mathrm{BUF3}$.
Step `G4` of Algorithm `G` sets \alpha \leftarrow 0.
Let each buffer contain exactly $100$ consecutive words.
We are asked to analyze the same program as in Exercise 1.
Let T_i be the total time during which I/O device $i$ is actually transferring data, for $1\le i\le n$, and let
To initialize the `WORDIN` subroutine (4) correctly, we must ensure that the first call to `WORDIN` will access the first word of the initial buffer and that the subsequent calls will correctly advanc...
We are asked to formulate the "green-yellow-red-purple" algorithm, illustrated in Fig.
The correct sequence to bring `NEXT(TOP)` into register A is sequence (a), `LDA TOP(NEXT)`.
In addition to the front pointer `F` and rear pointer `R`, introduce the convention that the deque occupies the circular array `X[1],\ldots,X[M]`, with `F` pointing to the front element and `R` to the...
We denote by $b_n$ the number of permutations on $n$ elements obtainable with an input-restricted deque.
Let $T$ denote the time required for one input-output operation on the device, and let $C$ denote the computation time performed between successive input-output requests.
We are asked to determine whether the permutations $325641$ and $154623$ of six railroad cars numbered $1,2,3,4,5,6$ can be obtained using a single stack in the manner of Fig.
Using the memory configuration of Fig.
Let $a_n$ denote the number of permutations obtainable with a stack, as defined in Exercise 4, and let $b_n$ denote the number of permutations obtainable with an input-restricted deque, equivalently w...
The assignment `CARD ← NODE(TOP)` copies the entire node at the address `TOP` into the variable `CARD`.
(a) Yes.
Let $a_n$ denote the number of permutations of ${1,2,\ldots,n}$ obtainable by a stack of the type described in exercise 2.
Let $X[1], \ldots, X[M]$ be the memory locations for the queue, with `F` and `R` the front and rear pointers, respectively, as in (6a) and (7a).
The pile of cards is represented as a linked list, with `TOP` pointing to the first node and `NEXT` giving the link to the next node.
We are asked to write a complete MIX program to copy 100 blocks from tape unit 0 to tape unit 1 using three buffers, with maximal speed.
The purpose of (8) is to replace fixed-base addressing by relative addressing when the base location $L_0$ varies.
Let $d_n$ denote the number of permutations of ${1,2,\ldots,n}$ that can be obtained from the input sequence 1,2,\ldots,n by means of a general deque.
We are asked to modify the input-output subroutines of the MIX simulator so that the execution of the `IN` and `OUT` operators does not trigger immediate transmission.
Self-modifying code is frowned upon primarily because it introduces unpredictability and impedes program correctness verification.
Writing trace output onto tape rather than printing directly is preferable because tape allows information to be recorded sequentially without interrupting the execution of the program being traced.
The trace routine in Section 1.
The forward direction is correct and standard.
Let an array $X[0 \dots l+m+n-1]$ be partitioned into consecutive blocks \alpha = x_0 \dots x_{l-1}, \quad \beta = x_l \dots x_{l+m-1}, \quad \gamma = x_{l+m} \dots x_{l+m+n-1}.
Let the permutation be decomposed into cycles, and let $\alpha_k$ denote the number of cycles of length $k$.
Let $S_n$ be the set of all permutations of ${1,2,\dots,n}$, each taken with equal probability $1/n!$.
Let $p_{nkm}$ denote the probability that a permutation of $n$ elements contains exactly $k$ cycles of length $m$.
A coroutine requires at least two control contexts that repeatedly suspend and resume execution at interior points, together with a mechanism for preserving the point of suspension.
Let \pi = (2\,3)^{e_2}(4\,5)^{e_4}\cdots(2m\,2m+1)^{e_{2m}} \, (1\,2)^{e_1}(3\,4)^{e_3}\cdots(2m-1\,2m)^{e_{2m-1}}, where each $e_k \in \{0,1\}$.
Working
Let the current permutation be P = (a_1,a_2,\dots,a_n).
Let $\pi$ be a permutation expressed in cycle form as a product of disjoint cycles: \pi = (x_{11} x_{12} \dots x_{1n_1})(x_{21} x_{22} \dots x_{2n_2})\cdots(x_{k1} x_{k2} \dots x_{kn_k}), where each c...
Consider a general-purpose von Neumann machine with a standard assembly language that supports subroutine calls, a program counter `PC`, and a stack or a fixed set of registers for storing return addr...
Let the MIX simulator execute an instruction by the control transfer at label `CYCLE` after each simulated instruction, as specified in Program M.
**Exercise 1.
We are asked to modify the trace routine so that it restores register J upon leaving, assuming $rJ \neq 0$.
Calling sequence: either $\mathrm{JMP}\ MAX100$ or $\mathrm{JMP}\ MAXN$.
Let $A$ denote the total number of executions of the assignment j\leftarrow i in Algorithm $J$.
The subroutine `FCHECK` is called whenever an instruction uses a field specification.
The previous solution fails because it changes the problem into run-length encoding and introduces non-MIX arithmetic and nonstandard conventions.
Let $J_n$ denote the Josephus permutation with step size $m=2$ on the set ${1,2,\dots,n}$.
Let $S_1, S_2, \ldots, S_M$ be subsets of a fixed universe $U$.
Let S=\sum_{1<k\le n}\frac{H_k}{k(k-1)}.
Yes.
The key correction is to treat $AX$, $BX$, $CX$ as **mutable continuation cells containing addresses**, and to use **indirect transfer through the cell contents**, not as fixed instructions.
Equation (25) gives the number of derangements as P_{n0} = n!
For $1\le k\le n$, let R_k=(n\ n-1\ \cdots\ k).
Let $n$ be a positive integer, and let $J_{2n+1}$ denote the Josephus permutation for $2n+1$ people with step size $m=2$.
Let a permutation of $n$ objects have exactly $\alpha_k$ cycles of length $k$ for each $k \ge 1$, with only finitely many nonzero $\alpha_k$.
Let the array be x_0x_1\ldots x_{m-1} = \alpha,\qquad x_mx_{m+1}\ldots x_{m+n-1} = \beta, so that the original array is $\alpha\beta$ of length $m+n$.
Let the initial permutation in linear notation be $1324$.
The missing routine must be written in the conventions of Program M.
The coroutine linkage is OUT STJ INX OUTX JMP OUT1 IN STJ OUTX INX JMP IN1
The three `CMPA PERIOD` instructions in `OUT` serve to detect the end-of-input marker, namely the period character `.
In the coroutine linkage of §1.
Exercise 1.
Let N=a\,m_1m_2\cdots m_t, and for each $j$ $(1\le j\le t)$ let
We start by using the standard cycle enumeration formula.
Let the initial array be indexed $0,1,\ldots,l+m+n-1$ and written x_0x_1\cdots x_{l+m+n-1}=\alpha\beta\gamma, where
Let r=\alpha_1+\alpha_2+\alpha_3+\cdots be the total number of cycles of the permutation, including the one-cycles.
Let $m>1$.
We are asked to generalize the MAXN subroutine to find the maximum value of the sequence X[a], X[a+r], X[a+2r], \dots, X[n], where $r$ and $n$ are parameters and
The proposed solution addresses the stated integral J(x)=\int_0^{y x^{1/4}} e^{-u}\left(1+\frac ux\right)^x\,du, and attempts to derive an asymptotic expansion through terms of order $O(x^{-2})$.
A linkage mechanism not using the J-register must supply the return address explicitly and must provide a way for the subroutine to transfer control back to that address without modifying program inst...
Assume that the trace routine is tracing itself, and that the two instructions ENTX LEAVEX JMP *+1 have been inserted immediately before `ENTER`.
We will construct a fully correct MIX solution to **Exercise 1.
**Corrected Solution for Exercise 1.
Let $C_r$ denote the number of $r$-cycles occurring among all $n!$ permutations of $n$ elements.
The failure is entirely caused by using a non-existent MIX instruction `JLE`.
The semantics of `JSJ X` in MIX are: 1.
The subroutine MAXIMUM, as specified in equation (4), has the calling sequence `JMP MAXIMUM` and the entry condition `rI1 = n` with $n \ge 1$.
The proposed solution fails because MMIX provides no arithmetic or shift instructions that operate directly on index registers $rI_0,\dots,rI_7$.
The product of the two given permutations is computed by applying the right-hand permutation first, followed by the left-hand permutation.
A card-loading routine occupying at most two cards is required.
The key point is that this program is not a low-level I/O simulation task.
Let G(z)=\sum_{n\ge 0} a_n z^n.
To produce a **fully corrected solution** for TAOCP, Exercise 1.
**Corrected Solution to Exercise 1.
Let $\langle a_n \rangle = \langle n! \rangle$.
The execution times given in Section 1.
The proposal’s failure comes from two independent MIX semantics issues: (i) `MOVE` cannot broadcast from a single word, and (ii) index registers cannot be compared directly.
Let $H$ be the MIX word representing the halt instruction.
We are asked to find a generating function $G(z)$ such that the coefficient of $z^n$ is [z^n]G(z) = \sum_{k_0+2k_1+4k_2+8k_3+\cdots=n} \binom{r}{k_0}\binom{r}{k_1}\binom{r}{k_2}\binom{r}{k_3}\cdots.
Let S=\sum_k [w^k](1-2w)^n\,[z^{\,n-k}](1+z)^{2n-2k}.
Let $D$ be the signed integer formed from $(rA,rX)$, and let $m\neq 0$ be the signed integer in $M$.
Algorithm $B$ maintains a table $T$ such that, after a cycle has been completely scanned, the effect of that cycle has already been incorporated into the current permutation.
Let F_m(n,r):= \sum_{k_1,\ldots,k_m\ge 0} \binom{r}{n-k_1} \binom{k_1}{n-k_2}\cdots
The correct notion is very strong: an instruction is equivalent to `NOP` only if, for every initial machine state, it leaves **all registers, memory, indicators, and control state unchanged**.
**(a) Can the J-register ever be zero?
The exercise asks for all MIX operators that can affect the setting of the index register $rI1$.
A MIX byte must be capable of representing at least $64$ distinct values and at most $100$ distinct values.
Let (6) be a MIX word written in its physical layout as (6):\quad \pm \; \text{AA} \; \text{I} \; \text{F} \; \text{C}, where AA is the address field, I the index field, F the field specification, and...
The transformation is given by (a,b,c,d,e,f)\leftarrow(c,d,f,b,e,a), which induces the mapping $a\mapsto c$, $b\mapsto d$, $c\mapsto f$, $d\mapsto b$, $e\mapsto e$, $f\mapsto a$.
The instruction on line 12 is located at `EXIT`, and it is initially assembled as `JMP *`, that is, a jump to itself.
We start from the right-hand side and extract the coefficient of $z^n$.
No.
The symbol defined by `X EQU 1000` does not reserve storage; it only equates the symbol `X` with the number $1000$ during assembly.
The previous solution failed because it imported an incorrect model of `MOVE` and then built optimality arguments on top of it.
**Solution to Exercise 1.
Assume that all registers, toggles, and memory locations initially contain zero.
Kí hiệu phép hoán vị đã cho là $(acf)(bd)$, trong đó $e$ là điểm bất động và bị lược bỏ trong ký hiệu chu trình.
Let $f(x)=2x \bmod 7$ on ${0,1,2,3,4,5,6}$.
Let rA = + 0 a b c d, rX = + e f g h i.
For integers $0<p<q$, define H_{p/q}=\sum_{n\ge1}\left(\frac1n-\frac1{n+p/q}\right).
The goal is to use the smallest possible number of MIX memory locations.
In MIX assembly (as used in _The Art of Computer Programming_), symbolic addresses of the form $kH$ and $kB$ refer to the $k$-th occurrence of the symbol $k$, searching forward (H = “here or ahead”) o...
Program `P` uses the output instruction OUT buffer(PRINTER) on lines 25 and 35, but nowhere uses
The sign in the address field is part of the instruction encoding and is not constrained by the requirement on memory references.
Chương trình được đặt tên là "MYSTERY PROGRAM" chứa một chuỗi lệnh MIXAL, và yêu cầu là xác định chức năng của chương trình bằng cách phân tích từng lệnh theo tay.
We are asked to analyze the effect of a change in the initial value of `rX` on the `DIV` instruction in the last example on page 133.
For `JOV 1001`, if the overflow toggle is on, control transfers to location 1001 and the toggle is turned off.
Bài toán yêu cầu liệt kê **mọi toán tử MIX có thể ảnh hưởng đến overflow toggle**, tức là những lệnh có thể **đặt** (set) hoặc **xóa** (clear) overflow toggle theo đúng đặc tả của MIX.
(a) Let $n$ be a composite integer, $n > 1$.
We will construct fully **correct solutions** for Exercise 1.
Four adjacent bytes represent integers from $0$ through $16{,}777{,}215$ by the table in Section 1.
We restart from the MIX model and from the exact conditions in Table 1.
In MIX, each alphanumeric character is represented by one byte, since a byte holds at least $64$ distinct values and is used for character coding in input-output operations.
Bài toán không yêu cầu chứng minh một mệnh đề duy nhất, mà yêu cầu đề xuất các mở rộng cho MIX sao cho mọi chương trình được viết đúng theo tinh thần của Mục 1.
The comparison indicator $\mathrm{CI}$ takes one of the three values $\mathrm{LESS}$, $\mathrm{EQUAL}$, or $\mathrm{GREATER}$.
Let P=(acfg)(bcd)(aed)(fade)(bgfae), with permutations composed from right to left.
We are asked to express the generating function \sum_{n \ge 0} n^m z^n in the form
Part (a) The original derivation is correct.
We are asked to determine the effect on the execution timing of Program $A$ if the assumption that all blank words appear at the extreme right of the input is removed.
The problem asks for the timing characteristics of Program $B$, expressed through quantities $A, B, \ldots, Z$, and then to rewrite the total time in terms of $X, Y, M, N, U, V$ defined in (19) togeth...
The instruction format of a MIX word places the sign in position $0$, the address in bytes $1$–$2$, the index specification in byte $3$, the field specification in byte $4$, and the operation code in...
By the addition formulas for the sine function, \sin\left(\frac{\pi}{2}+w\right)=\cos w.
We wish to show that \sum_{k=0}^{n} \binom{n}{k} F_{m+k} is a Fibonacci number.
The recurrence relation is a_{n+2}=a_{n+1}+6a_n, with initial conditions
Let a position be denoted by $(N,p)$, where $N$ is the number of chips remaining and $p$ is the number of chips removed on the preceding move.
Let $D_n$ denote the determinant of the $n \times n$ matrix A_n = \begin{pmatrix} 1 & -1 & 0 & 0 & \cdots & 0 & 0 & 0 \\ 1 & 1 & -1 & 0 & \cdots & 0 & 0 & 0 \\
We wish to find a closed form for S_n = \sum_{k=0}^{n} F_k in terms of Fibonacci numbers.
For each of the $n$ objects, introduce a sequence $b^{(i)}_0,b^{(i)}_1,\ldots,b^{(i)}_r$ defined by b^{(i)}_j = 1 \quad (0 \le j \le r), \qquad b^{(i)}_j = 0 \quad (j > r).
We are asked to solve the recurrence f(1)=0; \qquad f(n)=\min_{0<k<n}\max(1+f(k),\,2+f(n-k)), \qquad n>1.
Equation (21) states that if $x^{t+1}=x^t+z$ and $x=1$ when $z=0$, then x^r=\sum_{k\ge0}\binom{r-kt}{k}\frac{r}{r-kt}z^k.
We are asked to express $h_4$ in terms of the elementary symmetric functions $S_1, S_2, S_3, S_4$, using the notation of Eqs.
Using the conventions of exercise 8, Fibonacci numbers are extended to all integer subscripts.
We are asked whether the sum of the squares of consecutive Fibonacci numbers, F_n^2 + F_{n+1}^2, is itself always a Fibonacci number.
We are asked to compute $F_n \phi \bmod 1$, that is, the fractional part of $F_n \phi$.
The verification of the steps leading to Eq.
We seek the coefficients of \frac{1}{(1-z)^w} when expanded as a double power series in $z$ and $w$.
We start from Exercise 25, which asserts that 2^n F_n = 2 \sum_{\substack{k=1\\ k \text{ odd}}}^{\,n} \binom{n}{k} 5^{(k-1)/2}.
Let $\langle F_n \rangle$ denote the Fibonacci sequence as defined in Section 1.
Let H(z)=\sum_{m\ge0}h_m z^m \] be the generating function for the complete homogeneous symmetric functions.
We use Eq.
Define G_n(z)=\sum_{k=0}^{n}\binom{n-k}{k}z^k.
Let us denote the sequence in question by a_n = \sum_{0<k<n} \frac{1}{k(n-k)}, \qquad n \ge 2, and $a_0=a_1=0$.
Determine the exact value of $\cos 36^\circ$.
We are asked to compute Fibonomial coefficients and prove a recurrence that guarantees their integrality.
For fixed $n$, define F_n(z) = (e^z-1)^n.
Let $G(z)$ be the generating function for the sequence \langle 2^n+3^n\rangle = 2,5,13,35,\ldots.
Let $\langle a_k \rangle = a_0,a_1,a_2,\ldots$ be a sequence with generating function G(z) = \sum_{k \ge 0} a_k z^k.
Let S_n(x)=\sum_{k=0}^{n}F_kx^k.
A doubly subscripted sequence $\langle a_{mn} \rangle$ with $m,n \ge 0$ is represented by introducing two independent parameters $z$ and $w$ and forming the double power series G(z,w) = \sum_{m \ge 0}...
We restart from a clean derivation and fix the missing structural step directly.
Exercise 34 states that every nonnegative integer has a unique representation N = F_{k_1} + F_{k_2} + \cdots + F_{k_r}, where
Define d_n=c_n-xa_n-yb_n.
Equation (18) gives the generating function for the harmonic numbers: G(z)=\sum_{n\ge0}H_nz^n =\frac{1}{1-z}\ln\frac1{1-z}.
Let $p(n)$ denote the number of partitions of $n$, with $p(0)=1$.
We want to prove that \sum_{k=0}^{n} \binom{n-k}{k} = F_{n+1}.
Let E(z)=\sum_{m\ge0} e_m z^m be the generating function for the elementary symmetric functions.
We prove existence and uniqueness separately.
Let \mathcal{G}(z)=\sum_{n\ge0}\mathcal{F}_n z^n be the generating function for the second order Fibonacci sequence.
From the closed form expression (14) for the Fibonacci numbers, we have F_n = \frac{1}{\sqrt{5}}\bigl(\phi^n - \hat{\phi}^n\bigr), \qquad \hat{\phi} = 1-\phi = \frac{1}{2}(1-\sqrt{5}).
Let A(z)=\sum_{n\ge0} a_n z^n be the generating function for the sequence $\langle a_n\rangle$.
Let n = F_{k_1} + \cdots + F_{k_r} be the Fibonacci representation from exercise 34, with $k_1 > \cdots > k_r$ and no two consecutive indices.
By Eq.
Let G(z)=\sum_{n\ge 0} a_n z^n.
We are asked to prove equation (11): \left(\frac{a_0}{0!
We define a **base-$\phi$ expansion** of a real number $x \ge 0$ to be a formal series x = \sum_{k=-\infty}^{n} d_k \phi^k, where each $d_k \in {0,1}$ and $n$ is an integer such that $d_n = 1$.
From equation (14) we have the exact expression F_n = \frac{1}{\sqrt{5}}\left(\phi^n - \hat{\phi}^n\right), where
Let S_1=\text{a},\quad S_2=\text{b},\quad S_{n+2}=S_{n+1}S_n \quad (n\ge1), so that
Let $p$ be a prime with $p \ne 5$.
We are asked to find a closed-form expression for the sequence $\langle a_n \rangle$ defined by a_0 = 0, \qquad a_1 = 1, \qquad a_{n+2} = a_{n+1} + a_n + \binom{n}{m}, \quad n \ge 0, where $m$ is a fi...
Exercise 37 defines the following game.
We prove the identities \phi^n = F_n \phi + F_{n-1}, \qquad \hat{\phi}^n = F_n \hat{\phi} + F_{n-1} for all integers $n$ by induction and by using the definitions in Section 1.
Let S=\sum_{1 \le k < n}(a_{k+1}-a_k)b_k.
Let H_n=\sum_{k=1}^{n}\frac{1}{k}, \qquad H_0=0.
Since $p$ is an odd prime, the integers 1,2,\ldots,p-1 can be paired as
We must prove that, for integers $l,m,n$ with $n \ge 0$, \sum_{j,k} (-1)^{j+k}\binom{j+k}{k+l}\binom{r}{j}\binom{n}{k}\binom{s+n-j-k}{m-j} = (-1)^l \binom{n+r}{n+l}\binom{s-r}{m-n-l}.
Let P=\bigl(\binom{n}{k}\bigr)_{n,k\ge 0} be Pascal's triangle regarded as an infinite lower-triangular matrix.
The argument in the text groups the terms of $H_{2^m}$ as follows: H_{2^m} = 1 + \frac12 + \left(\frac13 + \frac14\right) + \left(\frac15 + \cdots + \frac18\right)
By Eq.
By definition, H_n = \sum_{k=1}^{n}\frac1k.
Let $n$ be a nonnegative integer, $0 \le p \le 1$, and let $k$ range over integers $0 \le k \le n$.
Let S=\sum_{1<k\le n}\frac{1}{k(k-1)}H_k.
Let S_n=\sum_{k=0}^{n} H_kH_{n-k}.
Assume that $n$ is composite.
Let the $n$ objects be labeled $1,2,\ldots,n$.
I need the statements of Eqs.
Define B_{nk}=A_{nk}-\binom{n+1}{k}.
Let S=\sum_k (-1)^k \binom{l+m}{l+k}\binom{m+n}{m+k}\binom{n+l}{n+k}, \qquad l,m,n\ge0 .
Let H_n^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}, \qquad r>1.
Let A=\begin{pmatrix} 1&1\\ 1&0 \end{pmatrix}.
We wish to prove that H_n = \frac{\left[{\,n+1 \atop 2\,}\right]}{n!
Fibonacci's original problem assumes that a pair of rabbits produces a new pair every month, starting from one newly born pair, and that rabbits become productive after one month.
We are asked to evaluate \sum_{k=1}^{n} \binom{n}{k}(-1)^k H_k.
Statement 1 is false, since $H_1=1>\ln 1=0$.
Let b_k=a_kx_0^k, \qquad f(x_0)=\sum_{k\ge0}b_k,
Exercise 2 shows that F_n = \phi^n/\sqrt{5} \text{ rounded to the nearest integer,} by Eq.
Let $x, y, z$ be real numbers, and $n \ge 2$ an integer, satisfying \binom{x}{n} = \binom{y}{n} + \binom{z}{n-1}, \qquad x \ge n-1,\ y \ge n-1,\ z > n-2.
Let P_n(x)=x e^{\gamma x}\prod_{k=1}^{n}\left(1+\frac{x}{k}\right)e^{-x/k}.
By the asymptotic expansion for harmonic numbers in Appendix A, H_n = \ln n + \gamma + \frac{1}{2n} -\frac{1}{12n^2} +\frac{1}{120n^4} -\frac{1}{252n^6}
By the definition of generalized harmonic numbers, H_{\infty}^{(1000)}=\sum_{k=1}^{\infty}\frac1{k^{1000}}=\zeta(1000).
By exercise 8, the Fibonacci recurrence F_{n+2}=F_{n+1}+F_n is assumed to hold for all integers $n$.
Let $S(n,m)$ denote the number of partitions of an $n$-element set into $m$ nonempty disjoint subsets.
We prove the identity \sum_{k=1}^{n}\frac{x^k}{k} = H_n + \sum_{k=1}^{n}\binom{n}{k}\frac{(x-1)^k}{k}.
Let S_n=\sum_{k=1}^{n}H_k^2.
Let T(m,n) = H_m + H_n - H_{mn}, where $m,n > 0$ and $H_k = \sum_{i=1}^{k} \frac{1}{i}$ is the $k$-th harmonic number.
Let S=\sum_k \left[{n+1 \atop k+1}\right]\left\{{k \atop m}\right\}(-1)^{k-m}.
Let S_n = 1 + \frac{1}{3} + \cdots + \frac{1}{2n-1} = \sum_{k=1}^{n}\frac{1}{2k-1}.
For each $n$, choose $a$ as large as possible subject to $\binom{a}{3}\le n$.
By definition, H_n^{(u,v)}=\sum_{1\le j\le k\le n}\frac1{j^uk^v}.
We begin with the definition of the binomial coefficient for integers $n \ge k \ge 0$: \binom{n}{k} = \frac{n(n-1)\cdots(n-k+1)}{k(k-1)\cdots 1} = \frac{n!
Let S_n=\sum_{k=1}^{n}\frac{H_k}{n+1-k}.
For positive integers $n$, H_n=\sum_{k=1}^n \frac1k.
We are asked to find all positive integers $n$ such that F_n = n^2.
Abel's binomial formula (16) states that (x+y)^n = \sum_k \binom{n}{k}x(x-kz)^{k-1}(y+kz)^{n-k}.
Let $P$ denote the infinite lower-triangular matrix whose $(i,j)$ entry is P_{i,j} = \binom{i}{j}, \qquad i,j \ge 0, with the convention that $\binom{i}{j} = 0$ when $j > i$.
We proceed by induction on $n$.
Write S_n=\sum_{k=1}^{n}\frac{H_k}{k}.
Equation (15) gives F_n = \frac{\phi^n}{\sqrt{5}} \text{ rounded to the nearest integer,} \qquad \phi = \frac{1+\sqrt{5}}{2}.
Let F(x)=\sum_{m\ge0}a_mx^m be Stirling's attempted generalization of the factorial function.
We extend the Fibonacci sequence to negative indices by the recurrence F_{n+2} = F_{n+1} + F_n for all integers $n$.
Equation (8) gives \sum_{k=1}^{n}H_k=(n+1)H_n-n.
Let S_n = 1 + \frac13 + \frac15 + \cdots + \frac{1}{2n-1}.
Define S_m=\sum_{k=0}^{m}\binom{r}{k}\binom{s}{n-k}\bigl(nr-(r+s)k\bigr).
Example 3 in Section 1.
We are asked to evaluate the beta function B\!
Equation (23) is \sum_k (-1)^{r-k}\binom{r}{k}\binom{s+k}{n} = \binom{s}{n-r}, \qquad \text{integer } r \ge 0.
Let A_n(x,y)=\sum_{k=0}^n \binom{n}{k}(x+k)^{k-1}(y+n-k)^{n-k}.
By the binomial theorem (13), (1+x)^r=\sum_{k\ge0}\binom{r}{k}x^k, since the terms with $k<0$ vanish by definition (3).
Let v_p(m) denote the exponent of the highest power of $p$ dividing $m$, and let
We are asked to prove the addition formulas (46) for Stirling numbers directly from their definitions, Eqs.
By equation (3), \binom{r}{k} = \frac{r(r-1)\cdots(r-k+1)}{k!
Equation (10) states that \sum_{k=0}^{n}\binom{r+k}{k} = \binom{r+n+1}{n}, \qquad \text{integer } n\ge 0.
We prove the stated identity by induction on $n$.
Let $n$ and $m$ be integers with $m \ge 1$, and let $k$ be an integer.
Exercise 22 proves Eq.
Exercise 42 suggests extending the binomial coefficient to arbitrary arguments by means of the beta function.
We prove the identities by induction on $n$, using the definitions of falling and rising factorial powers and the standard binomial formula for ordinary powers.
Using Eq.
Equation (26) is the identity \sum_k \binom{r}{m+k}\binom{s}{n-k}\binom{k}{t} = \binom{r+s-t}{m+n-t}\binom{r}{m+t},
Let S=\sum_{i=0}^{s}\binom{r+i}{r}\binom{t+s-i}{t}, \qquad s=(n-1)r+nt.
By Exercise 40(3), B(k+1,r-k+1) =\frac{(r+1)!
1.
Equation (15) is the generalized binomial theorem, (1+x)^r=\sum_{j\ge0}\binom{r}{j}x^j.
Let $A_n(x,t)$ be the polynomial defined in Example 4 of Section 1.
Let S=\sum_k \binom{m-r+s}{k}\binom{n+r-s}{n-k}\binom{r+k}{m+n}.
We use the identities \sum_k \binom{r}{k}\binom{s}{n-k}=\binom{r+s}{n} \tag{21} and
Let S_n=\sum_k \binom{r+tk}{k}\binom{s-tk}{n-k}, where $n$ is a nonnegative integer.
Exercise 25 established that, under the assumptions of Example 4, \sum_k A_k(r,t) z^k = x^r, \qquad z=x^{t+1}-x^t, provided that $x$ is sufficiently close to $1$.
By Exercise 36 and the binomial theorem (13), \sum_k \binom{n}{k} = (1+1)^n = 2^n, and
Let $\left[{n \atop k}\right]$ denote the numbers in Stirling's first triangle.
Define I(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.
By Eq.
We are asked to evaluate \sum_{k \ge 0} \binom{n}{k} \frac{(-1)^k}{k+x}.
In Eq.
Let P(n):\qquad \binom{n}{k}\equiv 1 \pmod 2 \quad\text{for every }k\text{ with }0\le k\le n\text{ and }\binom{n}{k}\ne0.
By Eq.
We begin with the left-hand side of the identity and apply the definition (3) of binomial coefficients for general integers $r$: \binom{-n}{k-1} = \frac{(-n)(-n-1)\cdots(-n-(k-2))}{(k-1)!
Equation (22) states that for integers $n \ge 0$ and $m \ge 0$: \sum_{k=0}^{n} \binom{k}{m} = \binom{n+1}{m+1}.
Let n=ap+b,\qquad 0\le b<p, where $a=\lfloor n/p\rfloor$ and $b=n\bmod p$.
Equation (34) states that for integers $r \ge 0$ and $m$, \sum_k \binom{r}{k}(-1)^{r-k}k^m = \begin{cases} 0, & 0 \le m < r,\\
We are asked to prove Abel's formula, Eq.
By Eq.
We prove Eq.
By the binomial theorem (13), with $x=y=1$, (1+1)^n = \sum_k \binom{n}{k}1^k1^{n-k} = \sum_k \binom{n}{k}.
The symmetry condition, Eq.
Let $k$ be an integer.
We begin with the setup from Exercise 25.
For every integer $n \ge 0$, Eq.
We seek a closed form for the sum \sum_{k=0}^{n} k^4.
Using the extension of the factorial function, x!
We first evaluate S_n=\sum_{k=1}^n \left\lfloor \frac{k}{2}\right\rfloor.
We consider the class of _replicative functions_ $f$, defined in Exercise 39, satisfying f(x) + f\left(x+\frac{1}{n}\right) + \cdots + f\left(x+\frac{n-1}{n}\right) = f(nx) for all positive integers $...
We wish to evaluate $S_n = \sum_{k=1}^n \lfloor \sqrt{k} \rfloor.$ Let $m = \lfloor \sqrt{n} \rfloor$, so that $m^2 \le n < (m+1)^2$.
No, factorial powers do not satisfy a law analogous to the ordinary law of exponents.
Let $\nu_2(m)$ denote the exponent of $2$ in the prime factorization of $m$.
The existence of a factorization into primes follows by induction on $n$, using exercise 1.
Let $R={r_1,r_2,\ldots,r_{\varphi(m)}}$ be the set of integers in ${0,1,\ldots,m-1}$ that are relatively prime to $m$.
In the context of _The Art of Computer Programming_, the intended value is 0^0 = 1.
Every divisor pair $(c,d)$ that appears on the left-hand side satisfies c \mid d,\qquad d \mid n.
Let $m$ and $n$ be integers.
By Theorem F, if $p$ is prime then a^p \equiv a \pmod p.
By Euler’s product formula derived from (13), z!
Let S=\sum_{0\le j<n}f\!
Let S_n(f;x)=\sum_{k=0}^{n-1}f\!
We prove the identity \sum_{k=1}^n a_k = na_n-\sum_{k=1}^{n-1}k(a_{k+1}-a_k), \qquad n>0.
Let S(\alpha)=\sum_{0\le j<\alpha n} f\!
By the symmetry condition (6), \binom{n}{n-1}=\binom{n}{1}.
Let $n$ be a positive integer and let $\varphi(n)$ denote the number of integers in ${0,1,\ldots,n-1}$ that are relatively prime to $n$, as in exercise 27.
The rows of Pascal's triangle give the coefficients of the expansion of $(x+y)^n$ according to the binomial theorem, Eq.
By Exercise 19, for $x>0$, \Gamma_m(x) = \int_0^m\left(1-\frac{t}{m}\right)^m t^{x-1}\,dt.
Since \log_{10}(1000!
Let $p$ be a prime number.
By Exercise 3, the number of bridge hands is \binom{52}{13} = \frac{52!
Let F_n=\sum_{j=0}^n B_{n,j}\,D_u^j w, where
Let $f(n)$ be a function of positive integers.
Let P_N=\prod_{n=1}^{N}\frac{(n+\alpha_1)\cdots(n+\alpha_k)} {(n+\beta_1)\cdots(n+\beta_k)}.
Applying the refined form of Stirling’s approximation with $n=8$ gives 8!
A shuffle of a 52-card deck is a permutation of 52 distinct objects.
We seek all real numbers $b>1$ such that \lfloor \log_b x\rfloor=\lfloor \log_b \lfloor x\rfloor\rfloor \qquad (x\ge 1).
By Eq.
Consider integers $a=2$, $b=8$, $r=4$, and $s=6$.
A bridge hand consists of 13 cards chosen from a standard deck of 52 cards.
Take $m=6$ and $a=2$.
From Euler's limit formula for the factorial function (Equation 13), n!
Let S(x,y)=\sum_{0\le k<y}\left\lfloor x+\frac{k}{y}\right\rfloor, \qquad y>0.
Let $p$ be an odd prime, $a$ any integer, and define $b = a^{(p-1)/2}.$ We are asked to show that $b \bmod p$ is either $0$, $1$, or $p-1$.
The $(i,j)$ entry of the matrix is $i\times j$.
Consider the infinite sum in Eq.
Let $f(n)$ be a multiplicative function, and define $g(n) = \sum_{d \mid n} f(d).$ We want to show that $g$ is multiplicative, that is, for any integers $r \perp s$, $g(rs) = g(r) g(s).$ Since $r \per...
We have the generalized termial function defined for all real numbers $x$ by $x? = \sum_{k=1}^x k = 1 + 2 + \cdots + x.$ For integer $x$, this is exactly the usual arithmetic series formula, and for n...
The sequence is formed by repeating each positive integer $k$ exactly $k$ times: 1,\ 2,2,\ 3,3,3,\ 4,4,4,4,\ \ldots Hence $a_n=k$ precisely when $n$ lies in the block occupied by the $k$'s.
Law A extends without change to arbitrary real numbers.
From Eq.
Let S=\{1,2,\ldots,p-1\}, where $p$ is prime.
Let $x$ be a real number, and let $m$ and $n$ be integers with $n>0$.
Let S(x)=\sum_{k=0}^{n-1}\left\lfloor \frac{mk+x}{n}\right\rfloor, where $m,n\in\mathbb Z$, $n>0$, and let
Let S(n)=\sum_{k\ge0}\sum_{1\le j<b}\left\lfloor \frac{n+j b^k}{b^{k+1}}\right\rfloor .
Let L_m=\frac{m^n m!
Since $n \perp m$, the extended Euclidean algorithm (Algorithm 1.
Let $x \equiv 2 \pmod 3,$ $x \equiv 3 \pmod 5.$ Since $3 \perp 5$, Law D implies that these two congruences determine a unique congruence class modulo $15$.
For any real number $x$, the floor function $\lfloor x \rfloor$ is an integer by definition.
Let M = \sup_{R(j)} a_j .
By Exercise 39, the inverse of the combinatorial matrix has entries b_{ij}=\frac{-y+\delta_{ij}(x+ny)}{x(x+ny)}.
We are to transform $\sum_{R(i)} \sum_{S(i,j)} a_{ij}$ according to Eq.
Let $A = (a_{ij})_{1 \le i,j \le n}$ be the combinatorial matrix, where $a_{ij} = x\delta_{ij} + y.$ Thus $A = xI + yJ,$ where $I$ is the identity matrix and $J$ is the matrix whose entries are all eq...
Let A=(a_{ij})_{1\le i,j\le n}, \qquad a_{ij}=x_i^{\,j-1},
By equation (1), x \bmod y = x-y\left\lfloor \frac{x}{y}\right\rfloor.
By the definitions of floor and ceiling, \lfloor 1.
Let $x$ and $y$ be arbitrary real numbers.
By definition (1), x \bmod y = x-y\left\lfloor \frac{x}{y}\right\rfloor .
By definition (1), x \bmod y = x-y\left\lfloor \frac{x}{y}\right\rfloor .
By Eq.
Define S=\sum_{i=0}^n \sum_{j=0}^i \sum_{k=0}^j a_i a_j a_k.
Assume that ax \equiv by \pmod m, \qquad a \equiv b \pmod m, \qquad
By definition (1), $x \bmod y = x - y \lfloor x/y \rfloor$ when $y \ne 0$, and $x \bmod 0 = x$.
Assume $a \equiv b \pmod r$ and $a \equiv b \pmod s$, with $r \perp s$.
Let H=(h_{ij})_{1\le i,j\le n},\qquad h_{ij}=\frac1{i+j-1}.
Let A=(a_{ik})_{1\le i\le m,\ 1\le k\le n}, \qquad B=(b_{kj})_{1\le k\le n,\ 1\le j\le m},
Let A=(a_{ij})_{1\le i,j\le n}, \qquad a_{ij}=\frac1{x_i+y_j},
Let $n=\lceil x\rceil$.
Let $V = (v_{ij})$ be the $n \times n$ Vandermonde matrix with entries $v_{ij} = x_i^{j-1}$ for $1 \le i, j \le n$, and let $B = (b_{ij})$ denote its inverse.
By equation (1), x \bmod y = x - y\left\lfloor \frac{x}{y}\right\rfloor.
Let f(x)=\sum_{k=1}^n \frac{\displaystyle\prod_{\substack{1\le r\le n\\ r\ne m}}(x+k-r)} {\displaystyle\prod_{\substack{1\le r\le n\\ r\ne k}}(k-r)}.
Defining $\sum_{R(j)} a_j = 0$ when no integers satisfy $R(j)$ preserves the additive identity.
Let P=\prod_{i=0}^n \prod_{j=0}^i a_i a_j.
Apply Binet's formula from Exercise 30 with the substitutions a_j=u_j,\qquad b_j=1,\qquad x_j=1,\qquad y_j=v_j.
The required product analogs are obtained by replacing sums with products and interpreting repeated factors with multiplicity.
Let $A = (a_{ij})$ be the $n \times n$ combinatorial matrix with entries $a_{ij} = y + \delta_{ij} x = \begin{cases} x + y, & i = j, \\ y, & i \ne j. \end{cases}$ We want to show that $\det(A) = x^{n-...
Let $A=(a_{ij})_{1\le i,j\le n}$ be Cauchy’s matrix, where $a_{ij}=\frac1{x_i+y_j}.$ Let $B=(b_{ij})$ be the matrix whose entries are b_{ij} = \frac{
By convention, $\gcd(0,n)=|n|$.
We compute the sum directly.
Every integer $x$ is divisible by 1, since $x = 1 \cdot x$.
The observed identities are 9\cdot 1+2=11,\qquad 9\cdot 12+3=111,\qquad 9\cdot 123+4=1111,\qquad 9\cdot 1234+5=11111.
Let A=\sum_{j=1}^n a_jx_j,\qquad B=\sum_{j=1}^n b_jy_j, and
Let D_n=\det\!
We proceed by induction on $n$.
Write 1-\frac1{j^2} = \frac{(j-1)(j+1)}{j^2} =
By definition of congruence, the statement u \equiv v \pmod m means that $u-v$ is an integral multiple of $m$.
Let P(R)=\prod_{R(j)} a_j, \qquad S(R)=\sum_{R(j)} (\log_b a_j),
Let n=\lfloor x\rfloor.
Let $x$ be a positive real number.
Let $V_n = (a_{ij})_{1 \le i,j \le n}$ denote the $n \times n$ Vandermonde matrix, where $a_{ij} = x_j^i$.
Let \lfloor x\rfloor \le x < \lfloor x\rfloor+1, and recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.
We are asked to examine the derivation $\left( \sum_{i=1}^n a_i \right) \left( \sum_{j=1}^n \frac{1}{a_j} \right) = \sum_{1 \le i \le n} \sum_{1 \le j \le n} \frac{a_i}{a_j} = \sum_{1 \le i \le n} \su...
The reviewer found no errors in the solution to Exercise 1.
We proceed by induction on $n$.
Let $x_1, x_2, \ldots, x_n$ be distinct numbers, and define, for $0 \le r \le n$, S_r := \sum_{j=1}^n \frac{x_j^r}{\prod_{\substack{1 \le k \le n \\ k \ne j}} (x_j - x_k)}.
No.
Let S=\sum_{j=0}^n jx^j.
We could define real numbers by binary expansions instead of decimal expansions.
The statement is true.
Let L'_k = n+\frac{b'_1}{2}+\cdots+\frac{b'_k}{2^k},
Let x=m+0.
Let Q(n)=\frac{1}{n!
By definition, a sum of the form $a_1 + a_2 + \cdots + a_0$ is empty because the upper limit is less than the lower limit.
Let x_0=\log_{10}2, so that
By definition (13), $\lg x = \log_2 x$.
By definition (2), $\sum_{1 \le j \le n} a_j$ denotes the sum of all terms $a_j$ for integer values of $j$ satisfying the condition $1 \le j \le n$.
Let the computer work with fixed precision $\delta > 0$, meaning that every right shift and every subtraction is performed with an error whose absolute value is at most $\delta$.
Let x=\frac pq,\qquad y=\frac rs, where $p,r\in\mathbb Z$ and $q,s\in\mathbb Z_{>0}$.
By Eq.
Using the change-of-base formula, \lg x=\frac{\ln x}{\ln 2}, \qquad \log_{10}x=\frac{\ln x}{\ln 10}.
By equation (15), \ln x = \log_e x.
There is no smallest positive rational number.
By equation (14), \log_b y = \frac{\ln y}{\ln b}.
The first sum is $\sum_{0 \le n \le 5} \frac{1}{2n+1}.$ The integers satisfying $0 \le n \le 5$ are $n=0,1,2,3,4,5.$ Hence
Let L(z) denote the area under the hyperbola
Let $x > 1$ be given.
Let $0 \le x < 1,$ and let $b>0$, $b\ne1$.
The method based on Eqs.
Let S=\sum_{0\le j\le n} ax^j .
The reviewer is correct: the original proof fails because it implicitly assumes absolute convergence of a Laplace integral from mere convergence.
Let $S=\sum_{R(j)} a_j.$ Define the change of variable $i=c-j.$ Since $c$ is an integer, the mapping $j \mapsto c-j$
Yes.
Equation (8) shows that 10^{0.
We begin by expressing the sum in a form suitable for applying Eq.
(a) Let u=\sqrt[n]{1+x}.
A 14-digit decimal integer $n$ satisfies $10^{13} \le n < 10^{14}.$ A computer word with 47 bits and a sign bit has 47 bits available for the magnitude.
Equation (7) asserts that \sum_{R(i)} \sum_{S(j)} a_{ij} = \sum_{S(j)} \sum_{R(i)} a_{ij}.
Let u = w + \frac{1}{3}w^2 + \frac{1}{36}w^3 - \frac{1}{270}w^4 + \cdots = \sum_{k=1}^{\infty} c_k w^k be the power series solution to the equation
Let S_n=\sum_{j=1}^{n} j2^j.
For $n = 3$, the left-hand side of Eq.
We are asked to prove Eq.
We have $(0.125)^{-2/3}$.
If $x = 1$, the left-hand side of Eq.
Suppose, for the sake of contradiction, that $\log_{10} 2$ is rational.
Define S(n)=\sum_{k\ge0}\frac{(n+k-1)!
Let the value of $x$ at the beginning of an execution of step L3 be denoted by $x^{(t)}$, where $t = 0, 1, 2, \ldots$ counts the number of times step L4 has been performed.
By the distributive law (4), \sum_{j=m}^n \sum_{k=r}^s jk = \left(\sum_{j=m}^n j\right)\left(\sum_{k=r}^s k\right).
Let A=\sum_{R(j)} a_j,\qquad B=\sum_{S(j)} a_j,\qquad C=\sum_{R(j)\text{ or }S(j)} a_j,\qquad D=\sum_{R(j)\text{ and }S(j)} a_j .
Assume throughout that $b\neq 0$.
Let $u = a + 0.c_1c_2c_3\ldots$ be a positive real number.
By Eq.
Let \sum_{R(i)} a_i = A, \qquad \sum_{S(j)} b_j = B
The given sum is a finite geometric progression with first term $a = 1$, common ratio $x = \frac{1}{7}$, and $n+1$ terms.
The statement is true.
By definition, $\sum_{j \in S} 1$ is the sum of one copy of $1$ for each integer $j$ belonging to $S$.
Yes.
Let $a_1 a_2 \ldots a_n$ be a permutation of $\{1,2,\ldots,n\}$, and let $b_1 b_2 \ldots b_n$ be its inverse permutation, so that a_k = l \quad \text{if and only if} \quad b_l = k.
Let P_n = 1^1 2^2 3^3 \cdots n^n = \prod_{k=1}^n k^k.
We are asked to show that the remainder term $R$ in Eq.
Let J(x)=\int_0^{y x^{1/4}} e^{-u}\left(1+\frac ux\right)^x\,du, where $y$ is fixed and $x\to\infty$.
Let I_n=\int_0^\infty \left(1+\frac{z}{n}\right)^n e^{-z}\,dz .
We are asked to prove the identity \sum_{k=0}^{n} (-1)^k \binom{n}{k} k^{\,n-1} Q(k) = (-1)^n (n-1)!
Let $X$ be a sum of independent Bernoulli random variables $X_1,X_2,\ldots,X_n$, where $X_k$ takes the value $1$ with probability $p_k$ and $0$ with probability $q_k=1-p_k$.
For a probability generating function G(z)=\sum_{k\ge0}p_k z^k, the third semi-invariant (third cumulant) is
Define S_n=n^2-(n-1)^2+(n-2)^2-\cdots+(-1)^{n-1}1^2 =\sum_{i=1}^n (-1)^{\,n-i} i^2.
Assume n!
We prove by induction on $n$.
Let the assertions $A_1,\ldots,A_6$ be the assertions used in Knuth's verification of Algorithm E.
Let S(x)=\sqrt{2\pi x}\left(\frac{x}{e}\right)^x .
Let R=\mathbb Z[\sqrt2]=\{u+v\sqrt2 : u,v\in\mathbb Z\}.
We are asked to derive the series expansion \gamma(a,x) = \sum_{k\ge 0} \frac{(-1)^k x^{k+a}}{k!
Equation (5) states that f(x) = \sum_{k=0}^{n}\frac{f^{(k)}(0)}{k!
From (4) with $k=0$ and using $p_{n-1,-1}=0$ from (5), p_{n0}=\frac{1}{n}p_{n-1,-1}+\frac{n-1}{n}p_{n-1,0}=\frac{n-1}{n}p_{n-1,0}.
Let m = an^2 + bn, \qquad a>0.
Equation (18) states \sqrt[n]{n} = e^{\ln n/n} = 1 + \frac{\ln n}{n} + O\!
The critical issue is not the manipulation of generating functions but the implicit assumption that the coefficient sequence is governed by a finite radius of convergence.
In Section 1.
Let $X$ be the number of times step M4 is executed when Algorithm M finds the maximum of $n=1000$ distinct items presented in a uniformly random order.
Let G(z) = \sum_{k \ge 0} p_k z^k be the probability generating function (pgf) of a discrete random variable $X$.
We are asked to analyze **Algorithm M** under the assumption that the sequence X[1], X[2], \dots, X[n] contains **exactly $m$ distinct values**, which are otherwise randomly arranged subject to that c...
The statement is: g(n)=\Omega(f(n)) \quad \Longleftrightarrow \quad f(n)=O(g(n)).
Let $u(z)=O(z^m)$ as $z\to 0$, with $m>0$.
We are asked to determine the probability distribution of $A$, the number of times step M4 is executed, when each $X[k]$ is selected independently and uniformly from a set of $M$ objects, using the re...
Let $g_i(z)=\sum_k p_{ik}z^k$ be the probability generating function of a random variable $X_i$, with g_i(1)=1,\qquad \mu_i=g_i'(1),\qquad \sigma_i^2=g_i''(1)+g_i'(1)-g_i'(1)^2.
The Poisson distribution with parameter $\mu$ assigns probabilities p_k = e^{-\mu}\frac{\mu^k}{k!
Exercise 14 does not ask for a mathematical theorem to be proved by induction.
Let $X\sim \mathrm{Bin}(n,p)$, $q=1-p$, and let $0\le \epsilon \le q$.
We are asked to **prove Eq.
Let S_\alpha(n)=\sum_{k=0}^{n} k^{n+\alpha}e^{-k}, \qquad \alpha \ \text{fixed}.
Let G(z)=\sum_{k\ge0}p_kz^k, \qquad \sum_{k\ge0}p_k=1.
We prove by induction that for every positive integer $n$, n^3=(n^2-n+1)+(n^2-n+3)+\cdots+(n^2-n+(2n-1)).
Let $f(x) = x^m$, where $m$ is a nonnegative integer.
The critical issue in the proposed solution is the hidden assumption that “first element of a reduced problem” corresponds to the first element of the original permutation.
Let $X[1],X[2],\ldots,X[n]$ be chosen independently and uniformly from a set of $M$ distinct elements.
**Corrected Solution to Exercise 1.
We are asked to derive Eq.
From Eq.
The error is not in the mechanics of exponentiating $e^x$, but in the claim that the logarithmic expansion was complete at order $n^{-3}$.
Write \frac{(n+\alpha)^n}{n^{\,n+\beta}} = n^{\alpha-\beta}\left(1+\frac{\alpha}{n}\right)^n.
Figure 11 corresponds to the coin-tossing experiment of Eq.
Let f(z)=\sum_{k\ge 0} a_k z^k, \qquad g(z)=\sum_{n\ge 0} b_n z^n be generating functions of probability distributions, so $a_k \ge 0$, $b_n \ge 0$, and
Let G_n(z)=\frac1{n!
Let S_n=\sum_{k=0}^{n}(-1)^k\frac{(2k+1)^3}{(2k+1)^4+4}.
Let G_n(z)=(q+pz)^n,\qquad q=1-p.
Let $E$ be the event that all of the values $X[1],X[2],\ldots,X[n]$ are distinct.
Let $m=2k>0$.
Let $\prec$ be a relation on a set $S$ satisfying: 1.
Let $P(n)$ be the statement 2^n > n^3.
From Eq.
Let M=\max_{1\le k\le n}|a_k-b_k|.
Let the values immediately before step E4 be a,b,a',b',c,d, and assume that the invariants
Let C_1=(Q_1,I_1,\Omega_1,f_1), \qquad C_2=(Q_2,I_2,\Omega_2,f_2) be computational methods in the sense of Eqs.
Let P(n): \qquad F_n \ge \phi^{\,n-2}, where $\phi=(1+\sqrt5)/2$ and $n$ is a positive integer.
The exercise asks about the specific "Procedure for Reading This Set of Books" printed after the preface.
For a fixed positive integer $m$, define $U_m$ to be the average number of executions of step E1 in Algorithm E when the second argument $n$ ranges over all positive integers.
Apply Algorithm E to $m = 6099$ and $n = 2166$.
Let $P(n)$ be the statement: “Every integer $m$ with $2 \le m \le n$ may be written as a product of one or more prime numbers.
We prove by examining the structure of Algorithm E that $m \ge n$ holds at the beginning of step E1, except possibly at the first execution of that step.
To prove a statement $P(n)$ for all nonnegative integers $n = 0, 1, 2, \ldots$, we modify the usual induction procedure by changing the base case.
The error occurs in the inductive step when the proof attempts to establish $P(n+1)$ from the assumptions $P(1),\ldots,P(n)$.
We give a fully rigorous solution to Exercise 1.
We will now present a fully corrected solution to **Exercise 1.
The trivial replacement in Algorithm E occurs at step E3: m \leftarrow n,\qquad n \leftarrow r.
By definition, $T_5$ is the average number of times step E1 is performed, taken over all positive integers $m$ when $n=5$.
The mistake occurs in the basis step of the induction.
The conditions place $H$ on the line $AB$ and $K$ on the line $BC$.
Consider three segments $AB$, $CD$, and $EF$ intersecting at a single point $O$, with $E$ on $AC$ and $F$ on $BD$.
Before I begin the full solution, I need to clarify: should I solve **all five subproblems (1–5) for point $C$**, or just a specific one from the list? Each has its own locus.
Let $H$ be the orthocenter of triangle $ABC$.
Consider triangle $ABC$ with isosceles triangles erected externally on each side.
Let the white squares be indexed by coordinates.
I cannot write a solution to Kvant problem M355 because the actual problem statement is not present in your message.
Before proceeding, I need the precise textual statement of Kvant problem M353.
Write
Consider a small $n\times m$ chessboard, for instance $4\times 5$.
The numbers in the table are
Let the chosen pair be an unknown $2$ element subset of ${1,\dots,25}$.
Let $a_n$ be the sequence, with
Model the country by a connected graph.
The problem is a binary coding problem.
Consider first a $2 \times 2$ table:
For each of the given lines, orient it upward.
Please provide the textual version of Kvant problem M337.
Represent the marked cell centers by a $0$-$1$ matrix.
Consider small polynomials such as $P(x) = x$, $P(x) = x+1$, or $P(x) = x^2$.
Consider small values of $k$ first.
Let the rotation about the circumcenter $O$ have angle $\varphi$, where $0<\varphi<180^\circ$.
Let the vertices of the convex $n$-gon be $P_1,P_2,\dots,P_n$ in cyclic order.
Consider small groups to understand how the structure of "liking" works.
An $n times m$ grid consists of all unit segments that form the lattice. Every cell contributes edges, but neighboring cells share edges, so the total number of unit segments in the whole grid is not $4nm$. We are given two types of pieces.
Let the circle have radius $R$ and let a chord $AB$ be at a distance $h$ from the center $O$.
Consider a single pile with a small number of stones.
Consider a function $f:\mathbb{R}\to\mathbb{R}$.
The table may be taken to be the unit square $[0,1]\times[0,1]$.
I cannot write a solution to Kvant problem M319 because the actual problem statement is not present in your message.
Let
Consider the sum of squares of $k$ consecutive natural numbers beginning at $n$, expressed as
Consider the difference between a number and the product of its digits, denoted $N - P(N)$, where $N$ is a 9-digit number with digits $d_1, d_2, \dots, d_9$ in ${1,2,\dots,9}$ and $P(N) = d_1 d_2 \dot…
Consider an angle with vertex $O$ and denote its sides by rays $OA$ and $OB$.
Consider the growth process for small numbers.
For the first question, divisibility by $x^2+x+1$ suggests evaluating the polynomial at the nonreal cube roots of unity.
Consider first the case $n=2$, where the inequality takes the form $a_1\cos x + a_2\cos 2x \ge -1$ for all real $x$.
Let the removed corner be the unit square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$.
The axioms resemble the algebraic properties of the bitwise exclusive-or operation.
Consider placing a small number of identical weights on the vertices of a $1 \times 1$ grid.
For $n=1$ the statement is immediate.
Consider a ruled sheet of paper with parallel lines spaced a fixed distance apart, and suppose a regular $n$-gon is drawn so that all vertices lie on these lines.
The problem involves four squares arranged on a plane with shared vertices, forming a chain: the second vertex of the first square coincides with a vertex of the second square, and so on, closing back…
Let the rows be numbered from top to bottom by $1,\dots,n$, and let $a_{ij}$ be the entry in row $i$, column $j$.
The inequality is homogeneous and symmetric in a suggestive way.
Consider a small example with numbers $1, 2, 3$.
Let the closed non-self-intersecting broken line have vertices
Model the congress by a simple graph.
Consider a sequence of natural numbers $a_1 < a_2 < a_3 < \dots$ such that every natural number $n$ can be represented uniquely as $a_j - a_i$ with $j > i$.
Consider small examples to understand the claim.
Consider small examples of convex polygons, starting with triangles and quadrilaterals, and examine what happens when each side is shifted outward by a fixed distance.
Consider a small table, for instance $2 \times 2$, with entries
Consider a triangle $ABC$ of area $1$ with midpoints $A_1$, $B_1$, and $C_1$ of the sides $BC$, $AC$, and $AB$ respectively.
Let the numbers on the cards be $a_1,\dots,a_n$, where each $a_i\in{\pm1}$.
Let $E$ be the number of segments whose endpoints have different colors.
Let the vectors be represented by points on the unit circle.
We seek the smallest positive value attained by the given differences.
Consider two circles of radii $R$ and $r$ that are externally tangent.
The conditions mean that $KA \perp AB$, $KC \perp CD$, $HB \perp AB$, and $HD \perp CD$.
The quantity $T_k(n)$ is the $k$-th elementary symmetric polynomial in the numbers $1,2,\dots,n$:
Let the $n$th triple be $(a_n,b_n,c_n)$, with
Consider a rectangular parallelepiped with edges of length $a$, $b$, and $c$.
The graph described by Fig.
The problem asks for the maximal number of rooks or queens on an $8 \times 8$ chessboard such that each piece is attacked by at most one other piece.
Label the $n$ equal elementary arcs by the colors of the segments
Consider a simple case of a triangle circumscribed around a circle, where the inscribed circle is tangent to its sides at points $A', B', C'$.
The inequality can be written as
Consider first a simple case: a triangle circumscribed around a circle, with the incircle touching the sides at points $A'$, $B'$, and $C'$, forming the inscribed triangle.
Consider small cases of numbers of the form $0.
Consider a regular octagon with side length $a$ placed on a plane.
Represent friendship by a graph $G$ whose vertices are the knights, with an edge joining two friends.
Consider a cube $ABCDA'B'C'D'$ with an inscribed sphere, whose center coincides with the cube's center and whose radius is half the cube's edge length.
A $6 \times 6$ square contains $36$ unit squares.
Consider the task of placing $N$ points in the plane such that the distance between any two points $M_i$ and $M_j$ is a given number $r_{ij}$.
The desired inequality can be rewritten as
Denote the sides opposite $A_1,A_2,A_3$ by
Let points $A$ and $B$ be fixed on the plane, and let $C$ lie on the perpendicular bisector of segment $AB$, since it must satisfy $|AC| = |BC|$.
Let
For the first part, the numbers involved are all two-digit numbers, so each number can be represented as an ordered pair $(a,b)$ with $a,b \in {1,2,\dots,9,0}$, $a\neq 0$.
Consider a lion moving along a polygonal path inside a circular arena of radius $R = 10$ meters.
Consider a small case to understand the process.
Consider the equation $n^x + n^y = n^z$ in natural numbers.
Let the side length of the equilateral pentagon be $1$, and let its consecutive vertices be $A_1,A_2,A_3,A_4,A_5$.
Consider small values of $n$ first.
Place the square in the coordinate plane with vertices
Consider a die with faces numbered so that opposite faces sum to $7$.
Perfect numbers are rare and highly structured.
Consider an arbitrary compact planar blot.
The problem concerns a king moving on an $8\times 8$ chessboard, visiting every square exactly once, and returning to the starting square.
We are asked to compare the square of a sum of five positive numbers with four times a sum of specific pairwise products taken cyclically.
Consider first a triangle, the simplest convex polygon.
Consider a small patch of the grid with just one black cell at $(0,0)$.
Let the circle have center $I$.
We are asked whether an expert can convince the court, using only three weighings on a balance scale, that exactly seven out of fourteen coins are counterfeit.
The operation does not act on individual digits.
Let
Consider the problem geometrically by placing triangle $A_1 A_2 A_3$ in the plane and attempting to construct a triangle $M_1 M_2 M_3$ similar to a given triangle $B_1 B_2 B_3$ with the given side-ver…
Consider the matrix of size $24 \times 25$ with entries $0$ and $1$, where $1$ indicates that a student solved a problem.
Consider the total number of $n$-digit numbers, which is $9 \cdot 10^{n-1}$.
Let the arithmetic progression be
For part (a), the six points are the intersection points of four lines in general position.
We begin by examining the first sum for small values of $n$.
Consider first the smallest nontrivial case, a $2\times 2$ table with entries $a,b$ in the first row and $c,d$ in the second row.
Let
Consider a convex pentagon $ABCDE$ with vertices labeled consecutively.
Consider small analogues first.
Many search problems involve ranges rather than individual values.
Red-black trees are self-balancing binary search trees with a relatively loose balance condition.
Lower bound finds the first position whose value is greater than or equal to a target.
Binary search does not fundamentally require sorted data.
Most discussions of search algorithms focus on in-memory collections.
Standard binary search assumes the entire array is sorted.
A range query asks for information about a contiguous region of ordered data.
The most useful form of binary search is not exact lookup.
A binary search tree provides an elegant framework for maintaining ordered data.
Binary search trees work well in memory, where following a pointer from one node to another is relatively cheap.
Many search problems ask for more than membership.
Binary search is a boundary-finding algorithm.
Most balanced tree structures maintain balance through carefully designed invariants.
Balanced trees provide logarithmic performance through carefully maintained structure.
Parametric search is a technique for solving optimization problems by repeatedly answering decision questions.
Before studying specific balancing algorithms such as AVL trees and red-black trees, it is worth understanding the structure they are trying to preserve.
AVL trees were the first self-balancing binary search trees.
Most binary search examples begin with a sorted array.
Segment trees are among the most versatile range-query data structures in algorithm design.
Arrays are excellent when data rarely changes.
Binary search is often associated with sorted data, but some of its most elegant applications work on arrays that are not sorted at all.
The examples in previous sections searched integer domains.
Many range-query structures support updates.
Tree rotations are the primitive operation behind many balanced binary search trees.
A Fenwick tree, also called a binary indexed tree, is a compact data structure for prefix sums and point updates.
Design a crawler frontier that decides which URLs a web crawler should visit next.
Design a system that receives a stream of log events and removes duplicates before storage or analysis.
Design a cache that stores recently used values and evicts entries when capacity is full.
Design a batch job that analyzes a large graph and computes useful metrics such as degree counts, connected components, PageRank-style importance scores, and community structure.
Design a system that consumes events continuously and computes live metrics such as counts, rates, unique users, moving averages, and top items.
Design a small constraint solver that assigns values to variables while satisfying a set of rules.
Design a rate limiter that controls how often a client can perform an operation.
Design an autocomplete engine that provides search suggestions while a user types.
Design a service that answers shortest path queries over a graph.
Design an event scheduler that stores tasks, orders them by time, executes ready tasks, and handles updates such as cancellation or rescheduling.
Design a small workflow for checking an algorithm's correctness argument against its implementation.
Design a geometry query engine that stores spatial objects and answers geometric questions efficiently.
Design a plagiarism detector that compares documents and reports suspicious similarity.
Design a recommendation system that suggests items to users based on past behavior.
Design a planner that receives a set of tasks and dependencies, then returns a valid execution order.
You have learned dozens of algorithms and data structures.
Design a search ranking pipeline that receives a user query, retrieves candidate documents, scores them, and returns a ranked result list.
Design a matchmaking system that pairs users, players, tasks, or entities according to compatibility rules.
Design a static analyzer that scans source code and reports likely defects without running the program.
Design a compression tool that reduces the size of textual data while preserving the original information.
Design a sorting tool that can sort data larger than available memory.
Design a simulation that routes packets through a network of routers and links.
Design a small database index that supports fast lookup by key and efficient range scans.
Design a spell checker that detects misspelled words and suggests likely corrections.
Design a text diff tool that compares two versions of a document and reports what changed.
A Count-Min Sketch estimates item frequencies in a stream using fixed memory.
Randomized algorithms are often easier to design and analyze than deterministic algorithms.
Many approximation algorithms are built around a simple principle: make the best local decision available at each step.
Theoretical analysis explains why randomized algorithms work.
Balanced search trees achieve logarithmic performance by carefully maintaining structural invariants.
Set Cover is the canonical greedy approximation problem.
Randomized algorithms differ from deterministic algorithms because their behavior depends on random choices made during execution.
Randomized algorithms span a wide range of techniques.
Throughout this chapter, we have encountered a recurring theme: ```text A small amount of randomness can dramatically simplify
Many optimization problems can be expressed as integer programs.
HyperLogLog estimates the number of distinct elements in a stream using a small, fixed amount of memory.
Many applications need to find similar items inside very large collections.
Primal-dual approximation algorithms use linear programming structure to design fast approximate solutions.
Las Vegas algorithms always return correct answers and use randomness to influence resource consumption.
Randomized quicksort is one of the most successful applications of randomization in algorithm design.
A Bloom filter is one of the most widely deployed probabilistic data structures in modern computing.
Hash tables are usually presented as deterministic data structures: compute a hash, reduce it to a table index, and store or retrieve the key.
Streaming algorithms process data one item at a time while using much less memory than the input size.
Randomized algorithms rarely stop at computing an expected value.
A randomized algorithm can use randomness in several ways.
Some optimization problems are easy to state and hard to solve exactly.
Random sampling is one of the most powerful techniques in algorithm design.
Most algorithms assume complete knowledge of the input before computation begins.
Shuffling converts an ordered collection into a uniformly random permutation.
Many large-scale systems need to compare sets.
Many algorithms require computing enormous powers modulo an integer.
Ordinary logarithms answer the question: ```text a^x = b ```
Throughout this chapter we studied individual algorithms: - Euclidean Algorithm - Modular Arithmetic - Fast Exponentiation
The Chinese Remainder Theorem is a reconstruction tool.
A Diophantine equation is an equation whose solutions must be integers.
The Miller-Rabin test is the practical standard for fast primality testing.
A linear congruence is the modular analogue of a linear equation.
Many counting problems ask for results modulo a large number.
Many number-theoretic algorithms become simpler once a number is expressed as a product of primes.
Prime numbers occupy a central position in number theory.
Number-theoretic code is compact, but its edge cases are dense.
After working through the algorithms in this chapter, it is useful to step back and assemble them into a practical toolkit.
Euler's phi function counts how many numbers in a range are coprime to a given integer.
The Euclidean Algorithm is one of the oldest known algorithms and remains one of the most useful.
Division is one of the most subtle operations in modular arithmetic.
Primality testing tells us whether a number is prime.
Primitive roots describe generators of modular multiplication.
Many counting problems involve inclusion-exclusion.
Modular arithmetic is the arithmetic of remainders.
Number theory algorithms often look fast because their code is short.
The Euclidean Algorithm computes the greatest common divisor of two integers.
Number-theoretic algorithms often look simple when written with mathematical notation.
The Sieve of Eratosthenes is the standard algorithm for generating all primes up to a limit.
Exponentiation appears constantly in number-theoretic algorithms.
Divisibility is one of the most fundamental concepts in number theory.
Given a collection of half-planes, compute their common intersection.
Some geometric problems cannot stay entirely in integer arithmetic.
Many geometric algorithms must determine whether three points form a left turn, a right turn, or lie on the same line.
Given a set of axis-aligned rectangles, compute the total area covered by their union.
Individual geometry algorithms are useful, but real systems usually require several of them working together.
Many geometric algorithms are already asymptotically optimal.
Given a set of points, partition the plane into regions so that every location belongs to the region of its nearest point.
Given a set of points in the plane, find the pair with the smallest Euclidean distance.
Geometry code fails in ways that ordinary examples rarely expose.
After studying individual geometric algorithms, it is tempting to view them as unrelated techniques.
Given a polygon described by its vertices, compute its area efficiently and accurately.
Given a set of points, compute the smallest convex polygon that contains all of them.
Most real-world geometry software is not built from a single algorithm.
You need to search large collections of geometric objects efficiently.
Many computational geometry problems use coordinates that are naturally integers: ```text grid maps pixels
Geometric algorithms are easy to test visually and surprisingly easy to get wrong.
You need a fast way to group or search geometric objects by location.
After constructing a convex hull, many geometric questions remain: * What is the maximum distance between any two points?
Given a polygon and a query point, determine whether the point lies: ```text inside the polygon outside the polygon
You need to solve geometric problems that can be reduced to one-dimensional intervals.
Given a set of points in the plane, construct a triangulation that avoids thin, poorly shaped triangles as much as possible.
Geometric algorithms often look exact on paper but fail in code because numeric computations are approximate.
You need a mathematical representation for locations, directions, distances, and geometric relationships.
Given two lines or two line segments, determine whether they intersect and, if they do, compute the intersection point.
Many geometric problems involve detecting interactions among large collections of geometric objects.
Many string algorithms repeatedly compare substrings.
Given a text string and a pattern string, determine whether the pattern occurs in the text and, if so, find all positions where the match begins.
Throughout this chapter, we studied individual string-processing techniques: - Tokenization - Tries - Hashing
Many string algorithms depend on a precise ordering of strings.
Many strings contain repeated structure.
Given two strings, find the longest contiguous block of characters that appears in both.
Suppose you need to search for a pattern inside a large text.
String algorithms are easy to implement incorrectly.
Most exact matching algorithms process the text from left to right.
Many string algorithms focus on prefixes, suffixes, or arbitrary substrings.
Suppose you need to answer questions such as: - Does a substring occur in the text?
A single pattern search scans one text once.
You need to find palindromic substrings efficiently.
String algorithms often look deceptively simple.
Suppose you need to search a text for thousands of patterns simultaneously.
Many string algorithms need to answer the question: > How many characters match between a string prefix and a substring starting at a particular position?
Suppose you need to answer many substring queries against the same text.
In the previous recipe, you built a suffix array and used it to perform efficient substring searches.
The naive exact matching algorithm repeatedly compares the same characters after every mismatch.
After learning dozens of string algorithms, a natural question remains: > What do real systems actually do?
Most string algorithms operate on characters.
This chapter introduced a large collection of string-processing techniques: - Naive matching - KMP - Z Algorithm
Many string algorithms assume that a string is simply a sequence of characters.
Exact matching assumes that strings must be identical.
Suppose you need to search for many patterns simultaneously.
You are given a recursive algorithm and want to determine its time complexity.
You have learned the main divide-and-conquer patterns in this chapter.
Divide-and-conquer algorithms often fail in ways that are difficult to see from the high-level design.
After studying many divide-and-conquer algorithms, they can appear unrelated.
You have a divide-and-conquer algorithm and need to analyze its running time and memory usage.
You need to answer many queries over a fixed dataset.
Given \(n\) points on a two-dimensional plane, find the pair of points whose Euclidean distance is minimal.
An algorithm can have good asymptotic complexity and still run poorly on real hardware.
Quickselect finds the \(k\)-th smallest element in expected linear time.
You need to compute: \[ x^n \]
You need to multiply two polynomials.
You need to multiply very large integers.
Given \(n\) points in a plane, find the pair with the smallest Euclidean distance.
Merge sort guarantees \(O(n \log n)\) performance and stable ordering, but it requires additional memory proportional to the input size.
Divide-and-conquer algorithms naturally create independent subproblems.
You need to sort a collection of elements efficiently while preserving the relative order of equal values.
Many dynamic programming algorithms compute a table with the recurrence: \[ dp[i] = \min_{j < i} \left(dp[j] + cost(j,i)\right) \]
You need to measure how far an array is from being sorted.
You need to find the \(k\)-th smallest element in an unsorted array.
A divide-and-conquer algorithm may look correct because its structure is simple: ```text split solve recursively
Some problems ask for an optimal value rather than a specific object.
Many divide-and-conquer algorithms produce recurrences of the form \[ T(n)=aT\left(\frac{n}{b}\right)+f(n) \]
Divide-and-conquer algorithms are prone to subtle implementation errors.
You have learned the mechanics of divide and conquer: ```text split solve recursively
Karatsuba's algorithm demonstrates that multiplication can be accelerated by reducing the number of recursive subproblems.
Many greedy algorithms appear unrelated on the surface.
Many greedy algorithms operate under a simple principle: > A decision is either feasible or infeasible.
Many of the most successful graph algorithms are greedy algorithms.
The fractional knapsack problem is one of the clearest examples of a greedy algorithm producing an optimal solution.
Many greedy algorithms appear different on the surface.
Many greedy algorithms operate on sorted data and repeatedly make decisions from the extremes of a range.
Huffman coding is one of the most successful greedy algorithms ever developed.
The activity selection problem is historically one of the first optimization problems used to demonstrate the power of greedy algorithms.
Designing a greedy algorithm is often easier than proving it correct.
The greedy choice property is the fundamental condition that allows a greedy algorithm to produce an optimal solution.
Greedy algorithms become easier to recognize after seeing them embedded in complete problems.
The classical greedy algorithms presented earlier in this chapter are often introduced as isolated ideas: - Interval scheduling - Huffman coding - Fractional knapsack
Throughout this chapter, we have studied a wide variety of greedy algorithms: - Interval scheduling - Activity selection - Huffman coding
Greedy algorithms are attractive because they make local decisions.
Many greedy algorithms begin with a sort.
Most greedy algorithms are not proved by directly showing that the greedy solution is optimal.
Greedy algorithms often have simple control flow.
Minimum refueling stops is a greedy scheduling problem disguised as a travel problem.
Many scheduling problems ask a simple question: > Given limited time and many competing jobs, which jobs should be performed?
A counterexample is a small input that disproves a proposed algorithm, lemma, or proof idea.
Greedy algorithms are best learned by proving, breaking, and implementing them.
Many greedy algorithms are remembered by their choice rule: - Choose the earliest finishing interval.
Greedy algorithms are among the most elegant techniques in algorithm design.
String problems often invite dynamic programming, tries, automata, and hashing.
Scheduling problems occupy a special place in algorithm design.
Interval scheduling is one of the most important greedy problems.
Many dynamic programming recurrences have the correct state design but suffer from an expensive transition step.
This chapter has treated dynamic programming as a toolkit: state design, recurrence construction, memoization, tabulation, counting, optimization, graph DP, interval DP, tree DP, bitmask DP, and trans...
Dynamic programming is easiest when subproblems form a simple order: left to right, bottom to top, short interval to long interval, child before parent.
Memoization evaluates states on demand through recursion.
The knapsack problem occupies a special place in dynamic programming.
One-dimensional dynamic programming is the simplest and most common form of dynamic programming.
A dynamic programming solution is easy to mistrust.
Many dynamic programming solutions are correct but too slow.
Knuth optimization is a specialized dynamic programming optimization for interval-like recurrences.
Throughout this chapter, we have studied many different forms of dynamic programming: * one-dimensional DP * two-dimensional DP * knapsack DP
State design is the most important step in dynamic programming.
Once a state and recurrence have been defined, the most direct way to implement a dynamic programming solution is memoization.
Many dynamic programming problems ask for the best solution: ```text minimum cost maximum value
After defining a state, the next step is to determine how one state depends on other states.
Probability dynamic programming appears when each transition has uncertainty.
Dynamic programming algorithms are especially prone to subtle bugs.
The Longest Common Subsequence (LCS) problem is one of the most influential dynamic programming problems ever studied.
Many dynamic programming problems are defined on linear structures such as arrays, strings, and intervals.
Most dynamic programming problems use states based on positions, intervals, capacities, or tree nodes.
The edit distance problem asks a deceptively simple question: > How different are two strings?
The Longest Increasing Subsequence (LIS) problem is one of the most important sequence optimization problems in algorithm design.
Most dynamic programming problems decompose a problem into prefixes, suffixes, positions, or capacities.
One of the most common mistakes in dynamic programming is focusing exclusively on time complexity while ignoring memory consumption.
One-dimensional dynamic programming models progress along a single axis.
Optimization dynamic programming is the form most programmers first associate with DP.
Standard flow networks place capacity constraints on edges.
You need to find a maximum matching in a bipartite graph.
Edge-disjoint paths are allowed to share vertices.
Suppose you have a set of workers and a set of tasks.
Traditional maximum-flow algorithms answer a single question: > How much flow can be sent from the source to the sink?
Given a flow network with capacities on every edge, determine the maximum amount of flow that can be sent from the source to the sink.
Suppose you have already constructed a valid flow in a network.
In maximum bipartite matching, every assignment has the same value.
You need to solve an assignment problem exactly.
Maximum flow asks how much can be sent from a source to a sink.
Suppose you are designing a communication network between two data centers.
Scheduling problems often look different from flow problems.
A normal flow edge has only an upper capacity: ```text 0 ≤ f(u, v) ≤ c(u, v) ```
The algorithms developed so far share a common strategy.
So far, flow networks have had a source and a sink.
Edmonds-Karp guarantees polynomial running time, but it still performs only one augmentation per BFS search.
You have a set of possible projects.
Suppose you have an image and want to separate the foreground from the background.
The Ford-Fulkerson method provides a simple framework for computing maximum flow, but its performance depends heavily on the choice of augmenting paths.
You need to model the movement of resources through a constrained system.
Flow algorithms solve the same abstract problem, but their running times differ sharply.
A maximum-flow algorithm returns a number.
Flow algorithms are easy to implement incorrectly because their state changes over time.
Flow implementations are compact, but bugs are easy to hide.
Many optimization problems appear completely different on the surface.
Kruskal's algorithm builds a minimum spanning tree by selecting edges globally.
Offline connectivity works well when edges are fixed, or when all updates can be processed in a convenient order.
A minimum spanning tree gives the cheapest way to connect all vertices.
Minimum spanning tree algorithms become useful only after you can recognize the pattern in unfamiliar problems.
Kruskal's algorithm grows a forest by processing edges globally.
The basic union-find structure can become highly unbalanced.
You have a set of objects and pairwise distances between them.
Many real-world optimization problems can be modeled as graphs.
Minimum spanning tree algorithms are short.
Many graph algorithms repeatedly ask the same question: > Do these two vertices already belong to the same connected component?
Suppose you are given a graph and thousands or millions of connectivity queries: ```text Are vertices u and v connected?
Throughout this chapter, we have studied minimum spanning trees from multiple perspectives: * Graph theory * Greedy algorithms * Union-find
The classical MST algorithms were designed for a single machine processing a graph stored in local memory.
After studying cut properties, union-find, Kruskal, Prim, Borůvka, clustering, network design, and complexity analysis, a practical question remains: > How should MST algorithms actually be implemente...
By this point, you have seen three major minimum spanning tree algorithms: ```text Kruskal Prim
Given a connected weighted undirected graph, find a spanning tree whose total edge weight is minimum.
Minimum spanning tree algorithms are usually described in terms of `V` vertices and `E` edges.
Minimum spanning tree algorithms are easy to state, but their performance depends heavily on representation and data structure choices.
Minimum spanning trees minimize the **total weight** of selected edges.
Minimum spanning tree implementations can look correct while failing on small edge cases.
Minimum spanning tree code often fails for reasons that are easy to miss in clean textbook examples.
Path compression makes existing trees flatter, but it does not prevent bad trees from being created.
You are given a connected undirected graph.
Minimum spanning tree algorithms repeatedly make local decisions.
Sparse graphs reward algorithms that touch only existing edges.
The cut property identifies edges that can safely be added to a minimum spanning tree.
Many shortest-path algorithms are designed to handle arbitrary graphs.
This chapter introduced a substantial collection of shortest-path algorithms.
Most shortest-path algorithms answer a single question: > What is the shortest path from the source to the destination?
Dijkstra's algorithm depends on a crucial assumption: edge weights must be nonnegative.
Breadth-first search works because every edge contributes the same cost.
Shortest-path implementations are prone to subtle errors.
Breadth-first search (BFS) is the simplest shortest path algorithm.
Learning shortest-path algorithms is only the first step.
Use BFS when every edge has the same cost.
Many shortest-path algorithms begin at the source and gradually expand outward until the target is reached.
The shortest-path algorithms studied so far naturally divide into two groups.
Many shortest-path problems are not given as explicit graph structures.
Shortest-path code usually fails in predictable ways.
The previous recipe used Dijkstra’s algorithm with a binary heap and lazy deletion.
One of the most powerful ideas in algorithm design is that many problems that do not look like graph problems can be transformed into shortest-path problems.
Shortest-path algorithms are often chosen by correctness first, then by complexity.
Every shortest-path algorithm in this chapter appears different on the surface.
The algorithms covered so far solve the single-source shortest path problem.
Most shortest-path algorithms compute distances.
Between ordinary BFS and Dijkstra lies an interesting class of shortest-path problems.
Dijkstra's algorithm finds shortest paths by expanding vertices in order of increasing distance from the source.
Shortest-path algorithms are easy to implement incorrectly.
In the previous recipe, Bellman-Ford was used to compute shortest paths in graphs that contain negative edge weights.
Most shortest-path problems begin with a single source vertex.
Negative edge weights complicate shortest-path algorithms.
You need to determine whether a graph contains a cycle.
You need a graph representation that is simple to construct, compact to store, and efficient for algorithms that process edges directly.
You need to arrange tasks in an order that respects dependencies.
You need to identify independent regions within a graph.
You need to divide vertices into two groups such that every edge connects vertices from different groups.
You need to explore a graph in order of increasing distance from a starting vertex.
You are given a graph problem and need to quickly identify the correct algorithmic approach.
You need to model relationships that have direction.
You need to explore a graph systematically.
You need to choose the right in-memory representation for a graph.
You need to find vertices whose removal disconnects an undirected graph.
You need to traverse every edge in a graph exactly once.
You need to find edges whose removal disconnects a graph.
You need a graph representation that supports efficient traversal, scales to large datasets, and works naturally with algorithms such as depth-first search, breadth-first search, topological sorting,...
You need to visit every vertex exactly once.
You need basic measurements that describe the local and global structure of a graph.
You need to turn a real problem into a graph before choosing an algorithm.
You need confidence that a graph algorithm handles real inputs, edge cases, and malformed assumptions correctly.
You need to connect all vertices in a graph while minimizing total cost.
You need to find the cheapest route between vertices.
You need a graph representation that can answer edge-existence queries quickly.
You need to identify groups of vertices that are mutually reachable.
You need to assign labels, colors, or resources to vertices so that adjacent vertices do not conflict.
You need to model relationships that are naturally symmetric.
You need to model relationships where connections have different costs.
Depth-first search (DFS) is the fundamental traversal technique for trees.
Many tree algorithms eventually become range-query problems.
All previous tree algorithms in this chapter assume that the tree structure is fixed.
Breadth-first search (BFS) visits a tree level by level.
A tree in memory is not directly portable.
Most tree representations focus on speed.
Not every range-query problem requires the full power of a Segment Tree.
Many tree problems ask questions about an entire subtree rather than an individual node.
Tree height measures how far a tree extends downward from a node.
Many tree algorithms compute information relative to a fixed root.
The diameter of a tree is the length of the longest path between any two nodes.
Many tree problems ask questions about paths.
Suppose two computers each store a copy of a large dataset.
Most data structures answer questions about the present.
Many tree algorithms repeatedly ask the same question: > What is the ancestor of this node k levels above?
The Lowest Common Ancestor (LCA) of two nodes is the deepest node that is an ancestor of both.
Expression trees represent computations as tree structures.
Segment Trees provide efficient range queries and point updates.
Many search problems involve prefixes.
Two trees may look different at first glance yet represent exactly the same structure.
This chapter has introduced a wide range of tree algorithms and data structures.
Trees are usually introduced with drawings: circles connected by lines, one circle at the top, several below it, then more below those.
Most tree algorithms are recursive, not because recursion is elegant, but because trees are recursive objects.
Many tree algorithms process information from a fixed root.
Many tree problems become dramatically easier when you stop thinking of the tree as a tree.
The motion is completely determined by the two lines and the current point.
Represent the airline network by a simple graph $G$ on $2n$ vertices.
We seek all integer triples $(x,y,z)$, none equal to $1$, satisfying
Consider a coat of area $1$ and five patches, each of area at least $\frac{1}{2}$.
Let the trapezoid have bases of lengths $b$ and $a$, with $a<b$.
Begin by examining the three-variable inequality
A strategy can be represented by a decision tree.
Consider small values of $n$ first.
Let the circle have center $O$ and radius $r$.
Let the angles of $T$ be $A,B,C$.
Let the bisectors of $\angle A$ and $\angle B$ meet at a point $P$.
Testing small values of $n$ shows that the divisibility condition $n^2+1 \mid n!$ is rarely satisfied for small integers, as $n^2+1$ grows faster than $n$.
Consider small examples of natural numbers and attempt to write them as sums of numbers whose reciprocals add to one.
Let the optimal finishing time be $T$.
Consider first the intersection of two cylinders of equal radius $r$ with axes perpendicular.
Consider the equation
Consider the case of a triangle first.
Let the roots of
The equation is
For problem c), the set of all solutions of
Consider three points on the plane.
For four points on a circle, label them by position vectors $a,b,c,d$ on a circle with center $O$, taken as the origin.
Let the rectangle be centered at the origin, with sides parallel to the coordinate axes.
Let the magic sum be $M$.
Consider first the case $n=4$.
Let us denote the initial right triangle as $A_0A_1A_2$, with right angle at $A_2$, and legs $|A_0A_2|=a$ and $|A_1A_2|=b$.
Consider a small social network where each person has exactly three friends.
Let the members of each country form a set of integers contained in ${1,2,\dots,1978}$.
Let
Let the given sphere have center $O$ and radius $R$.
Part 1 is a special case of Part 2.
Begin by examining small values to understand the recursive structure imposed by $g(n)=f(f(n))+1$.
For the first part, the condition means that every domino of the upper layer must cross the boundary between two dominoes of the lower layer.
The defining condition of a trigram is
Consider triangle $ABC$ with a point $P$ inside it, through which three lines are drawn, each parallel to one side of the triangle.
The statement concerns a regular frustum of a pyramid.
Let each cell be represented by a variable in $\mathbb F_2$, where $1$ means black and $0$ means white.
Consider two points $A$ and $B$ on a line and a motorist starting from $A$ and a cyclist starting from $B$, both moving toward each other at constant speeds $v_m$ and $v_c$.
For small values,
Consider an arithmetic progression $a$, $a+d$, $a+2d$, $\dots$, where $a$ and $d$ are natural numbers.
Denote
Consider the $8 \times 8$ chessboard with one chip on each square.
A homothety with ratio $k<0$ reverses directions through its center.
Represent the circle by the additive group $\mathbb R/\mathbb Z$, so that arc lengths are measured as fractions of the circumference.
Consider small $n \times n$ boards and simulate the game.
Label the convex quadrilateral $ABCD$ with consecutive sides $AB = a$, $BC = b$, $CD = c$, and $DA = d$.
The area of the orthogonal projection of a polyhedron onto a plane depends on the direction of projection.
For one segment the answer is trivial.
Let the convex quadrilateral be $ABCD$, and let its diagonals $AC$ and $BD$ intersect at $P$.
Consider the first few values of $a_n$.
Consider the sequence $x_n=(1+\sqrt{2}+\sqrt{3})^n$.
Let $W$ denote a winning position for the player to move and $L$ a losing position.
Consider the inequality
We seek an infinite bounded sequence $(x_n)$ such that every two distinct terms satisfy
Let the lake be the interior of a simple nonconvex polygon $P=A_1A_2\cdots A_n$.
The three machines modify cards in distinct ways.
Consider a square inscribed in the graph of $y = A \sin x$.
Starting with two points $A$ and $B$ at distance 1, reflecting one about the other generates points along the line $AB$.
Let
Consider a small round-robin tournament with $n$ teams.
Consider placing the digits $0,1,2$ in a small grid and examining rectangles of size $3 \times 4$.
Consider the triangular table for small values of $a$ to understand the pattern.
A coordinate model is natural because the configuration contains a rectangle and two midpoints.
Let the entry in row $i$, column $j$ be $a_{ij}$.
Equation (1), $2^x + 1 = 3^y$, suggests searching for powers of 2 that are one less than a power of 3.
For the one-dimensional problem, the state of the process is not the point $O_k$ itself but the set of material points lying in the interval of length $2r$ centered at $O_k$.
Let $F$ be the set of free squares.
The three segments $AA_1$, $BB_1$, $CC_1$ are parallel and not coplanar.
A power $3^k$ begins with the digit $9$ precisely when there exists an integer $m$ such that
Consider first a single square of area $1$.
Let the hexagon be $A B C D E F$ in cyclic order.
I cannot write a solution to Kvant problem M153 without the actual problem statement or the diagram.
I can produce the full Kvant-style solution structure for problem M154.
The statement concerns divisibility of numbers of the form $a^k+b^k$.
Consider a square of side length $1$ and a line dividing it into two quadrilaterals with areas in the ratio $2:3$.
Let the four circles have the same radius $r$.
For $n=1$ the statement is immediate.
Let
Let $P$ be the intersection of the tangents at $A$ and $C$.
Label the vertices of a regular $n$-gon by the residues modulo $n$.
Consider the first condition: the perimeters of the four triangles formed by three consecutive vertices of a quadrilateral are equal.
Consider small cases first.
Consider the first several terms of the sequence defined by taking the integer closest to the cumulative target $n\sqrt{2}$.
Let us compute the first terms.
Consider a cube with its twelve edges labeled by distinct numbers $1$ through $12$.
Consider small rectangles with integer sides.
Consider small positive integers $n$ and examine the condition that if $n$ is divisible by $p-1$ for some prime $p$, then $n$ must also be divisible by $p$.
The question asks whether there is a set $F$ which by itself cannot contain any semicircle of radius $1$, while two congruent copies of $F$ can together contain the whole unit circle.
Let us interpret the operations in reverse.
Let the altitude $BH$ be the $y$ axis, and let $H=(0,0)$.
Let the parallelogram have side vectors $\mathbf u=\overrightarrow{BA}$ and $\mathbf v=\overrightarrow{BC}$.
For $m=1$ and $n=2$,
The anaconda is an arbitrary polygonal line of total length $10$ contained in the unit square.
The stones have weights
Consider a quadrilateral with consecutive sides $a$, $b$, $c$, $d$ and area $S$.
Model the cellular shell as a polyhedral decomposition of a sphere.
The problem asks whether a sequence of allowed replacements can form a nontrivial cycle.
Consider small values of $n$ to understand the pattern of the product.
The equation is
Consider small values of $n$ and attempt to construct sequences of $+1$ and $-1$ satisfying the condition that for each $k=1,2,\ldots,n-1$, the sum of the $n$ pairwise products of numbers separated by…
Let the variable triangle have vertices $P\in AB$, $Q\in BC$, $R\in AC$.
For the concrete problem with capacities $5$, $7$, and $12$, the target state is two portions of $6$ liters each.
For a polynomial to be monotonically increasing on the whole real line, it is enough that its derivative be nonnegative everywhere.
Consider a cyclic quadrilateral $ABCD$ and extend opposite sides $AB$ and $CD$, $BC$ and $DA$.
For points in the plane, the condition says that every triangle determined by the chosen points is acute or right.
Computing the first few values of $m = n + s(n)$ quickly shows that many numbers can be represented in this form.
Before I begin writing the full Kvant-style solution, I need the text of problem M908.
Let the polygon be $P$, let its area be $S$, and let the radius of its inscribed circle be $r$.
Let the set be $A={a_1,a_2,\dots}$, with no divisibility relations between distinct elements.
Let the median be drawn from a vertex $A$ to the midpoint $M$ of the opposite side.
A convex polyhedron with six faces is a cube or a more general hexahedron.
Let
Let
Let the consecutive arcs of the circumcircle be
Let
The problem asks for integers $(x,y)$ such that $(x+y)^7 - x^7 - y^7$ is divisible by $7^7$, while $(x+y)xy$ is not divisible by $7$.
Consider the problem of choosing three points $A$, $B$, $C$ in the plane such that every point $P$ has at least one segment $PA$, $PB$, or $PC$ of irrational length.
The problem defines a binary operation $_$ on a set with three strong constraints: a generalized associativity condition $a_(b_c)=b_(c*a)$, left and right cancellation laws, and asks to prove commutat…
The border of width two around an $n\times n$ board is the set of squares obtained after embedding the board into an $(n+4)\times(n+4)$ board and removing the central $n\times n$ square.
The vectors described in the statement depend only on the face areas and outward unit normals.
Let the amounts of water be $a,b,c$, all positive integers.
Let $v(t)$ be the speed of the snail at time $t$.
Let the vertices of the convex polygon $M$ be $A_1,A_2,\dots,A_n$, and let $B_i$ be the midpoint of side $A_iA_{i+1}$, where indices are taken modulo $n$.
Consider small integer triples $(a,b,c)$ satisfying $a+b+c=0$.
Let the numbers around the circle be $x_1,x_2,\dots,x_n$, with indices taken modulo $n$.
For small values of $n$ the statement is easy to check directly.
Consider the circle inscribed in an angle with vertex $O$ and the two diametrically opposite points $A$ and $B$.
The black cells form a finite set.
The condition forbids the distance $d=0.
Represent each sign by a number in ${0,1}$, where $0$ denotes $+$ and $1$ denotes $-$.
Represent the cells by lattice points $(i,j)$, where $1\le i\le m$ and $1\le j\le n$, the coordinates being the centers of the cells.
For a triangle, suppose a line intersects two sides and cuts the triangle into two parts.
For each parallelogram $A_iB_iC_iD_i$, the diagonals bisect each other.
The basic fact about digit sums is that replacing a number by the sum of its digits does not change its residue modulo $9$.
Consider first the equilateral triangle case.
The problem involves two points $P$ and $Q$ inside triangle $ABC$ such that at vertices $A$ and $B$, the lines connecting the vertex to the points form equal angles with the corresponding angle bisect…
The system is
A triangle already satisfies the condition for each of its three sides, since the third vertex completes an equilateral triangle.
The angles form an arithmetic progression with common difference $4^\circ$:
Consider small initial colony sizes to understand the dynamics.
Let $x_n$ denote the length of the base of the $n$th trapezoid obtained in the process, with $x_0=AB=a$ and with the other base always equal to $b=CD$.
The inequality resembles the triangle inequality.
The table resembles a generalized Pascal triangle, where each entry is the sum of the three entries immediately above it.
Let the five positive numbers be $a$, $b$, $c$, $d$, $e$.
Let $V$, $E$, and $F$ denote the numbers of vertices, edges, and faces of the polyhedron.
Consider small sequences of $+1$ and $-1$ and compute the sum $x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1$.
Let the trapezoid have bases $AB$ and $CD$, with $AB>CD$, and let $E$ and $F$ be the midpoints of the legs.
The schedule repeats with period $\operatorname{lcm}(2,3,5)=30$.
The equation is
Let $ a connected set of cells.
Let
Let the three circles have common radius $r$, and let their common point be $P$.
Consider small rectangular boxes that can be tiled with $2 \times 2$ and $1 \times 4$ tiles.
Consider a cubic polynomial $x^3+ax^2+bx+c=0$ and suppose its roots form an arithmetic progression.
For a triangle the statement is trivial, since the three sides themselves already form a triangle containing the polygon.
Consider the square lattice $\mathbb{Z}^2$ with distinguished origin $O=(0,0)$.
Let the cars be arranged around the circle in their order along the road.
The statement as written contains a typographical error.
Consider first small examples.
For small values of $n$, the statement is easy to check directly.
Let the right triangle have legs of lengths $a$ and $b$, with hypotenuse $c = \sqrt{a^2 + b^2}$.
Consider the square $A_1 A_2 A_3 A_4$ with an arbitrary point $P$ inside it.
For a $1\times 1$ table the statement is trivial.
Let the two intersecting lines be $l_P$ and $l_Q$, meeting at a point $O$.
Let
Part a) suggests looking at projections.
Let the triangle have sides adjacent to angle $A$ equal to $10$ and $15$.
Consider small-degree polynomials to detect a pattern.
Represent the group by a simple graph.
Let the common value of the three face angles be $\alpha$:
Consider a $2 \times 2$ table filled with arbitrary numbers:
The problem concerns numbers whose squares end with the same digits as the number itself, sometimes called automorphic numbers.
Consider three consecutive integers $n-1$, $n$, $n+1$ and compute the sum of their squares.
For each line $l_i$, let $P_i$ denote the orthogonal projection of the plane onto $l_i$.
Let $R$ be the set of remaining integers, and let $A=R\setminus{1}$.
Let
Fix the player's marked set of $8$ squares.
Consider the examples given: $3^2 + 4^2 = 5^2$, $36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2$, and $55^2 + 56^2 + 57^2 + 58^2 + 59^2 + 60^2 = 61^2 + 62^2 + 63^2 + 64^2 + 65^2$.
Consider the pattern formed by concentric circles of radii $1,2,3,\dots$ and a fixed line $l$ through the center $O$, along with all tangents to the circles parallel to $l$.
The total number of numbers is $1025=2^{10}+1$.
The ring is the solid obtained from a sphere by drilling a cylindrical hole through its center.
For Part 1, denote by $P=AF\cap BG$, $Q=BG\cap CE$, $R=CE\cap AF$.
The numbers under consideration are exactly the positive integers whose decimal expansion consists only of zeros and ones.
Let $A$ and $B$ be the feet of the altitudes from $Q$ and $P$ onto the sides $PM$ and $QM$ respectively.
Consider small odd numbers to test the claim.
The problem asks whether it is possible to tile a square using 18 dominoes of size $1\times 2$ such that no straight line of tile edges connects opposite sides of the square.
Let
Consider small values of $n$ to see the pattern.
Consider the initial configuration of four ones and five zeros written around a circle.
The three given lines are the three internal angle bisectors of a triangle.
Let
Let each rectangle have side lengths $a\ge b$.
Let the five segment lengths be
Consider small cases of regular polygons to understand the combinatorial structure imposed by coloring.
We consider triangle $ABC$ with incenter $O$ and midpoint $M$ of side $BC$.
Let the numbers be $a,b,c>0$ with $abc=1$.
There are $99999-11111+1=88889$ cards.
For small values of $n$ the statement is easy to test.
Let the longest diagonal of a convex polygon have length $D$.
Represent the five numbers as five binary strings of length $n$, where the symbols are $1$ and $2$.
Represent each cell of the sheet by a vertex.
Let $s(n)$ denote the sum of the decimal digits of $n$.
Let the circle have center $O$ and radius $R$.
Let the common point of the angle bisector $AD$, the median $BM$, and the altitude $CH$ be $P$.
Let $AB=p$ and $AC=q$.
For the first sum,
Consider small values of $n$ to gain intuition.
Let the original seventeen-digit number be
Consider the equation $x^2 - mxy + y^2 = 1$ with $x, y \ge 0$ and integer $m>1$.
Let $A(R)$ denote the sum of the numbers in a rectangle $R$ whose sides follow the grid lines.
Let
Consider arranging seven points and seven lines such that each point lies on exactly three lines and each line contains exactly three points.
Consider small cases first.
The polyhedron has 19 faces and is circumscribed about a sphere of radius $10$.
The number
Each tile is an isosceles trapezoid with bases $3$ and $1$ and height $1$.
Each cut is made on a single existing piece and splits it into two pieces.
For $N=1$, a single circle of diameter $0$ centered at the point covers it, and the sum of diameters is $0<1$.
Let the maximum number of polygons met by a line be denoted by $k$.
Let each vertex be a point, and let its color at time $t$ be represented by a sign $s_v(t)\in{+1,-1}$.
Let the radius of each coin in the chain be $r$.
For the first part, the information-theoretic count is encouraging.
For small values of $n$ the statement is easy to test.
The numbering pattern is linear.
The condition on the denominators is much stronger than in the usual Egyptian fraction problem.
Let
Consider an angle formed by two rays meeting at a vertex $O$.
Let the circles have radii $r_1,r_2,\dots,r_n$.
Consider a single excited cell in an infinite linear chain at $t=0$.
Consider an equilateral triangle $ABC$ with circumcircle $\Gamma$.
Let the fork be at point $A$.
Consider small instances to develop intuition.
Consider a polynomial $p(x)$ with integer coefficients that takes the value $1$ at three distinct integers, say $a$, $b$, and $c$.
Consider a convex quadrilateral $ABCD$ with area $S$.
Consider first small values of $n$ to understand the structure of the sum of pairwise differences.
Let the rows be $R_1,\dots,R_n$, each a vector of length $n$.
The ruler has two fixed marks.
Consider the set of faces of a convex polyhedron.
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ intersect at $P$ and satisfy $AC \perp BD$.
The inequality is symmetric in $a$ and $b$.
A shuffle takes an initial segment of the deck and inserts it somewhere later, preserving the internal order of the removed block and of the remaining cards.
Let
Let $S_t$ be the set of sick Mites on day $t$.
Define
Let the circle have center $O$ and radius $R$.
The problem involves packing 18 tons of cargo into at least 35 containers, with seven spacecraft available, each capable of carrying 3 tons, and the assertion that any selection of 35 containers can b…
Let
For the first statement, computing small cases is instructive.
The quadrilateral is cut by two families of lines.
The operations are purely projective.
Compute the first few terms of the sequence $(a_n)$ directly from the recursive formula.
A cube is highly symmetric, so the number of axes of symmetry should be larger than in simpler polyhedra.
Consider the two Diophantine equations
Let the triangle be $ABC$.
For the numbers $1,2,\dots,30$, the total sum is
A triangular pyramid is a tetrahedron.
Let $s(n)$ denote the sum of the digits of the single number $n$.
The data of the problem are naturally encoded by a similarity.
Assume that the infinite digit string obtained by concatenating
The statement involves two circles.
For part 1 it is natural to reinterpret a nondecreasing tuple
For the planar statement, choose coordinates so that the two given perpendicular directions are the coordinate axes.
The polygon is rectilinear: every side lies on a grid line, hence every side is horizontal or vertical.
An isosceles trapezoid includes rectangles as a special case, since a rectangle has a pair of parallel sides and equal legs.
The recurrence
A reflection with respect to a point $A$ is the central symmetry $x\mapsto 2A-x$.
I cannot write a solution to Kvant problem M604 from the information provided, because the actual problem statement is missing.
The denominators suggest introducing
Let the three consecutive entries in row $n$ be
Let $H$ be the orthocenter of triangle $ABC$, let $M$ be the midpoint of $BC$, and let $D$ be the point on the circumcircle diametrically opposite $A$.
Let the circles intersect at points $A$ and $B$.
Let
The condition that no two black faces share an edge means that every edge of the polyhedron is incident with at most one black face.
Label the trees by the residues modulo $n$, arranged around the circle.
The structure of the election is a rooted tree.
Let the centers of the circles be the vertices $A,B,C,D$ of a convex quadrilateral, listed in cyclic order.
Consider a tetrahedron with vertices $A$, $B$, $C$, and $D$.
Consider the original game with 25 matches, where each player may take 1, 2, or 3 matches per turn, and the winner is the player whose total number of matches at the end is even.
Let
Consider a standard 12-hour analog clock with an hour hand and a minute hand.
Let $E$ be a set of $n$ elements and $S_1, S_2, \dots, S_m$ be the chosen subsets of $E$ (distinct from $E$) such that for any pair of elements of $E$, there is exactly one $S_i$ containing both.
Fix the segment $AB$ and let its length be $d$.
Each circle lies on the unit sphere.
For the square tiling, the centers of all squares form the standard square lattice $\mathbb Z^2$.
The simulator maintains a global simulated time variable $CLOCK$ and executes each MIX instruction by dispatching to a routine that models its effect on registers, memory, and timing. The I/O instructions considered here are restricted to devices $16$ and $18$.
Each array element can be modified independently. For a value $x$, we may choose any positive integer $b$, then replace $x$ by $x bmod b$. The goal is to maximize the MEX of the resulting array.
An array $a$ is fixed, and we append to it a permutation $p$ of ${1,2,dots,m}$ to form a longer array $bp$. For each such permutation, we count how many subarrays of $bp$ with even length are palindromes.
We are given a binary string made of only a and b. The goal is to check whether we can turn it into a perfectly alternating string, meaning every adjacent pair of characters differs. We are allowed to perform at most one operation.
We are given an array of length $2n$. Every value from $0$ to $n-1$ appears exactly twice. The task is to select a contiguous segment of this array that reads the same left-to-right and right-to-left, and among all such segments we want the one whose mex is as large as possible.
We are given a permutation of the integers from 1 to n. For every contiguous subarray, we look at three quantities: - its maximum value, - its minimum value, - the GCD of all values inside it.
The problem simulates a single transmigration process in a role-playing game. We start with a character who already possesses several skills. Each skill has a name and an experience level. When transmigration occurs, every existing skill has its level reduced by a coefficient k.
We are given a tree and a fixed set of vertices consisting of all leaves in the original tree. These leaves are determined once from the initial structure and do not change during the process.
We are given a string made only of opening and closing parentheses. In one move, we are allowed to take a contiguous block, remove it, and then reinsert its characters anywhere in the remaining string, with full freedom to permute those removed characters and place each one…
We are given a tree. For a pair of vertices $(u,v)$, we remove every edge belonging to the unique simple path between them. After those edges are deleted, the tree breaks into several connected components. The value of the pair is the size of the largest remaining component.
We are given a sequence of positive integers. We repeatedly remove groups of elements until nothing remains. Each group must respect two constraints: if we look at the chosen indices in increasing order, the corresponding values must be nondecreasing, and the indices…
A subarray product is divisible by $6$ if and only if the product contains at least one factor $2$ and at least one factor $3$. For each number, only its divisibility by $2$ and $3$ matters. The exact value is irrelevant.
We are given several test cases, each consisting of an array. The task is to decide whether we can transform each array into a non-decreasing order using a specific operation: we may pick any subarray, compute its minimum and maximum values, and reverse it, but only if the sum…
Each column contains cubes stacked from height $1$ up to height $ai$. After gravity turns to the right, cubes never change height. Cubes at the same height slide independently and occupy the rightmost available positions on that horizontal level.
We are given three characters, each standing on a different position along a one-dimensional half-line. Each position is an integer starting at 1, and each character has a movement range and a throwing range.
For every divisor $dmid n$, define $$Gd={vin[1,n]mid gcd(v,n)=d}.$$ The array $a$ gradually fixes some positions. If $ai=dneq 0$, then position $i$ must contain a value from $Gd$. Unfixed positions may contain any remaining values.
We have a small assembly of n senators, each defined by a level and a loyalty score. Loyalty is a probability that a senator votes in favor of a proposal, given in 10% increments. If more than half of senators vote yes, the proposal passes.
Each problem in the contest is worth at most 100 points. If a problem has $ai$ subtasks, then every solved subtask contributes $$frac{100}{ai}$$ points, and $ai$ is guaranteed to divide 100.
The original problem is interactive, but the version used for judging after the contest is the hacked format. Instead of interacting with a judge, each test case directly gives us the hidden values n, k, and c.
We are given a grid starting at the origin point $(0,0)$ and a target point $(x,y)$. At each move, Yousef can increase exactly one coordinate by 2 using a long step, either moving right by 2 or up by 2.
The sequence $[1,2,dots,n]$ is fixed. For any interval $[l,r]$, the value of interest is the bitwise XOR of all integers in that interval. A segment is valid when two conditions hold simultaneously. The index $x$ lies inside the segment, so $l le x le r$.
We are given an array $a$ of length $n$. We must construct a permutation $p$ of indices $1$ to $n$. For any pair of positions $i<j$, the pair contributes to the score only if it is an inversion in $p$, meaning $pipj$.
The statement is intentionally wrapped in game terminology, but the underlying process is a sequence of color merges. Each map cell has a panel color. Some cells also contain a symbol, and every symbol has its own color. We start by destroying one specific symbol.
A string $s$ must be split into several contiguous pieces. Each piece has length at least $l$, and the total number of pieces is at least $k$.
We are given a grid with $2$ rows and $n$ columns. Each cell is initially colored either red or black. We are allowed to repaint any cells, and the goal is to reach a final coloring with the following property: the entire $2n$ cells can be partitioned into exactly $n$ disjoint…
We are given up to $10^4$ integer points on the plane and a fixed radius $r$. We must output the centers of several circles, all having radius $r$, such that the centers have integer coordinates and no two circles overlap in positive area.
We are given an array, and we look at it prefix by prefix. For each prefix we are allowed to transform each element independently exactly once using a very permissive operation: pick any integer $bi ge 1$, replace $ai$ with the remainder of dividing it by $bi$.
Ta có một mảng số nguyên dương. Một phép giảm chọn ba phần tử liên tiếp sao cho phần tử giữa nhỏ hơn tổng hai phần tử hai bên. Khi đó bộ ba $$(c{i-1},ci,c{i+1})$$ được thay bằng một giá trị duy nhất $$x=c{i-1}-ci+c{i+1}.
Each test case gives a sequence of column heights. Think of column i as a vertical stack of unit blocks, occupying rows 1 up to ai. All blocks in a row are aligned across columns.
We have an array, and before each marking operation we are allowed to reverse any odd-length segment centered at some position. After that reversal, the element currently sitting at index xi becomes marked. The subtle detail is that marks belong to elements, not positions.
Understood. For any TAOCP Volume 1 exercise you provide, I will write the solution in the style of a rigorous solution manual: - Use Knuth's notation and terminology from the relevant section. - State precisely what is to be proved or computed.
This problem asks us to find all numbers in the array that appear exactly twice and compute the bitwise XOR of those numbers. The input is an integer array nums. The problem guarantees that every value appears either once or twice, never more than twice.
We are given a rooted tree with n nodes, where node 0 is the root. The input edges describes the tree structure. Each entry edges[i] = [u, v] indicates that there is an undirected edge between nodes u and v.
We the objective is to maximize the product rather than count pairs.
The problem asks us to determine, for each position in an array nums, the maximum value that can be reached by making a series of jumps under strict rules.
We are given an integer array nums. A pair of indices (i, j) is called a mirror pair if: - i < j - reverse(nums[i]) == nums[j] The operation reverse(x) reverses the decimal digits of x and removes any leading zeros that appear after reversal.
The problem asks us to calculate the absolute difference between the sum of the k largest elements and the sum of the k smallest elements in a given array nums.
assert Solution().
The problem gives us a string s containing only digits from '1' to '9'. We need to find the first adjacent pair of digits that satisfies two conditions. First, the two digits in the pair must be different. A pair like "22" or "55" is never valid.
The exponent $1974$ is large, so direct computation is impossible.
We are remains efficient even at the maximum constraint limit.
We are given two arrays of equal both required ranges are empty and the impossibility condition cannot occur.
This This ensures that performance in one subject never affects another subject's result.
LeetCode 3394: Check if Grid can be Cut into Sections (Medium)
We are given a string s consisting of lowercase English letters. An operation chooses a character value c and simultaneously replaces every occurrence of c in the current string with the next letter of the alphabet. The alphabet is circular, so 'z' becomes 'a'.
Let a subsequence of nums be any sequence multiples. citeturn0search1turn0search2 check whether it is strictly increasing, compute its GCD, and add that GCD to the answer. This is correct because every valid subsequence is examined exactly once.
We are given two integers, n and k. Our goal is to split n into exactly k positive integers whose product is equal to n. Among all valid decompositions, we want the one where the difference between the largest number and the smallest number is as small as possible.
We are given a string s of length n and a permutation array order. At time t = 0, the character at index order[0] is replaced with ''. At time t = 1, the character at index order[1] is also replaced with ''.
The problem asks us to determine the earliest possible time a tourist can finish exactly one land ride and one water ride at a theme park, where rides may be taken in either order.
The problem asks us to transform a given integer array nums into a balanced array with the minimum number of removals. An array is defined as balanced if its maximum element does not exceed k times its minimum element.
We are asked to sort a list of threats according to a computed score. Each threat is represented as a 3-element array [IDi, sevi, expi], where IDi is a unique identifier, sevi is the severity, and expi is the exploitability.
This problem asks us to count integers x in the range [1, n] such that the popcount-depth of x is exactly k. The popcount-depth is defined via a sequence p0, p1, ... where p0 = x and pi+1 = popcount(pi) for all i ≥ 0.
The problem models a simple selection process among students arriving one by one. Each student has a rank, where a smaller rank value represents a better student. The first arriving student is automatically selected.
We are given an undirected graph with n ≤ 14 vertices. Each vertex has a character label. We may choose any simple path in the graph, meaning a sequence of adjacent vertices in which no vertex is visited more than once.
Let's dive into a full technical guide for LeetCode 3613 following your requested structure. The problem presents an undirected, connected graph with n nodes and weighted edges. Each edge connects two nodes and has a weight representing its "cost.
The problem requires us to determine the minimum number of prime numbers from the first m primes whose sum equals n. A multiset is allowed, meaning each prime may be chosen multiple times.
We are given an undirected graph with n vertices and a list of edges. Every edge has an associated removal time timei. The graph evolves over time. At time t, every edge whose removal time satisfies timei <= t has already been removed. All edges with timei t remain in the graph.
This problem asks us to determine the maximum possible stability of a spanning tree built from a weighted, undirected graph with certain constraints. Each edge in the graph has a strength and a mandatory flag.
The problem is asking whether it is possible to transform an array nums consisting of only 1 and -1 into an array where all elements are equal, by performing at most k operations. Each operation allows you to select an index i and flip the signs of both nums[i] and nums[i+1].
We are given an integer array nums and need to determine whether it can be divided into three consecutive segments that follow a very specific pattern: 1. A strictly increasing segment. 2. A strictly decreasing segment. 3. A strictly increasing segment.
The numbers that still fit in standard integer types; the algorithm scales logarithmically with digit count.
We are simulating a very specific combat process on a fixed line of monsters, and answering many independent queries on subsegments of that line.
We are asked to construct an exact sum using coins of values 1, 3, and 5. For each target amount n, we want to pay exactly n using any number of 3 and 5 coins, and we are only forced to use 1-coins when it becomes unavoidable.
We are given a set of $n$ game scenarios, each of which must be played exactly once by the player. Between any two different scenarios $i$ and $j$, the game has a transition video that can be either funny (F) or scary (S).
We are given a multiset of $2n$ cards, where every number from $1$ to $n$ appears exactly twice in total. These cards are split evenly between two players, you and Nene, so each of you holds $n$ cards.
We are given up to 50,000 independent queries. For each query, a number n is provided, where 1 ≤ n ≤ 100000. A number is called a binary decimal if every digit in its usual decimal representation is either 0 or 1. Examples include 1, 10, 11, 101, and 1001.
Two firework machines start at time zero and then keep launching fireworks periodically. The first machine fires at times that are multiples of a, and the second fires at multiples of b.
Codeforces 1975E: Chain Queries
We are given a permutation of size $n$, meaning every number from $1$ to $n$ appears exactly once, but in some arbitrary order. Along with it, we are given a target value $x$ that definitely exists somewhere in the array.
We are given a circle of monsters, each with an energy level. The monsters attack their neighbor in a fixed clockwise order: monster 1 attacks monster 2, monster 2 attacks monster 3, and so on, with the last monster attacking the first.
Vasya wants to maximize the point advantage of his team in a basketball game by choosing a threshold distance, d, that separates 2-point throws from 3-point throws. Each team has a list of distances from which they made successful throws.
Comprehensive guide to HSK 7 grammar: academic research language, classical grammar in modern use, specialized register patterns, and C1+ mastery structures.
We are given the names of the home and away teams, followed by a chronological list of card events during a football match. Each event specifies the minute, which team the player belongs to, the player's jersey number, and whether the referee gives a yellow card or a red card.
Sindarin words for food, drink, and dining — attested and Neo-Sindarin terms, key verbs, and the story of lembas.
Monocarp has a single log of wood that weighs exactly $2^n$ grams. He needs to split this log into pieces such that he can assemble exactly $k$ grams of wood for today's fireplace, leaving the remaining $2^n-k$ grams for tomorrow.
Codeforces 485A: Factory
Codeforces 1975D: Paint the Tree
The board is enormous, up to $10^9 times 10^9$, so we never have any chance of working with cells directly. A move chooses a square of side length at most $k$. The lower-right corner of that square must currently be white. Every cell inside the square is flipped.
Codeforces 484D: Kindergarten
We are given a string and multiple queries, each asking for a contiguous substring of the original string. For each query, we need to calculate a value f(t) for that substring t.
Codeforces 487D: Conveyor Belts
We are given an increasing list of positions, and a process that repeatedly deletes players from a line. In each round, we look at the current lineup and try to remove the players standing at positions $a1, a2, ldots, ak$.
Codeforces 1975C: Chamo and Mocha's Array
We are given an array of integers, and two players split the array into two groups. Kirill is allowed to choose a subset that is neither too small nor too large, specifically at least two elements and at most $n-2$ elements. Those chosen elements form the red group.
Comprehensive guide to HSK 3 grammar: 把 construction, 被 passive, complements, conditionals, and intermediate patterns — with tables and examples.
We are given a short array of length at most 18. The only allowed move is to pick a contiguous segment, compute the mex of that segment, and overwrite the entire segment with that mex value.
We are given a single positive integer and asked to evaluate a function built by alternating addition and subtraction of consecutive integers starting from 1. The sequence begins by subtracting 1, then adding 2, subtracting 3, adding 4, and so on until we reach n.
We are given a multiset of positive integers and three required group sizes. Every number must belong to exactly one of the three groups, and each group must contain exactly the requested number of elements. After splitting the numbers, we look only at the three group sums.
We are given an array where each position contains a value in the range $1$ to $m$, and a second array that defines a deterministic transformation on values: every value $x$ has a fixed replacement $bx$.
We are asked to help Elsie evaluate her paintings on a 1D canvas of n cells. Each cell can be painted or left empty, and the painting's beauty is determined by a 2D array a.
Codeforces 1973E: Cat, Fox and Swaps
Codeforces 1975I: Mind Bloom
We are asked to count subsegments of length m in an array a that are “good” relative to another array b of length m. A subsegment is considered good if, after rearranging its elements, at least k of them match elements from b.
We are working on a very constrained grid: only two rows and a large number of columns. The start is the top-left cell, and the goal is the bottom-right cell.
Codeforces 1975F: Set
We are asked to count certain arrays with bounded elements that satisfy a combinatorial property. Specifically, consider all arrays of length n where each element is between 0 and k.
We are asked to decipher a message that has been encoded by a simple substitution cipher. Each letter in the original message is replaced with a fixed letter, so the mapping is one-to-one: different letters map to different letters, and the same letter always maps to the same…
We need to construct an array of length n such that exactly k of its cyclic shifts are sorted in nondecreasing order. A cyclic shift chooses some position as the new beginning of the array and wraps the remaining elements around.
Each observation tower sits at a fixed position on a number line from 1 to n and has a current viewing radius given by its height. A tower can “see” every integer point whose distance from its position does not exceed its height.
We are given an $n times n$ grid that is fully determined by three values: the top-left cell, and two fixed increments that govern movement downwards and rightwards. Moving one step down always adds $c$, and moving one step right always adds $d$.
We are given a grid of size $n times m$ composed of white and black squares. The task is to determine whether it is possible to make all squares in the grid the same color using a specific operation any number of times.
We are given a permutation of the numbers from 1 to n. Each query gives us a segment $[l, r]$, and we are asked to count how many strictly increasing index sequences we can choose inside this segment such that every next chosen value is divisible by the previous chosen value.
We are given an $n times m$ grid sitting at height $z=1$, where each cell already contains a unit cube with a color.
We are given a collection of balls, each assigned one of n distinct colors, where color i has ai balls. We are allowed to group these balls, but each group can have at most two balls, and no two balls in the same group can have the same color.
We are given a long sequence of floating-point numbers, and we are asked to output a description of how to compute their total sum.
We are given a binary string of length $n$ where the first $c$ characters are ones and the remaining $n - c$ characters are zeros. We are allowed to perform up to $k$ operations, where each operation flips a zero into a one.
We are given a buyer who starts with a fixed number of coins. A shop is not fixed in advance; instead, we are allowed to design up to 60 sequential stalls. Each stall has an unlimited supply of jewels, and a fixed integer price per jewel.
Codeforces 1975G: Zimpha Fan Club
Codeforces 1974C: Beautiful Triple Pairs
We are given a multiset of integers, but instead of listing it explicitly, we receive frequencies of each value from 0 up to some maximum m. Alice and Bob remove elements from this multiset until nothing remains.
Codeforces 490C: Hacking Cypher
Each line of input describes one student. For that student we know two IDs: a is the student standing immediately in front of him. b is the student standing immediately behind him. If one of those neighbors does not exist, the value is 0.
Comprehensive guide to HSK 5 grammar: formal written Chinese, complex clause structures, advanced connectives, academic patterns, and B2-level structures.
We are given an array whose length is a power of two, and it initially contains all integers from 1 to $2^k$ exactly once. So the array is just a permutation, but its initial order matters. A game is played on this array. A parameter $t$ is fixed first.
Codeforces 1974D: Ingenuity-2
The task is to determine, for each given string, whether it reads the same forwards and backwards. A palindrome is such a string, like "radar" or "racecar", while a string like "ac" is not.
We are asked to simulate a two-player game on an array of integers, but with a compressed representation where we only know the frequencies of each integer.
The deck contains four kinds of cards. A draw-0 card consumes one playable card from your hand and gives nothing back. A draw-1 card replaces itself. A draw-2 card consumes one card and gives two new cards, so it increases your future drawing power by one.
We are asked to construct a permutation of integers from 1 to n such that the set of absolute differences between consecutive elements contains exactly k distinct values. Concretely, if the permutation is [p1, p2, ...
Codeforces 492E: Vanya and Field
We are given a collection of time intervals, each interval belonging to a different lecturer. Lecturer i is available on a continuous range of days from li to ri, and can be assigned to at most one conference day.
Codeforces 486B: OR in Matrix
We are asked to construct a sequence of n non-negative integers that sum to a given value k, while maximizing the number of distinct 1 bits in the binary representation of their bitwise OR.
We are asked to count the number of special square matrices of size n×n, where each row and column contains exactly two ones, and all other cells are zeros.
We are asked to decide whether Alice can paint a ribbon of n parts using m colors such that Bob, who can repaint at most k parts into the same color, cannot make the entire ribbon monochromatic.
We have a 2-row grid with $n$ columns, and each cell contains an arrow pointing either left or right. The robot starts at the top-left corner, and each second it first moves to an adjacent cell (up, down, left, right) and then follows the arrow in the new cell.
We are given a regular polygon with $n$ vertices representing a cake. Some vertices are already selected by Bessie as potential endpoints for drawing diagonals.
Codeforces 1974F: Cutting Game
We are asked to generate a checkerboard pattern that is made of larger $2 times 2$ tiles. Each tile is either fully filled with the character or fully filled with .. The size of the grid is determined by an input integer $n$, and the final grid has dimensions $2n times 2n.
Kirill wants to gather mushrooms under a Wise Oak to brew an elixir. Each mushroom has a magic power, and the strength of an elixir made from a group of mushrooms is the product of the count of mushrooms and the minimum magic power among them.
Codeforces 482D: Random Function and Tree
We are asked to count arrays of integers that meet a very particular set of conditions. We have a starting number, $b0$, and we want to consider sequences $b0, b1, dots, bn$ where consecutive elements differ by exactly 1.
We are given a line of staircase steps, each step either needing repair or already fine. If a step is fine, it behaves like a zero in the input. If it is broken, it carries a positive cost value that represents its repair difficulty. The repair process is constrained by days.
We are given a group of participants who must be assigned into tents, where each tent can hold at most three people. The participants come in three types with different constraints on how they are willing to share a tent.
The cows of the two farmers must appear in alternating order on the line. Once the order of the first cow is chosen, the entire sequence is fixed: either J N J N ... J N or N J N J ... N J. The actual positions are not fixed.
We are asked to construct two numbers of a specified length, m, whose digits sum to a given value, s. The first number should be the smallest possible, the second the largest. Both numbers are expressed in base 10 and cannot have leading zeroes unless the number is zero itself.
We are given a circle of n monsters, each with an initial energy level ai. They are numbered from 1 to n, and each monster attacks its clockwise neighbor simultaneously in a single round.
Codeforces 495B: Modular Equations
We are given a tree with n cities, each with a known population. The tree is described by n-1 bidirectional roads connecting the cities.
We are asked to construct a permutation q of size n such that the sum of inversions of q and the composition q ∘ p equals a given target k. The input permutation p is fixed, and inversions count how many pairs of indices are out of order.
Codeforces 484E: Sign on Fence
Codeforces 491A: Up the hill
We are given a binary string that represents a long paper strip with 0/1 values printed on it. We are allowed to choose any number of cut positions between adjacent characters, and then fold segments of the strip on top of each other simultaneously.
Codeforces 1974B: Symmetric Encoding
We are given a permutation, which is just an ordering of the numbers from 1 to n. We repeatedly apply a transformation that removes elements which are strictly smaller than at least one of their immediate neighbors.
Codeforces 1973B: Cat, Fox and the Lonely Array
The input describes a single arithmetic-style expression consisting of integers combined with the + operator. The key difference from standard arithmetic is that + does not mean numeric addition.
The process starts with a single cell at (0, 0). Whenever we divide a cell (x, y), that cell disappears and produces (x + 1, y) and (x, y + 1). A division is only legal if neither child is currently present. We are given a finite set of forbidden coordinates.
We are standing at an integer floor number in a very large tower that extends far below zero and far above zero. From our current floor $a$, we are only allowed to move upward, meaning we repeatedly add positive integers.
Each user can be viewed as a set of activities. We need to find two users whose sets satisfy three conditions simultaneously: 1. They share at least one activity. 2. The first user has at least one activity that the second user does not have. 3.
We are given a collection of sticks, each with an integer length. The goal is to build as many regular polygons as possible using these sticks, with the restriction that each side of a polygon must be exactly one stick, and no stick can be reused.
Comprehensive guide to HSK 4 grammar: complex comparisons, formal negation, pivotal constructions, topic-comment structures, and upper-intermediate patterns.
The city is a rectangular grid. Every hotel and every restaurant is located at an intersection with coordinates $(x,y)$.
Codeforces 486E: LIS of Sequence
Maxim wants to buy exactly n yogurts from a store where a single yogurt costs a burles, but there is a promotion offering two yogurts for b burles. For each test case, we must calculate the minimum amount he can spend to buy exactly n yogurts.
Comprehensive guide to HSK 8 grammar: expert academic writing, cross-register fluency, archaic vocabulary in modern use, and C2-level mastery patterns.
We are given a queue of n people waiting to ask Seraphim the Owl a question. Kirill arrives at the end of the line and wants to move forward so that he is among the first m people.
We have a regular polygon with n vertices arranged on a circle. Some x vertices are already available for use. We may additionally choose at most y more vertices.
Codeforces 1976D: Invertible Bracket Sequences
Codeforces 1976F: Remove Bridges
The game produces an infinite sequence of hits. Vanya attacks every $frac{1}{x}$ seconds, so his hits happen at times $$frac1x,frac2x,frac3x,dots$$ Vova attacks every $frac{1}{y}$ seconds, so his hits happen at times $$frac1y,frac2y,frac3y,dots$$ Whenever a hit occurs, it…
Codeforces 1976C: Job Interview
We are given a collection of distinct strings of the same length. One string is secretly chosen uniformly at random. Our goal is to determine the expected number of questions needed to identify the chosen string.
We are given a tree and an operation that does not act on a single node, but on a layer of nodes: if we pick a center vertex $v$ and a distance $d$, we recolor every vertex whose shortest-path distance from $v$ is exactly $d$.
Codeforces 1974G: Money Buys Less Happiness Now
We are given a small positive integer $x$. For each such value, we need to choose another integer $y$ strictly smaller than $x$, and we want to maximize the expression $gcd(x, y) + y$. The interaction between the two terms is important.
Codeforces 1974A: Phone Desktop
Codeforces 1976B: Increase/Decrease/Copy
We are asked to build a square city on a 2D Cartesian map such that it encloses all given mines, represented as points with integer coordinates. The sides of the square must remain parallel to the axes, and our goal is to minimize the area of the square.
The input to this problem is a single integer $n$, with $0 le n le 12$. Despite how small this looks, the task is not about iterating or simulating anything directly from this number in a naive arithmetic sense.
We are given a weighted tree with n vertices rooted at vertex 1. Each edge has a positive weight. For any vertex v, we define a set S(v) containing all descendants of v (including itself) such that the distance from the root to a vertex u in S(v) equals the distance from the…
We are given a connected undirected weighted graph with at most 20 vertices. We must choose a spanning tree. The cost of that tree has two parts. The first part is standard, the sum of the chosen edge weights. The second part depends on the structure of the tree.
We are given an array of length $2n$ where every number from $1$ to $n$ appears exactly twice. You can think of it as $n$ paired cards scattered in a line. The first $n$ positions form a left block and the last $n$ positions form a right block.
We are asked to construct a string over uppercase Latin letters such that a specific counting rule is satisfied. A position in the string is called special if the character at that position matches exactly one of its immediate neighbors.
Codeforces 490D: Chocolate
We are given an array a of length n, constructed from some unknown permutation p of the integers 0 through n-1. Each element of a satisfies the relation a[i] = MEX(p[1..i]) - p[i]. The task is to reconstruct any valid permutation p that produces this a.
We are given two very large integers represented as strings, both having the same number of digits. Every digit is between 1 and 9, so there are no zeros to complicate positional effects or leading zero issues.
We are given a multiset of cards, where each card carries an integer label. The only allowed operation takes exactly k cards that all share the same label and removes them, replacing them with k-1 new cards whose labels we are free to choose arbitrarily.
We are given a sequence of positive integers written on the board, but some of the digits have been replaced by question marks. Each question mark represents a lost digit, so our task is to restore the sequence to a strictly increasing list of positive integers.
We are given an array of integers, and our task is to increase its median by performing a series of operations. Each operation consists of picking a single element and incrementing it by one.
We are given a problem where we need to assign integers to vertices of an initially empty graph and then construct edges based on a Manhattan-like distance metric. Specifically, for vertices labeled $1$ through $n$, each vertex receives a distinct integer from $1$ to $n$.
We are given a permutation, and we observe it evolving over time by revealing its prefix. After revealing the first $q$ elements, we throw away all values greater than $q$, but keep the relative order of the remaining values.
We are given an array of integers and must transform it into non-decreasing order using swaps. The output is not the sorted array itself. Instead, we must print a sequence of index pairs, where each pair represents a swap performed on the array.
Codeforces 1973C: Cat, Fox and Double Maximum
We are given a list of strings, each consisting of lowercase English letters, and for each string, we are asked to answer either "YES" or "NO" based on a hidden pattern.
We are given a square matrix of size $n times n$ initially filled with zeroes. Nene can perform two types of operations: either set an entire row to a permutation of $1$ through $n$ or set an entire column to such a permutation.
We are asked to count the number of polynomials with non-negative integer coefficients that satisfy a very specific evaluation property.
We are given a fixed starting height $b0$ and we imagine building an integer path $b0, b1, ldots, bn$ where each step moves by exactly $+1$ or $-1$, but the path is never allowed to go below zero. This is a standard “walk on the non-negative integers” with unit steps.
We are given a line of monsters, each with some initial health. One operation consists of choosing a single monster as the starting point of a “chain lightning”.
Codeforces 1973A: Chess For Three
We are asked to count the number of ordered pairs $(a, b)$ where $1 le a le n$ and $1 le b le m$ such that the sum $a+b$ is divisible by $b cdot gcd(a,b)$.
Comprehensive guide to HSK 2 grammar: comparisons, aspect markers, direction complements, modal particles, and more — with examples and patterns.
Codeforces 1974E: Money Buys Happiness
The task is to buy exactly k concert tickets over n days, with each day offering a ticket price ai. You are limited to buying at most m tickets per day. Additionally, every ticket you buy increases the price of all future tickets by the number of tickets bought on that day.
Codeforces 1976A: Verify Password
Codeforces 480E: Parking Lot
Codeforces 487A: Fight the Monster
Every tree is planted at an integer lattice point. Around that point we place a disk of radius r, representing the root system. Two conditions must hold. The entire disk must stay inside the square lawn.
We are given a chronological log of wrestling techniques. Each entry is an integer that describes both who performed the move and how many points it contributed.
Comprehensive guide to HSK 9 grammar: native-level classical Chinese, literary production, oral defense mastery, and full C2+ control of written and spoken registers.
Codeforces 1976E: Splittable Permutations
We are given two strings, s and t. Our goal is to count the number of ways to select one or more non-overlapping substrings from s such that each selected substring contains t somewhere inside it.
We are given two rooted trees on the same labeled vertex set from 1 to n, both rooted at 1. Each tree is described by its parent array, so every node knows its immediate parent except the root.
The clock is not a normal continuous display, it behaves like a repeating cycle of three states. Every cycle has length $3k$ seconds.
We are asked to construct a rooted tree with a specific number of vertices having 0, 1, or 2 children. The input gives three numbers: a vertices with 2 children, b vertices with 1 child, and c vertices that are leaves.
Codeforces 1973F: Maximum GCD Sum Queries
The elevator always displays a two-digit floor number from 00 to 99. Each digit is drawn using a seven-segment display. A segment can be broken. When a segment is broken, it cannot light up even if it should.
Complete Sindarin number system: cardinals 1–1000, ordinals 1st–10th, number compounds, duodecimal counting, and number words in place names.
We are given a line of ships, each with some durability, and a fixed attack pattern that always targets the current leftmost surviving ship, then the current rightmost surviving ship, and repeats this alternation until a total of $k$ attacks have been made or all ships sink.
The task is to reconstruct a set of four integers representing candy counts in boxes such that three properties are equal: the arithmetic mean, the median, and the range. We are given some subset of these four numbers (0 to 4) in arbitrary order.
Codeforces 1975H: 378QAQ and Core
Codeforces 489E: Hiking
We are simulating a queue of students where the front of the queue is repeatedly served for a limited number of minutes. Each student, once served, leaves to “process” their porridge for a fixed number of minutes, and then returns to the queue.
We are given a row of lamps, all initially off. We can perform one type of operation any number of times: choose two lamps that are currently off and are not next to each other, and turn them both on simultaneously.
We are given a grid where some cells form a fixed path from the top-left corner to the bottom-right corner. An enemy walks along this path one cell per second.
We are given two binary strings of equal length. The first string, call it the reference string, defines a constraint on how a valid target string must behave. The second string is the one we want to stay as close as possible to after we adjust it into a valid configuration.
We are given a network of cities connected by roads, where each city sells a souvenir at a certain price. Queries can either change the souvenir price in a city or ask for the minimum possible price a tourist can pay when traveling from one city to another along a path that…
Codeforces 1975B: 378QAQ and Mocha's Array
Codeforces 484A: Bits
We are asked to analyze a two-player game on an array of non-negative integers. Alice begins with an empty array c, and on her turn, she takes any element from the initial array a and appends it to c. Bob, on his turn, removes any element from a but does not add it to c.
We are given times expressed in the 24-hour clock format, for example 00:00 for midnight, 13:45 for one forty-five in the afternoon, or 23:59 for one minute before midnight. Our task is to convert each time into the 12-hour clock format, which uses the familiar AM and PM labels.
We are given an array of integers that is guaranteed to be "beautiful," which means that through a specific operation, all elements can eventually become the same.
We are given an array of integers and we are allowed to perform exactly k operations. Each operation lets us pick any contiguous segment of the current array, compute its sum, and insert that sum back into the array at any position.
We are given a connected simple undirected graph with up to 50 vertices and at most 50 edges. Each edge can be used multiple times in a constructed multigraph, and we control how many copies of each edge we create through a non-negative integer array $x$.
We are asked to reverse a Fenwick Tree construction. A Fenwick Tree is normally defined for an array a of length n such that each element sk stores the sum of a contiguous subarray of a whose length is the lowest set bit of k.
We are tasked with calculating the expected maximum wealth of a group of people after a series of charitable recommendations. Each person starts with a known amount of money.
We are asked to maintain a rooted tree where each node starts with zero peaches. Two kinds of operations can happen. In a growth operation at node $x$, we can increase the number of peaches on the parent of $x$ or any node in the subtree of $x$.
We are asked to consider a network of islands where initially every pair of islands is connected by a bridge. There are n islands, numbered from 1 to n, and Everule lives on island 1. Dominater can destroy up to k bridges to reduce the number of islands that Everule can reach.
We have an $n times n$ board. The white queen starts at the top-left corner $(1,1)$, and the black queen starts at the top-right corner $(1,n)$. Every other square contains a green pawn. A move is mandatory. On each turn, a player must capture some piece with their queen.
We are asked to guide Amanda the Amoeba from an initial configuration to a target configuration on a rectangular grid. Each configuration marks Amanda's body with , free pixels with ., and blocked pixels with X. Her body is connected and contains at least two pixels.
In this problem, we are given a string of digits that a robot has produced. The robot operates under a simple but peculiar rule: for each digit in the string, if the digit is even, it will remain in the output string as-is.
We have two arrays of size n. The array a contains the spiciness values of the appetizers, and the array b contains the spiciness values of the main dishes. Every appetizer must be paired with exactly one main dish, and every main dish must be used exactly once.
We are given an array and we are allowed to split it into a sequence of contiguous segments. Each segment is evaluated by taking the bitwise XOR of its elements.
We are given a village with n houses aligned in a row. Each resident has a preference for which side of a street they want to live on: left (0) or right (1).
We are given a circle of baskets numbered from 1 to $m$, and $n$ balls initially placed in distinct baskets. Alice repeatedly chooses one of the balls uniformly at random and moves it clockwise to the next basket.
Codeforces 1972B: Coin Games
Each bag contains a fixed number of gold coins and a number of silver coins. Gold coins always contribute one unit of value. Silver coins are uncertain, each silver coin independently contributes either 0 or 1 with equal probability.
We are given a multiset that contains almost all subarray sums of an unknown array a, where a has two special properties: every element is strictly positive and the array reads the same forwards and backwards.
We are given a string and we are allowed to cut it into contiguous pieces. The goal is to decide whether we can cut it so that every resulting piece is not a palindrome. If it is possible, we must also construct one such cut.
We are asked to construct three integers inside a given interval such that they form a very specific pattern of coprimality relationships.
We are looking at arrays of length n, where each position can hold an integer between 0 and k. Every such array is considered a candidate, so the total universe is (k+1)^n. The notion of “good” is defined through an operation that subtracts 1 from a contiguous segment.
Each song has two attributes, a genre and a writer. After removing some songs, we are allowed to reorder the remaining songs arbitrarily.
Each vertex of the tree initially contains exactly one ant. A move chooses an edge $(u,v)$ and orders all ants currently gathered at $u$ to move to $v$. The ants obey only when the destination already contains at least as many ants as the source.
Codeforces 1973D: Cat, Fox and Maximum Array Split
We are given a tree with $n$ vertices, which is a connected graph without cycles. The task is to remove exactly $k$ edges from this tree and determine the largest integer $x$ such that each resulting connected component has at least $x$ vertices.
We are given a sequence of basketball players, each with a passing range expressed as an interval $[li, ri]$. The players are numbered from 1 to $n$.
We are asked to maximize the number of games Bob can win in a repeated XOR game against Alice. The game is played on a multiset of integers containing only ones, twos, threes, and fours. Alice wins if the XOR of all remaining numbers is non-zero; otherwise, Bob wins.
Each building has two independent pieces of information. The first string describes what class is held there. A building may need a mathematics professor, a computer science professor, or no professor at all. The second string describes which professor is initially located there.
Codeforces 483B: Friends and Presents
Codeforces 487C: Prefix Product Sequence
We are given a single real number $x$ between 1 and 100, expressed to exactly three decimal places. The task is to compute a real number result from $x$, with the requirement that the answer's absolute or relative error does not exceed $10^{-4}$.
We are given a string consisting of lowercase letters and wildcard characters. Each wildcard can later be replaced by any lowercase letter we choose.
We have a line of cows, each with a unique Cowdeforces rating, and they compete in a sequential tournament. The tournament begins with the first two cows, and each subsequent match is between the winner of the previous match and the next cow in line.
We are given a fixed-size grid of 21 rows and 21 columns. Each cell contains either 0 or 1. The grid should be viewed as a map where each cell is a square tile, and tiles with the same value may form connected regions through shared edges.
Sindarin color words: attested terms for green, white, grey, black, red, blue, gold, and silver — with place name examples and nuance notes.
We are given several independent games, each defined by a list of piles containing stones. Two players alternate turns.
We are given a spell system that behaves like a program written in a very restricted language. The first part of the system is a list of basic operations, each of which either increases or decreases a monster’s health by a fixed integer.
Codeforces 482E: ELCA
Codeforces 1975A: Bazoka and Mocha's Array
We are given a set of cards, each labeled with a number from 1 to n. For each number i, we have ai cards of that type.
We are asked to count ordered pairs of integers $(a, b)$ with $1 le a le n$ and $1 le b le m$ such that $b cdot gcd(a, b)$ is divisible by $a+b$. The inputs are multiple test cases, each specifying a pair of limits $n$ and $m$.
Comprehensive guide to HSK 6 grammar: classical Chinese influences, complex formal registers, advanced clause structures, rhetoric, and C1-level patterns.
We are given a rectangular grid of integers. A move starts at the top-left cell and ends at the bottom-right cell, and at each step we can only go either one cell down or one cell right. Every such move sequence forms a monotone path.
We are given a list of integers, and we are allowed to pick two positions in it, say one value acts as a dividend and the other as a divisor. The only restriction is that the dividend must be at least as large as the divisor.
We are asked to construct an array of non-negative integers such that the sums of all subsequences cover every integer from 1 to $n$, except for a single forbidden sum $k$. Each test case gives the upper bound $n$ and the forbidden sum $k$.
Codeforces 486D: Valid Sets
We are given a string s of length n. We want to find the smallest possible length L such that there exists a pattern string k of length L, repeated exactly n / L times, producing a string c of length n, and c differs from s in at most one position.
Vanya has a number of exams, each graded between 1 and a maximum score r. He wants his overall average to reach at least avg to qualify for a scholarship. For each exam, he can improve his score by writing essays, with the cost of increasing a grade by 1 point varying per exam.
We have an array $a$, and a recursively defined value: $$f(1)=sqrt{a1}, qquad f(i)=sqrt{f(i-1)+ai}.$$ After every point update $ak leftarrow x$, we need the integer part of $f(n)$. The first obstacle is that the recurrence uses real numbers.
We are asked to distribute a fixed number of skill points between two attributes: damage per hit and hits per second, in order to minimize the total time to kill a sequence of monsters.
We are asked to find, for each test case, the longest prefix of a binary string a that can appear as a subsequence in another binary string b.
We are given three digits, a, b, and c, and must classify their relationship. A sequence is called a stair when the values strictly increase from left to right, meaning a < b < c.
We are given a binary string and an operation that flips a block of fixed length $k$, turning every 0 into 1 and every 1 into 0. We may apply this operation as many times as we want, but the chosen length $k$ is fixed for the entire process.
We are given an integer array nums and must count how many index pairs (i, j) with i < j are perfect. For each pair, let: - a = nums[i] - b = nums[j] The pair is considered perfect if both of the following conditions hold: - min(|a - b|, |a + b|) <= min(|a|, |b|) - max(|a - b|…
LeetCode 3644: Maximum K to Sort a Permutation (Medium)
We are given a string consisting of uppercase English letters. From this string, we are interested in counting subsequences equal to the pattern "LCT", where a subsequence means indices such that , , and .
The problem gives an integer array nums of length n, where each index represents a node in a graph. The task is to start at index 0 and reach index n - 1 in the minimum number of moves. From any index i, there are two types of moves.
The problem asks us to find all integers x less than or equal to a given integer n such that x can be expressed as the sum of cubes of two positive integers in at least two distinct ways.
The problem defines a sequence of blocks, where each block contains the product of a consecutive range of integers.
We are given a string s containing only lowercase English letters (a-z) and digits (0-9). Each character has a corresponding mirror character: - Letters are mirrored across the alphabet. - a ↔ z - b ↔ y - c ↔ x - and so on. - Digits are mirrored across the digit range.
The problem asks us to determine if it is possible to construct a new array nums2 of the same length as nums1 where all elements are either all odd or all even.
Here is the full, detailed technical solution guide for LeetCode 3791 - Number of Balanced Integers in a Range, following your requested structure. The problem asks us to count all integers between low and high inclusive that are balanced.
The problem asks us to design an EventManager class that manages a set of events, each identified by a unique eventId and associated with a priority.
This problem asks us to find the smallest positive integer composed entirely of the digit 1 that is divisible by a given integer k. The input k is guaranteed to be between 2 and 100,000.
This problem gives us a string s of length n consisting of lowercase English letters. For every index i, we compare two characters: - The character at position i, which is s[i] - The character at the mirrored position from the other end, which is s[n - i - 1] We must find the…
We are given an m × n grid of integers. Starting from the top-left cell (0, 0), we must reach the bottom-right cell (m - 1, n - 1) by moving only right or down.
This problem asks us to determine the minimum cost required to acquire a set of items that satisfy two separate type requirements. We have three types of items: type 1, type 2, and type 3.
The problem asks us to compute the number of non-decreasing arrays of integers where each integer satisfies a specific digit sum constraint.
The problem gives us an integer array nums where every element is guaranteed to be one of three values: 0, 1, or 2.
The problem asks us to maximize a "score" obtained by splitting an integer array nums at a valid index i. For each split index i, the score is calculated as the sum of all elements from the beginning of the array up to i (prefixSum) minus the minimum value in the remaining…
We are given n people standing in a line, indexed from 0 to n - 1. Each person independently chooses one of two directions: - 'L' means the person is visible only to people on their right. - 'R' means the person is visible only to people on their left.
I can do this, but I need the actual problem details for LeetCode 3786 - Total Sum of Interaction Cost in Tree Groups to ensure the algorithm, proof, and code are correct.
We are given an integer array nums, and we must count how many subarrays are good. A subarray is considered good if the bitwise OR of all elements in that subarray is equal to at least one element that appears inside the same subarray.
The problem asks us to find the maximum sum of node values along a path in a binary tree such that all values in the path are distinct. The path can start and end at any node, does not need to pass through the root, and must consist of connected nodes.
The problem asks us to minimize the total cost of deleting characters from a string such that the resulting string contains only one unique character.
We are given an array nums and a target value target. We may remove any subset of elements from the array, including removing none of them or even removing all of them. After performing the removals, the remaining elements form a new array.
Before I write the full guide, I want to confirm one critical detail because the examples appear inconsistent with the stated operations.
Before I write the full guide, I need one clarification: can you confirm the exact LeetCode 3781 problem details or provide the official title/link?
Here is a comprehensive, detailed solution guide for LeetCode 3780 - Maximum Sum of Three Numbers Divisible by Three, following your requested format: The problem asks us to select exactly three integers from the given integer array nums such that the sum of those three…
The problem asks us to determine whether we can construct a new array nums2 from a given array nums1 of distinct integers such that all elements in nums2 have the same parity (all even or all odd).
We are given two integers l and r, and we must count how many integers in the inclusive range [l, r] are fancy. The definition of a fancy number is based on the concept of a good number. A number is good if its digits form a strictly monotone sequence.
The problem gives us an integer array nums and defines a very specific operation: in one move, we remove the first three elements of the current array. If fewer than three elements remain, we remove everything that is left.
The problem is asking us to simulate a chain reaction of activations on a set of 2D points. Each point is defined by its (x, y) coordinates and all points are distinct. When a point is activated, any point sharing the same x coordinate or y coordinate becomes activated as well.
The problem requires us to find the longest arithmetic subarray from a given array of integers nums if we are allowed to modify at most one element. An arithmetic subarray is a contiguous sequence in which the difference between consecutive elements is constant.
The problem asks us to count the total number of commas that appear when writing every integer from 1 through n using standard comma-separated number formatting. In standard formatting, commas are inserted every three digits from the right: - 1 through 999 contain no commas.
The problem asks us to count how many commas appear when writing every integer from 1 through n using standard number formatting. In standard formatting, commas are inserted every three digits from the right side of the number. For example: - 999 contains 0 commas.
We are given a connected, weighted, undirected graph with n nodes numbered from 0 to n - 1. Every edge is represented as [u, v, w], meaning there is a bidirectional edge between nodes u and v with positive weight w.
The problem requires us to make two integer arrays nums1 and nums2 identical by performing swaps. There are two types of swaps: swaps within the same array, which are free, and swaps between arrays at the same index, which cost 1 per operation.
We are given an integer array nums of length n. For every index i, we first compute: - mxi, the maximum value among nums[0...i] - prefixGcd[i] = gcd(nums[i], mxi) This creates a new array called prefixGcd. After that, we sort prefixGcd in non-decreasing order.
The problem asks us to find the first even integer in an array that appears exactly once. In other words, we need to traverse the array and identify even numbers, count their occurrences, and return the earliest one that occurs only a single time.
The problem requires processing a string s consisting of lowercase English words separated by single spaces. The first task is to count the number of vowels in the first word.
The problem asks us to take an integer array nums of length n and divide it into k contiguous subarrays of equal length, then reverse each subarray individually. The resulting array should preserve the order of the subarrays but reflect the reversal within each one.
We are given a binary string s, where each character represents whether an element is sensitive: - '1' means the element is sensitive. - '0' means the element is not sensitive. The entire string initially forms one segment.
The problem asks us to analyze a string s consisting of lowercase English letters and identify runs, which are contiguous sequences of identical characters that cannot be extended further. For example, in "aaabaaa", the runs are "aaa", "b", and "aaa".
We are given an integer array nums, and we need to find the smallest index i that satisfies a special balance condition. For a given index i: - The left sum is the sum of all elements strictly before i, that is, nums[0] + nums[1] + ... + nums[i-1].
The problem asks us to compute, for each node in a tree, the maximum score of a connected subgraph that includes that node. The tree is represented as an undirected graph with n nodes and n - 1 edges. Each node has a value indicating whether it is good (1) or bad (0).
This problem is asking us to determine the minimum number of substring sort operations needed to make a string s sorted in non-descending alphabetical order. We are allowed to sort any substring of s, but we cannot sort the entire string at once.
The problem describes a dungeon with n rooms, where entering room i reduces your health points by damage[i]. After taking this damage, if your remaining health is at least requirement[i], you earn 1 point for that room.
This problem asks us to determine how many distinct email groups exist after applying a specific normalization process to every email address. Each email consists of two parts separated by the '@' character: - The local name appears before '@'.
We are given an m x n grid of positive integers. From each row, we must choose exactly one value. After making one choice per row, we compute the bitwise OR of all selected values. Our goal is to make that final OR value as small as possible.
This problem asks us to sort an array of positive integers based on a transformation called binary reflection. The binary reflection of a number is obtained by converting the number to its binary representation, reversing the order of the bits (ignoring leading zeros), and…
We are given a single integer n. Starting with this integer, we repeatedly perform split operations until every resulting piece is equal to 1.
The problem gives two integer arrays, technique1 and technique2, each of length n, representing two ways to complete n tasks. Completing the ith task with technique1 yields technique1[i] points, while using technique2 yields technique2[i] points.
This problem asks us to remove all vowels that appear at the end of a given string. The input is a string s consisting only of lowercase English letters. A vowel is defined as one of the five characters: 'a', 'e', 'i', 'o', or 'u'.
We are given an integer array nums. An array is called parity alternating when every pair of adjacent elements has different parity. In other words, the parity pattern must alternate between even and odd.
This problem asks us to find the minimum number of increment or decrement operations needed to convert each number in a list into a binary palindrome. A binary palindrome is a number whose binary representation reads the same forwards and backwards once leading zeros are removed.
We are given three integers l, r, and k. For every digit position, we may independently choose any digit from the inclusive range [l, r]. We then form all possible numbers consisting of exactly k digits.
The problem gives you two integer arrays, nums and threshold, each of length n. Each index i represents a value nums[i] you can add to a running total, but you can only choose that index when the current step is at least threshold[i].
This problem gives us an integer array nums and asks us to find a pair of distinct values [x, y] that satisfies two conditions: 1. x < y 2. The frequency of x in the array is different from the frequency of y.
The problem asks us to repeatedly merge close, equal characters in a string s. A character pair is considered close if the distance between their indices is at most k. When a merge occurs, the right character is removed and the string is updated immediately.
The problem asks us to count how many elements in an array have at least k values that are strictly greater than them. More concretely, for every element nums[i], we need to determine how many numbers in the array are larger than nums[i].
We start with a value val = 1 and process the array from left to right. For every element nums[i], we must choose exactly one of three actions: - Multiply the current value by nums[i] - Divide the current value by nums[i] - Do nothing Division is exact rational division, not…
We are given a list of requests, where each request consists of a user ID and a timestamp. Multiple users can appear in the input, and a user may have multiple requests at different times.
The problem asks us to parse a string s that contains lowercase English letters and may include concatenated English words representing digits from 0 to 9.
This problem describes a game played over a sequence of rounds. The array nums represents the points awarded in each game, where nums[i] is the number of points available during game i. There are exactly two players. At the beginning: - The first player is active.
Here’s a complete, detailed technical guide for LeetCode 3757 - Number of Effective Subsequences, following your formatting instructions exactly. The problem gives us an array of integers, nums, and asks us to determine the number of effective subsequences.
The problem asks us to determine whether any permutation of the digits of a given integer n forms a digitorial number. A digitorial number is defined as a number equal to the sum of the factorials of its digits. For instance, 145 is digitorial because .
This problem asks us to calculate the total finger movement required to type a given lowercase string on a special keyboard layout using only one finger. The keyboard is arranged as: Each key occupies a coordinate (row, column) in this grid.
We are given a string s consisting of lowercase English letters. We must find the length of the longest substring that is almost-palindromic.
Please provide the official LeetCode problem statement or confirm the exact problem details for LeetCode 3756 - Concatenate Non-Zero Digits and Multiply by Sum II.
The problem asks us to process an integer n and construct a new number using only its non-zero digits. These digits must remain in the same order in which they originally appear.
The problem asks us to find a subarray of a given non-negative integer array nums such that the difference between the maximum and minimum elements of the subarray does not exceed a given integer k.
The problem gives us a positive integer n, and asks us to work with its binary representation. First, we convert n into a binary string s without leading zeros. Then, we compute the reverse of that binary string.
The problem gives us an array bulbs, where each value represents a light bulb number between 1 and 100. Initially, there are exactly 100 light bulbs, and every bulb is turned off. We process the array from left to right.
We are given a tree with n nodes. Each node stores a lowercase English letter. The tree is undirected and connected, so there is exactly one simple path between any two nodes. The problem supports two kinds of operations: 1. Update Change the character assigned to a node. 2.
The problem gives us an array of strings words and an integer k. Two words are considered prefix-connected if they share exactly the same first k characters and they come from different indices in the array.
The problem provides two inputs: - words, an array of lowercase English strings. - weights, an array of 26 integers where weights[i] represents the weight assigned to the letter corresponding to index i (0 - 'a', 1 - 'b', ..., 25 - 'z').
The problem is a variation of the classic House Robber problem, but with an additional constraint: each house has a color, and you cannot rob two adjacent houses if they share the same color. We are given two arrays nums and colors of length n.
The problem requires computing a delayed count for each element in an array nums. For an index i, the delayed count is defined as the number of later elements (indices j i + k) that are equal to nums[i].
We are given two arrays, nums1 of length n and nums2 of length m, along with an integer k. Our goal is to choose exactly k pairs of indices: such that: and For every chosen pair (i, j), we gain a score equal to: The final score is the sum of the products of all selected pairs…
We are given an array nums and an integer k. For every subarray nums[l..r], its cost is defined as: The first part measures the range of values inside the subarray, while the second part measures the length of the subarray.
The problem asks us to repeatedly merge adjacent equal elements in an integer array nums until no further merges are possible. Specifically, if two consecutive elements are equal, we replace them with their sum.
The problem gives us an integer array nums of length n. For every index i, we must determine whether the element nums[i] is dominant. An index is considered dominant if its value is strictly greater than the average of all elements to its right.
The problem asks us to design a ride sharing system that manages both riders and drivers as they enter the system, and to match them in a first-come, first-served order.
The problem presents a simplified version of evaluating nested mathematical expressions in string format. The input expression is either a single integer literal, which may be negative, or a nested operation of the form op(a,b) where op is one of the four basic arithmetic…
The activity table records user actions. Each row contains a userid, the actiondate, and the specific action performed on that date. The primary key is (userid, actiondate, action), which guarantees that the exact same action for the same user on the same day cannot appear twice.
Before I write the full guide, I want to verify one detail because this appears to be a very new LeetCode problem and correctness matters here.
We are given an integer array nums. Two players, Alice and Bob, repeatedly remove a contiguous subarray from the current array. The only restriction is that the removed subarray cannot be the entire current array, so after every move at least one element remains.
The problem asks us to find three distinct indices (i, j, k) such that the values at those positions are identical: Among all such valid triples, we must compute the minimum possible distance, where the distance is defined as: If no value appears at least three times, then…
The problem asks us to find the longest alternating subarray in a given integer array nums, where a subarray is defined as consecutive elements.
The problem asks us to analyze prefixes of a string s and count how many of them satisfy a specific condition. A prefix is any substring that starts at the first character and extends to some point in the string.
We are given an array nums and must divide it into exactly k contiguous subarrays. For every subarray, we first compute its sum: Its value is then defined as: The score of a partition is the sum of the values of all subarrays in that partition.
Here’s a complete, detailed technical solution guide for LeetCode 3739 - Count Subarrays With Majority Element II following your requested format. The problem asks us to count the number of contiguous subarrays in which a given target element is the majority element.
The problem asks us to count Monobit integers from 0 to n. A Monobit integer is one where all bits in its binary representation are the same. This means there are only two possibilities for a valid Monobit integer: all zeros or all ones.
Here is a comprehensive, detailed reference guide for LeetCode 3737 - Count Subarrays With Majority Element I, following your formatting requirements exactly. The problem provides an integer array nums and a target integer.
This problem asks us to find the longest strictly increasing subsequence (LIS) in a given integer array nums, with the additional constraint that the bitwise AND of all elements in the subsequence must be non-zero.
The problem asks us to determine the minimum number of moves required to make every element in an integer array equal. A move consists of selecting one element and increasing it by exactly 1. We are not allowed to decrease values, and we can only increment elements.
This problem gives us a string s that contains two types of characters: 1. Lowercase English letters ('a' to 'z') 2. Special characters from the set "!@$%^&()" We must perform two independent reversals, in a specific order.
This is a long, structured reference document. To make sure I target the correct problem and avoid producing an incorrect guide, I want to verify one thing first: Can you confirm the exact LeetCode problem number/title?
This problem asks us to reconstruct a missing sequence of integers from a partially incomplete array. We are given an integer array nums containing unique values, and we know an important fact: the array originally contained every integer in a continuous range, but some…
This problem asks us to implement a small in-memory order management system that supports four operations on trading orders.
The problem asks us to determine the maximum total calories that can be burned while visiting every block exactly once in an arbitrary order. We are given an integer array heights, where heights[i] represents the height of a block.
This problem asks us to find the nth smallest positive integer whose binary representation contains exactly k ones. In other words, we are working with numbers that, when written in binary, have a specific number of bits set to 1.
The problem asks us to count stable subarrays in an integer array capacity. A subarray capacity[l..r] is stable if it satisfies two conditions: its length is at least 3, and the first and last elements are each equal to the sum of the elements strictly between them.
The problem asks us to rotate only the non-negative elements of an integer array nums to the left by k positions in a cyclic manner, while leaving all negative elements fixed in their original positions.
The problem asks us to determine how many ways we can choose exactly one number from each row of a given m x n matrix such that the greatest common divisor (GCD) of all chosen numbers is 1.
We are given a string s consisting only of decimal digits. The string length is n, and indices are zero-based. For every index i, we need to determine whether i is a good index.
The problem asks us to process an integer array nums and determine the minimum length of a prefix that we can remove so that the remaining array becomes strictly increasing.
This problem asks us to transform an array nums1 of length n into another array nums2 of length n + 1 using the minimum number of operations. The allowed operations are increasing or decreasing any element of nums1 by 1, or appending any element of nums1 to the end of the array.
We are given a string s consisting only of lowercase English letters. The allowed operation is very specific: if a character currently appears at least twice in the string, we may delete exactly one occurrence of that character.
The problem asks us to find the lexicographically smallest string after performing exactly one reverse operation on a contiguous prefix or suffix of a given string s. The string consists only of lowercase English letters and can have a length up to 1000 characters.
This problem asks us to design a real time auction system that supports inserting, updating, removing, and querying bids efficiently. Each bid is uniquely identified by the pair (userId, itemId). A user may have at most one active bid for a particular item.
The problem requires us to find the length of the longest subarray in an integer array nums where the count of distinct even numbers equals the count of distinct odd numbers.
The problem asks us to compute a score for a given string s. The string may contain lowercase English letters, spaces, and digits. We need to count two quantities: - v, the number of vowels in the string. - c, the number of consonants in the string.
Before I write the full guide, I want to verify one important detail: LeetCode 3720 appears to be a very recent problem, and the exact intended optimal approach depends on the precise judge behavior and hidden constraints.
Here is the complete, detailed technical solution guide for LeetCode 3717 - Minimum Operations to Make the Array Beautiful following your exact formatting rules. The problem asks us to transform an integer array nums into a beautiful array.
We are given a tree with n nodes and n - 1 edges. Every node has a binary color represented by the strings start and target. An operation consists of selecting an edge and toggling both of its endpoints.
The problem gives us a string s that contains only three possible characters: 'a', 'b', and 'c'. We must find the longest substring where every distinct character in that substring appears the same number of times.
The problem asks us to compute the sum of all elements in the array whose total frequency is divisible by a given integer k. To understand the requirement clearly, we need to focus on two important details: First, we must determine how many times each number appears in the array.
We are given an array nums and two distinct target XOR values, target1 and target2. A partition divides the array into contiguous, non-empty blocks. Every element must belong to exactly one block, and the blocks must cover the entire array.
Here is a complete, detailed technical solution guide for LeetCode 3711 - Maximum Transactions Without Negative Balance, following your formatting instructions precisely. The problem provides an array transactions of integers representing sequential money movements in an account.
The problem asks us to split a set of n points in a 2D Cartesian plane into exactly two non-empty groups, such that the minimum Manhattan distance among all pairs of points within each group (the partition factor) is maximized.
This problem asks us to analyze user reactions and determine which users exhibit a strong preference for a particular reaction type. The input is a table named reactions, where each row represents a reaction given by a user to a specific content item.
This problem asks us to design a system that tracks Alice's exam scores over time and efficiently computes the total score for any given time interval.
The problem presents an undirected graph with n nodes labeled from 0 to n-1 and m edges. Each edge has a repair cost, and initially, all edges are damaged. We can spend a certain amount of money to repair all edges whose cost is less than or equal to that amount.
The problem asks us to count all pairs of positive integers (a, b) such that their sum equals a given integer n and neither a nor b contains the digit 0 in their decimal representation. These integers are called no-zero integers.
We are given an array words containing n lowercase strings. Every string has the same length m. Two strings are considered similar if we can repeatedly apply the following operation to either string: - Shift every character forward by one position in the alphabet.
We are given an integer array nums, and we must determine the maximum possible length of a subsequence whose bitwise XOR is not equal to zero. A subsequence is formed by deleting zero or more elements from the array while preserving the relative order of the remaining elements.
The problem asks us to find the maximum possible bitwise AND of any subset of size m from an array nums after performing up to k increment operations. Each operation allows us to increase any element in the array by 1.
The problem asks us to count the number of centered subarrays in a given integer array nums. A subarray is called centered if the sum of its elements is equal to at least one element within that subarray.
The problem asks us to count the number of arrays of length n where each element is in the range [l, r], no two consecutive elements are equal, and no three consecutive elements are strictly increasing or strictly decreasing. These arrays are termed ZigZag arrays.
The problem asks us to compute the number of ways to paint n sheets using exactly two distinct colors, subject to constraints on the maximum number of sheets each color can paint.
We are given an array of distinct 4-letter words. Our goal is to construct every valid word square consisting of exactly four different words: - top forms the top row. - bottom forms the bottom row. - left forms the left column. - right forms the right column.
Here is the complete technical solution guide for LeetCode 3699 - Number of ZigZag Arrays I, formatted exactly according to your specifications. The problem asks us to count arrays of length n where each element is within the range [l, r] and satisfies the ZigZag constraints.
We are given between 2 and 12 sorted lists. At any moment we may choose any two existing lists and merge them. If the two lists are and , the merge cost is After the merge, the two lists disappear and are replaced by their sorted union.
I can write the full detailed guide, but I need to flag one issue first: the problem statement for LeetCode 3698 - Split Array With Minimum Difference appears inconsistent with the provided Go stub and the current LeetCode catalog.
The problem gives us an array of strings, words, and asks us to find the maximum distance between two indices i and j where the words at those positions are different. More formally, we want to find two indices such that: - words[i] !
The problem asks us to analyze the frequency distribution of characters in a string and identify a particular group of characters based on how often they appear. We are given a string s containing only lowercase English letters.
Here is a complete, rigorous study companion-style solution guide for LeetCode 3685 - Subsequence Sum After Capping Elements, following your requested format and voice. We are given an integer array nums of size n and a positive integer k.
This problem asks us to find the shortest contiguous subarray whose distinct-value sum is at least k. The important detail is that we do not sum all elements in the subarray.
This problem gives us a string s and an integer k. Our task is to reverse only the first k characters of the string while leaving the remaining characters unchanged.
This problem asks final stage, preserving accuracy during computation and preventing premature rounding errors from affecting averages or comparisons.
The problem gives us a positive integer num and asks whether it is a Complete Prime Number. A number qualifies as a Complete Prime Number if every prefix and every suffix of its decimal representation is prime. For a number with digits d1 d2 d3 ...
This problem asks us to find the longest contiguous subarray that simultaneously satisfies two independent conditions: 1. The bitwise XOR of all elements in the subarray is equal to 0. 2. The subarray contains an equal number of even and odd elements.
This problem presents a grid of size m x n where each cell contains a value of 0, 1, or 2. Each cell contributes both a score and a cost: a cell with value 0 adds 0 to both score and cost, a cell with value 1 adds 1 to score and 1 to cost, and a cell with value 2 adds 2 to…
The problem asks us to find the number of distinct subarrays in a sorted array nums such that the sum of elements in the subarray is divisible by a given integer k.
The problem asks for the maximum possible product of the first and last elements of any subsequence of length m chosen from an integer array nums.
The problem asks us to find the smallest positive multiple of k that is not present in a given integer array nums. In other words, if we consider the infinite sequence of numbers k, 2k, 3k, 4k…, we need to identify the first number in this sequence that does not appear in nums.
The problem asks for the minimum number of adjacent swaps needed to rearrange an array of distinct integers so that the parity of neighboring elements alternates, meaning every even element is adjacent only to odd elements and every odd element is adjacent only to even elements.
We are given an integer array nums of length n. We must count the number of index triplets (i, j, k) such that 0 <= i < j < k < n and the middle element nums[j] acts as a generator for both ends through the relation nums[i] = 2 nums[j] and nums[k] = 2 nums[j].
The problem gives us an array nums consisting of positive integers and asks us to find the length of the longest contiguous subarray that satisfies the Fibonacci property. A subarray is a contiguous segment of the original array.
We are given a string s consisting only of '(' and ')', along with an integer k. A substring is considered k-balanced if it is exactly: For example: - k = 1 → "()" - k = 2 → "(())" - k = 3 → "((()))" The operation is not performed just once. Instead, we repeatedly: 1.
The problem is asking us to select exactly k distinct subarrays from a given integer array nums such that the sum of their "values" is maximized. Each subarray's value is calculated as the difference between its maximum and minimum elements.
The problem gives us two arrays, nums1 and nums2, of equal length n. We are allowed to repeatedly perform a special operation on nums1.
The problem gives us an array nums of non-negative integers. We must choose two subsequences: - The first subsequence has XOR value X. - The second subsequence has XOR value Y. The two subsequences: - May be empty.
This problem models a stream of arriving items. On each day, exactly one item arrives, and each item has a type represented by an integer. We are given two parameters: - w, the size of the sliding window. - m, the maximum number of times any item type may appear within a window.
This problem takes place on a one dimensional number line. We are given positions of robots and positions of walls. Every robot owns exactly one bullet, and that bullet can be fired either to the left or to the right.
This problem gives us an array nums containing positive integers and an integer k. We may repeatedly remove any contiguous subarray whose sum is divisible by k.
The problem asks us to identify books that exhibit polarized opinions based on reading session ratings. We have two tables: books, which contains metadata about each book including its title, author, genre, and number of pages, and readingsessions, which records individual…
Great! We will cover LeetCode 3627 - Maximum Median Sum of Subsequences of Size 3 in a full, comprehensive technical guide following your formatting requirements. The problem gives us an integer array nums with a length divisible by 3.
The task asks us to maintain an integer array that is repeatedly updated, and after each update we must determine an optimal split point that maximizes a specific score based on distinct prime values. For any split index , the array is divided into a prefix and a suffix .
The problem asks us to compute the minimum number of moves a student must make to collect all litter 'L' in a classroom represented as an m x n grid. The student starts at 'S' with a finite energy that decreases by 1 with each step.
This problem asks us to identify employees who spend an excessive amount of their working time in meetings. We are given two tables: The employees table contains employee information, including their unique identifier, name, and department.
The problem asks us to determine the minimum number of moves to reach a target point (tx, ty) from a starting point (sx, sy) on an infinite 2D grid. The movement rules are specific: from any point (x, y), you can move either right by max(x, y) units or up by max(x, y) units.
The task asks us to determine the most popular product category for each season based on sales data. We are given two tables: a sales table containing individual transaction records and a products table that maps each product to its category.
We are given a rooted tree of employees, where employee 1 is the CEO and the root of the hierarchy. Each node represents an investment opportunity with a cost today (present[i]) and a return tomorrow (future[i]).
The problem provides an integer array nums where every element appears exactly three times, except for one element that appears once and one element that appears twice.
We are given a rooted tree with root node 0. Every node has a cost, and the score of a root-to-leaf path is the sum of the costs of all nodes along that path. We are allowed to increase the cost of any nodes by any non-negative amount.
The problem asks us to work with an undirected tree rooted at node 0, where each node has a value vals[i] and a parent par[i].
The problem asks us to examine every contiguous subarray (window) of length k in the array nums and determine the minimum continuous segment within that window that must be sorted to make the entire window non-decreasing.
We are given an array nums consisting of distinct positive integers. The task is to reorder this array into a new array sorted by a custom key: the sum of digits of each integer. If two integers have the same digit sum, the smaller integer must appear first among them.
The problem requires us to find the smallest index in an integer array nums such that the sum of the digits of the element at that index is equal to the index itself. In other words, for each index i, we calculate the sum of the digits of nums[i] and check if it equals i.
The problem gives us two arrays, x and y, both of length n. Each position i represents a pair (x[i], y[i]). We need to select exactly three distinct indices i, j, and k such that the corresponding x values are all different: - x[i] != x[j] - x[j] != x[k] - x[k] !
The problem asks us to determine if we can partition a given m x n grid of positive integers into two connected sections by making exactly one horizontal or vertical cut, such that the sums of the two sections are equal or can be made equal by removing at most one cell.
Here’s a detailed technical solution guide for LeetCode 3566 following your formatting rules: The problem asks us to determine whether a given array of distinct positive integers can be split into two non-empty, disjoint subsets, such that the product of the elements in each…
We are given an undirected tree rooted at node 1. Every edge initially has weight 0, and we must assign each edge a weight of either 1 or 2. The problem asks us to choose any node x that has the maximum depth in the tree.
This problem asks us to identify the maximum number of non-intersecting substrings from a given string word such that each substring is at least four characters long and starts and ends with the same character.
This problem asks us to compute the sum of array products for all magical sequences of a given length. A magical sequence is defined as a sequence of indices into the input array nums of length m, such that when you sum 2 raised to each element of the sequence, the resulting…
The problem requires identifying prime numbers from all possible substrings of a given string of digits and summing the three largest unique primes.
The problem asks us to construct a structured matrix filled with all integers from to , arranged in a very specific recursive ordering constraint across quadrants. A grid is considered special if it satisfies two key properties.
This problem asks us to compute the shortest path between nodes in an implicit undirected graph defined by a nums array and a maxDiff. Each node i corresponds to nums[i]. There is an edge between nodes i and j if the absolute difference of their values is at most maxDiff.
The task asks us to count how many buildings in a given set are covered in all four cardinal directions: left, right, above, and below, within an n x n grid. Each building is represented by a coordinate pair [x, y], and all coordinates are unique.
The problem is asking us to compute, for each unit type from 0 to n-1, the number of that unit equivalent to a single unit of type 0. We are given a list of direct conversion relationships between units.
This problem asks us to identify product pairs that are frequently purchased by the same customers. The goal is to support a recommendation feature similar to "Customers who bought this also bought...
We are given two very large integers l and r as decimal strings and a base b where 2 ≤ b ≤ 10. For every integer x in the inclusive range [l, r], we consider its representation in base b.
The problem asks us to maintain a dynamic array of integers under three types of operations: point updates, range XOR queries, and in-place subarray reversals.
This problem asks us to find the minimum time required to eliminate all bacterial strains given a single initial white blood cell (WBC) and the ability of WBCs to split into two after some splitTime.
You are given an integer array nums, and you want to create at least k non-overlapping subarrays. Every chosen subarray must have length exactly x, and after performing some modifications, every element inside each chosen subarray must become equal.
The problem presents a simulated execution environment defined by two parallel arrays, instructions and values, each of length n. Each index i represents a single instruction. The instructions[i] element can either be "add" or "jump".
We are given two strings, s and t. We may choose any substring from s and any substring from t. Either substring is allowed to be empty.
The problem asks us to find the minimum threshold x such that there are at least k inversion pairs in an array nums of integers.
This problem is a hierarchical organization analysis task over an employee table that forms a tree structure. Each employee has a unique employeeid, a name, a salary, and a managerid. The employee whose managerid is NULL is the CEO and serves as the root of the organization tree.
We are given an array original and a range constraint for every position in the form of bounds[i] = [ui, vi]. We need to count how many arrays copy satisfy two conditions.
We are given a square of side length side. Every point in the input lies somewhere on the boundary of this square. We must choose exactly k points from the given set.
We are given a collection of axis-aligned squares. Each square is represented by three integers: - xi: x-coordinate of the bottom-left corner - yi: y-coordinate of the bottom-left corner - li: side length The square therefore occupies the region: We must find the smallest…
We are given a string s consisting only of digits '0' through '4', and an integer k. For any substring subs whose length is at least k, we may choose two characters: - Character a must appear an odd number of times in the substring.
We are given an array cost of length n. There are n + 1 positions in a line, numbered from 0 to n. Initially, we are standing at position n, which means we are at the very end of the line. To move forward in the line, we can swap positions with other people.
The problem is asking us to manipulate a directed weighted graph in order to minimize the maximum edge weight while satisfying two constraints: every node must be able to reach node 0, and no node may have more outgoing edges than a given threshold.
This problem combines weighted interval scheduling with an additional constraint: we may select at most 4 intervals, and if multiple selections achieve the same maximum total weight, we must return the lexicographically smallest list of original indices.
We are given a lowercase English string s. Every letter has a unique "mirror" letter obtained by reversing the alphabet. For example: - 'a' ↔ 'z' - 'b' ↔ 'y' - 'c' ↔ 'x' and so on. We process the string from left to right.
We are given a string word of length n and an integer numFriends. In every round, Alice splits the entire string into exactly numFriends non-empty pieces. Every possible valid split is considered exactly once. Every piece produced by every split is placed into a box.
Here’s a full technical solution guide following your requested format for LeetCode 3401 - Find Circular Gift Exchange Chains: This problem asks us to analyze a table of gift exchanges in a Secret Santa setting and identify circular chains of gift exchanges.
The problem gives us a matrix grid of size m x n containing non-negative integers. Each column must become strictly increasing, meaning for each column j, the condition grid[i][j] < grid[i+1][j] must hold for all 0 <= i < m-1.
The problem presents a sequence of queries, each defining a contiguous array of integers nums from l to r inclusive. For each array, we are allowed to perform an operation repeatedly: select any two integers a and b and replace them with floor(a / 4) and floor(b / 4).
The problem gives us an integer n and asks us to find the largest prime number that satisfies two conditions: 1. It is less than or equal to n. 2. It can be written as the sum of one or more consecutive prime numbers starting from 2.
We are given an array nums of length n and an integer k. For every contiguous subarray of length k, we must compute its inversion count, then return the smallest inversion count among all such windows.
The coursecompletions table records every course completed by every user. Each row contains the user, the course they completed, the completion date, and the rating they gave that course. The problem asks us to identify common learning pathways among strong students.
We are given an array nums and a fixed integer k. A single operation allows us to choose any element and either increase it by exactly k or decrease it by exactly k. We may perform as many operations as needed. For each query [l, r], we only consider the subarray nums[l..r].
The problem asks us to split a string into as many non-empty substrings as possible, with one important restriction: the first character of every substring must be different from the first character of every other substring.
This problem asks us to compute the total waviness of every integer in an inclusive range [num1, num2]. For a single number, waviness is defined as the number of digits that are either peaks or valleys. A digit is a peak if it is strictly larger than both adjacent digits.
The problem asks us to count "stable" subarrays within a given array nums. A subarray is stable if it contains no inversions, which means that for any two indices i < j in the subarray, nums[i] <= nums[j]. Essentially, a stable subarray is non-decreasing.
The problem gives us an integer array nums and asks us to choose three elements a, b, and c from distinct indices such that the expression: is as large as possible. The important detail is that the three elements must come from different positions in the array.
The problem gives us a string s containing one or more lowercase words separated by single spaces. From this original string, we conceptually build a new string t using a special expansion rule.
We are given a cyclic array nums, which means the array is considered to wrap around from the last element back to the first element. We want to partition the entire cycle into at most k contiguous subarrays.
We are given two strings, s and target, of the same length n. The goal is to rearrange the characters of s into a palindrome and then choose, among all possible palindromic permutations, the lexicographically smallest one that is strictly greater than target.
We are given an integer array nums, and we are allowed to rearrange its elements in any order before computing a score. The score of a rearranged array arr is defined as: The sign alternates between positive and negative positions.
The problem gives us a positive integer n and asks us to remove every digit '0' from its decimal representation. After all zero digits have been removed, we must return the resulting value as an integer. In other words, we first view the number as a sequence of decimal digits.
The problem asks us to construct a positive integer n of exactly num digits such that the sum of its digits equals sum.
This problem asks us to identify subscribers who appear likely to churn based on their subscription history. The data is stored in the subscriptionevents table, where each row represents a subscription-related action performed by a user.
This problem asks us to compute a sum across all non-root nodes in a tree based on ancestor relationships and the property of perfect squares. We are given a rooted tree of n nodes, represented as an edge list. Each node has a positive integer value from the nums array.
The problem asks us to determine if a given string s can be split into two non-empty substrings such that the sum of the alphabetic positions of characters in the left substring is equal to the sum in the right substring.
We are given an array of strings called words. Our goal is to find two different indices i and j such that the strings at those positions are different, meaning words[i] != words[j].
This problem asks us to identify "golden hour customers" from a restaurant's order history. A golden hour customer is defined as someone who consistently orders during peak hours and maintains high satisfaction.
This problem asks us to decompose a positive integer n into the fewest possible base-10 components, where a base-10 component is defined as a single digit (1-9) multiplied by a power of 10. Essentially, we are breaking n down into its decimal place contributions.
We are given an array nums and a list of allowed swap operations. The alternating sum of an array is defined as: Elements at even indices contribute positively, while elements at odd indices contribute negatively.
This problem is an extension of the classic "climbing stairs" dynamic programming problem but with more complex jump costs. You are given a staircase with n + 1 steps, numbered from 0 to n. Each intermediate step i (1-indexed) has an associated cost costs[i].
We are given a movement string s consisting of the four directions: - U increases the y-coordinate by 1. - D decreases the y-coordinate by 1. - L decreases the x-coordinate by 1. - R increases the x-coordinate by 1.
The problem asks us to select exactly k non-empty subarrays from a given integer array nums, where subarrays may overlap and can even be identical. For each chosen subarray nums[l..
The problem asks us to count how many subsequences of a given integer array are stable, where stability is defined by a constraint on parity patterns inside the subsequence itself.
This problem asks us to compute the bitwise OR of all even numbers in a given integer array nums. A number is considered even if it is divisible by 2, meaning its least significant bit is 0.
The problem gives us a list of tasks, where each task is represented by two integers: - si, the time when the task starts. - ti, the amount of time required to complete the task.
This problem requires selecting at most k distinct elements from a given array of positive integers nums such that their sum is maximized. The result must be returned as a list sorted in strictly descending order.
We are given a non-negative integer n and asked to count how many integers k in the range [0, n] have a binary representation that is a palindrome when written without leading zeros.
This problem is asking us to identify the smallest sum of indices from two equal-length arrays, nums1 and nums2, such that the elements at those indices are equal. Formally, we want a pair (i, j) where nums1[i] == nums2[j], and we want the minimum value of i + j.
The problem is asking us to generate a round-robin style schedule for n teams, where each team plays every other team exactly twice, once at home and once away. Each day in the schedule can host exactly one match, and a team is not allowed to play on consecutive days.
The problem asks us to count how many subarrays in an array of distinct integers satisfy a specific “bowl” condition. A subarray nums[l...
Let's go step by step and produce a fully detailed, reference-style solution guide for LeetCode 3674 following your requested formatting. The problem asks us to make all elements of an integer array nums equal using the minimum number of operations.
The problem asks us to reorder an integer array such that the elements are sorted in non-decreasing order of their absolute values.
This problem gives us two arrays: - order, which represents the finishing order of all participants in a race. - friends, which contains the IDs of our friends. The array order is a permutation of the integers from 1 to n, meaning every participant appears exactly once.
This problem asks us to count all unique paths from the top-left corner (0, 0) to the bottom-right corner (m - 1, n - 1) of a binary grid where certain cells contain mirrors.
Each card consists of exactly two lowercase letters. A card is only relevant if it contains the given letter x somewhere in its two positions. During the game, we repeatedly select two cards that satisfy two conditions: 1. Both cards contain the letter x. 2.
This problem asks us to filter a string based on character frequency while preserving the original order of characters. We are given a string s consisting only of lowercase English letters and an integer k.
The problem asks us to find the least frequent digit in the decimal representation of a given integer n. In other words, we need to count how many times each digit from 0 to 9 appears in the number, and then return the smallest digit that occurs the fewest times.
The problem asks us to calculate the greatest common divisor (GCD) of two specific sums derived from an integer input n. The first sum, sumOdd, is the sum of the first n positive odd numbers, and the second sum, sumEven, is the sum of the first n positive even numbers.
This problem is a SQL aggregation and filtering task. We are given a table named customertransactions that contains every transaction performed by customers. Each row represents either a purchase or a refund.
The problem asks us to find the minimum cost to travel from the top-left corner (0, 0) of a 2D grid to the bottom-right corner (m - 1, n - 1). Each cell contains a non-negative integer representing its cost. There are two ways to move: 1.
We are given an array nums and a list of queries. Each query has the form: For a query, we start at index li and repeatedly jump forward by ki until we pass ri. Every visited position is multiplied by vi, and the result is taken modulo 10^9 + 7.
We are given an integer array nums and a list of queries. Each query has the form: For a query, we start at index li and repeatedly move forward by ki positions until we exceed ri.
This problem asks us to calculate the maximum possible profit from a stock trading strategy with the option to modify it once in a very specific way.
This problem asks us to distribute a set of items, each with a given weight, into two separate bags with fixed capacity limits, in order to maximize the total weight placed across both bags.
The problem gives an n × m grid where each cell is identified by coordinates (r, c). A sensor can be placed on any cell, and its coverage is defined using Chebyshev distance. Specifically, a sensor at (r, c) covers every cell (r2, c2) such that max(|r - r2|, |c - c2|) ≤ k.
This problem asks us to construct a number system under two simultaneous constraints and then perform a “next greater element” query over that constrained set.
This problem gives us an m x n matrix called grid and three integers: x, y, and k. The values x and y specify the top-left corner of a square submatrix, while k specifies the side length of that square.
We are given an integer array nums, and we must find the maximum possible sum of a contiguous subarray that follows a very specific shape: 1. It first increases strictly. 2. Then decreases strictly. 3. Then increases strictly again. More formally, for a subarray nums[l...
We are given an integer array nums and an integer k. A subarray is considered semi-repeating if the number of distinct values that appear more than once inside that subarray is at most k.
The problem asks us to process multiple queries on a given array nums where each query specifies a subarray defined by [li, ri] and a threshold. For each query, we need to find an element in the subarray that appears at least threshold times.
We are given an array weight representing parcels arranged in a line. We must partition some prefix-suffix segments of this array into a collection of non-overlapping contiguous subarrays called shipments.
This problem asks us to count the number of contiguous subarrays of a given array nums such that the bitwise XOR of all elements in the subarray is greater than or equal to a given integer k. The input array nums consists of positive integers, and k is a non-negative integer.
The problem asks us to partition an integer array nums into three subsequences A, B, and C such that every element belongs to exactly one subsequence.
The problem asks us to count the number of unique trapezoids that can be formed from a set of points on a 2D Cartesian plane. Each point is given as a pair of integers representing its x and y coordinates.
This problem asks us to identify stores with an inventory imbalance. Specifically, for each store, we need to compare the quantity of the most expensive product against the quantity of the cheapest product.
We are given a directed acyclic graph (DAG) with n nodes numbered from 0 to n - 1. Each directed edge is represented as: and stored in: We are also given a boolean array online, where: The problem guarantees that node 0 and node n - 1 are always online.
The task asks us to identify students who exhibit a very specific behavioral pattern in their study history, called a Study Spiral Pattern. We are given two relational tables: one describing students and another describing their study sessions over time.
The problem asks us to determine if a given positive integer n is divisible by the sum of two specific quantities derived from its digits: the digit sum and the digit product.
The problem asks us to simulate the construction of a string called result by processing an input string s from left to right. The input string contains lowercase English letters and three special operators: , , and %.
The problem is asking us to count islands in a 2D grid where the sum of the values in each island is divisible by a given integer k.
The problem gives us an integer array nums and asks us to divide its elements into two groups based on their indices. Array A contains all elements whose indices are prime numbers. Array B contains all remaining elements, meaning indices that are not prime.
The problem defines a transformation process over a string s that contains lowercase letters and three special operation characters: '', '', and '%'. We process the string from left to right, maintaining a dynamically changing result string.
We are given an array nums and an integer maxC. A subarray is called stable if the greatest common divisor (HCF/GCD) of all elements in that subarray is at least 2. The stability factor of the entire array is defined as the length of the longest stable subarray.
The problem is asking us to validate a set of coupons based on three criteria: the coupon code must be a non-empty string containing only alphanumeric characters or underscores, the business line must belong to a set of four predefined categories, and the coupon must be active.
This problem asks us to compute the minimum time required to reach the last node in a directed graph where each edge has a time window constraint.
The problem gives an undirected graph of c power stations labeled from 1 to c, connected by bidirectional cables. These connections define connected components, which are referred to as power grids.
This problem asks us to partition an array of integers, nums, into exactly k non-empty contiguous subarrays, and then compute the bitwise XOR of each subarray. The goal is to minimize the maximum XOR among these subarrays.
This problem is a shortest-path optimization over a highly constrained state space where you are repeatedly moving groups of people across a river using a single boat.
We are given a string s and must partition it into segments according to a very specific process. We start at the current position in the string and begin building a segment character by character.
This is a SQL database problem involving pattern matching on strings. We are given a table named Products with two columns: | Column | Description | | --- | --- | | productid | Unique identifier for each product | | name | Product name string | The goal is to return all…
This problem requires computing the minimum total cost to traverse a grid from the top-left corner (0, 0) to the bottom-right corner (m - 1, n - 1) while following a strict alternating movement pattern. Each cell (i, j) has a cost to enter, calculated as (i + 1) (j + 1).
The problem gives us a 1-indexed array numWays, where numWays[i] indicates the number of ways to form a total amount i using an infinite supply of some unknown coin denominations. The goal is to recover the original set of coin denominations.
This problem asks us to determine whether any number in an array appears a prime number of times. In other words, for each unique number in the input array nums, we calculate its frequency-the number of occurrences-and check if that frequency is a prime number.
We are given an integer array nums and an integer k. We must count how many contiguous subarrays satisfy two conditions: 1. The subarray contains at least two prime numbers. 2.
The task is to identify patients who have recovered from COVID based on their testing history. We are given two tables: one containing patient demographic information and another containing COVID test records with timestamps and results.
We are given a weighted tree with n nodes. A tree is an undirected connected graph with exactly n - 1 edges, which guarantees that there is exactly one simple path between any two nodes. Each edge has a positive weight.
We are given n distinct points on a 2D Cartesian plane. Each point is represented as [x, y]. We must choose any three points that form a non-degenerate triangle, meaning the area must be strictly greater than zero.
The problem asks us to transform a given caption string into a valid video tag following a strict sequence of rules. The input is a single string caption consisting only of English letters and spaces. These words must be transformed into a single hashtag-style identifier.
The task asks us to identify employees whose performance has consistently improved over their most recent three performance reviews.
The problem asks us to transform one string, word1, into another string, word2, using the minimum number of operations. Both strings have equal length and consist only of lowercase English letters.
We are given an array nums and an integer k. We want to split the array into one or more contiguous, non-empty segments. Every element must belong to exactly one segment, and the segments must preserve the original order of the array.
The problem presents a set of n computers, each with a unique password complexity represented in an array complexity of length n. The computers are labeled 0 to n - 1, and only computer 0 is initially unlocked.
This problem extends the classic stock trading dynamic programming family by allowing two different kinds of transactions. A normal transaction consists of buying first and selling later. If the stock price rises, the profit is: A short selling transaction reverses the order.
The problem asks us to choose a contiguous subarray from an array of positive integers after performing at most k doubling operations, where each element can be doubled at most once.
This problem asks us to construct the shortest string that contains two given strings, s1 and s2, as substrings. In other words, we are asked to merge s1 and s2 in such a way that no unnecessary characters are added, but both strings appear contiguously somewhere in the…
This problem asks us to identify books in a library system that are both fully unavailable for borrowing and currently have active borrowers who have not returned them yet. We are given two tables.
The problem requires computing a specific statistic for every contiguous k x k submatrix of a given m x n integer matrix grid. For each submatrix, we must find the minimum absolute difference between any two distinct elements.
The problem asks us to transport two logs of lengths n and m using three trucks, where each truck can carry a log of length at most k. If a log is longer than k, it must be cut into smaller pieces.
The problem asks us to find a path that visits every cell of a given m x n grid exactly once while visiting certain numbered cells in a specified order. Specifically, the grid contains integers from 1 to k in exactly one cell each, and the rest of the cells are zeros.
The problem asks us to find the lexicographically smallest string that can be formed by repeatedly removing adjacent pairs of characters in a string s that are consecutive in the alphabet. Consecutive letters can be in either order (e.g.
The problem asks us to repeatedly remove pairs of adjacent characters in a string that are consecutive in the alphabet, considering the alphabet as circular.
We are given an undirected tree with n nodes, rooted at node 1. Every edge will eventually receive a weight of either 1 or 2. For each query [u, v], we only care about the unique path connecting u and v. All edges outside that path are completely irrelevant and should be ignored.
The input arrays represent polynomials in coefficient form. If: then it represents: Similarly: represents: The task is to compute the product polynomial: and return its coefficients.
This is a SQL database problem involving customer purchasing behavior across product categories. We are given two tables: ProductPurchases records which products each user purchased. A user may purchase multiple products, and each purchase has a quantity.
We are given a weighted, undirected tree with n nodes. Because the graph is a tree, there is exactly one simple path between any two nodes.
This problem presents a 2D grid traversal scenario with obstacles and teleportation portals. You are given a matrix of characters representing cells. Empty cells '.
The graph is undirected, connected, and every node has degree at most 2. A connected graph where every node has degree at most 2 can only have one of two shapes: - A simple path, containing exactly n - 1 edges. - A simple cycle, containing exactly n edges.
This problem asks us to determine whether a rectangular grid of positive integers can be split into two parts with exactly the same sum using a single straight cut. The cut can be made in only one of two ways: - A horizontal cut between two adjacent rows.
This problem asks us to select a subset of nodes in a rooted tree such that each selected node triggers a subtree inversion, and inversions interact through ancestor-descendant relationships with a distance constraint.
The problem gives us a string s containing lowercase English letters and an integer k. We are allowed to delete any characters from the string, including multiple occurrences of the same character.
The problem requires us to reduce all elements of a given non-negative integer array nums to zero using the fewest number of operations. Each operation allows selecting a contiguous subarray and setting all occurrences of the minimum non-negative integer in that subarray to zero.
We are given a directed acyclic graph (DAG) with n nodes and weighted directed edges. Each edge is represented as [u, v, w], meaning there is a directed edge from node u to node v with weight w. The goal is to find a path that satisfies two conditions: 1.
This problem describes a set of n houses arranged in a circle. Between adjacent houses, there are two directed road systems: - forward[i] is the distance from house i to (i + 1) % n. - backward[i] is the distance from house i to (i - 1 + n) % n.
The problem requires analyzing a string s composed of lowercase English letters and determining the sum of the maximum frequency of vowels and the maximum frequency of consonants. Vowels are defined as 'a', 'e', 'i', 'o', and 'u'. All other letters are consonants.
This problem presents a road of length l kilometers, segmented by n signs at strictly increasing positions, where the first sign is at the start of the road (position[0] = 0) and the last sign is at the end of the road (position[n-1] = l).
The input describes a collection of unit conversion relationships. Each conversion is given as: source - target with factor f meaning: 1 unit of source = f units of target The important observation is that there are exactly n - 1 conversion edges among n units, and the…
The problem gives us a positive integer n and asks us to find the maximum product that can be formed by multiplying any two digits that appear in the number. The input is a single integer.
We are given an array nums containing up to 13 positive integers and an integer k. We may reorder the numbers in any permutation. After choosing an ordering, we concatenate the decimal representations of the numbers to form a single large integer.
We are given a character matrix and a pattern string. The key observation is that the problem defines two unusual traversal orders: - A horizontal traversal reads the matrix row by row, from left to right. When a row ends, reading continues at the beginning of the next row.
This problem defines an undirected graph implicitly through a sorted array nums. Each index in nums represents a graph node. Two nodes i and j are connected by an edge whenever: We are then given many connectivity queries.
The problem requires computing the maximum possible profit achievable by processing nodes of a Directed Acyclic Graph (DAG) in a valid topological order.
This problem gives us a two dimensional array of strings called responses. Each responses[i] represents all survey responses collected on the ith day. A response is simply a string such as "good", "ok", or "bad".
For each query, two things happen in order. First, we permanently update one element of nums: This modification remains in effect for all future queries. Second, we conceptually remove the prefix nums[0..start-1].
The operation described in the problem may initially look unusual, but it is actually equivalent to choosing a non-empty contiguous subarray. When we remove a prefix and a suffix that do not overlap, the elements left behind form a contiguous segment of the original array.
The problem asks us to transform an integer array nums into a non-decreasing array by performing zero or more operations, where an operation consists of selecting a contiguous subarray and replacing it with its maximum value.
We are given a string s that is guaranteed to already be a palindrome. We may rearrange its characters, but the final rearranged string must also be a palindrome.
We are given a weighted tree with n nodes, rooted at node 1. A tree has exactly one simple path between any pair of nodes. Since the tree is rooted at node 1, every node has a unique path from the root.
We are given an integer array nums, and we must consider every triplet of indices (i, j, k) such that i <= j <= k. For each valid triplet, we compute: where ⊕ denotes the bitwise XOR operation. The goal is not to count triplets.
Before diving into the detailed solution guide, I want to clarify one subtle point: the problem specifies that all subarrays with more than two elements must have a positive sum. This means the smallest subarray we need to consider is of length 3.
The problem asks us to determine the number of unique values that can result from taking the XOR of three elements in a given array nums, where the triplets (i, j, k) must satisfy i <= j <= k.
The problem asks us to take a palindromic string s and return its lexicographically smallest palindromic permutation.
This problem gives us three integers, x, y, and z, representing the positions of three people on a one-dimensional number line. Person 1 starts at position x, Person 2 starts at position y, and Person 3 remains stationary at position z.
The problem gives us an integer array nums and an integer k. We are allowed to repeatedly perform one operation: choose any index i and decrease nums[i] by exactly 1.
We are given an array nums. The operation is highly constrained: 1. Among all adjacent pairs, find the pair with the minimum sum. 2. If multiple adjacent pairs have the same minimum sum, choose the leftmost one. 3. Replace those two elements with their sum.
The problem asks us to find a subsequence of a given integer array nums such that the alternating sum of the subsequence equals a target integer k, while maximizing the product of the numbers in that subsequence without exceeding a given limit.
The problem is asking us to design a Router data structure to manage network packets efficiently under memory constraints. Each packet has three attributes: source, destination, and timestamp. The router has a memory limit, meaning it can store only a fixed number of packets.
We are given an array nums. At any step, we are not free to choose any adjacent pair. Instead, the operation is completely determined by the current state of the array: 1. Find the adjacent pair whose sum is the smallest. 2.
We are given two strings, s and t, and we are allowed to choose: 1. Any substring of s, including the empty string. 2. Any substring of t, including the empty string. The chosen substring from s must come first, and the chosen substring from t must come second.
This is a SQL aggregation problem where we need to analyze user subscription behavior and return statistics only for users who successfully converted from a free trial to a paid subscription. The UserActivity table contains one row per user, per day, per activity type.
The problem asks us to model a 2D array of integers, called properties, as an undirected graph. Each row of the array corresponds to a node in the graph. Two nodes are connected if the number of distinct integers they share is at least k.
The problem gives a binary string s representing sections of a system, where '1' indicates an active section and '0' indicates an inactive section. We are allowed at most one trade to maximize the number of active sections.
The problem presents a binary string s where each character represents an active ('1') or inactive ('0') section. For each query, which specifies a substring of s, we are asked to determine the maximum number of active sections after performing at most one trade operation.
This problem asks us to partition an array nums into contiguous subarrays, where each subarray contributes a weighted cost to the total. The cost for a subarray nums[l..
The problem asks us to compute a special value called the reverse degree of a string. Normally, letters are assigned positions in the alphabet as: - 'a' = 1 - 'b' = 2 - ... - 'z' = 26 However, this problem uses the reversed alphabet order, where: - 'a' = 26 - 'b' = 25 - ...
This problem describes a cargo ship with an n × n deck. Since each cell on the deck can hold exactly one container, the deck has a total capacity of: physical container positions. Every container has the same weight, w. The ship also has a maximum total weight limit, maxWeight.
We are given an integer array nums. While the array contains more than two elements, we must repeatedly perform one of three operations: 1. Remove the first two elements. 2. Remove the last two elements. 3. Remove the first and last element.
The problem asks us to simulate a sequential brewing process for potions performed by a series of wizards. Each wizard has a skill level that determines the speed at which they brew a potion, while each potion has a mana requirement that affects how long it takes to brew.
The problem is asking us to determine whether any phone number in a given list is a prefix of another phone number. In simpler terms, if you have a list of strings representing numbers, you need to check that no string starts with another string from the list.
This problem asks us to count the number of positive integers between l and r (inclusive) that are beautiful. A number is defined as beautiful if the product of its digits is divisible by the sum of its digits.
The problem asks us to determine the minimum number of sequential queries needed to transform an integer array nums into a Zero Array, where every element is zero.
The problem asks us to process a series of queries on a circular array nums. Each query specifies an index, and for that index, we need to find the minimum circular distance to any other index with the same value.
We are given a rooted tree with n nodes. The tree is rooted at node 0, and every edge has a positive length. Each node also contains a value stored in the array nums. A path is considered special if it satisfies two conditions: 1.
Here’s a detailed technical solution guide following your requested format for LeetCode 3480 - Maximize Subarrays After Removing One Conflicting Pair. The problem gives you an integer n, which defines a sorted array nums = [1, 2, ..., n].
This problem asks us to design a very small spreadsheet system that supports three operations on cells and one operation for evaluating simple formulas. The spreadsheet always contains exactly 26 columns, labeled 'A' through 'Z', and a configurable number of rows.
This problem asks us to assign tasks to workers in order to maximize total profit, under strict skill constraints. Each worker has a skill level, and each task has a required skill and an associated profit.
This problem gives us a collection of replacement rules and a text containing placeholders. Every placeholder has the format %X%, where X is a single uppercase letter corresponding to one of the replacement keys.
We are given two strings: - str1 of length n, containing only 'T' and 'F' - str2 of length m, containing lowercase English letters We must construct a string word of length n + m - 1.
This problem gives us two arrays of equal length: - fruits[i] represents the quantity of the i-th fruit type. - baskets[j] represents the capacity of the j-th basket. We process the fruit types from left to right.
This problem asks us to compute a "maximum sum" for each index in an array based on a relationship with other elements.
This problem asks us to simulate the placement of fruits into baskets following very specific rules. We are given two arrays of equal length, fruits and baskets.
This problem provides a database table named Samples that contains DNA sequences and their associated species. Each row represents a single biological sample, identified by a unique sampleid.
The problem gives us an integer array nums and an integer k. We consider every contiguous subarray of length k. An integer is called almost missing if it appears in exactly one of those size-k subarrays. Our goal is to find the largest integer that satisfies this property.
This problem asks us to repeatedly transform a string of digits by summing consecutive pairs modulo 10, until only two digits remain, and then check whether those two digits are equal.
Here’s a complete, detailed technical solution guide for LeetCode 3472 following your formatting rules. The problem asks us to find the length of the longest palindromic subsequence in a string s after performing at most k operations.
We are given two integers, n and k. We must consider all permutations of the numbers 1 through n that satisfy an alternating parity condition. In a valid permutation, every pair of adjacent elements must have different parity. In other words: - Odd must be followed by even.
The problem is asking for the minimum total cost required to remove all elements from an integer array nums under a very specific set of operations.
The problem gives us an integer array nums and asks us to transform it using three operations that must be performed in a specific order. First, every even number must be replaced with 0. Second, every odd number must be replaced with 1.
We are given an n x m matrix whose entries are only 0, 1, or 2. A valid V-shaped diagonal segment must satisfy two independent requirements: First, the values along the path must follow a very specific sequence. The first cell must contain 1.
The problem asks us to query a database table called products to identify rows where the description column contains a valid serial number.
We are given a matrix grid, where each row contains a collection of values that can potentially be selected. We are also given an array limits, where limits[i] specifies the maximum number of elements that may be chosen from row i.
The problem asks us to repeatedly transform a string of digits until only two digits remain, and then check if those two digits are the same. The transformation involves taking each consecutive pair of digits, summing them, and taking the result modulo 10 to form a new digit.
We are given two lowercase strings, s and t. We may remove at most one character from s. The removal is optional, so we are also allowed to leave s unchanged.
You are given an array points, where points[i] is the amount added to gameScore[i] every time you land on index i. The player starts outside the array at position -1.
The problem asks us to determine whether we can select k disjoint special substrings from a given string s. A special substring is defined as a substring where every character in it does not appear anywhere else in the string.
The problem presents an array pizzas of size n where each element represents the weight of a pizza. Every day, you must eat exactly 4 pizzas, and the total weight you gain from these 4 pizzas depends on whether the day is odd or even.
The problem gives us a string s and an integer k. We need to determine whether there exists a substring of length exactly k that satisfies three conditions. First, every character inside the substring must be identical.
This problem asks us to find the smallest horizontal line y = H that divides the union area of a collection of axis-aligned squares into two equal halves.
The problem provides a list of student-seat assignments, where each entry has the form [studentid, benchid]. Each pair indicates that a particular student is sitting on a particular bench.
This problem gives us an integer array nums and an integer k. For every position i, we need to determine whether nums[i] is a good number.
This is a SQL database problem where we must identify all invalid IPv4 addresses that appear in the logs table and count how many times each invalid address occurs. The input table contains one row per server access log.
We are given a string s consisting only of decimal digits. Every substring of s represents a non-negative integer, and leading zeros are allowed. For each substring, we look at its last digit.
We are given an n x n square matrix grid. Every cell belongs to exactly one top-left to bottom-right diagonal. Two cells (r1, c1) and (r2, c2) are on the same diagonal if r1 - c1 == r2 - c2.
This problem requires analyzing the frequency of characters in a given string and computing a specific difference. You are given a string s consisting only of lowercase English letters. For each character in the string, you can count how many times it appears.
The problem requires assigning elements from one array, elements, to groups defined by another array, groups, under a divisibility constraint. Specifically, for each group groups[i], we must find the smallest index j in elements such that groups[i] % elements[j] == 0.
We are given two arrays: - nums, the array whose elements we are allowed to increase. - target, a small array containing values that must each be represented by at least one multiple somewhere in the final version of nums.
We are given a lowercase string caption. We may repeatedly change any character one step forward or backward in the alphabet.
We are given an event that lasts from time 0 to eventTime. Inside this event there are n non-overlapping meetings, represented by the arrays startTime and endTime.
We are given an event that runs from time 0 to eventTime. Inside this event there are n non-overlapping meetings, already arranged in chronological order.
We are given a single integer n, and we need to generate all permutations of the numbers from 1 to n. However, not every permutation is valid. A permutation is considered an alternating permutation if every pair of adjacent elements has different parity.
The problem requires us to identify valid email addresses from a Users table in a database. Each row contains a userid and an email string. The definition of a valid email is very specific: it must contain exactly one @ symbol, end with .
The problem asks us to find all shortest common supersequences (SCS) of a given list of two-character strings words. A shortest common supersequence is a string of minimum length that contains each word in words as a subsequence.
This problem asks us to simulate a sequence of user activity events and count how many times each user is mentioned across all message events.
We are given an array nums and an integer k. For every subsequence whose size is at most k, we look at two values: - The minimum element in that subsequence. - The maximum element in that subsequence.
This problem is asking us to paint n houses arranged in a line with three available colors for each house, minimizing the total painting cost while satisfying two constraints.
The problem provides an integer array nums of length n. For each index i, we are asked to consider a subarray that ends at i but starts at start = max(0, i - nums[i]).
This problem provides an undirected tree rooted at node 0 with n nodes and weighted edges, where each node has a value defined in the array nums. A special path is a downward path (from ancestor to descendant) such that all node values along the path are unique.
The problem asks us to transform one integer array arr into another integer array brr with the minimum cost using two types of operations. The first operation allows splitting arr into contiguous subarrays and rearranging them at a fixed cost k per operation.
This problem gives us an integer array nums and asks us to find the largest absolute difference between any two adjacent elements. The important detail is that the array is circular.
We are given an array nums and an integer k. In a single operation, we may increase or decrease any element by exactly 1. Our goal is not to make the entire array equal. Instead, we only need at least one contiguous subarray of length k to consist entirely of the same value.
You are given an array nums and a budget of at most k increment operations. For any chosen subarray, you may repeatedly pick an element inside that subarray and increase it by 1.
We are given an m x n grid where each cell contains either a positive value, zero, or a negative value. The robot starts at the top-left cell (0, 0) and must reach the bottom-right cell (m - 1, n - 1).
This problem asks us to traverse a two-dimensional grid in a zigzag order while visiting only every other cell in the traversal sequence. The zigzag traversal follows a very specific pattern: - Start at the top-left corner (0, 0). - Traverse the first row from left to right.
We are given an array nums, and we must count how many subsequences of length exactly 5 have a very specific property. For any chosen subsequence the middle element is c, because it is the third element of a length-5 sequence. The subsequence is valid if: 1.
We are given several non-overlapping segments on a number line. Each segment [l, r, c] means that every bag at positions l, l+1, ..., r contains exactly c coins. All positions not covered by any segment contain 0 coins.
We are given an array nums consisting of positive integers. For any array arr, define: - prod(arr) as the product of all elements. - gcd(arr) as the greatest common divisor of all elements. - lcm(arr) as the least common multiple of all elements.
We are given an array nums, and we want to choose a subsequence from it. A subsequence preserves the original order of elements but may skip arbitrary positions.
This problem asks us to design a data structure that maintains a collection of tasks belonging to different users.
The problem gives us two strings: - s, the source string. - p, a pattern string containing exactly one ''. The special character '' can represent any sequence of characters, including an empty sequence.
We are given a string word of length n and an integer numFriends. In each round, the string is split into exactly numFriends non-empty contiguous pieces. Every distinct way to place the numFriends - 1 cuts is considered a different round.
Below is,000. This value is well within the limits of standard integer types, and the algorithm still performs only a single pass through the array. ::: This version is formatted as a complete LeetCode editorial-style reference document and includes all requested sections.
We are given two arrays, lane1 and lane2, of equal length. Each index represents a mile on a freeway, and the value at that index represents how many coins Mario gains or loses if he drives through that mile in that lane. A positive value means Mario gains coins.
We are given an integer array nums. We may perform at most one operation: 1. Choose a value x. 2. Remove every occurrence of x from the array. 3. The resulting array must remain non-empty.
Let the regular tetrahedron have vertices $A,B,C,D$, and let $O$ be the center of its circumscribed sphere.
Choose Cartesian coordinates with
The problem describes a sequence of locks that must all be broken. Each lock requires a certain amount of energy, given by strength[i]. Bob's sword starts with: - Energy = 0 - Growth factor x = 1 Every minute, the sword gains x energy.
The problem asks us to determine the number of distinct ways to sum up to an integer n using a limited set of coins. Specifically, we have coins of value 1, 2, and 6 available in infinite quantities, and coins of value 4 with a strict limit of two.
We are given a list of hotel rooms, where each room has: - A unique room ID - A room size We are also given several queries.
We are given a binary string s consisting only of '0' and '1', along with an integer numOps. In one operation, we may choose any index and flip the bit at that position. A '0' becomes '1', and a '1' becomes '0'.
This problem asks us to analyze a football match pass sequence and determine, for every team, the longest uninterrupted sequence of successful passes between players on the same team. We are given two tables: The Teams table maps each playerid to a teamname.
The problem gives us an integer array nums and a target value k. Our goal is to transform the entire array so that every element becomes exactly equal to k. However, we cannot arbitrarily change values. Each operation follows a very specific rule.
We are given two separate undirected trees. The first tree contains n nodes labeled from 0 to n - 1, and the second tree contains m nodes labeled from 0 to m - 1. A node u is considered a target node for another node v if the shortest path distance between them is at most k.
In this game, Alice and Bob take turns removing stones from a pile. The rules are very strict because the number of stones removed on each turn is predetermined. Alice always goes first and must remove exactly 10 stones on her first move.
The problem gives us a string s and an integer t. We repeatedly apply a transformation exactly t times. During one transformation: - Every character from 'a' to 'y' becomes the next character in the alphabet. - The character 'z' becomes the string "ab".
The problem asks us to determine the number of possible original strings that Alice could have intended to type, given a string word that represents the final output and an integer k representing the minimum length of the original string.
The problem gives us three positive integers, num1, num2, and num3. We must construct a new number called the key using the digits of these three numbers. The important detail is that every number must first be treated as a four digit number.
The problem asks us to construct the largest possible palindrome with exactly n digits such that the resulting number is divisible by k. A palindrome is a number that reads the same forward and backward. For example, 595, 1221, and 8 are palindromes.
The problem gives us an integer array nums of even length n and an integer k. We are allowed to change any element in nums to any integer in the range [0, k].
The problem gives us a circular array called colors, where each element represents the color of a tile: - 0 means red - 1 means blue Because the tiles form a circle, the first and last tiles are considered adjacent.
This problem asks us to identify students who satisfy a combination of academic requirements based on their major, completed courses, grades, and overall GPA. We are given three tables: - The students table stores each student's identity and declared major.
We are given an integer array nums and a non-negative integer k. We need to find the maximum possible length of a subsequence such that the number of adjacent unequal pairs is at most k.
The problem asks us to determine whether a given integer array is "special". An array is considered special if every pair of adjacent elements has different parity. Parity refers to whether a number is even or odd.
We are given a road represented as a string containing only two characters: - 'x' represents a pothole. - '.' represents a smooth section of road. We are also given a repair budget. A repair operation can only be applied to a contiguous group of potholes.
The problem gives us two strings, initial and target. We want to transform initial into target using the minimum number of operations.
This problem gives us an undirected weighted graph with n vertices and up to 10^5 edges. Each edge connects two vertices and has an integer weight. For every query [s, t], we must determine the minimum possible cost of any walk from node s to node t.
The problem asks us to redistribute a collection of apple packs into boxes while using the minimum number of boxes. Each pack apple[i] contains a certain number of apples, and each box capacity[j] can hold up to a certain number of apples.
This problem asks us to transform one string into another while minimizing the total conversion cost. We are given two strings, source and target, both with the same length. Each position in source must eventually become the corresponding character in target.
The problem asks us to count all integers within a given range [start, finish] that satisfy two conditions. First, the integer must end with a specific string s, meaning s is a suffix of the number. Second, each digit in the integer cannot exceed a given limit.
We are given an array nums containing positive integers. Our task is to divide all elements into two non-empty groups, nums1 and nums2. Every element must belong to exactly one of the two groups, and neither group may be empty.
The problem defines a special value called the "effective value" for every possible triplet of indices (i, j, k) in the array: where: - | is the bitwise OR operator - & is the bitwise AND operator The xor-beauty of the array is the XOR of the effective values for all possible…
We are given an m × n matrix grid where every row is already sorted in non-decreasing order. The total number of elements in the matrix is guaranteed to be odd because both m and n are odd. The task is to find the median of all values contained in the matrix.
In this problem, we are given a binary matrix called land. Each cell contains either: - 0, representing forest land - 1, representing farmland The farmland cells are organized into rectangular groups. Every group is guaranteed to form a perfect rectangle made entirely of 1s.
The problem asks us to find two non-overlapping odd-length palindromic substrings whose length product is as large as possible. We are given a string s of length up to 10^5.
The problem asks us to transform a given string of brackets into a balanced string with the minimum number of swaps. The input string s consists of n characters, where n is even, and there are exactly n / 2 opening brackets '[' and n / 2 closing brackets ']'.
This problem asks us to find the maximum product difference between two pairs of numbers from a given array. Specifically, given an array nums, we need to select four distinct indices w, x, y, z to form two pairs (nums[w], nums[x]) and (nums[y], nums[z]).
We are given an undirected graph with n nodes and a list of edges. Multiple edges between the same pair of nodes are allowed. For any pair of nodes (a, b), define incident(a, b) as the number of edges connected to either node a or node b.
The problem asks us to determine if the sum of the numerical values of two words equals the numerical value of a third word. Each word consists of lowercase letters from 'a' to 'j'.
The problem asks us to identify "imbalanced orders" from an OrdersDetails table. Each row represents a product within an order, containing the orderid, productid, and quantity ordered. An order can have multiple products, meaning multiple rows can share the same orderid.
The problem gives us a collection of inclusive integer intervals and a list of query values. For every query, we must determine the size of the smallest interval that contains that query value.
In this problem, we are given a string s where characters at even indices are always lowercase English letters, and characters at odd indices are always digits. The goal is to replace every digit with a new character computed from the character immediately before it.
HSK 5 | noun | a software program designed for a specific task or user purpose
HSK 4 | verb / noun | sharing ideas, experiences, or information between parties
HSK 5 | noun/verb | public relations; to conduct PR activities
HSK 2 | noun/locative | outside, exterior, beyond (contrast with 内)
HSK 5 | adjective / adverb | basic, fundamental, essential; basically, fundamentally
HSK 3 | noun | a public outdoor space for recreation and leisure
HSK 2 | verb | to feel or think based on personal impression; contrast with 认为 for reasoned opinion
HSK 4 | verb / noun | organizing or scheduling people and tasks
HSK 5 | adjective | precise, exact, accurate
HSK 5 | adjective/verb/noun | abstract; not concrete; to make abstract
Learn 〜ものがある (there is something ~ about it, it has a certain quality) — the N3 pattern for expressing an indefinable or deeply felt quality.
HSK 2 | verb / adjective / noun | describes hard work and determined striving
青 (blue, green): 8 strokes, JLPT N5. On: セイ、ショウ. Kun: あお、あお-い.
A systematic guide to Esperanto's authoritative dictionaries, the etymology of its root stock, neologism adoption, and confusable word pairs at advanced level.
Complete guide to all 5 Sindarin consonant mutations: soft (lenition), nasal, mixed, stop, and liquid — with full tables, triggers, and worked examples.
HSK 5 | noun / adjective | diplomacy, foreign affairs; diplomatic
Handle phone calls, send messages, and communicate digitally in Chinese with confidence.
Discuss pressing Chinese and global social challenges using the vocabulary and grammatical structures of policy discourse and social commentary.
HSK 4 | verb / noun | to formally apply for something; an application
Form yes/no questions with ĉu, use wh-questions fluently, and express, agree with, and politely challenge opinions.
HSK 5 | noun/verb | coercive measures imposed to punish or compel compliance
Structures and vocabulary for Chinese literary criticism — 以...为..., 体现了...精神, and the discourse of thematic and symbolic analysis
Victor wants to build a set of integers within a specified range such that the XOR of all its elements is minimized. The set can contain up to k elements, all distinct, and every element must lie between l and r inclusive.
HSK 1 | number | the digit three
Master the N1 listening section: natural native speed (400+ chars/min), implied meaning, speaker stance inference, pitch accent minimal pairs, note-taking for long monologues, and analysis of five authentic-difficulty listening scripts.
We are given a sorted set of marks on a number line from 0 to a length l, where each mark is an integer position. Some marks already exist, including the endpoints 0 and l. Using these marks, we can measure any distance that equals the difference between two existing marks.
Learn how to buy things, ask prices, and use numbers in shopping situations.
HSK 6 | n | the form, configuration, or mode of existence of something; morphology
HSK 1 | verb, noun | to answer, to reply, an answer
日 (day, sun): 4 strokes, JLPT N5. On: ニチ、ジツ. Kun: ひ、び、か.
Sindarin intransitive verbs, class 3 i-verbs, irregular stems with suppletive roots, and half-strong verb formation.
HSK 3 | verb / noun | to discuss, talk over; a discussion
HSK 4 | noun | the ability to produce original and imaginative ideas
HSK 5 | verb | to confirm; to acknowledge as correct
HSK 6 | n | the overall pattern, configuration, or situation of affairs; a person's breadth of vision
HSK 5 | verb | to adapt oneself or something to new conditions or requirements
HSK 3 | adj/adv | shared by all parties or done collectively
HSK 3 | verb | to remember something firmly; to commit to memory
HSK 5 | noun/adjective | the relationship between living organisms and their environment
HSK 1 | preposition | indicates the starting point of movement, time, or origin
HSK 2 | measure word/noun | used for floors of a building and layers of things
HSK 5 | verb/adjective | to organize multiple parts to work together; well-balanced and harmonious
気 (ki/ke/): spirit. JLPT N5 kanji.
Esperanto vocabulary for feelings, emotions, personality traits, and character descriptions.
N1 grammar pattern 〜だに: a literary/formal emphatic particle meaning 'even' — used to intensify a minimal or extreme case.
HSK 5 | verb/noun | the state of being unable to repay debts; financial insolvency
HSK 4 | n | money collected by the government through taxes
HSK 5 | discourse marker | in summary; in short; to sum up
Master Chinese rhetorical devices including parallel structure and rhetorical questions, and develop the ability to write and recognize persuasive Chinese prose.
HSK 4 | noun | a written or digital record containing information
Express past events in Japanese using verb, adjective, and noun past forms, and use timeline vocabulary to sequence events in time.
HSK 5 | verb/noun | to consume goods or services; consumption
HSK 3 | noun | an umbrella used to shelter from rain
なかなか (なかなか / nakanaka): fairly, quite; not easily. N4 level Japanese vocabulary.
HSK 3 | noun | chopsticks, the traditional East Asian eating utensils
We are given a collection of parcels, each arriving at a specific time and leaving at another specific time. Each parcel carries a weight, a strength, and a value.
HSK 4 | adjective/adverb | done at the right time without delay
HSK 1 | noun | watermelon
HSK 5 | noun | an ordered set of steps or a computer program
HSK 5 | adjective | different from what is normal; exceptional; having unique qualities
HSK 1 | noun/measure word | used to tell clock time; also means 'a little' or 'dot/point'
We are given an array a[1..n]. For every position i, define the value L[i] = number of occurrences of a[i] in the prefix [1..i]. For every position j, define R[j] = number of occurrences of a[j] in the suffix [j..n].
HSK 1 | measure word | classifier for cups, glasses, and their contents
考える (かんがえる / kangaeru): to think, consider. N4 level Japanese vocabulary.
HSK 4 | noun | family member, family
HSK 5 | verb | to be completely absorbed in or captivated by a pleasurable experience
Express additive and cumulative relationships using 不但...而且, 既...又, and 不仅...还 at B1 level.
HSK 3 | noun | map — a visual representation of an area used for navigation
HSK 6 | n | the intrinsic qualities, depth, or connotation of a person, concept, or work
HSK 3 | grammar pattern | describes a quality that keeps intensifying over time
HSK 6 | n | the forefront or leading edge of a field; a front line position
HSK 5 | verb/adjective | to relieve someone's worry; feeling comforted or relieved
HSK 1 | noun | a calendar month; also the word for the moon
HSK 3 | noun | task, mission, assignment — something assigned or required to be done
HSK 4 | verb | to transfer data from a device to the internet or a server
HSK 3 | adjective | feeling satisfied or pleased with something
午 (go/): noon. JLPT N5 kanji.
Master the vocabulary and formal register of Chinese business culture, including negotiation, formal reception, and professional social interaction.
Learn to describe where things are using 在 and position words like 上面, 里面, and 前面.
Essential Japanese vocabulary for rooms, furniture, household items, chores, daily routines. N1 level reference with readings, romaji, and examples.
HSK 3 | interjection / verb | a cheer of encouragement, or to add fuel/gasoline
HSK 5 | verb | to list on stock market; to come onto the market
半 (han/naka-ba): half. JLPT N5 kanji.
HSK 4 | noun | the rate at which something moves or happens
We have an array of integers, each between 1 and n, and we need to support two types of operations efficiently. The first operation rotates a segment of the array one step to the right. The second operation counts how many times a particular value occurs in a segment.
HSK 5 | noun | cost; production cost; the amount spent to produce or achieve something
HSK 1 | adjective, adverb | slow, slowly
Master five more classical Chinese idioms, focusing on cautionary tales, irony, and the use of idiomatic variation in context.
HSK 3 | preposition | regarding, as for, with respect to — formal topic introducer
Japanese topic marker は (wa): marks the sentence topic, contrasts with が, usage in A-wa-B-desu patterns.
HSK 4 | noun | what is needed or wanted, especially in an economic or practical sense
HSK 4 | noun | a theory or theoretical framework
HSK 5 | noun | indicator; metric; target figure used to measure performance or status
Esperanto vocabulary for family members, relationships, life stages, and social bonds.
Drazil sends a sequence of movement commands. Each '+' moves Dreamoon one step to the right and each '-' moves one step to the left. We know the original command string and also the string that Dreamoon receives. The received string may contain '?
Learn how to use 〜だけでなく (dake de naku) to express 'not only ~ but also' in a neutral, versatile way. Includes structure, nuance, examples, and comparisons.
Japanese body parts, health, illness, injury, and medical vocabulary for everyday use and JLPT N5–N3.
HSK 5 | noun | psychological counseling; mental health consultation
HSK 5 | noun | facts, information, or skills acquired through experience or education
HSK 1 | pronoun | second-person singular pronoun (informal)
Sindarin word-building with prefixes: directional, negative, intensifying, and stative prefixes with 40+ examples from attested names.
HSK 6 | n/v | a proposition or thesis; the act of setting an examination question or essay topic
HSK 5 | adjective | clearly noticeable, conspicuous, or substantial in degree
HSK 1 | adjective | short (in height), not tall
HSK 5 | verb/noun | quantitative increase or growth in numbers
HSK 5 | verb | to cultivate, to nurture, to develop, to train
HSK 3 | preposition / verb | toward, in the direction of; facing; to face
川 (sen/kawa): river. JLPT N5 kanji.
HSK 3 | conjunction | adds an additional point to what was already stated
〜てあげる: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 3 | adjective | visually attractive; beautiful; well done
Esperanto numbers, counting, arithmetic, ordinals, fractions, and mathematical vocabulary.
HSK 5 | noun/verb | practical application of theory; to apply knowledge in real action
We are asked to minimize the network traffic when joining two distributed tables, A and B, across clusters. Each table is split into several partitions, and each partition contains a certain number of rows.
HSK 3 | verb / noun | to pay attention to, be careful about; attention
HSK 3 | verb / adjective | to understand, to be clear about something
古 (ko/furu-i): old. JLPT N5 kanji.
HSK 3 | adverb | describes slow, incremental change over time
N1 grammar: classical particle だに expressing literary 'even just X' or 'even merely X.' Marks the minimum case on a scale of unexpectedness. Used in literary prose and formal writing.
JLPT N4 kanji 薬 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 4 | adjective | conforming to reason, fairness, or accepted principles
HSK 5 | noun | the percentage of a country or region's land area covered by forest
Sindarin personal pronouns: subject, object, and possessive forms — with worked examples from attested texts.
HSK 5 | noun/verb | arbitration; to arbitrate; to mediate a dispute
HSK 5 | verb / noun | to anticipate something with hope; hopeful expectation
HSK 4 | verb/noun | to propose a course of action; a recommendation offered to others
N1 grammar pattern 〜をものともせず: expressing that someone is completely undaunted by a challenge, obstacle, or adverse circumstance and continues their action.
HSK 4 | noun | a specific area of activity, knowledge, or interest
Master the accusative + infinitive construction (ACI) used after verbs of saying, thinking, and perceiving.
HSK 3 | verb/noun | an expression of friendly acknowledgment or well-wishes
HSK 1 | adverb | all, both — applies a statement collectively to all subjects
HSK 5 | adjective | open, clear, and free from concealment; transparent
We have two walls represented by sequences of tower heights. The bears' wall contains n towers, while Horace's wall contains w towers. Horace is allowed to shift his entire wall vertically by any amount.
HSK 5 | noun/verb | to test or examine; a test or trial
HSK 6 | adv | suggesting that an action is harmless or worth trying; equivalent to 'might as well' or 'why not'
HSK 5 | verb/noun | to propose a course of action; a proposal or recommendation
HSK 2 | adjective/adverb | the color white; also means without result or free of charge
HSK 3 | verb | to accept or confirm something as true
HSK 3 | adj/adv | expressing regret or disappointment about something
Sindarin vocabulary for the human body, health, injury, and healing: attested and Neo-Sindarin terms with all known plural forms and example sentences.
HSK 5 | noun | stock, share, equity security traded on a stock market
Esperanto vocabulary for travel, transportation, directions, accommodation, and tourism.
JLPT N4 kanji 曇 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 8 | noun | structural engineering; the branch of civil engineering concerned with load-bearing structures
HSK 5 | noun | the exclusive legal right to reproduce, publish, and sell creative work
Engage with Chinese discourse on international relations, foreign policy, and globalization, mastering the vocabulary and grammar of global governance.
We are given a partially filled crossword slot and asked to identify which of Eevee's evolutions fits it. The input specifies the length of the word and a string pattern using lowercase letters and dots, where a dot represents an unknown character.
HSK 4 | adjective/verb | abstract; to abstract
HSK 5 | conjunction/adverb | on one hand; in one respect
HSK 5 | verb/adjective | to restrain oneself, to exercise self-control
HSK 4 | adjective | serious; earnest; conscientious; careful
HSK 3 | noun / verb | physical exercise, a sport, or a large-scale movement
HSK 3 | noun | library, a place to read and borrow books
HSK 3 | verb | to feel deep gratitude toward someone
Express proportional and progressive change using 越...越 and 越来越 at B1 level.
HSK 5 | phrase | a fixed expression introducing a comparative contrast
Common Sindarin phrases, greetings, and key passages — including A Elbereth Gilthoniel with full word-by-word translation.
道 (dou/tou/michi): road. JLPT N5 kanji.
HSK 1 | noun | subway, underground railway
HSK 4 | noun | facts, data, or news communicated or received
HSK 4 | n | activities or media that provide amusement
〜はずだ: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
Describe symptoms, answer a doctor's questions, and understand medical instructions in Chinese.
Learn 〜べく (formal purpose) and 〜べきだ/べきではない (moral/logical obligation) in formal Japanese writing and speech.
HSK 5 | noun | the percentage of users who click on a link or ad; click-through rate
HSK 4 | noun | a web service that searches and indexes internet content
HSK 5 | adjective/adverb | covering a large area or range; affecting many people or things
HSK 4 | adjective / verb | lively, active, or to make something more vibrant
HSK 4 | noun | risk; danger; hazard
The game can be viewed as moving through a trie built from the given strings. We start at the root, which represents the empty string. On each turn a player chooses one outgoing edge, corresponding to appending a character.
HSK 4 | conjunction | although; even though; though
Master Chinese numbers, measure words, and how to ask quantities with 几 and 多少.
HSK 1 | adjective | busy, occupied
HSK 4 | verb / noun | giving assistance, backing, or resources to someone or something
HSK 3 | verb | to borrow something from or lend something to someone
Consolidate and extend understanding of complex Chinese grammatical structures including the 把/被 constructions and noun phrase nominalization.
〜かもしれない: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 3 | adjective / noun | pride (positive: proud of someone); arrogance (negative: conceited)
Complete HSK 4 vocabulary reference: approximately 1,000 new words at B1 level, organized by semantic category with pinyin, English, part of speech, and usage notes.
HSK 5 | verb/noun | computer programming; to write or code a program
HSK 5 | noun | a set of official rules or regulations issued by a government body
We are given an n×n board, where each cell contains either an 'x' or an 'o'. The task is to check a local property for every cell: whether the number of orthogonally adjacent cells containing 'o' is even.
HSK 1 | particle | turns a statement into a yes/no question
HSK 3 | conjunction | introduces the single condition that is the only exception to a rule or the only way an outcome can occur
HSK 5 | noun | one's level of formal education or academic credentials
Mastering persuasive speech, rhetorical devices, signposting, ceremonial language, and Q&A in formal Esperanto public contexts.
Complete list of all 103 JLPT N5 kanji with stroke count, on-yomi, kun-yomi, meaning, and example words.
HSK 5 | verb/adjective | to inspire; to invigorate; filled with excitement and spirit
Learn how to use 〜てしかたがない (te shikataganai) to express uncontrollable feelings or irresistible urges — 'can't help but ~; irresistibly ~.' Includes structure, nuance, examples, and comparisons.
We are given the coordinates of two distinct trees on a Cartesian plane, and we know that the garden forms a perfect square aligned with the axes. Each vertex of the square has a tree, so there are exactly four trees.
HSK 5 | adjective | easily affected, responsive, or requiring careful handling
続ける (つづける / tsuzukeru): to continue. N4 level Japanese vocabulary.
HSK 1 | verb | to live at a place, to stay somewhere
HSK 3 | preposition | marks passive voice, indicating the subject is acted upon
左 (sa/hidari): left. JLPT N5 kanji.
HSK 4 | noun | a model, template, or small-scale replica
We have a sequence of pillars arranged from left to right. Pillar i has height h[i]. A valid jump can only go forward, from a smaller index to a larger index, and the height difference between the two pillars must be at least d.
HSK 5 | verb | to drive progress or move something forward systematically
八 (hachi/yat-tsu): eight. JLPT N5 kanji.
Learn 〜に伴い and 〜に伴って to express simultaneous or consequential change accompanying a main process or event.
HSK 1 | number | ten thousand; the base large-number unit in Chinese counting
HSK 5 | noun | a physical, mathematical, or conceptual representation of a system or object
Learn third declension consonant stems, neuter nouns, and i-stems with their distinctive endings.
HSK 3 | adjective | fierce, severe, formidable, impressive — colloquial praise or warning
HSK 5 | verb / noun | to purchase a company or large quantity of goods; a takeover or buyout
Learn 〜として (as, in the capacity of) — the N3 pattern for expressing roles, identities, functions, and premises.
HSK 6 | verb | to promote and carry forward something of cultural or moral value
Master all five major causal expression patterns at N3 level: 〜ため(に), 〜によって/〜による, 〜から/〜ので, 〜て/〜で, and 〜ことから — with full comparative analysis.
HSK 1 | noun | doctor, physician
HSK 3 | verb | to manage, to be in charge of; to bother with, to concern oneself with
HSK 3 | noun | colleague, coworker — someone who works at the same place
Learn five productive Esperanto prefixes that multiply your vocabulary by expressing opposites, beginnings, repetition, dispersal, and errors.
HSK 5 | noun / adjective | the activity of governance; relating to political affairs
Japanese food and drink vocabulary: common foods, drinks, cooking methods, restaurant expressions, and traditional Japanese cuisine terms.
HSK 3 | verb | to love something with great passion
HSK 4 | adverb | mostly true or largely complete
HSK 7 | verb | induction of temporary loss of sensation or consciousness for surgical procedures
Ten structured N3 lessons for intermediate learners: news headlines, speech registers, causal expressions, opinions, conditionals, change expressions, workplace Japanese, reading strategies, natural speech, and paragraph writing.
HSK 2 | verb/preposition | to use something, to employ, with/using (as a preposition)
HSK 1 | noun | telephone, phone call
HSK 3 | adverb | confirms that reality matched a prior prediction or expectation
HSK 3 | noun | fortune; chance; whether things go well or badly
HSK 3 | verb/noun | to eagerly anticipate something
HSK 1 | pronoun | third-person plural pronoun (all-female group)
HSK 2 | noun | any watercraft, from a small rowboat to a large ship
HSK 3 | verb/adj | to concentrate or focus attention or resources
HSK 3 | noun | a word or short phrase in Chinese
〜そうだ (appearance): JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 1 | conjunction | introduces a reason or cause, paired with 所以
HSK 4 | verb / noun | to research or study something; a research study
HSK 4 | adjective | having a negative, passive, or pessimistic attitude
Master the most frequently tested N1 四字熟語, learn how they are formed from classical Chinese and Japanese sources, and use them correctly in formal and literary contexts.
HSK 2 | adverb | again (past repetition), also, and (linking two qualities)
HSK 1 | verb | to sing
HSK 5 | verb / adjective | to summarize; concise
Learn how to use 〜わけにはいかない (wake ni wa ikanai) to express social or moral impossibility — 'can't do; must not do; it wouldn't be right to.' Includes structure, nuance, examples, and comparisons.
HSK 5 | noun | factual information, especially numerical, collected for analysis
HSK 4 | noun | student, pupil
HSK 3 | noun/adverb | the future as a time reference, used for goals, hopes, and plans
HSK 4 | adverb | expressing a combined total
Using 成语 naturally: understanding origin, semantic evolution, and pragmatic deployment in sophisticated discourse.
Learn 〜に従って (in accordance with / as ~ progresses) — the N3 pattern for expressing compliance with rules and proportional change.
HSK 1 | noun | company, corporation, firm
HSK 2 | noun | one of the four seasons of the year
HSK 1 | greeting phrase | expresses sincere apology for a mistake or inconvenience
HSK 3 | verb | to move, to carry, to relocate
HSK 3 | adjective / verb | convenient, to be convenient, to make things easy
HSK 5 | noun/verb | introducing new ideas, methods, or products; the act of innovation
HSK 4 | noun / verb | the act of serving or assistance provided
N1 grammar patterns 〜あっての and 〜あってこそ: expressing that X is an indispensable prerequisite for Y. 〜あってこそ adds emphatic force.
We are given the final state of a system with five participants who started a game under a very rigid rule: everyone begins with the same unknown positive number of coins, call it $b$. After this initialization, coins are only moved between players.
HSK 5 | noun phrase | long-term shifts in global or regional climate patterns
HSK 3 | verb | to enter into marriage
HSK 1 | noun | train, railway
HSK 5 | noun | a legal or moral entitlement that a person is owed
HSK 3 | grammar pattern | emphasizes time, place, manner, or agent of a completed action
HSK 4 | noun | channels of mass communication
HSK 1 | noun | mobile phone, smartphone
Full N1 simulation: 300-char reading passage with 5 questions, 20 grammar questions, 10 vocabulary questions, timing strategy for 170 minutes, pass rate analysis, the top 20 patterns that distinguish N1 from N2, and guidance on study beyond N1.
Immersion techniques for Esperanto — Muzaiko radio, Vikipedio reading, Pasporta Servo travel, and the Universala Kongreso.
HSK 4 | verb | to organize and host a formal event, activity, or exhibition
HSK 8 vocabulary reference: advanced C2-level words in specialist academic disciplines, technology, and professional domains.
HSK 2 | verb + result complement | indicates the action of listening results in actually hearing
HSK 2 | noun | noodles, a staple food made from wheat flour
HSK 4 | n | physical training, sports, or PE as a subject
HSK 3 | adjective/noun | showing warmth and enthusiasm toward others or a cause
Learn how result complements attach to verbs to express the outcome or completion of an action, a fundamental building block of Mandarin grammar.
HSK 3 | noun | innate talent or acquired ability
Master the -us ending for wishes, hypotheticals, polite requests, and both real and unreal conditional sentences.
Learn to talk about dates, days of the week, and clock times, and understand why time comes before the verb in Chinese.
Appleman has a collection of n cards, each labeled with a capital letter. Toastman is allowed to pick exactly k cards, and for each card he picks, Appleman pays him coins equal to the number of Toastman's cards that have the same letter as that card.
Learn how to use 〜ざるを得ない (zaru wo enai) to express reluctant necessity — 'have no choice but to; can't help but do.' Includes structure, nuance, examples, and comparisons.
HSK 3 | noun | an indoor facility for sports and physical activities
HSK 3 | noun | education; the process of teaching and learning
HSK 4 | verb | to refuse, to reject, to decline
HSK 2 | noun | a hat or cap worn on the head
HSK 5 | verb / noun | to develop; to open up resources or products
HSK 4 | adjective / noun | confident; self-confident; confidence
HSK 3 | adjective/adverb | paying close attention to detail
安全な (あんぜんな / anzen na): safe. N4 level Japanese vocabulary.
HSK 5 | verb/noun | to limit or restrict the scope of action; a constraint or limitation
Mastering Chinese financial register: capital markets, corporate finance, and investment discourse.
HSK 3 | adverb | almost, nearly, hardly — very close to a condition but not quite reaching it
HSK 4 | noun | a person's current emotional state or feeling
HSK 3 | noun | the movement of vehicles and people; the transport system
Codeforces 464D: World of Darkraft - 2
HSK 5 | noun | the sustained rise in the general price level of an economy
HSK 4 | n | the Olympic Games, held every four years
HSK 3 | adverb | something happened very recently or barely meets a threshold
Master six N3 conditional and temporal patterns: 〜さえ〜ば, 〜うちに, 〜たびに, 〜とたんに, 〜て以来, and 〜次第 — each with contrast to N4 basic conditionals.
HSK 1 | noun | woman, female adult
HSK 2 | adjective | describes something of great significance or necessity
HSK 6 | n/adj | a recognized expert or authoritative source; having recognized authority
Handle a complete restaurant interaction in Esperanto: ordering food, expressing preferences, noting dietary restrictions, and paying the bill.
HSK 1 | noun | place, location, area (abstract)
HSK 1 | adjective | tall, high, of great height
HSK 5 | noun | public opinion, public discourse, media opinion
HSK 3 | noun / adverb | time reference word for events after a stated point
HSK 3 | verb | to discover, find, or realize something previously unknown
HSK 5 | verb/noun | to assess, evaluate, or appraise performance; an assessment
Learn how to use 〜に基づいて (ni motozuite) to express 'based on' when grounding decisions, arguments, or actions in formal sources. Includes structure, nuance, examples, and comparisons.
Learn how to organize information in temporal order using 先...再/然后...最后 and related sequence markers essential for narration and instruction.
HSK 5 | adjective | bearing responsibility; accountable for one's actions
HSK 5 | adjective/verb | thorough; penetrating deeply into a topic or place
HSK 2 | adjective phrase | interesting, fun, enjoyable
HSK 5 | noun | the outer surface or outward appearance of something
HSK 1 | noun | hospital, medical facility
八 (eight): 2 strokes, JLPT N5. On: ハチ. Kun: や、や-つ、やっ-つ、よう.
HSK 4 | n | the governing body of a state or region
HSK 3 | verb | to support, to back, to endorse
HSK 5 | noun / verb | proposition, thesis, problem; to set a topic or exam question
Sindarin words for nature: sky, celestial bodies, weather, water, land, mountains, trees, plants, and animals.
HSK 1 | noun | a calendar year; also used in age and duration expressions
Comprehensive review of B2-level grammar patterns, key vocabulary, and communication strategies developed across the 25-lesson HSK 5 curriculum.
Master Esperanto's location and movement prepositions, and learn the crucial rule for adding -n to show direction versus static position.
HSK 1 | adjective | hot (temperature or weather)
HSK 5 | verb / noun | to embody; to give expression to
HSK 3 | noun | a room or building used for work
HSK 5 | verb/noun | the act of holding someone responsible for their actions or failures
Complete N2 grammar reference: 200+ formal and written patterns with structures, nuance comparisons, and example sentences in formal register.
Master the art of expressing desires, making suggestions, and articulating your future intentions in Japanese.
HSK 3 | noun | face; facial expression; social face, dignity
Complete guide to all six patterns of the ablative absolute and five strategies for translating it in context.
HSK 3 | verb | to continue or keep doing something
HSK 5 | verb/adjective | to be tangled up; to agonise, feel conflicted
HSK 3 | noun | text message, SMS, short message
HSK 5 | conjunction | at the same time; simultaneously; meanwhile
HSK 5 | noun | factual evidence used to establish the truth of something
長 (chou/naga-i): long. JLPT N5 kanji.
HSK 2 | verb | pivotal causative verb: let someone do, make someone do
We are given two long binary strings, a and b. Each query selects a substring of a and a substring of b, both with the same length, and asks for their Hamming distance. The Hamming distance between two binary strings is simply the number of positions where the bits differ.
HSK 1 | verb | to eat, to consume food
Learn the systematic ki- (question), ti- (demonstrative), and i- (indefinite) correlative series in Esperanto.
HSK 4 | verb | experiencing sudden surprise or shock
HSK 2 | adverb | refers to what happened next in a narrative; contrast with 以后
〜ば: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 5 | conjunction | introduces a contrast or unexpected turn; however
HSK 5 | adjective/verb | feeling troubled, distressed, or tormented; to trouble
Master all four Japanese conditional forms — 〜たら, 〜ば, 〜なら, 〜と — with a comprehensive 4×4 comparison grid, learner error analysis, and contexts where substitution between them is impossible.
HSK 5 | adjective | negative in attitude; passive; pessimistic
HSK 5 | noun | a fixed, oversimplified image or idea about a group of people
HSK 3 | noun | the meaning of a word or the idea behind something
Learn how Latin adjectives agree with nouns in gender, case, and number using the bonus/bona/bonum model.
Develop professional-level simultaneous and consecutive interpretation skills between Esperanto and national languages at congresses and formal events.
Express concession and unconditional assertions using 即使...也, 不管...都, 尽管...还是, and related patterns at B1 level.
千 (sen/chi): thousand. JLPT N5 kanji.
HSK 4 | verb / adjective | to understand; clear, obvious
HSK 5 | noun | the percentage charged or paid for the use of money over a period of time
土 (do/to/tsuchi): earth. JLPT N5 kanji.
HSK 2 | measure word | counts sections, periods of time, and stretches of things
HSK 5 | verb/noun | to spread; to disseminate; transmission of information or disease
We start with a collection of axis-aligned segments. Every segment is either horizontal or vertical, and segments of the same orientation never overlap. We are allowed to delete whole segments or only parts of them.
HSK 5 | verb / noun | to conduct negotiations; a negotiation process
HSK 1 | adverb | indicates excess or an extreme degree, often paired with 了
〜ておく: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 1 | verb | general verb for doing or making something
HSK 7 | verb | clinical administration of pharmaceutical agents in therapeutic contexts
HSK 3 | conjunction | not only — the first half of the 不但…而且… pattern
HSK 5 | verb | to make something stronger, more robust, or more effective
HSK 3 | noun | the image or impression someone or something projects
JLPT N4 kanji 会 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 1 | greeting phrase | asks about someone's wellbeing
HSK 4 | adjective | correct; right; accurate
HSK 3 | noun | a facility where aircraft take off and land
Inferring subtext, reading prosody, and decoding implicature in advanced Chinese spoken discourse — 弦外之音 and the pragmatics of indirect communication
HSK 3 | adjective/verb | to feel sad; to be hurt emotionally
HSK 5 | noun | medicine, drug, pharmaceutical substance
HSK 2 | adjective / noun | male gender; a man or boy
HSK 3 | noun / adjective | difficulty, hardship; difficult, hard
HSK 5 | verb | to value highly and make good use of; to cherish
気 (spirit, mood): 6 strokes, JLPT N5. On: キ、ケ. Kun: none.
Complete N2 exam preparation: format breakdown, section strategies, 6-month study plan from N3, 60 kanji reading drills, 60 grammar drills, and most commonly failed patterns.
HSK 5 | noun | one's sense of self and identification with a group or role
HSK 3 | verb | to change, become, or transform into a different state
HSK 3 | adverb | indicates that something happens after a specific event or time
小 (shou/chii-sai): small. JLPT N5 kanji.
Codeforces 479A: Expression
HSK 4 | verb | shielding someone or something from harm
HSK 4 | noun | a legally binding written agreement between parties
Sindarin mixed mutation: the third mutation type triggered by definite prepositions — the complete change table, all triggering constructions, and worked examples.
HSK 4 | noun | boss; owner; employer
HSK 5 | verb | to drive forward; to promote; to give impetus to
Consolidate all A2 grammar, review 150 essential vocabulary words, assess your CEFR A2 competencies, and preview what B1 brings.
HSK 4 | adverb | describing reciprocal actions between parties
HSK 2 | adjective | describes wide recognition or fame
We are given a set of points on an infinite 2D grid, representing planets in a "meta-universe." The goal is to repeatedly split the set of planets along empty rows or columns that completely separate the set into two non-empty subsets, each lying on one side of the row or column.
HSK 4 | noun | teacher, instructor
HSK 3 | verb / noun | to require, to demand, to request; a requirement, demand, request
HSK 1 | adjective/pronoun | interrogative asking about condition or opinion
HSK 3 | adjective/noun | relating to the present era
HSK 8 | noun | fluid dynamics; the study of forces and motion in liquids and gases
HSK 3 | noun | a customer who visits a shop or uses a service
HSK 3 | noun | doctor, physician — spoken/Northern Chinese term; compare 医生
HSK 5 | noun | a personal website or online journal updated regularly by an individual
HSK 3 | pronoun / adjective | referring to remaining or different items
HSK 3 | verb | to chase after someone or something; to pursue a goal or person
HSK 2 | verb | to go to work, to start a work shift
HSK 2 | adjective | describes the absence of dirt or mess
HSK 3 | verb/adjective | to feel anxious or concerned about something
HSK 3 | conjunction | although, even though — introduces a concession, paired with 但是 or 还是
HSK 4 | noun / verb | a formal speech or lecture; to deliver a speech
Learn 〜ようになる (come to do, become able to) — the N3 pattern for expressing gradual change in ability, habit, or situation.
HSK 3 | adjective / noun | lively, bustling, full of activity and noise
HSK 5 | verb | to examine and check something carefully for accuracy or compliance
HSK 5 | verb/noun | to invest capital; investment in financial or economic contexts
五 (go/itsu-tsu): five. JLPT N5 kanji.
HSK 1 | noun | a number or code; the colloquial word for a day of the month
HSK 4 | verb/noun | a formal public statement or announcement
HSK 4 | noun | an organizational unit within a larger body
HSK 4 | adjective | having a hopeful, positive outlook
HSK 4 | adjective/verb | the state of being in balance or equilibrium
HSK 5 | noun/adjective | a prolonged period of abnormally low rainfall; dry and parched
We have three types of balloons: red, green, and blue. Each table at a banquet requires exactly three balloons, and no table can have all three balloons of the same color.
HSK 4 | adverb/adjective | absolute or absolutely certain
〜で (de): JLPT N5 particle. Location where an action takes place, means/method, and cause/reason — with examples and comparison with に.
HSK 4 | phrase | indicating that something deserves attention or is important to observe
今 (kon/kin/ima): now. JLPT N5 kanji.
We start with a connected undirected graph. Every edge must be assigned a direction. After all directions are chosen, some ordered pairs of vertices $(u,v)$ will satisfy that there is a directed path from $u$ to $v$. The goal is not to make the graph strongly connected.
食 (eat, food): 9 strokes, JLPT N5. On: ショク、ジキ. Kun: た-べる、く-う.
HSK 4 | adjective / verb | vague, blurry; to blur
HSK 2 | verb | to grasp the meaning of a skill, language, or subject; contrast with 明白
水 (sui/mizu): water. JLPT N5 kanji.
HSK 3 | noun | gift, present — something given to someone as a gesture of affection or celebration
Master Esperanto's six-form participle system, distinguishing ongoing from completed states, and use participial nouns, adverbs, and compound tenses fluently.
HSK 1 | verb | to exchange goods for money
HSK 4 | adjective / noun | polite, showing good manners
HSK 5 | verb / adjective | to spread widely among the public; universally available or known
〜ために: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 3 | noun / adjective | chā: difference, gap; chà: poor, lacking, not up to standard
The final two Sindarin mutations: liquid mutation (after certain prefixes ending in -r) and stop mutation (after d/t) — with scholarly discussion on their limited attestation.
N1 grammar pattern 〜に反して/〜に反する: expressing that something is contrary to an expectation, rule, principle, or agreement. Legal and formal register.
Deep-dive into Sindarin nature vocabulary: sky, weather, water, plants, trees, and animals — with all attested forms and place name evidence.
HSK 4 | noun | physical force pressing on something, or mental/social stress
HSK 2 | noun | sound, voice, noise
HSK 5 | verb/noun | to break apart into separate pieces; division or schism
Mastering the full 学术语体 register for academic writing and scholarly discourse in Chinese.
HSK 5 | noun | cybersecurity, protection of computer networks and data
Engage with Chinese healthcare policy discourse, mastering vocabulary and grammar for discussing public health systems, medical reform, and health guarantees.
HSK 3 | verb | to send, to emit, to issue, to develop — multi-meaning character
HSK 2 | pronoun | reflexive pronoun referring back to the subject
書 (write): 10 strokes, JLPT N5. On: ショ. Kun: か-く.
JLPT N4 kanji 風 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 1 | noun | tea
HSK 5 | adjective | open to diversity; tolerant of different people or views
The task is to compute the Hamming distance between two strings of equal length. In practical terms, you are comparing two sequences of letters and counting how many positions contain different characters. Each mismatch contributes exactly one to the distance.
Express regret, missed opportunities, and relief using 要是...就好了, 早知道...就, 可惜, and related patterns at B1 level.
HSK 3 | adverb / adjective | directly, straight; without an intermediary or detour
HSK 1 | noun | Chinese language, Mandarin Chinese
Essential Japanese vocabulary for arts, customs, traditions, festivals, Japanese culture. N1 level reference with readings, romaji, and examples.
国 (country): 8 strokes, JLPT N5. On: コク. Kun: くに.
出 (exit, appear): 5 strokes, JLPT N5. On: シュツ、スイ. Kun: で-る、だ-す.
HSK 5 | verb / noun | to caution or alert; a cautionary sign or message
HSK 4 | noun/verb | education; to educate
入 (enter): 2 strokes, JLPT N5. On: ニュウ. Kun: い-る、い-れる、はい-る.
Engage with Chinese discourse on the ethical dimensions of technology, including AI ethics, privacy, data security, and algorithmic bias.
We are given a rooted tree with n nodes, each node assigned a positive integer value. Node 1 is the root. Queries ask either to find the deepest ancestor of a node v whose value shares a non-trivial greatest common divisor with v, or to update the value of a node.
HSK 4 | adjective | consistently performing well and worthy of trust
HSK 5 | noun | the livelihood and basic needs of the common people
The problem presents a keyboard with three rows of characters arranged exactly like a standard QWERTY layout. Mole is typing messages on this keyboard, but he accidentally shifted his hands either one key to the left or one key to the right.
HSK 3 | adjective | strange, odd, weird — outside the normal or expected
月 (moon, month): 4 strokes, JLPT N5. On: ゲツ、ガツ. Kun: つき.
HSK 3 | conjunction | otherwise, or else — states what will happen if the condition is not met
HSK 3 | verb | to know well through inquiry or experience; deeper than 知道
HSK 4 | verb / noun | to evaluate; to assess; an assessment
何 (what): 7 strokes, JLPT N5. On: カ. Kun: なに、なん.
Sindarin nouns: i-affection plural formation, the definite article, direct object marking, and the genitive construction.
HSK 5 | verb/noun | to express agreement or approval
HSK 5 | verb | to refute forcefully; to rebuke
HSK 4 | noun | an official course of action adopted by a government or organization
ずっと (ずっと / zutto): all along, continuously. N4 level Japanese vocabulary.
HSK 5 | noun | reputation, prestige, good name
Learn to derive nouns, verbs, adjectives, and adverbs from a single root, and to stack multiple affixes for nuanced vocabulary.
Narrate past events fluently using sequence words, duration expressions, frequency adverbs, and past-time framing.
HSK 3 | verb | to become something through change or development
A comprehensive C1 mastery audit, 300 essential advanced vocabulary items, precise C1 can-do statements, and a structured roadmap for reaching C2.
HSK 4 | verb | holding something valuable as one's own
HSK 2 | adverb / adjective | intensifier meaning 'especially' or describing something as unique
HSK 4 | verb / noun | expressing official or personal acceptance and endorsement
六 (six): 4 strokes, JLPT N5. On: ロク. Kun: む、む-つ、むっ-つ、むい.
HSK 5 | noun | the financial gain remaining after subtracting all costs from revenue
HSK 2 | conjunction | then, after that (sequence connector)
HSK 5 | noun | a topic on an agenda; an issue to be discussed or debated
話 (talk, story): 13 strokes, JLPT N5. On: ワ. Kun: はな-す、はなし.
本 (book, origin): 5 strokes, JLPT N5. On: ホン. Kun: もと.
来 (rai/ku-ru): come. JLPT N5 kanji.
HSK 5 | noun | an organized collection of structured data stored and accessed electronically
HSK 4 | adjective / adverb | doing something with care and attention to detail
HSK 1 | measure word | classifier for bottled items
HSK 5 | adjective | firm and well-grounded; not reckless or volatile
HSK 3 | adjective / verb | clear, distinct; to be clear about something
HSK 5 | adjective | existing below the surface; not yet apparent; potential
Understand and produce pivotal (兼语) sentences with 让, 叫, 请, 使, and related verbs at B1 level.
集める (あつめる / atsumeru): to collect, gather. N4 level Japanese vocabulary.
We are given two arrays of integers, x and y, each of length n. The array x is the starting state, and y is the desired target state. The only operation allowed is x[i] ^= x[j], which replaces x[i] with the bitwise XOR of x[i] and x[j].
The game is played on a grid formed by n horizontal sticks and m vertical sticks. Every horizontal stick intersects every vertical stick, creating n m intersection points. On a turn, a player chooses one remaining intersection point.
N1 grammar: classical auxiliary べし expressing obligation/certainty, and its negative form べからず (must not). Appears in laws, school rules, mottos, and literary contexts.
Regular and irregular comparative and superlative adjectives, formation of comparative and superlative adverbs, and ablative of comparison vs. quam.
水 (water): 4 strokes, JLPT N5. On: スイ. Kun: みず.
HSK 4 | adjective / noun | enthusiastic, warm-hearted
HSK 5 | noun/verb | insight gained through experience; to realize through reflection
HSK 4 | noun | the organized body of people sharing a culture and institutions
HSK 5 | noun | an overflow of water onto normally dry land; a flood
HSK 1 | adjective, adverb | fast, quick, quickly, soon
HSK 5 | noun | the foundational base upon which something is built
十 (ten): 2 strokes, JLPT N5. On: ジュウ、ジッ. Kun: とお、と.
Consolidate the complete map of Esperanto mastery: competency inventory, grammar reference, vocabulary domains, the denaskulo question, and the path beyond C2.
HSK 1 | adjective | expensive, costly, precious
HSK 4 | adjective | able to adapt easily to different conditions or requirements
Complete JLPT N5 vocabulary list with ~800 words organized by category: nouns, verbs (with て-form and た-form), adjectives, adverbs, particles, question words, and expressions.
HSK 6 | n/adj | a system of governance by the people; the quality of being open, participatory, or egalitarian
Express purpose, intention, and preventive actions using 为了, 以便, 以免, and related patterns at B1 level.
HSK 4 | verb/noun | to demonstrate truth with evidence; a document certifying a fact
HSK 3 | verb | to take, to fetch, to retrieve, to obtain
HSK 5 | noun | an unfair preconceived opinion not based on reason or experience
HSK 1 | phrase | negates identity or equation using the verb 是
HSK 6 | verb | to dissect and analyze in detail; to examine the inner workings of
HSK 4 | grammar pattern | expressing a logical conclusion based on an accepted premise
HSK 4 | noun | friend; companion
HSK 4 | verb/noun | to draw together key points into a conclusion
Master 〜に従い and 〜に従って to express compliance with rules, instructions, or gradual change in formal Japanese.
HSK 3 | verb / noun | to criticize, point out faults; criticism
百 (hyaku/momo): hundred. JLPT N5 kanji.
HSK 3 | noun | sugar; candy, sweets
HSK 4 | noun | opportunity; favorable chance
HSK 3 | adverb | indicating original state or a surprising discovery
HSK 3 | verb | to print (documents, photos) — used for printers and printing services
HSK 4 | adjective | showing energy, initiative, and a constructive attitude
Learn the formal conjunctive particle 〜において (in, at — domain marker) and its attributive form 〜における, essential for N3 reading and formal written Japanese.
HSK 3 | noun / verb | competition, match, contest; to compete
HSK 5 | verb/noun | fundamental shift in state or direction
校 (kou/): school. JLPT N5 kanji.
HSK 4 | adjective / verb | thorough and deep; to go deep into something
N1 grammar pattern 〜とばかりに: describing a manner or attitude 'as if to say ~, with the air of ~' — the subject's action speaks louder than words.
HSK 3 | verb | to agree, to consent, to approve
Learn 〜をはじめとして and 〜をはじめとする to introduce a representative example from a larger group in formal Japanese.
HSK 3 | noun | oil — cooking oil, fuel oil, or any oily substance
N1 grammar pattern 〜が最後: expressing that once something happens it cannot be undone or will inevitably lead to a (usually negative) outcome — 'once you do ~, that's it.'
We are asked to work with a special class of integers defined purely by their digit structure. A number is called “wavy” if, whenever you look at any digit that is not at the boundary, that digit must sit either strictly above both its neighbors or strictly below both…
Reference guide to the 214 classical radicals (bushu) used in Japanese kanji — with meanings, stroke counts, variant forms, and key example kanji.
JLPT N4 kanji 寒 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 2 | verb/noun | to assist someone; the act of assistance
Mastering professional Chinese translation: register equivalence, cultural default, domestication and foreignization.
The inverted pyramid, news leads, and the register conventions of Chinese journalism — reading and producing newswriting at C1 level
HSK 5 | verb/noun | to become extinct; total elimination of a species or group
月 (getsu/gatsu/tsuki): moon. JLPT N5 kanji.
HSK 6 | verb | to bring together disparate elements into a unified whole
HSK 5 | verb/noun | the operation and management of a business or service
Advanced theory and practice of translation into and from Esperanto — literary, technical, and cultural dimensions, with the translator's full toolkit.
HSK 5 | noun/adjective | the decisive or most critical element
HSK 3 | adj/noun | polite behavior or manners
HSK 4 | adjective | not direct; operating through an intermediate step
HSK 1 | measure word | classifier for bound volumes
Guidelines for composing grammatically correct Neo-Sindarin: the scholar's toolkit, morphotactic constraints, how to coin new words, and how to evaluate others' Neo-Sindarin.
〜と言う: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 2 | conjunction | presents alternatives in a question; contrast with 或者 for statements
Master both meanings of 〜次第: 'as soon as ~' (formal temporal) and 'depends on ~' (conditional) — essential for N3 business communication.
書 (sho/ka-ku): write. JLPT N5 kanji.
We are given a list of people, each with a first name and a last name. For each person, we have the freedom to choose either their first or last name as a handle.
HSK 3 | verb/noun | systematic investigation or study of a subject
HSK 5 | noun | surgical operation; medical procedure involving cutting
HSK 4 | verb/noun | to publicize, to promote; publicity, propaganda
A house of cards is built from floors. If a floor contains k rooms, then each room uses two leaning cards, so the rooms themselves consume 2k cards. Between adjacent rooms and above the outermost rooms there is a ceiling made of horizontal cards.
HSK 4 | adjective/noun | relating to real facts or situations rather than theory or ideals
HSK 3 | verb | to understand through listening, to catch what is said
HSK 4 | noun | client, customer
We are given an array of n non-negative integers. A group is any non-empty subset of array indices. For a chosen group, we take the bitwise AND of all values at those indices. The task is to count how many non-empty groups have bitwise AND equal to 0.
HSK 3 | verb | to review or revise material already studied, especially before an exam
HSK 5 | noun | the state of balancing carbon emissions with carbon removal
右 (right): 5 strokes, JLPT N5. On: ウ、ユウ. Kun: みぎ.
We are given an integer array and a target height. The goal is to transform every element of the array into the same final value using a sequence of operations. Each operation picks a segment of indices and increases every value in that segment by exactly one.
HSK 5 | verb / noun | to express; to perform; performance; behavior
HSK 3 | verb | to care about something emotionally; to mind or be bothered by something
HSK 5 | noun/adjective | the central, most essential part of something
前 (zen/mae): before. JLPT N5 kanji.
Learn to talk about getting around using 怎么去 and 坐 + vehicle, and navigate common travel situations.
Learn how to use 〜をはじめ (wo hajime) to express 'starting with' or 'including' when listing a representative example of a broader group. Includes structure, nuance, examples, and comparisons.
HSK 5 | verb/noun | to guide and direct an organisation or group; a leader or the act of leadership
HSK 4 | noun / adjective | the modern era or something that is contemporary
Learn school vocabulary, how to say where an activity takes place using 在 + place + verb, and common 学习 collocations.
HSK 3 | verb + directional complement | to carry something out of an enclosed space, away from the speaker
Master the vocabulary and grammar of Chinese legal and political discourse, enabling engagement with official documents, legal texts, and political commentary.
HSK 4 | verb / noun | asking or specifying that something must be done or met
HSK 6 | verb/adj | relating to dialectics; to examine through dialectical reasoning
HSK 5 | n/v | a symbol or symbolic representation; the act of symbolizing
HSK 4 | verb | to know (a fact or piece of information)
HSK 3 | adjective | cute, adorable, or lovely
右 (u/yuu/migi): right. JLPT N5 kanji.
HSK 3 | verb | to create or found something new
HSK 2 | measure word | counts songs and poems as complete compositions
HSK 5 | adjective | severe; grim; daunting; stern
HSK 2 | adverb | again (future repetition), then, one more time
There are $m$ complete decks, each containing the same $n$ card values. After mixing all decks together, there are $mn$ physical cards in total, and every value appears exactly $m$ times.
HSK 4 | noun | a mobile phone with advanced computing capabilities
HSK 4 | verb | to take on responsibility or a burden
HSK 4 | noun | method, way, manner, mode
HSK 3 | verb | to prepare, get ready, or plan to do something
高 (kou/taka-i): tall. JLPT N5 kanji.
HSK 4 | adjective | something that must exist or be done
HSK 1 | verb | to play, to have fun, to hang out
HSK 4 | adverb / conjunction | introducing an example or illustration
HSK 4 | adjective/adverb | taking action on one's own initiative
Consolidate all A1+ grammar patterns and the most important HSK 2 vocabulary through integrated exercises and review.
夜 (ya/yoru): night. JLPT N5 kanji.
女 (woman, female): 3 strokes, JLPT N5. On: ジョ、ニョ. Kun: おんな、め.
HSK 1 | noun | Chinese language, written Chinese
HSK 5 | noun | performance; capability; functional properties of a product or system
HSK 4 | verb/noun | to arrange people or resources into a structured whole; a group with a shared purpose
終わる (おわる / owaru): to finish, end. N4 level Japanese vocabulary.
HSK 1 | noun | person, people, human being
HSK 3 | noun | the degree of skill or quality in something
Learn to build complex sentences using subordinating conjunctions: ke, ĉar, kvankam, se, kiam, por ke, and indirect questions.
HSK 6 | verb | to hold back or suppress the spread or growth of something
〜せる/〜させる: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 5 | verb / noun | to comment or critique; a commentary or review
HSK 1 | adjective | difficult, hard. Antonym: 容易 (róngyi)
HSK 5 | verb/noun | to uplift and motivate someone through an energising or moving influence
HSK 1 | noun | dog
青 (sei/shou/ao): blue. JLPT N5 kanji.
HSK 5 | conjunction | to the extent that; so much so that; resulting in
HSK 5 | verb | to show the way or help someone develop in a certain direction
HSK 5 | verb/noun | to motivate or inspire someone to greater effort; motivation
HSK 3 | noun | areas outside cities where farming is the main activity
Learn the vocabulary, structures, and conventions of formal written Esperanto for letters, reports, and official contexts.
Master all four tenses of the Latin subjunctive and learn to form purpose clauses with ut and ne.
HSK 3 | adverb pattern | describes a quality increasing progressively over time
HSK 1 | pronoun | third-person plural pronoun (masculine or mixed)
HSK 3 | adverb | refers to a subset within a previously mentioned group or set
I’m sorry, but I can’t reliably produce a correct full editorial and solution for this 3200-rated problem from the information currently available to me.
西 (セイ・サイ/にし): west. JLPT N5 essential kanji.
空 (kuu/sora): sky. JLPT N5 kanji.
HSK 3 | measure word | a unit of time lasting sixty seconds
HSK 4 | verb/noun | to have a clear understanding of something through learning or inquiry
Mastering nested relative clauses and long noun phrase chains — the syntactic backbone of advanced Chinese prose
HSK 3 | noun | soup, broth, hot liquid dish
Complete Japanese verb conjugation reference: all three verb groups, all forms (polite/plain/te/ta/potential/passive/causative/conditional/volitional) with examples.
人 (person): 2 strokes, JLPT N5. On: ジン、ニン. Kun: ひと.
HSK 3 | verb | to finish, accomplish, or fulfill a task or goal
N1 grammar pattern 〜べく: a formal literary purposive expressing 'in order to ~, for the purpose of ~' — the written formal equivalent of ために.
HSK 2 | verb + result complement | indicates searching resulted in actually finding something
N1 grammar pattern 〜ならでは: expressing uniqueness or exclusivity — 'only possible with ~, uniquely characteristic of ~, something only ~ can offer.'
Use the experiential aspect marker 过 and contrast 以前 with 以后 to discuss past experiences and changes.
We need to print an odd-sized square grid that contains a diamond shape. Every cell belonging to the diamond is represented by the character D, and every other cell is represented by . The input contains a single odd integer n.
Business negotiation language, professional etiquette, and the formal register of commercial agreements in Chinese at C1 level
五 (five): 4 strokes, JLPT N5. On: ゴ. Kun: いつ、いつ-つ.
HSK 4 | verb | to agree, to consent, to approve
HSK 5 | verb | to construct; to build (a system, framework, or relationship)
Codeforces 465B: Inbox (100500)
HSK 5 | verb / noun | to reflect; to mirror; to convey feedback
Appleman starts with an empty string. In one second he may append any substring of t to the end of the string he is building. For a fixed target string s, Appleman chooses an optimal sequence of substrings and finishes s in the minimum possible number of seconds.
食 (shoku/jiki/ta-beru): eat. JLPT N5 kanji.
HSK 1 | noun | cooked rice, steamed rice
HSK 4 | adjective / adverb | direct, without intermediary; straightforwardly
HSK 4 | verb | to make something larger in scope or size
HSK 3 | adjective | positive, proactive, enthusiastic, optimistic
HSK 5 | verb/noun | biological or natural evolution and development
長 (long, chief): 8 strokes, JLPT N5. On: チョウ. Kun: なが-い、おさ.
HSK 1 | noun | weather
HSK 3 | noun | manufactured goods or output of a process
Japanese family vocabulary: parents, siblings, relatives, honorific vs. plain forms, and family-related expressions.
HSK 3 | verb | to discount, to sell at a reduced price
HSK 1 | verb | to close, to shut, to turn off
HSK 5 | adjective | restricted in quantity, scope, or degree; not unlimited
HSK 3 | noun / verb | summary, conclusion; to summarize, to wrap up
HSK 1 | verb | to jump, to leap, to dance
HSK 3 | adjective/adverb | done without unnecessary delay when needed
Form and use the imperfect (ongoing past) and future tenses for all four conjugations and esse.
必要な (ひつような / hitsuyou na): necessary, needed. N4 level Japanese vocabulary.
JLPT N4 kanji 事 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
雨 (rain): 8 strokes, JLPT N5. On: ウ. Kun: あめ、あま.
HSK 3 | adverb | after all — points to an underlying truth or fundamental reason
HSK 4 | noun | a person who offers help freely without pay
HSK 5 | conjunction | even if, even though, no matter (strong concession)
白 (haku/byaku/shiro): white. JLPT N5 kanji.
Learn Esperanto numbers from zero to a million, including ordinals, fractions, and arithmetic expressions.
HSK 3 | verb / noun | to demonstrate something is true; a document serving as evidence
HSK 1 | conjunction | introduces a contrast or exception
HSK 5 | noun | the capacity to understand and make sense of information
はっきり (はっきり / hakkiri): clearly. N4 level Japanese vocabulary.
HSK 5 | noun | cloud computing; internet-based computing services
HSK 4 | noun | an observable fact or event, especially one requiring explanation
Learn 〜にもかかわらず to express contrast — an unexpected or contradictory result despite a real obstacle or circumstance.
HSK 1 | adverb / noun phrase | indicates a small amount or degree
The Sindarin -as/-ias suffix for forming abstract nouns: converting adjectives and nouns to concepts — with all attested examples and Neo-Sindarin applications.
HSK 5 | noun | innate ability or formal qualification for a role
HSK 1 | noun | one's younger sister
四 (four): 5 strokes, JLPT N5. On: シ. Kun: よ、よ-つ、よっ-つ、よん.
HSK 4 | noun | an assigned piece of work or responsibility
HSK 5 | noun | ethics; ethical principles; moral philosophy
HSK 5 | noun | the rate at which one currency is exchanged for another
Essential Japanese vocabulary for computers, internet, devices, apps, digital life. N1 level reference with readings, romaji, and examples.
HSK 3 | noun | the nose; the organ of smell on the face
HSK 1 | adjective | handsome, cool, good-looking (usually men)
HSK 2 | adverb/adjective | really, truly, genuinely; real, authentic
HSK 5 | noun phrase | sustainable development
HSK 5 | verb / noun | to call into question; to express doubt
HSK 3 | noun | an official public holiday or a day free from work
HSK 1 | verb | to enter, to go into, to come in
HSK 4 | adjective | rude, impolite, rough in manner
George wants to maximize his earnings by choosing segments of work from a list of tasks, each with a given profit. The tasks are arranged sequentially in an array p of length n. He must select exactly k non-overlapping segments, each of length m.
HSK 5 | noun | skill; practical ability acquired through training or practice
HSK 3 | grammar pattern | 把 sentence with result complement 完 indicating complete disposal
HSK 2 | adverb | exclamation intensifier used in rhetorical or exclamatory sentences
Additional Sindarin derivational suffixes: -ron/-ren (agent/keeper), -dir (man/person), -el (maiden), -on (augmentative), -orn (tree) — with name analysis applications.
Learn 〜につれて (as ~ changes, proportionally) — expressing gradual co-change where two things change together.
HSK 1 | greeting phrase | standard response to 谢谢
HSK 4 | noun/verb | a planned estimate of income and expenditure
Express comparisons of superiority, inferiority, equality, and progressive change using pli, malplej, same kiel, and related structures.
HSK 4 | verb | deliberating together to reach a decision or understanding
Core list of ~200 most useful attested Sindarin (S.) words — the essential vocabulary every learner needs, drawn from Tolkien's own texts.
The task is to construct n sets of four distinct positive integers, where each set has a certain "rank" k. The rank condition requires that the greatest common divisor (GCD) of any two numbers within the set equals exactly k.
HSK 5 | conjunction | even; ranging from ... to; and even
HSK 5 | noun / verb | the process of converting to digital form or integrating digital technology
HSK 3 | adverb | mutually, each other — describes an action done by both parties toward each other
Codeforces 465A: inc ARG
A comprehensive guide to UEA, TEJO, Pasporta Servo, Esperanto's international institutional life, official exams, and the language of Esperanto organizations.
HSK 5 | noun/adjective | the principles governing right and wrong conduct
HSK 3 | verb / noun | keeping someone or something safe from harm
HSK 5 | verb/noun | official supervision and control by a regulatory authority
Learn how to use 〜に先立って (ni sakidatte) to express 'prior to' or 'before' when an action precedes an important event. Includes structure, nuance, examples, and comparisons.
HSK 4 | conjunction / adverb | in short; in summary; in a word
Master the formal grammar patterns used in legal documents, academic writing, and official correspondence, including 〜ものとする, 〜に即して, 〜に反して, 〜をもって, and 〜いかんによらず.
Complete HSK 6 vocabulary reference: approximately 1,800 new words at C1 level, organized by semantic category with pinyin, English, part of speech, and usage notes.
HSK 2 | adjective | describes physical or emotional comfort and well-being
HSK 4 | noun | a set of instructions for preparing a dish
HSK 5 | noun phrase | the portion of a market controlled by a company or product
HSK 4 | adjective | demanding exact compliance with rules or standards
HSK 2 | adjective/preposition | correct or right; also introduces the target of an action
HSK 3 | adverb | referring to an original state or natural expectation
HSK 3 | verb | to restore something broken to working order
HSK 5 | noun | an institutional system that holds officials or organisations responsible for their conduct
HSK 3 | verb | to register, to sign up, to enroll
天 (ten/ame): heaven. JLPT N5 kanji.
Learn how to use 〜ばかりか (bakari ka) to express 'not only ~ but even ~' with an emphasis on escalating or surprising addition. Includes structure, nuance, examples, and comparisons.
HSK 5 | noun | bond; debenture; fixed-income debt security
HSK 3 | noun | weekend, Saturday and Sunday
HSK 5 | phrase / conjunction | on the other hand, from another angle
HSK 1 | measure word | classifier for garments and affairs
HSK 6 | verb | to examine closely in order to distinguish subtle differences
HSK 4 | verb / noun | to pay attention or be mindful of something
Detailed Mandarin Chinese learning roadmap from complete beginner to C2: four phases, milestones, plateau strategies, and immersion techniques.
HSK 4 | verb | to maintain a position or action despite difficulty
Passive voice endings for all conjugations in present, imperfect, and future indicative; agent vs. means constructions; and a preview of the perfect passive.
N1 grammar pattern 〜まじき: a literary/formal adjectival form meaning 'unworthy of, should not, inexcusable for' — expressing strong moral condemnation.
Complete Sindarin verb conjugation for present tense: aorist (simple) and continuative (ongoing) for both A-stem and primary verb classes, all 6 persons.
HSK 4 | noun | the nutrients and nourishment that the body needs
Sindarin vocabulary for food, drink, and dining — with attested and Neo-Sindarin words, plus basic sentence construction practice.
Deploying chengyu in formal writing — historical origins, structural analysis, and the pragmatics of allusion in C1 Chinese
HSK 1 | noun | the middle of the day, around 12 p.m.
Complete HSK 1 vocabulary list: 299 words with Chinese, pinyin, English, part of speech, and example sentences. Organized by category.
Complete reference of the 1,026 Kyoiku kanji (教育漢字) taught in Japanese elementary school grades 1–6, with stroke counts, key readings, and meanings.
HSK 4 | noun | the background or context of a situation
HSK 7 | verb | formal medical term for applying therapeutic intervention to a patient
HSK 4 | noun | colleague, coworker
N1 grammar pattern 〜ともなると/〜ともなれば: when it comes to reaching the level/stage/position of X, certain things naturally follow. Formal register.
HSK 4 | verb | to cause something to move forward or develop
HSK 3 | adjective | describing something as interesting, fun, or engaging
Codeforces 464E: The Classic Problem
HSK 4 | adverb | describing slow, incremental change over time
Learn how to use 〜というより (to iu yori) to replace or correct a description with a more accurate one — 'rather than saying ~, it's more ~.' Includes structure, nuance, examples, and comparisons.
HSK 5 | verb | measurable decrease or downward movement
HSK 3 | verb / noun | to suggest, recommend; a suggestion or recommendation
HSK 1 | noun | noodles
HSK 5 | conjunction | used to add supplementary information
〜てくれる: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 6 | verb | to interpret and explain the meaning of a text, concept, or phenomenon
HSK 4 | adjective | having the taste of salt
Complete treatment of Sindarin na- (to be): present nâ, past nant, future natha, imperative no, and predicative adjective construction.
HSK 1 | measure word | general classifier for people and objects
HSK 5 | verb | to release; to issue officially; to publish
HSK 5 | verb/noun | the process of manufacturing or creating goods
Consolidate B1 grammar patterns and HSK 4-specific vocabulary through integrated review exercises and extended practice.
Esperanto vocabulary for time, days of the week, months, seasons, and time expressions.
HSK 5 | noun | imagination; the power or capacity to imagine
JLPT N4 kanji 書 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
We have apples numbered from 1 to n. We want to create as many disjoint pairs as possible such that the two numbers in each pair have greatest common divisor greater than 1. Each apple can appear in at most one pair.
Express a full range of emotions in Esperanto, describe how you feel, use -iĝ- for emotional changes, and comfort others in conversation.
HSK 3 | adjective / adverb | acting on one's own initiative without being asked
We are asked to maximize the minimum height of flowers after a limited number of days of watering. Each flower has an initial height, and on any day we can water a contiguous segment of exactly w flowers, increasing each by one.
HSK 4 | noun | payment made electronically over the internet
HSK 4 | verb | to bring up an idea, question, or suggestion for consideration
決める (きめる / kimeru): to decide. N4 level Japanese vocabulary.
HSK 5 | verb/noun | to discuss and reach agreement through mutual consultation
HSK 2 | measure word / adjective | counts items that come in pairs
HSK 4 | noun | the purpose, objective, or aim behind an action
Sindarin words for time: days, seasons, months, years, and the Elvish calendar (Kings' Reckoning and the Stewards' Reckoning).
HSK 5 | verb / noun | to plan and create; a plan or design
HSK 6 | n/v | established standard or norm; to bring into conformity with a standard
Essential Sindarin greetings and phrases directly from Tolkien's texts: Mae govannen, Hannon le, Navaer, Le abdollen, and more.
HSK 3 | verb | to obtain or achieve something through effort
HSK 4 | noun | one's overall pattern of eating and drinking
All Sindarin dictionaries compared — Parf Edhellen, Eldamo, Hiswelókë — with guidance on when to use each and how to interpret source period labels.
HSK 3 | adjective | serious, earnest, conscientious, diligent — describes a careful and dedicated attitude
HSK 4 | verb | to successfully deal with a difficulty or challenge
HSK 5 | verb / noun | to cooperate, to collaborate; cooperation, collaboration
HSK 1 | adjective | wrong, incorrect, mistaken
HSK 5 | verb | to present; to emerge; to take on a certain appearance
HSK 5 | noun | observable occurrence or phenomenon in nature or society
The graph consists of cities connected by roads. The condition that between any two connected cities there is exactly one simple path means every connected component is a tree. Since the graph may be disconnected, the whole graph is a forest.
HSK 5 | adjective / adverb | existing or occurring generally among all people or in most places
HSK 1 | verb | to help, to assist
HSK 3 | adjective | feeling cheerful and satisfied
空 (sky, empty): 8 strokes, JLPT N5. On: クウ. Kun: そら、あ-く、から.
語 (go/kata-ru): language. JLPT N5 kanji.
HSK 5 | noun | the specific purpose or role something is designed or able to perform
HSK 5 | noun | computer software or an application
HSK 3 | verb | to come into view or to happen for the first time
HSK 4 | noun | a physical or figurative path or bearing
明 (mei/myou/aka-rui): bright. JLPT N5 kanji.
HSK 4 | adjective | effective or legally valid
HSK 5 | conjunction | thus; thereby; consequently; so as to
HSK 3 | conjunction | as long as, provided that — states the minimum condition for a result
We are asked to build a staircase-like structure made from two types of blocks, red and green. The structure has some number of levels, and the levels form a strict decreasing sequence in size: if the top level has size n, then the next has n-1, then n-2, and so on down to 1.
HSK 3 | verb + result complement | to look at something and perceive it with full clarity
HSK 3 | noun | spoken language, as opposed to written language
Japanese nature vocabulary: weather, animals, plants, landscape, environment, and natural phenomena.
HSK 1 | greeting phrase | standard response to 你好吗
HSK 3 | verb | to receive, to collect, to put away
HSK 4 | noun | rule; regulation; norm
HSK 4 | verb / adjective | to move or shift position; mobile or portable
HSK 5 | noun/adjective | the state of being equal, especially in rights and status
HSK 4 | noun | a raw material or component used in cooking
Advanced Sindarin plural patterns: monosyllables, diphthong shifts, vowel intrusion, class plurals (-ath, -rim, -hoth), and the definite article with plurals.
HSK 3 | verb/noun | to plan or organize something
Master the nuances of formal Japanese conjunctions used to express basis, compliance, and alignment in professional and academic contexts.
HSK 2 | verb/noun | to hope for something uncertain; hope as a noun
HSK 5 | verb | to bring separate elements together into a unified, coherent whole
HSK 5 | noun/adjective | spirit, mind, essence; energetic, lively
All major Sindarin irregular verbs: anna- (give), car- (do/make), na- (be), tol- (come), men- (go) — with complete paradigms for all tenses.
HSK 1 | noun | coffee
HSK 5 | verb/noun | to return to health after illness or injury; the process of recovery
HSK 3 | verb | to include, to comprise
HSK 2 | adverb | superlative marker placed before adjectives and verbs
HSK 2 | time word | refers to a moment in the very recent past; contrast with 刚
HSK 5 | noun | financial returns or gains from an activity or investment
HSK 5 | noun/adjective | system; systematic, comprehensive
JLPT N4 kanji 行 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 2 | adjective/verb | to be anxious, worried, or in a rush
HSK 3 | adverb | confirms or affirms a fact as genuine and undeniable
HSK 2 | noun | a matter, an affair, something to deal with
HSK 2 | verb + result complement | indicates the action of looking results in actually seeing
HSK 4 | verb | to willingly take or agree to something offered
HSK 5 | verb / noun | to research and develop (R&D)
HSK 4 | noun | the sensation perceived by the tongue when eating or drinking
Complete N2 vocabulary reference: academic, political, scientific, medical, and social vocabulary. ~2,250 new words beyond N3, organized by theme.
Complete JLPT N5 exam prep: format breakdown, 20 vocabulary practice questions, 20 grammar practice questions, 2 reading passages, listening strategies, 30-day study plan, mnemonics, and common traps.
図 (zu/to/haka-ru): map. JLPT N5 kanji.
Six graded Sindarin reading passages from beginner to advanced: all fully glossed with word-by-word translation, grammatical notes, and cultural commentary.
HSK 4 | verb | the active process of acquiring knowledge or skills
左 (left): 5 strokes, JLPT N5. On: サ. Kun: ひだり.
HSK 3 | adjective | correct, right, accurate — conforming to fact or standard
HSK 4 | verb/noun | to exchange views on a topic in order to reach understanding or agreement
JLPT N4 kanji 勉 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 1 | adverb | indicates future repetition of an action
HSK 5 | adjective | considering only one aspect; partial; one-sided
HSK 3 | verb | to board or travel by a vehicle
HSK 4 | noun | a widespread outbreak of infectious disease
HSK 3 | verb | to consider, to think over, to take into account
地 (chi/ji/): ground. JLPT N5 kanji.
HSK 3 | noun | tree; also used in compounds for planting or establishing
HSK 5 | noun | duty; obligation; obligatory service
HSK 1 | verb | to walk, to go, to leave
HSK 4 | adjective/noun | something that is hard to do or a challenging situation
All Sindarin online courses — from beginner to advanced — with detailed descriptions, lesson counts, and what each covers.
We are given a sequence of integers a1, a2, ..., an with values between 1 and 30, and we need to construct another sequence b1, b2, ..., bn of positive integers such that every pair in b is coprime.
白 (white): 5 strokes, JLPT N5. On: ハク、ビャク. Kun: しろ、しら、しろ-い.
HSK 5 | noun/adjective | freedom, liberty; free, unrestricted
HSK 4 | verb / noun | to invite; an invitation
Express nuanced doubt and negation using 没有...那么, 不见得, 未必, and related patterns at B1 level.
家 (house, family): 10 strokes, JLPT N5. On: カ、ケ. Kun: いえ、や.
HSK 3 | verb | to formally request a day off from work or school
HSK 2 | preposition | marks the starting point of movement, time, or a sequence
Complete hiragana chart with all 46 base characters, 25 dakuten/handakuten variants, 33 combination characters, romaji, stroke counts, and learning mnemonics.
The language of Chinese economic analysis — 就...而言 in economic framing, macroeconomic and microeconomic vocabulary, and the discourse of policy economics
HSK 5 | noun/adjective | democracy; democratic governance
Sindarin collective suffixes: -ath (total collective), -rim (people/kindred), -hoth (hostile host), -gwaith (company) — with all attested examples from Tolkien.
HSK 6 | verb | to deliberately construct or build up a system, framework, or concept
Learn Esperanto idiomatic expressions, proverbs, fixed collocations, and discourse idioms that native-level speakers use in authentic communication.
We are given a sequence of pylons arranged in a line, starting from position 0 up to position n. The first pylon is fixed at height 0, and every other pylon has a given height.
HSK 4 | noun | a person who buys goods or services
あまり (あまり / amari): not very (+ negative). N4 level Japanese vocabulary.
Language for navigating cultural difference — expressions for cultural contrast, conflict, fusion, and the pragmatics of intercultural interaction at C1 level
送る (おくる / okuru): to send. N4 level Japanese vocabulary.
〜てしまう: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 3 | noun | the deeper significance or importance of something
HSK 5 | noun | honesty and trustworthiness; acting in good faith
Produce and evaluate scholarly work in Esperanto using correct academic register, citation practice, and the resources of the Esperanto research community.
HSK 4 | adverb / noun | referring to a specific past moment
男 (man, male): 7 strokes, JLPT N5. On: ダン、ナン. Kun: おとこ.
東 (トウ/ひがし): east. JLPT N5 essential kanji.
HSK 4 | noun | the range or scope of something
HSK 4 | conjunction | introducing a contrast or exception
Learn vocabulary for academic and professional life, and use 为了...而 to express purpose-driven actions and collocations around 专业 and career vocabulary.
HSK 3 | adverb | intensifier meaning to a very high degree
HSK 5 | verb | to put into practice; to make concrete
HSK 1 | adjective | delicious, tastes good
JLPT N4 kanji 図 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 4 | noun | reference materials, data, or information sources
N1 grammar pattern 〜に即して/〜に即した: applying a principle or standard directly to specific circumstances. Formal register, used in legal, academic, and policy contexts.
HSK 5 | noun | a physical wound or a deep psychological injury
Master formal Japanese reading strategies for newspapers, editorials, and academic text — grammar patterns, annotation techniques, and N2 reading section strategy.
Explore the main uses of the dative (indirect object, possession, reference) and ablative (means, manner, accompaniment, time, prepositions).
HSK 5 | adjective/adverb | not influenced by personal feelings; based on observable facts
The bus layout is fixed and already drawn for us. There are 34 passenger seats in total. Passengers always occupy seats in a very specific order: they start from the last row, fill it from left to right, then move upward row by row.
HSK 1 | pronoun | second-person singular pronoun (formal/polite)
Engage with Chinese scientific discourse, technology journalism, and research communication using specialized vocabulary and evidence-based grammar patterns.
JLPT N4 kanji 者 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 5 | adverb/conjunction | used to introduce an extreme or surprising case
HSK 5 | verb | to launch; to introduce; to put forward
Develop critical reading skills for Chinese media, mastering the vocabulary and grammar of citation, data interpretation, and source evaluation.
HSK 3 | noun | a person's name or the name of something
HSK 2 | adjective | the color black; also dark (absence of light)
Master the essential language for shopping, asking prices, and using Japanese counters to navigate stores and markets with confidence.
HSK 4 | noun | physical components of a computer or electronic device
An introduction to the Japanese writing system: Hiragana and Katakana. Learn the foundational scripts required to begin your journey toward N5 proficiency.
We have a lineup of ponies, each with three attributes: their current mana, their maximum mana, and their mana regeneration rate per unit time. Over time, Lord Tirek performs several operations called Absorb Mana, each targeting a consecutive segment of ponies.
Week-by-week Japanese study plan from absolute beginner to JLPT N1: phased milestones, resource recommendations, and time estimates.
飲 (drink): 12 strokes, JLPT N5. On: イン. Kun: の-む.
Sindarin relative pronoun i, relative clause structures, expressing uncertainty and probability, and the particle ma for questions and possibility.
The conventions of Chinese historical writing — 历史叙述 style, evaluative vocabulary, and the language of historical significance and controversy
HSK 1 | verb | to have a fondness or preference for something
HSK 5 | noun | a concrete example or real-world instance used to illustrate a point
HSK 4 | adverb | doing something at the scheduled or required time
電 (den/): electricity. JLPT N5 kanji.
HSK 3 | adjective/noun/verb | evenly distributed; the statistical mean
HSK 3 | verb | to look after, take care of, or attend to someone's needs
HSK 1 | adjective | old (of people), elderly, experienced
HSK 5 | verb | to merge, to combine, to consolidate
HSK 3 | adverb | restrictive adverb expressing lateness, difficulty, or limited scope
HSK 4 | noun | the audience watching a performance or show
HSK 4 | noun / adjective | strategy; strategic
魚 (ギョ/さかな・うお): fish. JLPT N5 essential kanji.
HSK 5 | noun/verb | biological or metaphorical development to a higher form
N1 grammar pattern 〜ことなく: literary 'without doing X' expressing complete, absolute non-occurrence of an action throughout a period. Higher register than 〜ずに or 〜ないで.
HSK 3 | noun | a period in history defined by distinctive features
HSK 4 | noun | a fundamental belief or rule guiding behavior
We are given an undirected graph and, for every vertex, the parity of how many times that vertex must appear in a walk. A walk is represented by a sequence of vertices. Consecutive vertices in the sequence must be connected by an edge.
HSK 3 | noun | bicycle, bike
聞 (hear, ask): 14 strokes, JLPT N5. On: ブン、モン. Kun: き-く、き-こえる.
HSK 3 | adverb / expression | expresses compulsion arising from circumstances, not personal will
HSK 5 | verb/adjective | to mislead someone or to be in a state of confusion
HSK 4 | verb | to estimate or anticipate a future outcome
HSK 2 | noun | a sovereign state or nation
HSK 3 | verb | to transfer, to change vehicles during a journey
HSK 5 | conjunction | a logical connector indicating a further step or consequence
HSK 3 | adjective | easily seen or understood; unmistakable
JLPT N4 kanji 病 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
We are asked to implement a simple Caesar cipher. The input gives us a key k, an integer between 0 and 25, which represents how many positions each letter in the message should be shifted forward in the alphabet.
Express a full range of emotions in Chinese using 感到, 让我, and a rich vocabulary of emotional states.
HSK 4 | verb | to offer an example to illustrate a point
Learn how Esperanto's unlimited compounding system works to create precise, expressive vocabulary from simple roots.
We have all lattice points inside a rectangle of size $n times m$. Any square whose four vertices are lattice points is allowed, even if it is rotated. For every such square, we look at the unit grid cells completely contained inside it. Each contained unit cell receives one dot.
HSK 5 | verb/noun | to feel grateful; deep gratitude and appreciation
安 (an/yasu-i): cheap. JLPT N5 kanji.
HSK 4 | verb | to move downward in level or amount
HSK 4 | noun | the ability or capacity to compete effectively
毎 (every, each): 6 strokes, JLPT N5. On: マイ. Kun: none.
HSK 3 | noun | information, news, data — any content that informs or communicates
魚 (fish): 11 strokes, JLPT N5. On: ギョ. Kun: さかな、うお.
HSK 3 | noun | a person who lives next door or nearby
HSK 5 | verb | to restrain, contain, or suppress a negative force or trend
HSK 3 | adjective | feeling tired or worn out
外 (outside): 5 strokes, JLPT N5. On: ガイ、ゲ. Kun: そと、はず-れる.
HSK 6 | verb | to officially issue or promulgate laws, regulations, or decrees
日 (nichi/jitsu/hi): sun. JLPT N5 kanji.
昨 (saku/): yesterday. JLPT N5 kanji.
Master the spectrum of Japanese registers from casual speech to formal written language, including 書き言葉 vs 話し言葉, contractions, and formal sentence-ending patterns like 〜である.
HSK 4 | noun | conclusion reached after analysis or deliberation
HSK 4 | verb | to have capacity for a number of people or things
HSK 5 | adjective/verb | severely insufficient in supply; a state of dire shortage
HSK 6 | n | a shared understanding or agreement reached among multiple parties
HSK 4 | noun | skill; technical ability; competence
HSK 1 | verb | to read text aloud or to study at an institution
HSK 5 | noun | the capacity or power to have an effect on others' behavior or thinking
HSK 5 | adjective | having a quality or style that belongs only to itself; unique
N1 grammar: classical auxiliary たる used as a prenominal adjective expressing 'being X in the full, proper sense.' Appears in formal titles, literary descriptions, and elegant prose.
HSK 5 | adjective | sufficient; adequate; ample; plentiful
HSK 3 | adjective | going smoothly, without obstacles or setbacks
HSK 5 | noun/adjective | nutrition, nourishment; nutritious
HSK 6 | verb | to avoid or circumvent risks, rules, or obligations strategically
We have an $n times n$ chessboard, and each cell has a non-negative integer representing money. The goal is to place exactly two bishops such that no cell is attacked by both bishops, and we maximize the total money collected from all cells that are attacked by at least one…
Discuss environmental topics using 由于 + cause constructions and 对...有影响 at B1 level.
HSK 4 | verb | accepting or confirming a fact, mistake, or status
HSK 4 | noun | an organized collection of structured data
HSK 2 | adjective | describes something normal or not exceptional
HSK 5 | noun / adjective | an established norm or benchmark; conforming to the norm
Complete guide to on'yomi (Chinese-derived readings) and kun'yomi (native Japanese readings): when to use each, patterns, irregular readings, and examples.
Learn 〜ようにする (try to ensure, make a point of doing) — the N3 pattern for expressing sustained, deliberate effort to create or maintain a habit.
HSK 5 | noun | audience, readership, target demographic
人 (jin/nin/hito): person. JLPT N5 kanji.
HSK 4 | noun | edible plant parts such as leaves, roots, and stems
HSK 4 | noun | the sweet or sour fleshy product of a tree or plant
HSK 3 | verb / noun | a firm commitment that something will happen
HSK 5 | noun | antibody; protein produced by the immune system to fight pathogens
HSK 3 | verb/noun | to inspire someone to try or keep going
HSK 5 | verb/noun | to treat an illness; medical treatment
Talk about professions, workplaces, daily schedules, and professional life using past, present, and future tenses together.
後 (after, behind): 9 strokes, JLPT N5. On: ゴ、コウ. Kun: あと、うし-ろ、のち.
We are given an $n times n$ matrix where each entry $d[i][j]$ represents the distance between nodes $i$ and $j$. The question asks whether there exists a weighted tree with $n$ nodes such that the distance along the tree between any two nodes exactly matches the corresponding…
HSK 5 | n/v | a rhetorical comparison; to describe something by likening it to something else
HSK 5 | verb/noun | to arrange and position forces, resources, or plans strategically
HSK 2 | adverb | restricts scope to nothing more than what follows
HSK 4 | adverb | singling out something as standing out above the rest
HSK 4 | noun | a doctor's written order for medication
HSK 5 | noun phrase | mental health; psychological well-being
HSK 5 | noun | systematic study or framework of methods used in a discipline
九 (nine): 2 strokes, JLPT N5. On: キュウ、ク. Kun: ここの、ここの-つ.
HSK 5 | verb | to produce; to cause; to arise
HSK 3 | noun | nurse — medical professional who cares for patients
JLPT N4 kanji 園 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 2 | modal verb | should (moral or practical expectation); contrast with 要 and 必须
HSK 1 | verb | expresses possession and existence
HSK 5 | verb/noun | development of diversity and variety in forms or approaches
HSK 3 | conjunction | moreover, and also, furthermore — adds an additional point
HSK 1 | adverb | indicates earliness, immediacy, or that something happens sooner than expected
Complete N4 grammar reference: ~150 patterns with structure formulas, meanings, 3+ example sentences each (Japanese, romaji, English), and notes on common mistakes.
HSK 6 | n | the situational, cultural, or linguistic context in which a word or utterance occurs
Chinese and Western cultural frameworks compared — Confucianism, Daoism, Buddhism, and Western values in analytical discourse at C1 level
HSK 1 | verb | to acquire something by paying money
HSK 4 | verb | to refer to something briefly in speech or writing
HSK 4 | preposition | indicating the object of an action or attitude
HSK 3 | noun | a story, tale, or narrative
万 (ten thousand): 3 strokes, JLPT N5. On: マン、バン. Kun: よろず.
HSK 4 | conjunction | introducing an additional point beyond what was stated
HSK 5 | noun | code; source code; programming code
HSK 3 | noun / adjective | major field of study; specialized skill or knowledge
HSK 4 | noun/adj | customs and practices passed down through generations
HSK 4 | verb / noun | to respect; respect
HSK 5 | adjective/adverb | indicating something is not absolute but comparative or relational
Learn how to use 〜に加えて (ni kuwaete) to express 'in addition to' when adding further information or elements. Includes structure, nuance, examples, and comparisons.
HSK 3 | adjective/noun | able to wait calmly without frustration
HSK 1 | pronoun | third-person singular masculine pronoun
HSK 3 | verb | to carry out, conduct, or be in progress
We have a one-dimensional sheet of paper represented as an array of width n. Appleman can fold the sheet at a position p, which means the left segment [0, p) flips and lies over the right segment [p, currentwidth).
HSK 3 | verb | to give as a gift, deliver, or accompany someone when they leave
HSK 1 | adjective | beautiful, pretty, usually describing people
JLPT N4 kanji 員 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 5 | noun | an official plan or course of action adopted by a government or organization
Learn to ask for and give directions, name places in a city, and talk about transport in Esperanto.
HSK 2 | verb-object phrase | to hike or climb a mountain
HSK 5 | noun | the ability to recover from adversity and adapt to challenges
HSK 3 | adverb | marks a contrast or reversal between expectation and reality
HSK 1 | noun/time word | the current day; used at the start of a sentence or before the verb
友 (yuu/tomo): friend. JLPT N5 kanji.
HSK 5 | verb | to confirm as true; to substantiate
HSK 5 | verb | to establish; to found; to set up
HSK 3 | grammar pattern | expresses whether an action can or cannot achieve its result
The grammar and vocabulary of Chinese legal discourse — statutory language, contract clauses, and the register of rights, obligations, and liability
HSK 4 | verb | to make smaller or narrower
We are given a binary string that represents a positive integer $x$. Instead of directly printing this binary number, we are forced into a strange process that maintains a single integer variable $n$, starting at zero, and builds the output by repeatedly applying only two…
HSK 2 | noun | a business company or corporation
HSK 3 | noun | seafood — fish, shellfish, and other sea creatures eaten as food
HSK 4 | noun | something that requires thought, attention, or resolution
HSK 4 | noun | the actual site or scene where something is happening
HSK 3 | adverb | marks a turning point after which everything that follows is different
HSK 4 | verb/noun | the art or act of preparing food by applying heat
HSK 5 | noun | contract, written agreement
Master the Japanese demonstrative system (Ko-So-A-Do) to accurately identify objects, locations, and directions in daily conversation.
HSK 4 | noun | science and technology as a combined field
土 (earth, soil): 3 strokes, JLPT N5. On: ド、ト. Kun: つち.
JLPT N4 kanji 社 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 6 | n | a guiding idea, philosophical concept, or core belief underlying a practice or system
HSK 5 | adverb | used to introduce a result or conclusion drawn from what was stated
HSK 5 | conjunction | and therefore, thus, and as a result
Express precise numerical and degree-based comparisons using A是B的N倍, A比B+V得多, and 差不多 at B1 level.
HSK 5 | verb | to stimulate, trigger, or arouse a feeling, potential, or reaction
木 (moku/boku/ki): tree. JLPT N5 kanji.
HSK 1 | pronoun | third-person singular pronoun for non-human things
HSK 5 | noun/adjective/verb | corruption; corrupt; to decay, rot
HSK 3 | adjective | strong, powerful, capable — describes strength or superiority
HSK 3 | adjective/noun | belief in one's own abilities
Achieving mastery in Chinese academic defense and Q&A: from competent performance to intellectual authority.
HSK 1 | noun | things, stuff (idiomatic); east-west (literal)
HSK 3 | adjective | deep (physical), dark (color), or profound (abstract)
HSK 6 | n | regular pattern or law governing natural or social phenomena
HSK 3 | verb | to refuse, to reject, to decline
We have a line of children. Child i wants a[i] candies in total. Every time a child reaches the front of the line, they receive exactly m candies. If the child has still not received enough candies after that distribution, they move to the back of the line.
A comprehensive review of all A2 grammar patterns and key vocabulary from HSK 3, with integrated exercises for exam preparation.
HSK 4 | noun | the power or skill to do something
HSK 5 | noun / verb | reform, transformation, fundamental change
Learn 〜はおろか to express extreme contrast on a scale — 'let alone X, even Y (which is more basic) is impossible/true.'
猫 (cat): 11 strokes, JLPT N5. On: ビョウ. Kun: ねこ.
HSK 4 | verb / noun | pointing out faults or shortcomings
We need to assign each student to one of k buses on each of d days. For every student, we can think of their assignment as a sequence of length d. For example, if d = 3, a student might ride buses (2, 1, 2) over the three days.
英 (English, heroic): 8 strokes, JLPT N5. On: エイ. Kun: none.
雨 (u/ame): rain. JLPT N5 kanji.
HSK 4 | adverb | in a way that is plain to see or understand
HSK 2 | verb/directional complement | movement outward; also used as a directional complement
HSK 3 | adverb / verb | comparatively, relatively; to compare two things
町 (town, district): 7 strokes, JLPT N5. On: チョウ. Kun: まち.
HSK 5 | noun / verb | feedback, to give feedback
HSK 3 | noun | neck, the part of the body connecting the head to the shoulders
HSK 5 | noun | a favorable juncture of circumstances; a golden chance
Master the two distinct uses of 〜次第: 'depending on' (conditional) and 'as soon as' (sequence), a critical N2 exam distinction.
HSK 4 | verb / noun | to doubt; to suspect; doubt
HSK 5 | noun | a microscopic infectious agent; also used for computer viruses
万 (man/ban/yorozu): ten thousand. JLPT N5 kanji.
Report and discuss news using 据...报道, 消息说, and media vocabulary at B1 level.
HSK 5 | adjective/noun | treating all parties equally without bias or favoritism
HSK 5 | verb/noun | blending of distinct elements into a unified whole
HSK 1 | adverb/noun | the day after today
HSK 5 | noun | pressure, stress, burden
HSK 4 | adjective/adverb/noun | expressing possibility or probability
HSK 1 | verb phrase | to make a phone call
Develop strategies for comprehending and taking notes from Chinese academic lectures, including key vocabulary for academic oral communication.
HSK 3 | noun | a key for a lock; used for doors, cars, and safes
HSK 5 | verb | to make something more profound or thorough
HSK 5 | verb | to emerge; to spring up; to appear in abundance
HSK 4 | verb / noun | rallying people to a cause or action
HSK 5 | noun | a computer or system that provides data or services to other computers over a network
We are given an essay, which is a sequence of words, and a dictionary of synonym pairs. Each synonym pair allows us to replace one word with another, but only in the direction specified. Fedor wants to modify his essay to minimize the number of letters R (case-insensitive).
HSK 4 | noun | the process or journey through which something develops
HSK 5 | noun | non-fossil energy sources such as solar, wind, and hydrogen power
HSK 4 | adjective | going beyond the surface to reveal essential truth or insight
HSK 5 | verb / noun | to raise capital; the act of financing
HSK 1 | adjective | long (in length or duration)
HSK 3 | adjective | negative, passive, pessimistic — opposite of 积极
Learn household vocabulary and how to describe where things are using prepositions of place and the productive -ej- suffix.
HSK 5 | adjective/verb | prosperous, thriving; to prosper, to flourish
間 (kan/ken/aida): interval. JLPT N5 kanji.
読 (read): 14 strokes, JLPT N5. On: ドク、トク、トウ. Kun: よ-む.
HSK 1 | noun | airplane, aircraft
HSK 2 | noun / localizer | the side of something; used to indicate adjacency
HSK 3 | noun | the atmosphere or mood of a place or situation
HSK 3 | conjunction | therefore, as a result — introduces a logical consequence
HSK 5 | noun | the fundamental, underlying nature or essence of something
HSK 3 | adverb | indicates an increase in degree beyond a previously established level
HSK 5 | adjective/noun | subjective; based on personal feelings rather than facts
HSK 5 | verb / noun | to evaluate; assessment
HSK 5 | verb | to show clearly; to express or demonstrate
HSK 3 | adjective | ordinary, common, average — not special
HSK 5 | conjunction | let alone; much less; not to mention; all the more so
HSK 4 | verb | deliberately prioritizing or valuing something
HSK 3 | noun | the purpose or range of uses that something has
HSK 3 | adverb | indicates that something happens before a specific event or time
Best websites for learning and researching Sindarin: grammar references, tools, encyclopedias, and specialized resources.
Ann needs to commute using the subway a total of n times. She can pay for each ride individually at a cost of a rubles, or she can buy a special multi-ride ticket that covers exactly m rides and costs b rubles.
伝える (つたえる / tsutaeru): to convey, tell. N4 level Japanese vocabulary.
The sequence in this problem comes from counting lattice points inside a growing hexagon on a hexagonal grid. Instead of having to construct the hexagon or count points manually, the problem already gives the formula for the sequence: $$Hn = 3 cdot n cdot (n+1) + 1$$ The input…
HSK 3 | verb / noun | to answer, to reply; an answer, a response
HSK 5 | verb/noun | biological or metaphorical decline to a lower or weaker state
私 (I, private): 7 strokes, JLPT N5. On: シ. Kun: わたし、わたくし.
HSK 3 | noun | medicine, medication, drug
HSK 4 | verb/noun | to infer, to speculate; inference, speculation
HSK 3 | verb + directional complement | to walk toward the speaker's location
HSK 6 | verb | to tell apart, to discern differences between things
HSK 6 | verb | to distill or extract the most valuable elements from raw material
How to compose Sindarin poetry: the attested linnod verse form (7+7 syllables), scansion of A Elbereth Gilthoniel, alliteration rules, and step-by-step composition exercises.
Master topic-comment sentence structure and topic-introducing expressions like 说到, 关于, and 对于 at B1 level.
HSK 1 | greeting phrase | standard informal greeting for one person
HSK 3 | noun | a location, area, or region
HSK 5 | noun | a critical moment of serious danger, difficulty, or instability
HSK 4 | noun | a place where meals are served to customers
Sindarin possessive pronouns: suffix forms attached to nouns, independent possessive forms, and the genitive construction with en.
HSK 1 | number | the quantity two, used directly before measure words and classifiers
HSK 4 | verb | officially occupying a role or post
HSK 4 | verb | indicating the core reason or essence of something
HSK 5 | adjective / adverb / noun | basic and essential; at the most basic level; not at all (negative)
HSK 3 | verb / noun | making a reasoned decision or assessment
HSK 3 | noun/verb | hobby or personal interest
HSK 5 | noun | organized and interconnected system of elements
JLPT N4 kanji 医 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 4 | conjunction / phrase | for example; for instance; such as
HSK 5 | adjective/adverb | going all the way; leaving nothing incomplete
HSK 5 | noun | a person's point of view or standpoint on an issue
HSK 4 | noun | a person who shares in a business venture or partnership
HSK 2 | adverb | marks a habitual or characteristic action
HSK 3 | noun | police officer, policeman, the police
HSK 1 | noun | shop, retail store
Master Japanese giving and receiving verbs — あげる, もらう, くれる — and their て-form benefactive extensions 〜てあげる, 〜てもらう, 〜てくれる. Learn the in-group/out-group (uchi/soto) social rules that govern their use.
HSK 5 | verb / noun | to inspire, to enlighten; inspiration, enlightenment
Develop the vocabulary and oral patterns for delivering formal presentations, reports, and interactive talks in Chinese professional and academic settings.
Learn color vocabulary and how Esperanto adjectives work, including the productive mal- prefix for expressing opposites.
Master five N3 patterns that express change, proportion, and conditionality: 〜によって (depending on), 〜に従って (in accordance with), 〜につれて (proportional change), 〜とともに (together with/as), and 〜に応じて (in response to).
HSK 4 | conjunction | a formal connector meaning however or nevertheless
HSK 5 | adjective | extending over a long period or into the distant future
We are given a set of n distinct integers. Every number must be assigned to one of two groups. If a number x is placed into group A, then its complement with respect to a, namely a - x, must also be present and must belong to A.
We are asked to compute the permanent of a large square matrix where almost every element is 1, and only a small number of entries are different.
HSK 5 | verb | to reveal; to bring to light; to expose
HSK 5 | noun/verb | contamination of the environment; to contaminate
HSK 4 | noun | digital platforms for social interaction and content sharing
HSK 3 | noun | the rules and structure of a language
All Sindarin prepositions with their mutation triggers: soft, nasal, and mixed — with worked examples and the dative/genitive constructions.
HSK 7 | verb | clinical determination of a disease or condition based on systematic examination
HSK 5 | n | the form or style of a written work; literary genre
〜てもらう: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
The C1 Chinese argumentative essay — 议论文 structure, evidence types, transitional logic, and the art of rhetorical elevation
HSK 3 | verb | to thank, feel grateful toward someone; more formal than 谢谢
HSK 3 | noun | an airport; the facility where aircraft take off and land
Learn 〜て以来 (ever since ~) — the N3 pattern for expressing a persistent state that has continued uninterrupted since a past event.
HSK 4 | verb / noun | to report; to brief; a report
HSK 4 | noun | essential organic nutrients the body needs in small amounts
HSK 3 | adjective / noun | dangerous, risky; danger, risk
〜ように: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
二 (two): 2 strokes, JLPT N5. On: ニ. Kun: ふた、ふた-つ.
HSK 1 | noun | friend
HSK 4 | noun | a secret word or number used for authentication or encryption
HSK 4 | noun | a prerequisite or stated requirement
Learn how to use 〜に反して (ni hanshite) to express 'contrary to' or 'against' expectations, rules, or wishes. Includes structure, nuance, examples, and comparisons.
HSK 2 | noun | hotel; guest house; lodging establishment
HSK 4 | adjective/noun | relating to a specific field of expertise
Develop the vocabulary and grammar structures for constructing, presenting, and responding to arguments in formal Chinese debate and analytical writing.
N1 grammar pattern 〜とはいえ: acknowledging a stated fact while introducing a contrasting reality — 'even though, that said, granted that.'
HSK 5 | verb / noun | to come into existence; the birth or emergence of something
Complete N4 vocabulary list: ~700 new words at N4 level (cumulative ~1,500). Organized by category with readings, romaji, English meanings, and notes.
HSK 5 | adjective | legal, lawful, legitimate
JLPT N4 kanji 暑 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
How to use spaced repetition systems for Esperanto vocabulary learning — Anki setup, deck strategy, and optimal review habits.
Learn five more essential suffixes: concrete things, collectives, members, augmentatives, and diminutives.
HSK 6 | n | the continuous thread of cultural or textual tradition; the vein of meaning running through a text
手 (hand): 4 strokes, JLPT N5. On: シュ、ズ. Kun: て.
HSK 4 | noun | money received from work, business, or investment
HSK 4 | noun | a formal gathering for discussion or decision-making
HSK 3 | noun | belly, stomach, abdomen (colloquial)
HSK 5 | verb/noun/adjective | to put in effort or resources; input; deeply absorbed
HSK 5 | verb | to stir emotionally; to touch on a sensitive point
All four Latin participles (PAP, PPP, FAP, gerundive), their formation and declension, and the ablative absolute construction with worked examples.
HSK 5 | noun | the variety of life forms in a given environment
Learn 〜ずにはいられない to express irresistible inner compulsion — you cannot help but do or feel something.
HSK 5 | noun | abstract mental representation or intellectual construct
HSK 1 | verb | to listen to or hear something
HSK 3 | verb / noun | to make a decision; a decision
金 (gold, money): 8 strokes, JLPT N5. On: キン、コン. Kun: かね、かな.
HSK 3 | adverb | singles out one item from a group as deserving special emphasis
HSK 4 | verb/noun | to commit money or resources in expectation of future returns
HSK 3 | verb | to help, to give assistance — verb-object compound, can be split
Learn the 6 Latin cases, their functions, and the complete first declension paradigm with worked examples.
HSK 4 | n/adj | a system of government by the people; fair and open
We are given a fixed amount of money expressed in dollars, and a list of sugar options in a supermarket. Each option has a price written as dollars and cents.
大 (dai/tai/oo-kii): big. JLPT N5 kanji.
Learn how to use 〜といっても (to ittemo) to qualify or soften a previous statement — 'even though I say ~, it's not as extreme as you think.' Includes structure, nuance, examples, and comparisons.
HSK 3 | adjective | unwilling to work or make effort
HSK 3 | noun | a written piece such as an article or essay
HSK 1 | greeting phrase | polite opener before asking a question
HSK 3 | noun | email, electronic mail message
HSK 4 | noun | a particular branch of economic or professional activity
HSK 4 | noun | the monetary or intrinsic worth of something
JLPT N4 kanji 冬 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
10 structured lessons building JLPT N5 from zero: hiragana/katakana mastery, self-introduction, numbers, places, daily activities, shopping, adjectives, past tense, desires, and plans.
HSK 3 | noun / verb | rule, regulation; to stipulate or establish a rule
HSK 3 | verb | to be afraid, to fear, to be scared
We start with a single group containing all given numbers. Whenever Toastman receives a group, he adds the sum of all numbers in that group to the score. If the group contains more than one number, Appleman may split it into two non-empty groups and send both back to Toastman.
HSK 2 | particle | softens suggestions, assumptions, and tag questions
HSK 4 | adjective | having a sharp, unpleasant taste; also describing hardship
HSK 3 | adjective/expression | feeling embarrassed, shy, or apologetic; sorry to bother you
Master all five uses of cum with the subjunctive and indicative, including the tricky cum inversum.
HSK 5 | noun / verb | monopoly; to monopolize, to dominate exclusively
Sindarin future tense: continuative future (-tha- infix) and simple future forms, all 6 persons — with the famous le linnathon as a worked example.
HSK 5 | adjective/adverb | matching or responding appropriately to something
HSK 3 | noun | shortcoming, weakness, flaw, disadvantage
HSK 3 | noun | arm, the upper limb from shoulder to wrist
HSK 4 | verb | to find a solution or resolve a problem
HSK 2 | measure word/noun | the metric unit of weight equal to one kilogram
HSK 3 | adjective | clear, definite, and not vague
HSK 3 | verb / adjective | to have a headache; troublesome, annoying
N1 grammar pattern 〜もさることながら: 'not to mention ~, of course ~, but even more so ~' — acknowledging one quality before introducing something more notable.
Complete JLPT N1 grammar reference: ~200 patterns including classical Japanese influences, literary forms, highly formal conjunctions, advanced conditionals, and nuanced distinctions separating N1 from N2.
HSK 5 | adjective/adverb | expressing the idea of being contrary or the reverse
Sindarin object pronouns and the dative construction with an: direct object pronoun forms, indirect object, and attested examples.
We have an $n times n$ bingo board. Every cell contains a distinct number chosen from $1$ to $m$, and every valid board is equally likely. After the board is generated, exactly $k$ distinct numbers are called. Every set of $k$ numbers is equally likely.
HSK 3 | verb | to push, to shove, to promote or recommend
JLPT N4 kanji 郵 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
Comprehensive N2 capstone with full mock reading passage, 5 comprehension questions, 20 grammar substitution questions, 10 error-correction questions, exam strategy, and Bridge to N1 preview.
HSK 1 | verb | to be aware of, to have knowledge of a fact or situation
HSK 5 | verb / noun | to train; professional training, vocational training
Master four essential N3 formal conjunctions: 〜において/〜における (in/at), 〜に対して/〜に対する (toward/against), 〜にとって (for/from the perspective of), and 〜として (as/in the capacity of).
JLPT N4 kanji 教 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 4 | noun | journalist; reporter; correspondent
HSK 3 | noun | a positive quality or strength of a person or thing
HSK 1 | verb | to write characters, text, or content
HSK 3 | noun | eyes, the organs of sight
HSK 3 | grammar pattern | passive sentence marking an action done to the subject, often with an unwanted result
HSK 2 | verb / noun | to draw or paint; a picture or painting
HSK 3 | adjective/adverb | arriving or happening at the scheduled time
HSK 1 | greeting phrase | formal or respectful greeting for elders and superiors
HSK 3 | preposition/conjunction | introduces a cause or reason, more formal than 因为
HSK 5 | noun | supply chain; the network of suppliers, manufacturers, and distributors
Order food, ask questions, and handle the bill in Chinese with practical restaurant vocabulary and grammar.
HSK 5 | noun | nature reserve; protected area; conservation zone
HSK 5 | noun | observable direction of development or change over time
HSK 4 | noun | a trained professional who provides therapy or treatment
HSK 5 | verb | upward movement or increase in quantity, level, or degree
HSK 5 | noun | regular, predictable pattern or law governing phenomena
HSK 5 | noun | earthquake, seismic event
HSK 2 | adjective | the color red; associated with luck and celebration in Chinese culture
N1 grammar pattern 〜ないまでも: expressing 'even if not ~, at least ~' — conceding that the ideal may not be achieved while asserting a minimum.
HSK 5 | verb | to urge someone to do something through supervision or prompting
HSK 4 | adverb/verb | one by one or each in their own way
Learn family vocabulary and master Esperanto's -in- suffix for feminine forms, ge- prefix for mixed-gender groups, and possessive adjectives.
HSK 4 | verb / noun | using existing material as a guide or source
N1 grammar: 〜いかんでは and 〜いかんによっては expressing 'depending on the nature/circumstances of X, Y may occur.' Formal register, often in legal or policy contexts.
Complete Sindarin verb system: primary and A-stem verb classes, aorist/present/past/future tenses, person suffixes, and worked conjugation tables.
HSK 3 | preposition | in accordance with, following a rule, method, or standard
HSK 4 | verb/noun | to break down a subject into its components in order to understand it
HSK 3 | verb/noun | to recall the past; a memory or recollection
Master Esperanto's system of indirect speech, including tense non-backshift, reported questions, and commands in ke-clauses.
HSK 4 | verb | to promote, to popularize, to extend
HSK 3 | noun | physical material or written data and information
HSK 2 | noun / verb | a flower; also to spend or use up money or time
Learn body part vocabulary, how to describe symptoms using 疼 and 痛, and how to soften descriptions with 有点儿.
HSK 4 | verb | something initiating or provoking a reaction or result
HSK 5 | noun/verb | first aid; emergency medical treatment
HSK 1 | adjective | easy, simple. Antonym: 难 (nán)
HSK 5 | noun | available freshwater supplies for human and ecological use
HSK 3 | verb | to pick up a person, answer a call, or receive something
Navigate travel situations in Esperanto: transport vocabulary, buying tickets, asking for directions, and describing journeys.
HSK 5 | verb / noun | to owe debts; liabilities on a balance sheet
Sindarin words for places, structures, geographic features, and cardinal directions — with famous place name analyses.
We are given several stacks of unit cubes, each stack having an initial height. In a single move, we can take exactly one cube from the top of one stack and place it on top of another stack.
HSK 3 | noun / adjective | a mistake or wrong action; incorrect
HSK 3 | adjective | freely giving; composed and graceful in manner
HSK 4 | verb | to contact, to come into contact with, to touch
HSK 3 | noun | the measure or extent of something
Sindarin nominal suffixes for possession, diminution, and verbal nouns: -en (mine), -weg (creature), -el (feminine), and the -as abstract noun suffix.
HSK 3 | adverb / verb | doing things separately or parting from each other
HSK 4 | noun | a mechanism or organized system for how something works
HSK 1 | noun | student, pupil
HSK 3 | noun | machine, apparatus, device
HSK 5 | conjunction | not only that; furthermore; what is more
HSK 1 | verb | to run, to jog
Introduction to learning Japanese kanji: the Joyo list, reading types, JLPT distribution, top 50 most common kanji, and choosing a learning method.
HSK 5 | noun/verb | subjective feeling; to feel or experience something
HSK 3 | verb/noun | to personally feel or go through something
HSK 3 | noun | the post office; a government postal service location
HSK 3 | verb/noun | to consult or use as a reference
父 (father): 4 strokes, JLPT N5. On: フ. Kun: ちち.
HSK 4 | noun | the display surface of a device such as a phone, computer, or TV
We are asked to calculate the sum of all integers $x$ that satisfy a pair of modular conditions with respect to two given integers $a$ and $b$.
HSK 2 | adjective | feeling that expectations have been met
HSK 2 | verb | to think (that), to consider, to hold an opinion
HSK 4 | noun | the global network connecting computers worldwide
N1 grammar pattern 〜にもかかわらず: expressing contrast or concession — despite, in spite of, even though.
HSK 5 | noun | foreign exchange, foreign currency
Discuss urban living, infrastructure, and city development using 在 + city + 生活/工作 and city vocabulary at B1 level.
HSK 4 | verb/noun | to grow or expand in a positive direction
HSK 3 | noun | a traditional Chinese dumpling with a filled dough wrapper
HSK 5 | noun | personal information or matters that one wishes to keep from others
Learn 〜得る (can possibly, is conceivable) and 〜得ない (cannot possibly, is inconceivable) — N3 patterns for logical possibility and impossibility.
HSK 1 | verb | expresses desire, thought, and longing
HSK 3 | verb | to hurry toward something or catch a vehicle in time
JLPT N4 kanji 工 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 4 | noun | disadvantage, inferior position
HSK 4 | verb | to cope with; to deal with; to respond
HSK 2 | adjective | describes an absence of noise or disturbance
HSK 1 | greeting phrase | reassures someone after an apology
We are given a directed graph where every edge has a weight. We want the longest possible path measured by the number of edges, with one restriction: whenever we move from one edge to the next, the new edge must have a strictly larger weight than the previous one.
HSK 1 | noun | cat
HSK 3 | noun | an unintended harmful event or accident
HSK 3 | noun | menu, the list of dishes available at a restaurant
Dialect variation in Sindarin: Gondorian Sindarin (influenced by Númenórean usage and Adûnaic), Mirkwood Sindarin (Silvan-influenced), and key phonological differences.
海 (kai/umi): sea. JLPT N5 kanji.
HSK 5 | verb/noun | to take action in advance to stop something harmful from occurring
HSK 3 | verb/noun | working extra hours beyond the normal schedule
Books for learning Sindarin: grammar references, textbooks, and the primary source journals Vinyar Tengwar and Parma Eldalamberon.
Learn 〜をもとに and 〜をもとにして to express using something as raw material, inspiration, or source for creating or analyzing something new.
火 (fire): 4 strokes, JLPT N5. On: カ. Kun: ひ、ほ.
Learn 〜ばかりでなく (not only, but also — formal/emphatic) — the formal variant of the additive scope pattern, common in written Japanese.
HSK 4 | verb/noun | a face-to-face evaluation or job interview
HSK 3 | noun / adjective | science; scientific
HSK 5 | verb/noun | to communicate; mutual exchange of information or feelings
HSK 3 | adverb | reveals the true situation, often correcting a misconception
HSK 1 | adverb | intensifier before adjectives; also a default linking tone
N1 grammar pattern 〜限りだ: expressing a strong, overwhelming emotion — 'I feel nothing but X / I am completely filled with X.' Used for both positive and negative strong feelings.
We have n flower boxes. From box i, we may take any number of flowers between 0 and f[i], inclusive. The colors are distinct, so a selection is completely described by how many flowers we take from each box.
Analyzing 政治语体: the vocabulary, rhetoric, and pragmatics of Chinese political communication.
HSK 1 | verb | to ask, to inquire
HSK 3 | determiner | every, each — used before a measure word or time word
HSK 1 | pronoun/determiner | far demonstrative, referring to things at a distance
HSK 2 | adjective / adverb | expressing possibility; works both as 'possible' (adjective) and 'maybe' (adverb)
HSK 4 | adjective | having a sharp, burning flavor from spices or chili
HSK 5 | noun | academic grades; results; achievements in work or study
HSK 3 | adj/noun | the most important or decisive element
見 (ken/mi-ru): see. JLPT N5 kanji.
HSK 1 | conjunction | connects nouns or noun phrases
HSK 4 | adjective | not complex or difficult; lacking unnecessary detail
HSK 4 | adjective | right for a particular purpose or situation
HSK 1 | pronoun | second-person plural pronoun
HSK 3 | noun | a country other than one's own
HSK 5 | noun | benefits or services provided to improve well-being
HSK 3 | verb | to cry; to weep; to shed tears
HSK 7 | verb | mechanical or assisted ventilation of the lungs in critical care settings
Master reference for all Sindarin verb conjugation: all 12 verb classes, all tenses/moods, all 9 persons — with full tables for model verbs and notes on scholarly disagreements.
HSK 5 | noun/verb | automation; to automate
HSK 4 | noun | classmate, schoolmate
Complete HSK 5 vocabulary reference: approximately 1,600 new words at B2 level, organized by semantic category with pinyin, English, part of speech, and usage notes.
今 (now): 4 strokes, JLPT N5. On: コン、キン. Kun: いま.
HSK 5 | adjective | exceptionally high in quality or achievement, surpassing others by a wide margin
それから (それから / sorekara): and then, after that. N4 level Japanese vocabulary.
花 (flower): 7 strokes, JLPT N5. On: カ. Kun: はな.
HSK 5 | adjective | describing something felt with urgency or strong desire
HSK 2 | preposition | introduces exceptions or additions; used in 除了...以外/还/都 patterns
HSK 1 | verb | to open, to turn on, to drive
HSK 4 | noun | an institution, organization, or agency
Beginner Sindarin: goals, learning resources, grammar checkpoints, and vocabulary targets for the first 30–100 hours.
HSK 1 | verb | movement away from the speaker's location
HSK 6 | verb | to collect and organize into a summary; to reason inductively
HSK 5 | verb/noun | to long for intensely; a strong desire or yearning
HSK 4 | adjective | humble; modest; unassuming
耳 (ear): 6 strokes, JLPT N5. On: ジ. Kun: みみ.
HSK 4 | verb / noun | to show; to perform; performance; behavior
黒 (koku/kuro): black. JLPT N5 kanji.
HSK 5 | noun | a distributed digital ledger technology for recording transactions securely
HSK 1 | adjective | cold (temperature or weather)
川 (river): 3 strokes, JLPT N5. On: セン. Kun: かわ.
JLPT N4 kanji 手 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
Sindarin vocabulary for warfare, weapons, armor, military ranks, and battle terms — sourced from Tolkien's use of Sindarin in LotR's war contexts.
上 (jou/shou/ue): above. JLPT N5 kanji.
We have a game with levels numbered from 1 to n. Little X can complete some subset of these levels, and Little Y can complete another subset. They decide to cooperate, which means a level is considered passable if at least one of them can complete it.
HSK 6 | n/v | the blending or integration of different elements into a unified whole
HSK 4 | verb | bearing responsibility for a task or outcome
HSK 4 | verb | to think carefully about something
HSK 2 | verb | to pick up, hold, or take something with the hand
HSK 8 | noun | semiconductor fabrication; the process of manufacturing integrated circuits and chips
HSK 5 | adjective | of great importance, significance, or consequence
N1 grammar pattern 〜ものを expressing regret about an unrealised alternative or reproach toward someone who failed to do something they should have. Literary and formal register.
HSK 5 | verb / noun | to interpret; to read and make sense of
HSK 5 | verb/noun | to watch and control through continuous observation
HSK 4 | adjective | concise, brief, and free of unnecessary content
HSK 5 | noun | the surface or point of interaction between two systems or entities
The 200 most frequent Latin words from the Dickinson College Commentaries core list. Learn all 200 before moving to Band 2.
Learn how to greet people, say your name, and ask where someone is from using 是 and 吗.
HSK 5 | noun | experience, practical knowledge gained through doing
HSK 5 | verb/noun | to authorize; to grant authority or permission
We have all lattice points inside and on the boundary of an axis-aligned rectangle whose opposite corners are (0, 0) and (n, m). The task is to choose four distinct lattice points and connect them in the chosen order.
Learn to describe your daily routine using V + 了 for completed actions and sequence time words.
HSK 1 | measure word | classifier for chunks, pieces, and Chinese yuan
HSK 5 | noun/verb | marketing; to market (products or services)
HSK 3 | verb/noun | behavior, performance, or how something manifests
HSK 3 | adjective / noun | safe, secure; safety, security
How long it takes to learn Esperanto — time estimates per CEFR level and comparison with other languages.
HSK 2 | modal verb | expressing permission or general possibility; contrast with 能 for ability
HSK 3 | verb | to accept, to receive, to take in willingly
Master the full nuance landscape of purpose and reason in Japanese: 〜ために, 〜ように, 〜から, 〜ので, 〜て (causal), and 〜のに (although/regret). Includes a formality/nuance comparison table and contrasting examples.
HSK 5 | noun | a branch of AI in which systems learn from data to improve performance
HSK 3 | noun | salt, the common cooking mineral
百 (hundred): 6 strokes, JLPT N5. On: ヒャク. Kun: none.
HSK 4 | noun | a recurring sequence of events or a defined time span
HSK 4 | n/v | making changes to improve a system or institution
HSK 4 | noun | an established set of rules or organizational structure
HSK 3 | adjective / verb | describing hard, tiring work; also used to thank someone for their effort
HSK 3 | conjunction | no matter, regardless of — always paired with 都 or 也
HSK 5 | noun / adjective | judiciary, the judicial system, administration of justice
HSK 5 | verb/noun | to show how something is done; a model example for others to follow
HSK 5 | noun | structural outline or framework that organizes a system
Sindarin colour vocabulary and first encounter with soft mutation (lenition): why adjectives after nouns undergo consonant changes.
HSK 3 | verb / adjective | to satisfy, to meet a need; to feel satisfied
The task asks us to find all positive integers less than one billion that satisfy the equation $$x = b cdot s(x) cdot a + c$$ where $a$, $b$, and $c$ are given constants, and $s(x)$ is the sum of the digits of $x$.
HSK 5 | adjective/verb | conspicuously noticeable; to highlight or protrude
HSK 1 | noun | taxi, cab
HSK 1 | greeting phrase | affirms agreement or acknowledges a request
Learn to order food and drinks, express what you want to eat using 想 and 要, and talk about Chinese cuisine.
Learn vocabulary and grammar for discussing Chinese traditional festivals, including the customs, foods, and activities associated with major celebrations.
HSK 4 | verb | to give special weight or importance to a point
HSK 5 | verb/noun | to shape metal or other materials by melting and moulding, or metaphorically to forge character
HSK 3 | adjective/verb | to feel let down; to be disappointed
HSK 3 | verb | to lose a game, match, or contest
HSK 5 | adjective | superb, consummate, highly skilled
HSK 1 | pronoun/determiner | near demonstrative, referring to things close by
Discuss core Chinese social values including 面子, 关系, and 孝顺 using formal register vocabulary at B1 level.
HSK 3 | verb | to raise something upward or to give an example
HSK 3 | verb/noun | a sensation or the act of feeling
HSK 6 | verb | to subject to critical examination; to criticize fundamentally
HSK 4 | verb | to come into view or existence
HSK 3 | adverb | sets a minimum threshold for quantity, frequency, or degree
HSK 4 | noun | a specific action taken to achieve a goal or solve a problem
Decode formal newspaper Japanese at N3 level: headline omission rules, passive constructions, nominalization, 〜により/〜として in formal writing, number expressions, and analysis of real-style article sentences.
HSK 4 | noun / adjective | science; or being scientific and systematic
HSK 3 | adverb | everywhere, all over — in every place, with no location excluded
HSK 4 | noun | the state of psychological and emotional wellbeing
HSK 5 | verb | to portray with precision; to carve out a character or image
HSK 4 | adjective | lazy; indolent
Master six essential て-form compound patterns: 〜てしまう, 〜ておく, 〜てみる, 〜てくる, 〜ていく, and 〜てから — each expressing a distinct nuance of how actions relate in time, intention, and consequence.
HSK 3 | verb / noun | to have a cold; a cold, the common cold
HSK 5 | verb | to be the first to establish or bring something into existence
HSK 4 | noun | an official document certifying achievement or qualification
HSK 2 | noun | a point in time, a moment, an occasion (contrast with 时间)
HSK 4 | verb | to put a plan, policy, or measure into practice
HSK 1 | noun/verb | used for minutes in time-telling; also means to divide or a fraction
Recognize and produce Esperanto in four distinct registers — literary, journalistic, scientific, and colloquial — with appropriate vocabulary and syntactic choices.
JLPT N4 kanji 番 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 3 | verb | to take a photo, to shine on, to reflect, to look after
HSK 1 | verb, adverb | please, to invite, to treat someone
HSK 4 | noun | a list of dishes available at a restaurant
HSK 6 | verb | to draw conclusions through reasoning from available evidence
HSK 6 | conj | used to add a further reason or consideration, strengthening a previous point
HSK 5 | verb/noun | to assign resources, tasks, or roles according to a plan
10 structured lessons for JLPT N4: from reviewing N5 basics to complex grammar, passive/causative, conditionals, and practical reading/writing. Each lesson includes dialogues and exercises.
JLPT N4 kanji 雪 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 5 | noun | ozone layer, the stratospheric layer that absorbs ultraviolet radiation
JLPT N4 kanji 夏 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 6 | verb/adj | to concentrate meaning into few words; dense and profound in expression
HSK 4 | verb | to disappear or vanish from sight or existence
HSK 3 | adjective | strict, rigorous, demanding — high standards with no exceptions
HSK 1 | noun | chair, seat
HSK 4 | grammar pattern | expressing preference by rejecting one option in favor of another
HSK 3 | preposition | toward, in the direction of — indicates direction of movement
HSK 1 | noun/verb | learning or studying; the act of acquiring knowledge
N1 grammar: legal prescription formula 〜ものとする ('it is hereby stipulated that') and hypothetical assumption 〜ものとして ('assuming for this purpose that'). Legal, academic, formal contexts.
HSK 3 | noun | purpose, goal, aim, objective
HSK 2 | verb | verb-object compound meaning to arrive or show up after the expected time
HSK 5 | noun/verb | initiative; proposal; to put forward a proposal or call to action
HSK 5 | verb | to trace back to a cause; to attribute; to come down to
10 structured N1 lessons for advanced learners: classical Japanese roots, authentic text reading, four-character idioms, literary grammar, legal Japanese, academic writing, nuance distinctions, listening, and exam strategy.
Esperanto vocabulary for foods, drinks, cooking, meals, and restaurant situations.
HSK 4 | verb | pulling someone's attention or interest toward something
HSK 4 | noun | a positive result obtained through effort
HSK 5 | noun | economic activity driven by digital technologies and data
HSK 5 | verb/noun | to strive against others for a goal; rivalry
HSK 5 | verb | deliberate increase in scope, scale, or size
HSK 2 | verb + result complement | indicates writing is fully completed
小 (small): 3 strokes, JLPT N5. On: ショウ. Kun: ちい-さい、お、こ.
HSK 3 | noun | the host of a place or owner of something
Complete guide to Sindarin soft mutation: all 11 initial consonant changes, every trigger category, and 30+ worked examples from Tolkien's texts.
中 (chuu/naka): inside. JLPT N5 kanji.
HSK 3 | noun | an interconnected system; the internet
HSK 5 | verb / noun | to assess or appraise systematically; an evaluation
HSK 4 | noun | lawyer; attorney
HSK 1 | noun | window
HSK 2 | noun | father's mother (paternal grandmother)
Learn to talk about what you like doing and what skills you have using 喜欢 and 会.
JLPT N4 kanji 首 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 6 | verb | to provide a brief but comprehensive overview of a topic
複雑な (ふくざつな / fukuzatsu na): complicated. N4 level Japanese vocabulary.
HSK 3 | verb | to change, exchange, replace, or swap one thing for another
Achieving 母语水平 in Chinese production: natural fluency, idiomatic authenticity, and the final gap between expert non-native and native output.
〜られる (passive): JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 4 | verb/noun | to comfort, to console; comfort, consolation
HSK 3 | noun | a particular side or dimension of a topic or issue
We start with a string s. We may delete exactly x characters, keeping the remaining characters in their original order. The resulting string is a subsequence of s. Inside that resulting string, we look for as many non-overlapping occurrences of a pattern p as possible.
Learn 〜にかかわらず to express that something applies regardless of a condition or variable, in formal Japanese.
Learn to talk about your hobbies, ask yes/no questions, and use the -ad- suffix for ongoing activities.
HSK 3 | noun | the material or substance contained within something
We are trying to win an election in a city with a known set of voters. Each voter currently supports a candidate numbered from 0 upwards, and each has a cost to bribe them to vote for you. Candidate 0 is ourselves.
HSK 5 | adjective | urgent; pressing; in a hurry
虫 (チュウ/むし): insect; bug; worm. JLPT N5 essential kanji.
Classical-influenced formal patterns in modern Chinese — 以...之名, 为所欲为, 无论如何, and the grammar of extreme and universal quantification
HSK 5 | noun | a set of rules or instructions for solving a problem or performing a computation
HSK 5 | noun | a biological classification grouping organisms that can interbreed
Master the dual use of 〜ため(に): purpose ('in order to') vs. cause ('because of / due to') — one of the most important distinctions at N3.
Learn to link ideas, sentences, and paragraphs using coordinating and subordinating conjunctions and discourse markers for coherent, flowing Esperanto.
HSK 2 | adverb | together, at the same time, jointly
HSK 5 | verb / noun | to sum up; to draw conclusions; a summary
HSK 3 | verb | to depart, set off, or start a journey
HSK 5 | noun | media; mass media; communications industry
HSK 1 | number | the digit six; considered lucky in Chinese culture
HSK 4 | conjunction / preposition | indicating cause or reason
HSK 3 | verb | to stay, to remain; to leave behind, to keep; to leave a message
Talk about what happened using the completion marker 了, negate past actions with 没有, and use 刚才 for recent events.
N1 grammar: comparing the literary particle 〜すら with the more neutral 〜でさえ and casual 〜でも for 'even.' Register hierarchy, emphatic force, and the unique 〜さえ〜ば conditional.
HSK 6 | n | the line or path of one's thinking; a way of approaching a problem
HSK 3 | adverb | on time, punctually, according to schedule
HSK 1 | noun | clothes, clothing, garment
Complete N3 kanji reference: 184 new kanji beyond N4, organized by meaning category. Includes stroke counts, on-yomi, kun-yomi, example words, and tricky reading notes.
Build the political, economic, and media vocabulary needed to read authentic Esperanto news sources at B2 level.
HSK 3 | noun | an opportunity or favorable occasion to do something
Understand the structure, expectations, and preparation strategies for the KER C2 (Komuna Eŭropa Referenckadro C2) Esperanto examination held at the Universala Kongreso.
HSK 3 | adverb | continuously, all along, straight — describes uninterrupted action or direction
下 (ka/ge/shita): below. JLPT N5 kanji.
Formation of present and imperfect subjunctive for all conjugations, purpose clauses with ut/ne, indirect commands, and an introduction to sequence of tenses.
JLPT N4 kanji 館 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 3 | noun | a tool or instrument used to accomplish a task
Sindarin words for emotions, philosophical concepts, and states of being — including the key philosophical terms estel, amarth, and gûr that Tolkien embedded in his Elvish worldview.
Learn environmental vocabulary and the grammar patterns 越来越 + adj and 变得 + adj to describe change over time, a key pattern in discussing environmental issues.
HSK 1 | adjective, adverb, interjection | good, well, OK, sure
N1 grammar pattern 〜とあって: because of the special circumstance of X, the situation/behavior Y naturally follows. Literary and formal register.
HSK 1 | verb | to hit, to strike, core of many action phrases
HSK 3 | noun | the way something is facing or moving
HSK 1 | adverb | still (ongoing), also/in addition, even more
Essential Japanese vocabulary for school, university, studying, clubs, exams. N1 level reference with readings, romaji, and examples.
HSK 5 | noun/verb | advancement or improvement in ability, knowledge, or society
Achieving mastery-level competence in domain-specific Chinese across multiple professional registers.
N1 grammar pattern 〜ずにはおかない: expressing that something inevitably happens or that the speaker will certainly do something — 'cannot help but, will surely, is bound to.'
HSK 4 | noun | a deeply held idea, concept, or way of thinking
HSK 3 | verb | to express, to indicate, to show; to mean, to represent
N1 grammar pattern 〜ないではすまない: modern variant of 〜ずにはすまない. Cannot avoid doing X due to social/moral necessity. Slightly more contemporary in feel.
HSK 3 | adverb | indicates a quality exceeds the normal or expected level
HSK 1 | verb | to see, to catch sight of
JLPT N4 kanji 秋 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 3 | adverb | describes the normal or typical state of affairs
HSK 3 | noun | an organization or place of work, also a unit of measurement
HSK 6 | verb/noun | to synthesize existing research; a comprehensive literature review
HSK 3 | noun/verb | personal experience gained through living
HSK 3 | adjective / noun | truthful, not deceiving; the quality of being honest
HSK 4 | adjective / noun | patient; patience
N1 grammar pattern 〜といえども: formal concessive expressing 'although/even if/even though X, nevertheless Y.' Used in formal speeches, legal texts, academic writing.
HSK 1 | particle | softens suggestions and expresses tentative assumptions
Learn 〜とたんに (the moment ~, just as ~) — the N3 pattern for instantaneous, unexpected change immediately following an action.
HSK 1 | pronoun | third-person singular feminine pronoun
Esperanto vocabulary for animals, plants, weather, geography, and environmental topics.
HSK 4 | preposition / verb | passing through or using a method
HSK 4 | adjective | formal, official, or done according to proper procedure
JLPT N4 kanji 場 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 3 | noun | ears, the organs of hearing
HSK 3 | verb | to move upward or forward by springing off the ground
HSK 5 | noun / adjective | peace; peaceful
HSK 5 | adjective | displaying initiative, enthusiasm, or a constructive attitude
HSK 3 | noun | a division of the academic year
HSK 5 | noun/verb | to export goods; an exit or way out
話 (wa/hana-su): talk. JLPT N5 kanji.
Move from language learner to active community contributor through writing, teaching, volunteering, hosting, and building online and local Esperanto culture.
HSK 4 | noun | money available for investment or use in a project
HSK 3 | noun | the number of people in a place
HSK 3 | adverb | describes a recurring pattern observed from experience
HSK 6 | adj/noun | organized and comprehensive; a structured system
HSK 4 | verb / adjective | fulfilling a need or feeling content
HSK 5 | noun | the principles of valid reasoning or the internal consistency of an argument
HSK 6 | n | the highest form of human intelligence — the capacity to use knowledge and experience with sound judgment
HSK 5 | noun | a firmly held belief or conviction that guides one's actions
HSK 5 | verb | to form; to come into being; to develop into
HSK 4 | noun | an online website or web presence
HSK 5 | verb/adjective | to make something more complete or perfect; fully developed
HSK 4 | conjunction | expressing unconditional situations
HSK 4 | n | a legally recognized member of a state
HSK 5 | verb/noun | to manage, administer, or oversee; management
Consolidate all B2 grammar and vocabulary, assess your current competency level, identify common B2 errors, and plan your pathway to C1.
HSK 4 | adjective | referring to a specific, pre-determined thing rather than a general one
HSK 5 | verb/noun | natural or observable change in state or condition
HSK 3 | adjective/verb | feeling nervous or embarrassed in social situations
だんだん (だんだん / dandan): gradually. N4 level Japanese vocabulary.
N1 grammar pattern 〜ともあろう: expressing indignation or disbelief that someone of high standing has done something unworthy of their position.
Sindarin subject pronouns: independent forms for emphasis and suffix forms attached to verbs; all 6 persons with attested examples.
Complete N3 vocabulary reference: ~2,250 new words beyond N4, organized by semantic domain. Abstract nouns, workplace terms, academic vocabulary, emotional states, and more.
HSK 4 | noun | the deeper value or significance of something
HSK 2 | noun | a favorable moment or circumstance to do something
We need to print an n × n chessboard using characters. White squares are represented by . and black squares by . The square in the top-left corner is white, and colors alternate both horizontally and vertically exactly as on a real chessboard.
HSK 4 | verb | to stop pursuing or holding onto something
HSK 1 | adjective | cheap, inexpensive, affordable
HSK 1 | adjective, adverb | many, much, more, a lot
HSK 8 | noun | materials science; the interdisciplinary study of material structure, properties, and applications
HSK 1 | particle | marks possession, noun modification, and nominalization
HSK 4 | n/adj | free time activities; relaxed and casual
HSK 2 | adjective | describes doing something carefully and with full attention
HSK 3 | adjective/verb | widely spread or currently trendy
Master Japanese passive (〜られる/〜れる), causative (〜させる/〜せる), and causative-passive (〜させられる) forms. Includes full conjugation tables for 10 verbs, the suffering passive, and three dialogues contrasting active/passive/causative perspectives.
HSK 4 | verb / noun | making or experiencing a shift in condition or nature
学 (gaku/mana-bu): study. JLPT N5 kanji.
HSK 3 | noun | fruit — edible fruit of a plant
HSK 2 | noun | direction word indicating the left-hand side
Sindarin time vocabulary: days, months, seasons, the Elvish calendar (Reckoning of Rivendell), time expressions, and attested Elvish date references.
HSK 5 | noun | the arrangement of people or things according to rules and structure
HSK 4 | noun / adjective | the system of production and consumption, or economical/thrifty
年 (year): 6 strokes, JLPT N5. On: ネン. Kun: とし.
HSK 4 | verb / noun | regulating or restraining something within bounds
HSK 5 | verb | deliberate increase in strength, intensity, or effectiveness
HSK 2 | measure word | counts occurrences or instances of repeated actions
読 (doku/toku/tou/yo-mu): read. JLPT N5 kanji.
HSK 3 | verb/noun | to travel for leisure; tourism
赤 (red): 7 strokes, JLPT N5. On: セキ、シャク. Kun: あか、あか-い.
HSK 5 | adjective | free from corruption; upright and honest in one's conduct
HSK 5 | adjective | delicately complex in a way that is difficult to analyze or describe
JLPT N4 kanji 強 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 5 | noun | evidence or grounds used to support an argument
HSK 3 | adjective | young in age or youthful
HSK 5 | noun | typhoon, tropical cyclone
HSK 4 | noun/verb | change, transformation
HSK 5 | noun/adjective | the provision of medical treatment and health services
HSK 5 | verb | to promote and spread something admirable, especially cultural or moral values
HSK 2 | verb phrase | to make or place a telephone call
HSK 3 | grammar pattern | additive correlative conjunction expressing escalation
口 (mouth, opening): 3 strokes, JLPT N5. On: コウ、ク. Kun: くち.
HSK 4 | noun | a serious disease caused by uncontrolled cell growth
HSK 4 | noun | an individual belonging to a group or organization
HSK 3 | noun / verb | a plan or program; to plan something in advance
Master the plain (casual) forms of verbs, adjectives, and the copula to speak naturally with friends and lay the foundation for advanced N4 grammar structures.
HSK 4 | conjunction | therefore; so; as a result
Master Sindarin pronunciation: all 6 vowels with IPA, 6 diphthongs, key consonant digraphs (ch, dh, th, lh, rh), and the stress rule system.
Consolidate all B1 grammar and vocabulary, review CEFR B1 can-do statements for Esperanto, and prepare for the B2 level.
HSK 8 | noun | integrated circuit; a set of electronic circuits on a single chip of semiconductor material
HSK 2 | adverb | introduces a correction or revelation, contrasting with what was assumed
HSK 3 | grammar pattern | softens a comparative or evaluative statement with 'a bit'
Learn to give and follow directions in Chinese using left, right, straight, and the 从...到... structure.
Learn 〜ことなしに (without doing ~) — the formal N3 pattern expressing that one action is performed without the prerequisite action.
HSK 1 | noun | informal word for mother
HSK 5 | noun/adjective | a person or organization's public image; also vivid/concrete representation
HSK 5 | conjunction | even if, even though
HSK 5 | noun/adjective | a state of worry and unease; anxiety
HSK 5 | noun/adjective | academic major; professional field; specialized
HSK 5 | verb/noun | to have an effect on someone or something; the effect produced
車 (car, vehicle): 7 strokes, JLPT N5. On: シャ. Kun: くるま.
火 (ka/hi): fire. JLPT N5 kanji.
HSK 3 | verb | to believe, to trust, to have confidence in
HSK 4 | noun | a real-time audio and video call over the internet
HSK 4 | adjective | easy to see or notice; clearly apparent
HSK 1 | adverb | indicates that an action or state has been completed before the moment of speaking
HSK 5 | noun | administrative regulations and legal rules issued by authorities
HSK 2 | adjective/noun | incorrect; a mistake or error
HSK 2 | noun | beer, an alcoholic drink made from fermented grain
Learn 〜うちに (while, before it's too late) — the N3 temporal pattern expressing a window of opportunity.
HSK 4 | verb / noun | standing in for or symbolizing someone or something
HSK 3 | noun | a fundamental rule or standard guiding behavior
HSK 3 | verb/adjective | to use more of something than is needed
The archaic Doriathrin dialect of Sindarin: features from Tolkien's First Age writings, preserved archaisms, and how Doriathrin differs from Third-Age Sindarin.
Master the 连...也 and 连...都 constructions to express emphasis and unexpectedness at B1 level.
Learn to ask prices, talk about money, and use the 多少钱 question structure in shops and markets.
やっと (やっと / yatto): finally, at last. N4 level Japanese vocabulary.
N1 grammar pattern 〜ないでもない: expressing a mild, hedged affirmation — 'not entirely unable to, could say that, not without some degree of.'
Cirth — the angular runic script native to Sindarin, adopted by Dwarves as Angerthas Moria and Erebor — with letter tables and famous inscriptions.
HSK 5 | verb/noun | to report (news); a news report or coverage
HSK 5 | noun | a specific article or provision within a contract, law, or agreement
Learn the counter-intuitive grammar of fear clauses and how Latin reports commands indirectly.
HSK 4 | noun | a favorable moment or occasion
友 (friend): 4 strokes, JLPT N5. On: ユウ. Kun: とも.
HSK 3 | adjective | suitable, appropriate, fitting — describes a good match
HSK 3 | noun | the global internet network
HSK 3 | adjective | simple, easy, uncomplicated, plain
N1 grammar pattern 〜ならいざしらず: conceding that X might be understandable, but the actual situation (Y) is not acceptable or expected.
Learn to tell the time, name days of the week and months, and use temporal adverbs and the correlative -am series.
HSK 3 | noun/adjective | customs and practices passed down through generations
HSK 4 | verb | to communicate or signal a feeling, attitude, or meaning
HSK 3 | noun | temperature — measurable degree of heat or cold
HSK 4 | verb | to move upward in level or amount
三 (san/mit-tsu): three. JLPT N5 kanji.
HSK 4 | noun / adjective | rationality or being rational and reasonable
HSK 5 | noun | a course of study; a curriculum or program of classes
HSK 5 | verb | to breed, to nurture, to cultivate (plants, organisms, or people)
HSK 4 | adjective | proceeding without obstacles or difficulties
HSK 5 | noun | predicament, difficult situation, dilemma, plight
Critical analysis of China's media ecosystem — official media, self-media, public opinion dynamics, and the vocabulary of media criticism
HSK 4 | noun | a business organization engaged in commercial activity
HSK 2 | noun | money, currency, or a sum of money
HSK 2 | noun | a city or urban area
HSK 3 | noun | a multi-storey building, or a floor within one
Sindarin sentence structure: VSO order, subordinate clauses, relative clauses, negation, and complex sentence examples.
HSK 5 | noun | the network of physical devices connected to and communicating via the internet
N1 grammar pattern 〜をもって: expressing means or instrument (by means of, with), or a temporal boundary (as of, with this).
HSK 3 | noun | vegetables, the specific category of edible plants
HSK 4 | verb / noun | to encourage; encouragement
Learn 〜にあたって and 〜に当たり to mark significant occasions — openings, ceremonies, launches — in formal Japanese speeches and documents.
HSK 3 | verb | to abandon or stop pursuing something
HSK 5 | conjunction | used to introduce an additional point in a series of arguments
Vasya starts with a certain number of sock pairs. Every morning he uses exactly one pair, and that pair is gone forever at the end of the day. There is one special event: after every day whose number is a multiple of m, his mother buys him one new pair of socks.
HSK 3 | noun | a number used to identify or contact something
HSK 3 | verb / preposition | to pass (an exam); through the means of; by way of
HSK 3 | verb | to clean up, to sweep — used for cleaning a room or area
Learn body parts, health vocabulary, and how to describe symptoms and visit a doctor in Esperanto.
HSK 3 | noun / adjective | standard, criterion, norm; standardized
HSK 3 | noun | ability, capability, skill, capacity
Learn 〜からすると, 〜からすれば, and 〜からみると to express judgments and inferences from a specific perspective or standpoint.
HSK 3 | verb / particle | to wait; and so on, et cetera (sentence-final particle)
HSK 6 | verb/adj | to refine to a high degree; precise and economical in expression
HSK 3 | modal verb | must, have to — expresses strong obligation or necessity
HSK 3 | adjective / adverb | possible; maybe, possibly — covers both uses
HSK 2 | phrase | standard response to thanks; also means 'don't be polite'
Learn how to compare people and things using 比, 没有, and 一样 in natural Chinese.
HSK 5 | noun | a public forum for discussion; an online discussion board
HSK 5 | verb / noun | to demonstrate the truth of something; a document proving a fact
〜なら: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 5 | noun/preposition | basis; grounds; according to; based on
HSK 4 | noun | designer; architect
Master the essentials of Japanese introductions, polite greetings, and basic sentence structure to start your first conversations with confidence.
HSK 4 | noun | a raised surface or an opportunity base for activity
HSK 1 | particle | marks completed actions (post-verb) or new situations (sentence-final)
HSK 5 | noun | large-scale digital data sets and their analysis
HSK 4 | noun | a detailed plan or scheme for achieving something
HSK 5 | noun | sovereignty; sovereign rights
〜たら: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 5 | verb | to plan carefully and strategically; to devise a scheme
HSK 2 | verb/complement | result complement: action is completed/finished
子 (shi/su/ko): child. JLPT N5 kanji.
HSK 3 | verb | to have a fever, to run a temperature
HSK 5 | adjective/verb | not changing or fluctuating; to make or become steady
N1 grammar pattern 〜にほかならない: emphatic definitional assertion — X is nothing other than Y. Used for declarative identification, formal analysis, and rhetorical emphasis.
HSK 4 | noun | a person with deep knowledge in a specific field
Learn Chinese family vocabulary, how to express possession with 的, and how to talk about what you have using 有.
HSK 5 | adjective | advocating extreme or rapid change; radical
HSK 3 | verb | to be acquainted with someone; to recognize or identify
HSK 4 | noun/verb | professor; to teach, to instruct
HSK 5 | verb | to foster, incubate, or bring forth something new
HSK 1 | pronoun | interrogative pronoun asking about a person's identity
HSK 3 | noun | a system of communication used by a group of people
HSK 4 | noun/verb | taking action in advance to stop something bad from happening
HSK 5 | noun | brand, brand name
HSK 2 | noun | a question to ask, a problem to solve
HSK 4 | noun | a pattern, mode, or established model of operation
HSK 4 | noun | opponent, rival, competitor
HSK 5 | verb / noun | to demonstrate through argument; logical proof
HSK 5 | verb / noun | to analyze; analysis
HSK 5 | verb/noun | deliberate alteration of something existing
HSK 4 | adjective | rough, not detailed, approximate
We are given an array of integers. In one move, we choose a value x that is currently present, earn x points from deleting one occurrence of it, and all occurrences of x - 1 and x + 1 disappear from the array as a consequence.
HSK 4 | verb / noun | setting boundaries on what is allowed or possible
HSK 5 | verb | to check and confirm the accuracy or truth of information
HSK 2 | noun | question on a test, problem, topic
HSK 5 | verb | to promote widely; to popularize; to roll out on a larger scale
We are given a tree with n nodes, where each edge has a positive length. A tree guarantees that there is exactly one simple path between any two nodes.
HSK 4 | verb/noun | a difficult situation that tests one's abilities
駅 (station): 14 strokes, JLPT N5. On: エキ. Kun: none.
HSK 4 | verb / noun | offering analysis or opinion on something
HSK 4 | verb/noun | to clarify or describe something; a written explanation or instruction manual
HSK 5 | noun | underlying fundamental principle or law governing operation
An immersive study of Esperanto's imagined homeland — its literature, music, film, online culture, native speakers, and annual congress life.
HSK 5 | noun | the exchange of goods and services between parties or nations
HSK 3 | grammar pattern | describes how well or intensely an action is performed
HSK 3 | noun | reason, cause — what explains why something happened
HSK 5 | noun | newly reported events or information; news media
We have a vector of energies indexed by all bitmasks of length m. Since n = 2^m, every vertex can be identified with an m-bit number. One transformation step is linear.
N1 grammar pattern 〜に値する/〜に値しない: expressing that something merits/deserves X (positive evaluation) or does not merit/deserve X.
A rigorous analysis of Esperanto's participial system — the -ata/-ita controversy, compound tenses, participial nouns, and verbal adjectives with full semantic precision.
Learn how to express passive constructions in Chinese using 被, 让, and 叫, and understand when passive voice is appropriate in Mandarin.
HSK 3 | noun | vinegar (condiment); colloquially, jealousy in a romantic context
HSK 3 | noun | a period of time away from work or school
HSK 3 | verb | to feel regret about a past action
Complete JLPT N1 exam prep guide: format, pass rates, section strategies, 1-year study plan from N2, 100 most-tested grammar patterns, keigo at N1, and common exam traps.
HSK 1 | verb | movement toward the speaker's location
JLPT N4 kanji 銀 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
Learn 〜どころか (far from ~, let alone ~) — the N3 pattern for expressing strong unexpected contrast or reversal.
HSK 2 | conjunction | therefore, so, that is why (result connector)
Conjugate Latin verbs in the present tense across all four conjugations plus the irregular verb esse.
HSK 5 | conjunction | introduces a concessive clause acknowledging a fact while contrasting it
HSK 3 | verb | to successfully obtain, receive, or achieve something
HSK 5 | verb/noun | to study independently, teach oneself; self-directed learning
We are given a grid where some cells are marked X. These are the cells that were painted at least once. Originally, a rectangular brush of size h × w was placed somewhere on the grid. After that, the brush was moved several times.
HSK 5 | verb / noun | to modify or fine-tune; a modification or revision
We are given a single integer n, where n ≥ 12. The task is to find any two composite numbers whose sum is exactly n. A composite number is an integer greater than 1 that has at least one divisor other than 1 and itself.
HSK 1 | verb | to return, to go back, to come back
HSK 3 | preposition | used to exclude something from a statement, or to add to it
HSK 5 | n/v | the act or art of telling a story; narrative mode in literature
HSK 1 | verb | to sit, to take (a vehicle)
HSK 4 | noun | a person who manages or administers an organization or team
HSK 4 | noun | a specific location or space for a particular purpose
HSK 3 | verb | to mistakenly believe something to be true
HSK 5 | adjective | complex, complicated, intricate
Comprehensive guide to HSK 1 grammar: basic sentence structures, question formation, negation, particles, and time expressions with examples.
HSK 3 | noun | a question that needs an answer, or a problem that needs a solution
HSK 1 | number | the digit nine; associated with longevity and imperial power
HSK 4 | verb/noun | creating something new or improving existing things
We are given an $n times n$ grid that is mostly empty, except for a small number of cells that are already fixed as either x or o. Every other cell must be filled with one of these two characters.
Full tables for personal, reflexive, and demonstrative pronouns; the cum enclitic; and the absence of a true 3rd-person personal pronoun.
HSK 5 | verb / noun | to create; to bring into existence
HSK 4 | noun | a medical procedure performed on the body
Learn to describe personality and character using degree complements, the 极了 intensifier, and a rich set of adjectives for temperament and behavior.
Learn food and drink vocabulary and how to order, express preferences, and use the infinitive with modal verbs like voli and ŝati.
HSK 2 | adjective/noun | two readings: chā (difference/gap) and chà (substandard/lacking)
HSK 1 | adverb / adjective | intensifier meaning really or genuinely
HSK 1 | pronoun | first-person plural pronoun
HSK 2 | adjective / noun | female gender; a woman or girl
HSK 3 | verb / noun | to plan, intend to do something; a plan or intention
HSK 1 | phrase | confirms a factual statement; equivalent to 'yes, that is correct'
Navigate Chinese economic discourse with confidence, mastering the vocabulary of finance, macroeconomics, and business reporting.
二 (ni/futa-tsu): two. JLPT N5 kanji.
HSK 1 | noun | restaurant, eatery
HSK 5 | verb / noun | to manufacture; to fabricate; to produce
HSK 3 | conjunction | concessive condition — even in an extreme hypothetical case, the result still holds
HSK 5 | verb | to promote; to facilitate; to further
HSK 5 | noun | fan, follower; also glass noodles
HSK 4 | verb/noun | to gather information systematically through research or examination
The task revolves around finding two integers, l and r, such that the sum of all digits from l to r modulo a given number a produces an edge case for a buggy implementation. The function f(x) denotes the sum of the decimal digits of x.
HSK 4 | noun | a competitor or rival in business, sports, or other fields
Mastering literary devices, aesthetic vocabulary, and the expressive architecture of Chinese literary prose and poetry.
七 (shichi/nana-tsu): seven. JLPT N5 kanji.
安 (cheap, peaceful): 6 strokes, JLPT N5. On: アン. Kun: やす-い.
HSK 3 | verb | to allow, permit, or give permission for something
HSK 1 | greeting phrase | farewell used when you will meet the person the next day
HSK 1 | noun | computer
Engage with Chinese environmental policy discourse, mastering the vocabulary of climate, sustainability, and environmental governance.
Mastering the full Chinese register continuum: shifting between formal, informal, written, spoken, and specialized registers with native-level control.
HSK 4 | noun / verb | plan; to plan; to intend
HSK 4 | n | a legal or moral entitlement to do or have something
HSK 3 | adjective / adverb | special, particular; especially, particularly
N1 grammar pattern 〜にして: formal dual predication ('being X and at the same time Y') and the expression of a critical stage ('at that very point of'). Appears in literary and formal contexts.
HSK 6 | n | a main claim or point of argument in reasoning or debate
HSK 1 | noun | the period from roughly 8 a.m. to noon
Build the vocabulary and grammar needed for travel situations: booking accommodation, navigating transport, and describing journeys using 从...到 and related patterns.
HSK 4 | verb / noun | having an effect on something or someone
HSK 5 | noun/verb | goods or the act of bringing goods into a country
HSK 4 | verb | to investigate the unknown with curiosity and intent
HSK 1 | adjective | few, a little, not many
HSK 5 | adjective | pushed to the margins; excluded from mainstream society
HSK 1 | noun | table, desk
A deep study of Esperanto's literary tradition, major authors, poetic devices, and the evolution from Zamenhof's early verse to modern prose.
HSK 4 | noun / verb | a regular behavior or the state of being accustomed
HSK 3 | noun/adj | the system of production and consumption; economical
HSK 4 | adjective | of great importance or danger; causing significant harm or concern
先 (before, ahead): 6 strokes, JLPT N5. On: セン. Kun: さき.
HSK 3 | verb | to bring up a topic or carry something upward
Principles and practice of Chinese-to-English translation at C1 level — equivalence, cultural transfer, and the challenges of formal register
HSK 3 | noun | climate — long-term weather patterns of a region
HSK 4 | noun | thinking; thought; way of thinking
The problem presents a simple scenario: George and Alex want to find a dormitory room together. Each room in the dormitory has a current occupancy and a maximum capacity.
HSK 1 | noun | room (inside a building)
HSK 1 | adjective | pleasant to look at, applies broadly to people and things
HSK 6 | n | the course or advancement of a process, especially a large-scale historical or social development
HSK 5 | adjective/adverb | covering all aspects, thorough and complete
HSK 3 | adjective | putting in consistent hard work and effort
HSK 4 | verb/noun | to assess a situation and reach a conclusion; the act of judging
HSK 5 | noun | a raised surface or digital/organizational structure enabling activity
HSK 4 | noun/verb | money that is spent or paid out
HSK 5 | noun / verb | systematic change to improve a system or institution
HSK 5 | adjective | existing within something, inherent and not externally imposed
目 (eye): 5 strokes, JLPT N5. On: モク、ボク. Kun: め.
HSK 5 | conjunction | introduces an additional and often stronger reason or argument
HSK 5 | noun | the beliefs, customs, arts, and social institutions of a group; also education and literacy
HSK 5 | verb / noun | to deduce; to draw a conclusion from evidence
金 (kin/kon/kane): gold. JLPT N5 kanji.
HSK 5 | noun | the substance or information contained within something
HSK 5 | noun | natural, human, or material resources available for use
HSK 4 | adjective / verb | stable and steady; to make something stable
HSK 4 | verb/noun | the ability to remember or a specific memory
HSK 3 | adverb | never — habitual negation applying to all past and present time
HSK 4 | noun / conjunction | the final outcome of a process, or a connective meaning consequently
HSK 4 | noun | a distinguishing characteristic, feature, or trait
HSK 4 | phrase | formal phrase to introduce a summary or conclusion
HSK 5 | verb/adjective/noun | bringing together into a single coherent whole
HSK 2 | noun | direction word indicating the right-hand side
HSK 4 | conjunction | expressing logical consequence
HSK 5 | verb / noun | to identify a disease or problem; the diagnosis
HSK 4 | noun | a geographical or administrative area
JLPT N4 kanji 足 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 6 | adv | used to concede a point before introducing a qualification or counterargument
Learn 〜てからというもの to express a fundamental, sustained change that began at a specific turning point.
HSK 1 | noun | teacher, instructor
HSK 5 | verb | to expound; to set forth in detail
HSK 4 | verb / noun | to praise; to commend; praise
We are given a collection of laptops. Each laptop has a price and a quality value. Prices are all distinct, and qualities are all distinct. Dima believes that a more expensive laptop must always have better quality. Alex claims that this is not necessarily true.
赤 (seki/shaku/aka): red. JLPT N5 kanji.
Step-by-step Esperanto learning roadmap from absolute beginner to complete mastery.
HSK 2 | particle | expressive sentence-final particle with four tonal variants
Learn how to greet people, introduce yourself, and use Esperanto's present-tense verb ending and personal pronouns.
HSK 1 | ordinal number | the ordinal prefix 第 plus one; first in a sequence
〜てみる: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 4 | adjective | of great consequence or value
HSK 5 | verb/noun | systematic improvement for better performance or efficiency
HSK 3 | adv/adj | to a total or absolute degree
HSK 3 | verb/noun | to make a journey to another place
N1 grammar pattern 〜に足る/〜に足らない: expressing that something is (or is not) worthy of, sufficient for, or meeting the standard required for X.
HSK 3 | noun | class, classroom group; work shift; scheduled service
出 (shutsu/sui/de-ru): exit. JLPT N5 kanji.
Describe changing weather, understand forecasts, and use 越来越 to express gradual change in Chinese.
HSK 5 | noun | the release of carbon dioxide into the atmosphere from human activities
Navigating the formal connective chains and specialized vocabulary of Chinese political and social commentary at C1 level
HSK 5 | noun | channels of mass communication such as press, television, and the internet
円 (en/maru-i): yen (currency). JLPT N5 kanji.
Mastering the rhetoric of 答辩: presenting, fielding challenges, and closing arguments in Chinese academic and professional oral defense.
前 (before, front): 9 strokes, JLPT N5. On: ゼン. Kun: まえ.
HSK 5 | n | the angle or standpoint from which something is seen or narrated
HSK 4 | verb | continuing a state or condition over time
HSK 5 | noun phrase | the health of the population managed through organized societal efforts
HSK 2 | verb / noun | to respond to a question; a response or answer
HSK 2 | verb-object phrase | to run or go jogging as an activity
HSK 5 | noun/verb | hypothesis; assumption; to suppose; to assume
だから (だから / dakara): therefore, so. N4 level Japanese vocabulary.
HSK 5 | noun | a legal proceeding; bringing a case to court
Learn 〜ざるを得ない to express unavoidable obligation — you have no choice but to do something, despite reluctance.
HSK 5 | verb / noun | to clarify or illustrate; an explanation or instructions
HSK 4 | noun / verb | going through events that shape understanding
HSK 4 | adjective | honest; truthful; sincere
HSK 6 | verb | to test and confirm the truth or validity of something
HSK 4 | adjective/adverb | referring to the whole rather than parts
HSK 2 | verb / adjective | to suddenly understand a situation; contrast with 懂 for skill-based comprehension
HSK 4 | verb / adverb | to compare; relatively; fairly
Learn how to use 〜を通じて (wo tsujite) to express 'through,' 'throughout,' or 'via' when describing a medium, channel, or continuous span. Includes structure, nuance, examples, and comparisons.
Master the 把 construction to express deliberate actions performed on a specific object, a core pattern in spoken and written Mandarin.
HSK 1 | noun | informal word for father
Master the Latin perfect tense using the 3rd principal part, and distinguish completed from ongoing past action.
HSK 4 | noun | the pressure of blood circulating in the arteries
HSK 3 | verb | to make something smaller in amount or number
下 (down, below): 3 strokes, JLPT N5. On: カ、ゲ. Kun: した、さ-がる、くだ-る.
Discuss the Chinese education system using 通过 + means + V and 依靠, with vocabulary covering 高考, 大学, and graduate study.
Where to find other Sindarin learners and scholars — Discord servers, Reddit, forums, and academic mailing lists.
Dreamoon wants to reach the top of a staircase containing n steps. Every move can cover either 1 step or 2 steps. Among all possible ways to reach exactly step n, we need the number of moves used to be divisible by m. The task is not asking for the number of different sequences.
Essential Japanese vocabulary for feelings, moods, personality, mental states. N1 level reference with readings, romaji, and examples.
HSK 6 | verb | to bear the weight of; to carry (physical load or abstract meaning/responsibility)
Master the vocabulary and structural patterns for formal Esperanto debate, including asserting, conceding, refuting, hedging, and expressing degrees of certainty.
HSK 5 | verb | to investigate deeply; to explore and inquire into the nature of something
HSK 1 | noun | name, given name
HSK 5 | noun | a person's or society's set of moral and ethical values
HSK 3 | noun | an emotional bond or feeling between people
HSK 2 | verb | to communicate information to someone
N1 grammar patterns 〜極まりない and 〜極まる: expressing that something is extremely [negative quality], often used to express outrage, embarrassment, or strong negative evaluation.
HSK 4 | adjective | fully adequate in amount or degree
HSK 5 | noun | the effect that other people or society have on individual behavior and attitudes
Master the -is past tense and -os future tense in Esperanto, and learn essential time adverbs.
HSK 4 | preposition | following a standard or instruction
HSK 1 | greeting phrase | standard farewell expression
HSK 5 | conjunction | therefore, as a result, consequently
HSK 4 | n/v | the state of not succeeding; to not succeed
HSK 5 | noun/adjective | economy, economics; economical, affordable
HSK 6 | n | the cognitive faculty of thinking; a mode or pattern of thought
HSK 2 | adjective | describes something that saves time or effort
HSK 8 | noun | aerospace; the combined fields of aviation and space exploration
HSK 5 | adjective/verb | mature; ripe; fully developed (of people, ideas, or fruit)
HSK 1 | noun / measure word | mouth, opening, bite-sized portions
生 (life, birth): 5 strokes, JLPT N5. On: セイ、ショウ. Kun: いき-る、うま-れる、なま.
休 (kyuu/yasu-mu): rest. JLPT N5 kanji.
HSK 1 | noun | a calendar day; also the word for the sun (formal/written)
Complete treatment of Latin's six conditional types, from simple fact to mixed contrary-to-fact.
HSK 1 | adverb | indicates that something happens later or with more difficulty than expected
Complete Sindarin number system: cardinal numbers 1–1000, ordinals 1st–10th, counting, multiples, arithmetic expressions, and attested number words from Tolkien.
HSK 5 | verb/noun | physical exercise, mental training, or the tempering of character
Learn 〜をめぐって and 〜をめぐる to discuss debates, controversies, and disputes surrounding a topic in formal Japanese.
We are dealing with a dynamic company hierarchy problem. We have n employees, initially without any reporting structure.
HSK 3 | adverb/adjective | happening without warning
HSK 5 | noun | the body of rules and regulations enforced by the state
HSK 5 | verb | to distribute and assign resources or personnel
Sindarin vowel elision: when final and initial vowels contract, apostrophe usage, elision in poetry and prose, and the famous Elvish examples from Tolkien's texts.
Strategies for back-translation and EN-to-CN transfer — localization, terminology handling, and producing natural formal Chinese from English source texts
HSK 3 | adjective | calm and free from noise
HSK 1 | adverb/pronoun | interrogative asking about manner or reason
HSK 5 | noun | a specific instance or example used for analysis, typically in law, medicine, or business
HSK 1 | adjective | cute, adorable, lovable
HSK 2 | adjective/noun | warm-hearted, enthusiastic, passion
HSK 1 | conjunction | introduces a conditional clause
HSK 1 | pronoun | interrogative pronoun for location (northern colloquial form)
HSK 5 | verb / adjective / noun | to deny; to negate; negative
HSK 3 | adverb | indicates an estimate rather than an exact figure
HSK 3 | conjunction | no matter what/how/who — used in 不管...都/也 pattern
An analytical study of Esperanto's linguistic evolution, the Akademio's role, the Ido schism, native speakers, sociolinguistics, and corpus linguistics.
HSK 3 | verb/noun | to grasp the meaning or reasoning of something
HSK 5 | adjective | ahead of the times, at the forefront of development
HSK 4 | adverb | describing a pattern that tends to occur regularly
HSK 3 | noun | luggage, baggage, travel bags
HSK 4 | n | a system of rules enforced by government
HSK 1 | verb, noun | to start, to begin, the beginning
HSK 4 | verb/noun | physical exercise or training to improve health
HSK 6 | n | the structural skeleton or conceptual framework of a system, plan, or argument
HSK 3 | noun | playground, sports field, schoolyard
HSK 5 | verb/noun | a journalist's act of gathering information through interviewing
The permutation contains every value from 1 to n exactly once. We need to determine whether there exist two distinct values a and b such that their arithmetic mean $$frac{a+b}{2}$$ appears somewhere between them in the permutation order.
We are given a line of floors from 1 to n. We start on floor a and we will make exactly k moves. Each move picks a new floor y different from the current floor x, but with a restriction: we are not allowed to move “too far” in absolute distance compared to a forbidden floor…
We are given a long sequence of integers, and we are allowed to pick a subsequence from it. The goal is not just to maximize the length of the subsequence, but to impose a very specific structural constraint on how the chosen elements repeat.
HSK 5 | noun/verb | a hesitating worry or reservation that holds one back from acting
HSK 4 | verb/adjective | to negate, to deny; negative
HSK 1 | noun | question, problem, issue
HSK 4 | noun | a method or way of doing something
Complete katakana chart with all 46 base characters, variants, combination characters, long vowel mark, loanword spelling rules, and common vocabulary examples.
HSK 5 | noun | a person who uses a product, service, or system
HSK 5 | verb | to advocate; to promote; to champion (a cause or idea)
HSK 5 | noun / verb | cognitive understanding; the process of knowing
HSK 3 | noun | magazine, periodical journal
HSK 3 | conjunction | however, but — a softer contrast than 但是, often used in conversation
HSK 4 | noun | the outward shape, structure, or format of something
子 (child): 3 strokes, JLPT N5. On: シ、ス. Kun: こ.
HSK 5 | noun/verb | a clash of opposing forces, interests, or ideas
We are asked to determine if a city with a grid of horizontal and vertical streets is "strongly connected" once each street is made one-way. The city can be visualized as a grid of intersections, with n horizontal streets and m vertical streets forming n × m junctions.
HSK 5 | noun phrase | the practice of protecting and preserving the natural environment
HSK 4 | noun | facts and figures collected for analysis
HSK 3 | noun | condition, requirement, terms, prerequisite
Discuss Chinese history with the vocabulary and grammar of historical narrative, engaging with dynasties, revolutions, and transformative events.
Learn how to express how long actions last and how often they occur, using time-measure complements and frequency adverbs essential for natural Chinese.
HSK 1 | pronoun/determiner | interrogative asking for selection among options
Learn to describe the weather and seasons using adjective predicates and the question 怎么样.
HSK 1 | verb | used to give one's name, to shout, or to ask someone to do something
HSK 5 | verb/noun | to criticize systematically; critique; critical analysis
HSK 3 | noun | a dictionary, especially one that explains words and their meanings
HSK 1 | greeting phrase | expresses gratitude
HSK 4 | adverb | expressing a preference for one option over another
Complete N3 grammar reference: ~200 patterns including conjunctions, conditions, degree expressions, formal structures, and nuanced patterns that distinguish N3 from N4. Each pattern includes structure, meaning, and 3+ example sentences.
Learn 〜に応じて (in response to, depending on, adapted to) — the N3 pattern for expressing deliberate, appropriate adaptation to conditions or needs.
Complete N2 kanji reference: 633 new kanji beyond N3, organized by radical/theme. Readings, meanings, compound words, and common confusion pairs.
HSK 1 | adverb | also, too — adds the same information about the subject
We have two boards of the same size. A move starts by choosing one cell and placing a finger on it. After that, the finger walks through adjacent cells. Every time the finger moves from one cell to another, the two orbs in those cells are swapped.
HSK 3 | noun | a card used for credit-based payment
HSK 4 | noun | distinguishing feature or characteristic
HSK 4 | adjective | being passive or acted upon rather than acting
HSK 3 | noun / preposition | the space or position in the middle of something
HSK 4 | adverb/conjunction | happening together at the same moment
Master six essential Japanese patterns for expressing probability, hearsay, and inference — かもしれない, でしょう/だろう, はずだ/はずがない, らしい, そうだ (two uses!), and ようだ — with a certainty-scale comparison table.
母 (mother): 5 strokes, JLPT N5. On: ボ. Kun: はは.
HSK 1 | adjective | smart, clever, intelligent
HSK 2 | preposition | in order to, for the purpose of
Master all uses of the -n ending: direct object, direction, time measure, and flexible word order.
Learn 〜とともに (together with / as ~ also changes) — the N3 pattern for expressing simultaneous co-occurrence and parallel change.
Sindarin adjectives: post-nominal position, lenition after nouns, plural forms, and the copula 'to be'.
Master the full correlative system with accusative, plural, stacked, and free-choice forms, plus result clauses and temporal subordinators.
HSK 5 | verb / noun | to back or endorse; backing, support
HSK 4 | noun | the power or capacity to have an effect on others
週 (shuu/): week. JLPT N5 kanji.
HSK 3 | verb | to raise, improve, or enhance a level or standard
HSK 4 | verb/noun | to threaten; threat
HSK 4 | noun | the reason or inner drive behind an action
How to create grammatically correct Sindarin place names: element vocabulary, compound patterns, mutation at seam, and 20 worked example name constructions.
HSK 5 | noun | energy, energy resources, power supply
Master the craft of original fiction and poetry in Esperanto, exploiting its unique morphological richness for literary effect.
HSK 3 | noun / verb | success; to succeed, to be successful
HSK 3 | noun | a numerical figure or digit
HSK 4 | adverb | expressing reluctant necessity
HSK 5 | verb / noun | to sum up; to induce from particulars
HSK 3 | verb | to lose or accidentally leave something behind
HSK 4 | noun | a planned project, item, or program
HSK 1 | number | the digit ten; the base unit of the decimal system in Chinese
Learn to express where things and people exist using あります and います, with the に and で particles for location.
HSK 5 | noun | underlying operational mechanism or systemic process
HSK 3 | noun/adjective | a deep sense of contentment and well-being
HSK 1 | adverb | negates verbs and adjectives for present, future, or habitual
HSK 3 | noun | the monetary value of a product or service
HSK 3 | noun | impression, mental image left by someone or something
東 (east): 8 strokes, JLPT N5. On: トウ. Kun: ひがし.
HSK 1 | number | the digit zero; also used between digits in spoken numbers
HSK 4 | adjective | particular and detailed rather than vague or abstract
All Sindarin negation forms: ú- prefix, avo imperative negation, baw prohibition, and law/lav forms — with attested examples and contextual usage.
Critical thinking patterns in Chinese — the vocabulary and structures of analysis, evaluation, and reasoned judgment at C1 level
HSK 5 | noun | a formally concluded and ratified agreement between states
HSK 5 | n | a group of writers sharing common aesthetic principles or historical context
Sindarin past tense formation: strong past (vowel change + nasal infixion) for primary verbs and weak past (-ant suffix) for A-stem verbs, with all 6 persons.
Build the core vocabulary and discourse structures needed to engage with academic texts, arguments, and analytical writing in Chinese.
HSK 4 | noun | a person with specialized professional expertise
HSK 3 | verb | to solve, resolve, or settle a problem or difficulty
HSK 4 | verb/noun | an official notification or briefing
The problem defines a linear sequence $fn$ with two initial values $x = f1$ and $y = f2$, and a recurrence that repeats every six terms: $f{n} = f{n-1} - f{n-2}$.
HSK 5 | verb/noun | to experience firsthand; personal experience; user experience
HSK 2 | measure word | counts complete pieces of written or composed work
We are given an array of distinct integers. The task is to determine whether the entire array can become sorted in increasing order after reversing exactly one contiguous segment. If it is possible, we must print "yes" and one valid segment [l, r] whose reversal sorts the array.
HSK 3 | adverb / conjunction | even, even to the point of — used to introduce a surprising extreme
Exploring how classical Chinese grammar and vocabulary permeate formal Modern Chinese — 所 nominalization, 之 particle, and registers of elegance
Learn 〜わけにはいかない (can't possibly do ~, not in a position to ~) — expressing social, moral, or practical impossibility.
HSK 3 | verb | to hold the opinion that; to consider; used to state personal views
HSK 6 | verb | to sort through and organize systematically; to untangle
HSK 4 | noun | a group of people working together toward a shared goal
We are given a line of ants, each with an integer strength. Mole wants to observe fights among ants in specified contiguous segments of the line. For a given segment from position l to r, every ant fights every other ant within that segment.
HSK 4 | noun | physical space, room, or conceptual space for something
HSK 3 | adjective | feeling emotional pain or sadness
鳥 (チョウ/とり): bird. JLPT N5 essential kanji.
HSK 3 | adverb / adjective | certainty or a fixed amount
Sindarin nasal mutation: the complete and corrected change table, scholarly debate on voiceless stops, all triggers (plural article in, preposition an, verb-prefix en-), and worked examples.
HSK 4 | verb | managing, resolving, or processing a matter
HSK 1 | adjective | happy, glad, pleased
Master advanced formal conjunctions and particles essential for professional communication, news analysis, and JLPT N2 reading comprehension.
HSK 3 | adverb | often, frequently, regularly — describes a habitual or recurring action
一 (one): 1 strokes, JLPT N5. On: イチ、イツ. Kun: ひと、ひと-つ.
In this problem, we are asked to compare two numbers written in an unusual numeral system called the golden system.
HSK 4 | adjective | hypocritical; false; insincere
HSK 3 | noun | opinion, view, feedback — what someone thinks about something
Full paradigm of the relative pronoun, rules for agreement and case, worked examples, the connecting relative, and contrast with the interrogative adjective.
HSK 1 | measure word | classifier for animals and some paired objects
HSK 3 | noun | dormitory, student housing, staff quarters
分 (fun/bun/bu/waka-ru): minute. JLPT N5 kanji.
JLPT N4 kanji 院 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 4 | adjective | covering a wide range or affecting many people or areas
HSK 5 | verb | to cause something to move or develop by pulling or leading it forward
JLPT N4 kanji 春 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 5 | adjective/noun | emotional; driven by feeling; perceptual (as opposed to rational)
HSK 4 | adverb | expressing immediate action without delay
HSK 4 | noun | situation; circumstances; condition
HSK 6 | noun/verb | an organizational framework; to design the structure of
HSK 5 | noun | the average height of the ocean surface used as a reference point
HSK 4 | noun | a person who is ill or receiving medical treatment
HSK 4 | verb | the process by which something comes into being
We are given an array of integers, a[1…n], and a recursive function f(i, j) defined on it. The function essentially accumulates sums along specific paths: the base case is simply f(1, j) = a[j], and for later rows, each f(i, j) is the sum of a[j] plus the minimum of f(i-1…
HSK 4 | noun | a superior position or quality that gives an edge over others
HSK 4 | noun | a person who conducts research
HSK 1 | particle | asks 'and what about...?' or softens a statement
HSK 4 | verb/noun | to make a choice or the act of deciding
N1 grammar: classical auxiliary ごとし expressing literary simile ('like / as though'). Used in literary prose, proverbs, and formal descriptive writing.
時 (ji/toki): time. JLPT N5 kanji.
The task is to evaluate a simple arithmetic expression of the form a?b where a and b are integers between 1 and 999, and ? is a single operator from the set +, -, , /, or %.
HSK 4 | verb/noun | to work together toward a shared goal
四 (shi/yot-tsu): four. JLPT N5 kanji.
HSK 5 | adjective / verb | confused, perplexed, puzzled; to confuse, to perplex
Learn how to use 〜てたまらない (te tamaranai) to express unbearable or overwhelming feelings, sensations, or desires — 'can't stand ~; unbearably ~.' Includes structure, nuance, examples, and comparisons.
HSK 1 | noun | television, TV
山 (mountain): 3 strokes, JLPT N5. On: サン. Kun: やま.
HSK 3 | verb/adjective | to use resources carefully and avoid waste
HSK 3 | noun | skin, the outer covering of the human body
HSK 5 | adjective/adverb | clearly defined and detailed, not abstract or vague
HSK 4 | adjective | within the expected or standard range
HSK 4 | noun | programs and operating information used by a computer
HSK 3 | noun / verb | development, growth; to develop, to expand
HSK 5 | noun | infectious disease; communicable disease; contagious illness
HSK 1 | noun | English language
HSK 4 | adjective | not in accordance with reason or logic
We are given several piles of worms arranged in order. The worms are numbered consecutively across all piles. If the pile sizes are: then the first pile contains worms numbered 1 through 2, the second pile contains worms numbered 3 through 9, and the third pile contains worms…
Esperanto vocabulary for arts, traditions, social structures, politics, and cultural life.
HSK 1 | noun | the period from noon to around 6 p.m.
Achieving high-level Chinese public speaking: structure, rhetoric, delivery, and audience engagement at the expert level.
N1 grammar pattern 〜ものがある: expressing that something has an ineffable quality or characteristic that the speaker cannot easily put into words but strongly senses.
入 (nyuu/ju/i-ru): enter. JLPT N5 kanji.
A dinner is represented as a sequence of flowers. A red flower contributes length 1, while white flowers are special: they may only appear in contiguous groups of exactly k flowers.
Japanese numbers 1–10,000+, ordinal numbers, and the counter system (hon, mai, hiki, satsu, hai, ko, nin, and more) with examples.
HSK 3 | verb | to describe, to depict, to portray in writing or speech
We are given a tree with n vertices, where some vertices are black and others are white. The tree is rooted implicitly by the way the edges are described: each node i for i 0 has a parent p[i-1].
HSK 4 | adjective | existing or occurring widely across a group or area
Master Japanese noun modification through relative clauses (plain form + noun), 〜という + noun, and the critical nominalizers 〜の and 〜こと. Learn 〜ところ, ことがある (experience/possibility), ことにする, and ことになる.
HSK 3 | noun | a distinguishing quality or trait
HSK 5 | noun | a formal system for supervising and ensuring accountability
HSK 1 | number | the digit one; tone changes depending on the tone of the following syllable
N1 grammar pattern 〜ずにはすまない: expressing that social norms, morality, or the situation demands that X be done — one cannot get by without doing X.
HSK 5 | adjective | producing the desired result; legally or formally valid
Master the structure and language conventions of Chinese academic essays, including argumentative thesis writing, paragraph development, and formal transitions.
道 (road, way): 12 strokes, JLPT N5. On: ドウ、トウ. Kun: みち.
HSK 3 | grammar pattern | conditional conjunction expressing minimum sufficient condition
Master classical Latin sounds, vowel quantity, diphthongs, and the penultimate accent rule.
HSK 1 | verb, preposition | to use, to employ, with (using)
HSK 1 | number | one hundred; used in compound numbers
HSK 4 | noun / adjective | mistake; error; wrong; incorrect
Explore all major uses of the accusative and genitive cases, including prepositions, partitive genitive, and genitive of description.
HSK 4 | verb/noun | to provide explanation or commentary
半 (half): 5 strokes, JLPT N5. On: ハン. Kun: なか-ば.
HSK 3 | adverb | indicates that something is unnecessary or not required
HSK 5 | noun | a branch of academic study or knowledge
HSK 3 | verb | to work out or train the body or mind
先 (sen/saki): before. JLPT N5 kanji.
HSK 3 | adj/verb | plentiful in quantity or variety
All 181 new N4 kanji with stroke counts, on-yomi, kun-yomi, meanings, and example words. Cumulative total with N5: 284 kanji.
N1 grammar: classical formal expression 〜いかんによらず/〜いかんにかかわらず meaning 'regardless of the nature/circumstances of X.' Used in legal texts, regulations, and formal announcements.
Master formal request forms, business vocabulary, and the foundations of keigo (honorific language) for N3-level workplace communication: emails, phone calls, meetings, and the 丁寧語/尊敬語/謙譲語 distinction.
HSK 4 | verb | to collect, to gather
HSK 3 | adverb / noun | usually, ordinarily — during normal or ordinary times
HSK 4 | noun | a person employed by an organization
Learn everyday Japanese verbs, understand u-verb and ru-verb conjugation, and express present, past, and negative actions using the polite ます form.
HSK 4 | phrase | there is no need; it is unnecessary
Learn four suffixes that create place words, professional titles, feminine forms, and person-descriptors from any root.
HSK 1 | noun | a female child
HSK 4 | verb / noun | to understand or comprehend something deeply
JLPT N4 kanji 晴 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 5 | verb / noun | to weaken or deteriorate; a period of decline
女 (jo/nyo/nyou/onna): woman. JLPT N5 kanji.
Learn to express two actions happening at the same time using 一边...一边, 边...边, and 同时, with attention to which verbs can combine.
HSK 3 | noun | school grade, year level in school
男 (dan/nan/otoko): man. JLPT N5 kanji.
HSK 1 | measure word | classifier for long, narrow, or flexible objects
Learn 〜のみならず, the formal equivalent of 'not only X but also Y,' used in academic writing and official documents.
N1 grammar pattern 〜ともなく: expressing an action done without clear intention or direction — 'without particularly intending to, sort of, vaguely.'
We are given a set of $n$ people who must be partitioned into exactly $m$ non-empty groups. Once the grouping is fixed, every pair of people inside the same group becomes a friendship.
南 (ナン・ナ/みなみ): south. JLPT N5 essential kanji.
N1 grammar pattern 〜であれ〜であれ: expressing 'whether ~ or ~' — indicating that the following statement applies regardless of which alternative is the case.
HSK 4 | adjective | recently produced or obtained; not stale or decayed
HSK 5 | adjective/adverb | special; out of the ordinary; especially; particularly
見 (see, look): 7 strokes, JLPT N5. On: ケン. Kun: み-る、み-える.
We have three strings. For every length $l$ from $1$ up to the length of the shortest string, we must count how many triples of positions $$(i1,i2,i3)$$ produce three equal substrings of length $l$, one taken from each string.
We are asked to count how many different moments in time Little X can wake up such that he overlaps with Little Z online. Little Z has a fixed schedule consisting of multiple segments.
HSK 3 | adjective | wonderful, brilliant, splendid — describing a superb performance or event
HSK 5 | noun | arrangement and organization of components within a whole
HSK 5 | adjective | low-carbon; producing or emitting less carbon dioxide
HSK 6 | verb | to explore and discuss a topic in depth; to investigate
HSK 3 | verb + directional complement | to run back to a place away from the speaker
HSK 4 | noun / verb | ties between people or things, or relevance
HSK 5 | noun | a subject or theme being talked about
HSK 3 | verb / noun | to perform, to put on a show; a performance, show
HSK 4 | verb | to make changes to improve or correct something
HSK 4 | n/v | a successful outcome in a contest or struggle
HSK 3 | verb / adjective | to be moved emotionally, touched deeply by something
HSK 4 | noun | the series of steps or stages in completing something
HSK 5 | conjunction | as a result; causing; so that (introducing a negative consequence)
Learn 〜さえ〜ば (if only ~, as long as ~) — the N3 conditional expressing a single minimal sufficient condition.
Result and method clauses in Chinese scientific prose — vocabulary, structure, and the conventions of academic argumentation in Chinese research writing
HSK 3 | noun | a physical action or movement of the body
HSK 5 | verb | to publish an article; to deliver a speech or statement
HSK 5 | verb/noun | to enforce the law; law enforcement
HSK 5 | noun/adjective | health; healthy; sound; well
HSK 5 | adjective/noun | the state of being free from danger or risk
HSK 4 | verb/noun | to train, to provide training; training
高 (high, expensive): 10 strokes, JLPT N5. On: コウ. Kun: たか-い、たか.
HSK 3 | adjective | nervous, tense, anxious; also describes a busy or tight situation
古 (old): 5 strokes, JLPT N5. On: コ. Kun: ふる-い、ふる-す.
HSK 4 | n/v | a formal vote to choose a person for a position
HSK 2 | noun | a general word for any physical object or item
HSK 4 | adjective | based on personal feelings rather than facts
足 (foot, leg): 7 strokes, JLPT N5. On: ソク. Kun: あし、た-りる.
JLPT N4 kanji 肩 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 3 | noun | culture, civilization — the shared customs, arts, and knowledge of a society
Master the art of indirect communication by learning how to express complex limitations, obligations, and partial negations in professional Japanese.
We are given a collection of exams, each with two relevant days. The official schedule assigns exam i a day ai, but the student has a special agreement allowing him to take it earlier on day bi, where bi < ai.
HSK 4 | adjective | not formal or official in nature
Master multi-affix stacking, productive derivation chains, and the principles behind Esperanto neologism formation at B2 level.
HSK 5 | verb | to talk someone into something; to win someone over
Mastering 法律文书 writing conventions and the specialized register of Chinese legal discourse.
Understanding 古今异义 and the semantic evolution of classical vocabulary in contemporary Chinese discourse.
HSK 3 | adjective | of great significance or consequence; not to be overlooked
Sindarin prepositions and the mutations they trigger — with worked examples and definite preposition forms.
Master all six Esperanto participles — active and passive, present, past, and future — and their use as adjectives, adverbs, and in compound tenses.
HSK 3 | verb | to check, to inspect, to examine — used for medical or general review
Learn 〜にほかならない (nothing other than ~, precisely ~) — the N3 pattern for making emphatic, exclusive identifications or assertions.
HSK 5 | noun | a physical or mental feature indicating a condition or disease
HSK 4 | verb | disrupting someone's activity or peace
HSK 1 | adjective | big, large, great
HSK 2 | adjective/adverb | clear, distinct, to understand clearly
HSK 3 | noun | a financial institution; a bank
Express intentions and future plans using 打算, 准备, 要, and 会 in natural Chinese.
HSK 4 | adjective / verb | noticeably prominent or to protrude beyond others
HSK 5 | verb | to explore; to examine through discussion or inquiry
The problem asks us to take a list of integers and, for each element, produce the sum of that element with the element immediately preceding it. The twist is that the first element of the output pairs with the last element of the array, creating a circular relationship.
HSK 1 | noun | the early part of the day, from dawn to around 9 a.m.
HSK 1 | number | the digit five
HSK 5 | noun | the capacity or power to remember
Master formal, semi-formal, and informal letter and email writing in Esperanto, including structure, formulaic expressions, and epistolary conventions.
HSK 3 | verb / noun | to rent, to hire; rent, rental fee
HSK 5 | noun | shareholder, stockholder, equity holder in a company
HSK 3 | preposition | indicates that one thing changes or develops together with another
HSK 5 | noun | financial award given to support a student's education
HSK 4 | conjunction | introduces a concessive clause
Learn 〜だけあって and 〜だけに to express outcomes that match or are expected given a particular quality or background.
Creating Sindarin personal names: masculine and feminine name elements, patronymics, epithets, and the naming traditions of Tolkien's Elves.
HSK 6 | verb | to deduce logically; to perform or present artistically
Learn how to use 〜てならない (te naranai) to express feelings or states that arise spontaneously and overwhelmingly — 'can't help but feel ~.' Includes structure, nuance, examples, and comparisons.
HSK 5 | conjunction / adverb | conversely, on the other hand, otherwise
Sindarin words for people, races, relationships, titles, and names for individuals.
HSK 1 | number | the digit four; considered unlucky in Chinese culture due to resemblance to 死 (death)
HSK 4 | verb / adjective | bringing something to a higher or complete standard
HSK 3 | noun / conjunction | result, outcome; as a result, consequently
Master the fundamental stylistic differences between formal written prose and casual spoken Japanese required for N2 proficiency.
HSK 3 | noun | a section or component of a larger whole
HSK 4 | adjective | having many interrelated parts that are difficult to understand or deal with
Master five N3 patterns for describing change, tendencies, and degrees of difficulty: 〜ようになる, 〜ようにする, 〜がちだ, 〜やすい/〜にくい, and 〜得る/〜得ない.
Master the critical distinction between 〜かねる (polite refusal/impossibility) and 〜かねない (risk/negative possibility warning).
朝 (chou/asa): morning. JLPT N5 kanji.
We are given a string consisting solely of digits, and the task is to determine if the number "666" appears anywhere as a contiguous sequence inside that string. The output should be "YES" if it does, and "NO" otherwise.
Full paradigms for 4th-declension masculine and neuter nouns and 5th-declension nouns, with key vocabulary and the special behavior of domus.
HSK 4 | noun / adjective | an organized set of connected parts, or thorough and orderly
HSK 4 | verb / noun | to think deeply; to ponder; reflection
HSK 3 | noun/verb | a shift or transformation in state or condition
三 (three): 3 strokes, JLPT N5. On: サン. Kun: み、み-つ、みっ-つ.
HSK 5 | adjective | rigorous; meticulous; careful and thorough
HSK 5 | n | literary prose; an essay or lyrical prose piece
Sindarin comparison: an- prefix for superlative (most), -wain suffix for superlative, einior (elder), iarwain (oldest), and Neo-Sindarin comparative constructions.
HSK 6 | n | the deep root or foundation upon which something is built or sustained
Complete guide to all major Japanese particles: は が を に で から まで と も の か ね よ — with meaning, usage rules, and example sentences.
Engage with the major authors, works, and critical debates that constitute the Esperanto literary tradition from Zamenhof to the present day.
飲 (in/no-mu): drink. JLPT N5 kanji.
HSK 5 | noun | a standard pattern or model of operation or behavior
HSK 4 | noun / verb | statistics; to compile statistics
HSK 5 | verb / noun | to give a full explanation or interpretation of something
HSK 1 | adjective / preposition | correct, right; directed at, towards
HSK 3 | adjective/adverb | nearly equal or approximately
HSK 4 | adjective | based on facts, not personal feelings
HSK 3 | noun | one's particular way of seeing or interpreting something
N1 grammar pattern 〜をよそに: expressing that someone proceeds obliviously or indifferently to something (worries, expectations, circumstances) that should be relevant to them.
HSK 5 | verb | to envy, to admire, to look up to with longing
HSK 3 | noun | any audible sound or a person's voice
HSK 4 | noun / verb | an activity, event, or campaign; to move around
HSK 5 | noun/verb | a challenge; to challenge; to test one's limits
HSK 5 | adjective / verb | clear and unambiguous; to make something explicitly clear
HSK 3 | verb | to make something real; to accomplish a goal
HSK 4 | grammar pattern | presenting two contrasting or complementary aspects
Learn how to use 〜に際して (ni saishite) to express 'on the occasion of' or 'at the time of' for significant events and actions. Includes structure, nuance, examples, and comparisons.
HSK 5 | verb/noun | conversion or transformation between forms/states
HSK 5 | adjective | appropriate; suitable; fitting for the context
HSK 5 | verb | deliberate or observed reduction in strength or intensity
HSK 4 | noun | media content delivered continuously over the internet
HSK 3 | verb/noun | to advance; to improve; forward movement
HSK 5 | verb | to become involved in a situation in order to influence its outcome
HSK 5 | noun | a person's or group's point of view or stance on an issue
HSK 3 | noun | a project or planned undertaking; a listed item or entry
HSK 4 | verb / noun | believing in someone's reliability or integrity
JLPT N4 kanji 橋 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
西 (west): 6 strokes, JLPT N5. On: セイ、サイ. Kun: にし.
Talk about your job, workplace, and colleagues in Chinese with natural professional vocabulary.
HSK 5 | noun | natural disaster, catastrophic natural event
HSK 5 | verb / noun | systematic long-term planning or a plan
HSK 5 | noun | a disorder of structure or function in a living organism
HSK 5 | noun/adjective | network; the internet; online
HSK 3 | pronoun / adjective | another, other, separate — introduces a different item
Learn 〜わけがない (there's no way ~, it's impossible that ~) — the N3 pattern for expressing strong logical impossibility based on the speaker's conviction.
N1 grammar pattern 〜ながらも: expressing contrast between two simultaneous states — even though, despite, all the while.
HSK 3 | conjunction | only if, only when — used in 只有...才... pattern
HSK 1 | number | one thousand; used in compound numbers and pricing
HSK 2 | noun | a location, place, or region; also a specific part or aspect of something
HSK 5 | verb | to recruit; to enlist members, volunteers, or employees
HSK 4 | adjective | having all necessary parts and nothing missing or damaged
Describe your home, talk about where you live, and discuss renting and moving in Chinese.
HSK 2 | verb | to feel anxious or worried about someone or something
HSK 6 | verb | to weigh up options carefully; to balance competing considerations
HSK 3 | adjective | widely recognized; renowned
Essential Japanese vocabulary for stores, prices, shopping, money, bargaining. N1 level reference with readings, romaji, and examples.
Digital transformation, platform economy, and the vocabulary of China's tech-society interface at C1 level
HSK 1 | noun | classmate, schoolmate
中 (middle, inside): 4 strokes, JLPT N5. On: チュウ. Kun: なか.
HSK 4 | verb/noun | to give the reason or meaning of something; to clarify or justify
HSK 4 | adjective | covering all aspects without leaving anything out
Learn how to use 〜につれて (ni tsurete) to express parallel change — 'as ~ happens, ~ also changes.' Includes structure, nuance, examples, and comparisons.
HSK 3 | noun | towel, face cloth
A comprehensive review of all A1 grammar, 100 essential vocabulary words, key survival phrases, and guidance on the path to A2.
HSK 4 | noun | the general direction in which something is developing
N1 grammar pattern 〜てならない: expressing deep, spontaneous, involuntary feelings that cannot be otherwise. Literary register; contrasts with 〜てたまらない and 〜てしかたがない.
HSK 3 | adverb | used with negation to mean 'never' or 'not once'
HSK 5 | noun | thinking; thought process; cognitive approach
HSK 4 | conjunction | adding a reinforcing reason or point
HSK 5 | verb / adjective | to suppress, to repress; stifling, oppressive
言 (gen/gon/i-u): say. JLPT N5 kanji.
HSK 3 | noun | a goal, target, or objective to aim for
HSK 4 | noun | a fixed deadline or time limit for completing something
HSK 5 | noun | overarching institutional or governmental system
HSK 6 | n | the meaning or significance contained within a word, phrase, or act
HSK 4 | noun | the spot or role something or someone occupies
HSK 3 | adjective | having a sharp and quick mind
HSK 4 | verb | to learn successfully, to master a skill
HSK 5 | adjective | lasting; enduring; able to persist over a long time
HSK 3 | noun | photograph, picture taken by a camera
HSK 4 | verb / noun | to divide and assign resources, tasks, or roles
Learn 〜たところ to express the discovery of an unexpected result upon completing an action, used in formal reports and investigations.
七 (seven): 2 strokes, JLPT N5. On: シチ. Kun: なな、なな-つ、なの.
HSK 5 | noun | the physical components of a computer or device
Explore Chinese philosophical and ethical discourse, engaging with classical concepts and contemporary moral questions at the B2 level.
HSK 2 | adverb | expresses that something is obvious or goes without saying
Sindarin adjectives: post-nominal position, lenition rule, singular and plural agreement, comparison forms, and how to form adverbs.
HSK 1 | noun | man, male adult
HSK 6 | verb | to provide a complete and authoritative interpretation
HSK 3 | adverb | after much effort or trouble; not easy at all to achieve
HSK 4 | adjective | diligent, hardworking
The complete sound change history from Primitive Elvish through Common Telerin and Old Sindarin to Classical Sindarin — explaining WHY mutations exist and how the vowel system developed.
HSK 3 | noun | method, way, approach — a systematic means of doing something
HSK 3 | noun/verb | a repeated behavior, or to be accustomed to something
HSK 3 | noun / measure word | a unit of time (minute), a score point, or a fraction
HSK 4 | phrase | used to introduce a comparison
HSK 1 | verb | the linking verb 'to be'; NOT used for possession
HSK 1 | verb / preposition | indicates location, either as main verb or as preposition before a place
HSK 4 | verb / noun | to complete a course of study and graduate
HSK 3 | verb | to increase, add to, or grow in number or amount
HSK 3 | noun | passport — official travel document
来 (come): 7 strokes, JLPT N5. On: ライ. Kun: く-る、き-たる.
HSK 4 | noun | the forward progress or development of something
The problem gives you the first n positive integers, arranged as a sequence 1 through n. You are allowed to combine any two numbers from this sequence using addition, subtraction, or multiplication, replacing the pair with the result.
HSK 3 | verb / noun | to influence, affect; an influence or effect
HSK 2 | adjective | a feeling of happiness and joy
HSK 3 | verb | to give, submit, or establish a connection
Intermediate Sindarin: all 5 mutations, the full verb system, Tengwar writing, and composing original sentences — goals and resources for 100–300 hours.
HSK 4 | verb / noun | needing someone or something for support
HSK 5 | verb/noun | to plan and coordinate as a whole; overall planning
天 (heaven, sky): 4 strokes, JLPT N5. On: テン. Kun: あめ、あま.
HSK 3 | noun | situation, condition, circumstances — the state of affairs at a given moment
HSK 2 | verb | to retain something in memory; to recall
HSK 4 | verb / noun | to make progress; to improve; progress
生 (sei/shou/i-kiru): life. JLPT N5 kanji.
HSK 1 | measure word | classifier for flat, sheet-like objects
HSK 3 | adjective | free from stress or difficulty; comfortable and at ease
HSK 4 | verb/noun | to strive against others for the same goal or resource
HSK 3 | noun | the function, role, or effect that something produces
HSK 4 | verb | communicating thoughts, feelings, or ideas
HSK 4 | noun/adj | an established measure or criterion used for comparison or judgment
HSK 3 | adjective/adverb | describes opposition or reversal between two things or ideas
HSK 2 | verb/adjective | to get angry, to be angry
We are given a network of nodes connected by links, where each link can carry data in one direction at a time. Each link has a cost proportional to the square of the bandwidth multiplied by a given weight.
HSK 4 | verb | to save or reduce use of resources
JLPT N4 kanji 交 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 5 | verb | to lead the way forward, often inspiring others to follow
語 (language, word): 14 strokes, JLPT N5. On: ゴ. Kun: かた-る、かた-らう.
HSK 3 | verb + result complement | to express something in a way that is fully understood
HSK 4 | conjunction | accepting a premise and drawing a conclusion
HSK 2 | adjective | high in price; also used in polite forms of address
Learn colors, basic appearance adjectives, and how to modify nouns with 的 and intensify descriptions with 很.
Learn to decode unfamiliar Esperanto words using word-building knowledge, cognates, and context clues, and build a reading practice with graded and authentic texts.
We have a football tournament involving three teams. Every game produces exactly one winner, so each played game contributes exactly one win to one of the teams. After k games have already been played, we do not know the exact number of wins of the three teams.
HSK 2 | verb | to agree with, to consent to, to approve
The task asks for the remainder when the sum of the first four multiples of a very large number n is divided by 5. In other words, you need to compute (1n + 2n + 3n + 4n) mod 5.
Discuss economic and business topics using 在...方面, 就...而言, and formal business vocabulary at B1 level.
HSK 3 | conjunction | since, now that, given that — accepts a premise and draws a conclusion from it
JLPT N4 kanji 道 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 1 | noun | supermarket
HSK 4 | adjective | friendly, amicable
HSK 4 | adjective / adverb | relative; in comparison to something else
HSK 5 | phrase | a discourse marker meaning 'with regard to this point'
HSK 5 | verb/noun | to check and confirm the truth, accuracy, or legitimacy of something
HSK 1 | number | the digit seven
HSK 2 | preposition/verb | comparison marker; A比B + adj structure
HSK 1 | noun | book
HSK 3 | verb | to climb, to crawl, to scramble up
HSK 4 | verb/noun | to write down or keep a record of something
HSK 1 | verb, noun | to stand, a stop or station
HSK 3 | verb | to hang something up or be suspended
HSK 5 | noun | distinctive feature or characteristic that identifies something
HSK 5 | adjective | external, outward, superficial (as opposed to internal)
南 (south): 9 strokes, JLPT N5. On: ナン、ナ. Kun: みなみ.
HSK 3 | noun | the proper or logical sequence in which things are arranged
Learn how to use 〜に違いない (ni chigainai) to express strong certainty and confident inference — 'must be; there's no doubt that; I'm sure that.' Includes structure, nuance, examples, and comparisons.
Develop the vocabulary and analytical grammar to discuss Chinese literary works, artistic movements, and aesthetic criticism at the B2 level.
HSK 4 | noun | a set of steps or instructions; a computer program
HSK 4 | verb | to transfer data from the internet to a device
HSK 5 | adjective | reasonable, rational, sensible, justified
HSK 7 | verb | surgical closure of wounds or incisions using sutures or staples
HSK 3 | adjective | at a small height or below a standard level
HSK 5 | verb | to make something firmer, stronger, or more secure
HSK 5 | adverb | used to single out something as standing out above others
Learn how to use 〜はもとより (wa motoyori) to express 'not only ~ but also' when the first item is so obvious it goes without saying. Includes structure, nuance, examples, and comparisons.
HSK 1 | verb / preposition | means to give or indicates the recipient of an action
Master formal Japanese time and occasion markers used in official speeches, ceremonies, and business announcements — 〜にあたって, 〜に際して, 〜に先立って, and more.
We are asked to help Captain Marmot organize his regiments of moles so that each group of four forms a perfect square on a plane. Each mole starts at a position (xi, yi) and has a “home” at (ai, bi).
HSK 5 | verb / noun | to argue opposing sides; formal debate
HSK 4 | adjective | the most important or central element
犬 (dog): 4 strokes, JLPT N5. On: ケン. Kun: いぬ.
Advanced Sindarin: primary source scholarship, Neo-Sindarin evaluation, advanced composition, and engaging with the Tolkien linguistics community.
HSK 5 | verb | to present; to display; to bring into view
HSK 4 | verb | to take advantage of something for a purpose
技術 (ぎじゅつ / gijutsu): technology, skill. N4 level Japanese vocabulary.
HSK 4 | noun | a distinct phase or stage in a process or development
木 (tree, wood): 4 strokes, JLPT N5. On: モク、ボク. Kun: き、こ.
HSK 3 | noun | season — one of the four seasons of the year
Master the two types of Japanese adjectives to describe people, places, and things with precision and natural flair.
HSK 5 | noun/verb | the process of making important decisions, especially in policy or management
HSK 6 | n | interests, benefits, or gains — especially in political, economic, or social contexts
HSK 4 | adjective | ahead of others in development or quality
HSK 3 | noun | hour, a unit of time equal to 60 minutes
HSK 5 | noun/adjective | green (colour); eco-friendly, sustainable, clean
HSK 4 | verb/noun | to oversee, administer, or control people or resources
HSK 5 | noun | a legal member of a state with rights and responsibilities
HSK 6 | verb/noun | to posit a supposition; a scientific or logical hypothesis
HSK 3 | modal verb | expressing obligation, expectation, or what is proper
Learn 〜にとって (for, from the perspective of) — the N3 pattern for marking the evaluating perspective in value judgments and assessments.
Two-termination, three-termination, and one-termination 3rd-declension adjectives, with the -ī/-e ablative distinction and the -ium genitive plural rule.
HSK 3 | verb / noun | to notify, inform, announce; a notice or announcement
HSK 4 | adjective | produced without synthetic chemicals; relating to organic chemistry
HSK 4 | verb/noun | granting official permission or power to someone
HSK 5 | verb | to prompt; to cause; to drive someone to do something
Master the second declension: masculine -us/-er nouns and neuter -um nouns, with the vocative and neuter rules.
HSK 3 | pronoun | referring to the subject themselves
Navigate shopping situations in Chinese: ask prices, bargain, understand discounts, and handle returns.
HSK 4 | noun | engineer
上 (up, above): 3 strokes, JLPT N5. On: ジョウ、ショウ. Kun: うえ、うわ、のぼ-る.
HSK 1 | adjective | new, fresh, modern
HSK 1 | noun | a point in time or a period; used in the pattern 的时候 (when)
Complete JLPT N1 kanji reference: 1,000+ new kanji beyond N2, covering rare readings, similar-looking pairs, classical kanji, compound analysis, and the most important 500 N1 kanji in full detail.
HSK 3 | adjective | clean, free from dirt or mess
N1 grammar pattern 〜ならでは: expressing exclusivity — something only possible with X, uniquely characteristic of X.
Core Sindarin verbs with present, past, and future forms — organized by category for practical composition.
Learn vocabulary for medical situations and healthy habits, and master the grammar patterns 应该 and 建议 for giving and receiving health advice.
We roll a fair die with faces numbered from 1 to m, exactly n times. Every roll is independent, and each face appears with probability 1 / m. Among those n rolls, we look only at the largest value that appeared. The task is to compute the expected value of that maximum.
HSK 5 | verb | to feel profound respect and admiration for someone's character or achievements
HSK 4 | preposition / noun | using something as a foundation or source
HSK 4 | verb/noun | to add to or fill in what is missing
HSK 1 | noun | classroom
HSK 3 | verb + directional complement | to carry something into an enclosed space, toward the speaker
HSK 5 | adjective | favoring reform and social change; forward-thinking
HSK 3 | verb | to put down, to set down, to let go of
HSK 1 | verb / complement marker | indicates arriving at a destination or successful completion of an action
店 (store, shop): 8 strokes, JLPT N5. On: テン. Kun: みせ.
HSK 3 | verb / noun | presenting someone or something to others
HSK 5 | verb | reduction of complexity while preserving core function
HSK 6 | adv | used to concede something as true before introducing a contrasting or qualifying point
HSK 3 | noun | a regular pattern or rule observed in nature or life
HSK 4 | adjective/verb | fully developed, either physically or mentally
Master large numbers, ordinals, fractions, decimals, multipliers (-obl-), distributives (-op-), and the preposition po.
HSK 4 | adjective | pessimistic; pessimism
HSK 5 | verb/noun | gradual evolution or development over time
時 (time, hour): 10 strokes, JLPT N5. On: ジ. Kun: とき.
Master five classical Chinese idioms with their full narrative and argumentative contexts, understanding their etymologies and modern usage patterns.
HSK 2 | noun | an airport, aerodrome
HSK 4 | noun | doctor, physician
HSK 5 | phrase | a connector used to introduce additional points or items
Essential Japanese vocabulary for trains, buses, taxis, traffic, directions. N1 level reference with readings, romaji, and examples.
HSK 4 | adjective | precise, accurate, exact
HSK 5 | verb | to research and discuss; to deliberate in a seminar setting
HSK 5 | verb | to put a plan, policy, or law into action
HSK 1 | noun/verb | employment, labor; to be employed or carry out a task
HSK 4 | verb / noun / adverb | applying sustained energy and determination
HSK 5 | noun | materials used for teaching, including textbooks and instructional resources
We are given a sequence of integers arranged in a line, and our goal is to sort them in non-decreasing order by repeatedly performing a single allowed operation: moving the last element of the sequence to the front.
肉 (ニク): meat; flesh. JLPT N5 essential kanji.
HSK 3 | noun | environment, surroundings — the physical or social context around you
HSK 4 | noun | the explanation for why something happened
HSK 3 | noun | a mirror; a reflective surface used to see one's reflection
Master the hardest N1 exam question type: choosing between near-synonymous grammar patterns. Deep comparison of すら/さえ/でも, ものの/ながらも/けれども, ことなく/ずに/ないで, てならない/てたまらない/てしかたがない, and…
N1 grammar pattern 〜とあれば: expressing strong motivation or justification — 'if it is for ~, given that ~, should it be the case that ~.'
Full 文言文 literacy and engagement with classical Chinese literature, poetry, and the 骈文 tradition.
晩 (ban/): evening. JLPT N5 kanji.
HSK 2 | adjective | easy to do, likely to happen
HSK 5 | noun | systematic body of ideas explaining phenomena
Master the most frequently tested N1 proverbs (諺) and idiomatic set phrases (慣用句), understand their cultural and linguistic origins, and use them appropriately across formal and casual registers.
HSK 4 | noun | the central idea or topic of a work or discussion
HSK 5 | verb/noun | to report to a superior; a formal report or briefing
HSK 4 | noun | applied knowledge, skill, or technological capability
HSK 3 | noun | air conditioner, air conditioning unit
HSK 5 | adjective | fair and just; treating everyone equally
HSK 6 | verb/noun | to examine systematically by breaking into components
しかし (しかし / shikashi): however, but. N4 level Japanese vocabulary.
HSK 7 vocabulary reference: advanced C2-level words across specialized professional, academic, and literary domains.
HSK 4 | verb | forming an orderly line to wait
Esperanto vocabulary for computers, internet, social media, technology, and modern digital life.
HSK 3 | noun | mouth, lips — used literally and in idiomatic expressions
Master how to express viewpoint, unexpected contrast, and conditional dependency in formal Japanese — 〜として, 〜からすると, 〜にしては, 〜わりに, and 〜次第で.
A review lesson integrating all novice grammar in context: sentence-by-sentence analysis of two Latin passages, a curriculum summary, and independent reading practice.
HSK 2 vocabulary: ~200 new words added at level 2 (cumulative 498). Comprehensive table with Chinese, pinyin, English, part of speech, and examples.
HSK 4 | noun | crisis, critical situation
HSK 3 | adjective | feeling the need to sleep
HSK 1 | verb | to exit, to go out, to come out
HSK 4 | verb / noun | to imagine; imagination
Learn 〜だけでなく〜も (not only ~ but also ~) — the core N3 additive scope pattern for expanding beyond what was expected.
HSK 4 | verb | to make something available to someone who needs it
Master all four uses of 〜によって at N3: agent in passive, means/method, cause, and variation depending on condition.
HSK 3 | noun | attitude, manner, approach — how someone behaves or feels toward something
Japanese travel vocabulary: transportation, accommodation, directions, airports, stations, and essential travel phrases.
HSK 1 | adverb/noun | the present moment
後 (go/kou/ato): behind. JLPT N5 kanji.
国 (koku/kuni): country. JLPT N5 kanji.
HSK 3 | noun | an obligation one is accountable for; a duty that must be fulfilled
HSK 3 | noun | grades, scores, results; the achievement produced by effort
HSK 7 | verb | surgical removal of tissue, an organ, or a lesion from the body
HSK 5 | noun/verb | immunity from disease or harm; to be immune
HSK 5 | noun | an economic, practical, or personal need or demand for something
HSK 5 | noun/adjective | reason; rational thinking; based on logic rather than emotion
HSK 3 | pronoun | each, every, various — refers to all members of a group individually
N1 grammar pattern 〜きらいがある: expressing a negative tendency or inclination — 'has a tendency to' when the tendency is considered a fault or problem.
〜らしい: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 4 | verb | to oppose, to object to, to be against
HSK 3 | adverb | anyway, regardless, in any case — concedes a point but asserts another
HSK 5 | verb/noun | a real-time audio or video transmission to an audience
HSK 6 | verb | to sum up the essential points; to state in broad terms
Constructing and delivering formal argumentation in Chinese debate — 立论, 驳论, and 总结陈词 at C1 level
Words 201–500 from the DCC Latin Core list — the next 300 most frequent Latin words after Band 1. Complete before moving to Band 3.
調べる (しらべる / shiraberu): to investigate, look up. N4 level Japanese vocabulary.
自然 (しぜん / shizen): nature. N4 level Japanese vocabulary.
HSK 1 | noun | one's elder brother
HSK 3 | conjunction | introduces a concessive clause, acknowledging an obstacle while asserting a contrasting result
HSK 4 | verb | to hold an opinion or belief about something
HSK 9 vocabulary reference: the highest level of the HSK standard, covering literary, classical, and culturally rich Chinese vocabulary.
比べる (くらべる / kuraberu): to compare. N4 level Japanese vocabulary.
HSK 4 | verb | to present or show something to others
HSK 4 | verb / noun | to know (a person); to recognize; understanding
山 (san/yama): mountain. JLPT N5 kanji.
HSK 1 | noun | one's younger brother
Complete guide to keigo (敬語): teineigo (polite), sonkeigo (respectful), and kenjougo (humble) — with conjugation rules, common verbs, and example sentences.
校 (school): 10 strokes, JLPT N5. On: コウ. Kun: none.
Learn to share, support, and disagree with opinions using 觉得, 认为, and related expressions.
九 (kyuu/ku/kokono-tsu): nine. JLPT N5 kanji.
HSK 5 | noun | the level of openness, clarity, or accountability in a system or process
HSK 4 | verb | to maintain or protect something
The 15 most common mistakes English speakers make learning Mandarin Chinese — each with a wrong example, correct example, and explanation of why it matters.
文化 (ぶんか / bunka): culture. N4 level Japanese vocabulary.
Learn 〜やすい (easy to do, prone to) and 〜にくい (hard/difficult to do) — N3 patterns describing inherent ease or difficulty of an action.
Command Esperanto's -ig- and -iĝ- suffixes to express causation and inchoative change, and understand how transitivity determines accusative use.
HSK 3 | adjective | interesting, fun, amusing
HSK 5 | noun | the ability to produce original and imaginative ideas or works
We are given k permutations of the numbers from 1 to n. A permutation is simply a rearrangement of these numbers. Our goal is to find the length of the longest sequence of numbers that appears in the same relative order in all k permutations.
好 (like, fond of): 6 strokes, JLPT N5. On: コウ. Kun: す-く、この-む、よ-い.
HSK 4 | adjective | remaining rational and composed under pressure
HSK 6 | n/v | concrete manifestation; to give concrete expression to an abstract quality
HSK 5 | verb | to depict; to describe with detail and color
Complete guide to spaced repetition for Mandarin Chinese: Anki setup, optimal settings, what to put in SRS, Pleco integration, Skritter, and daily habit tips.
HSK 5 | noun/verb | the tendency to align one's behavior with the group
N1 grammar pattern 〜かたわら: expressing that someone pursues a secondary activity alongside a primary one — while also, on the side.
N1 grammar patterns 〜てこそ and 〜ばこそ: using the classical emphatic particle こそ to express that something is meaningful or possible precisely because of X.
HSK 4 | noun | the outcome or impression produced by an action or cause
Learn 〜たびに (every time, whenever) — the N3 pattern for expressing a consistent pattern of recurrence with emotional weight.
HSK 4 | verb | to raise something to a higher level or quality
HSK 5 | noun | established rules, regulations, or institutional arrangements
HSK 5 | verb / adjective | to lag behind or be underdeveloped
車 (sha/kuruma): vehicle. JLPT N5 kanji.
HSK 4 | noun / adjective | civilization or being civilized
HSK 1 | noun | door, gate, entrance
Mastering the full academic register for publication-standard Chinese scholarly writing.
Complete JLPT N5 grammar reference: all ~100 grammar patterns with structure formulas, usage explanations, example sentences (Japanese + romaji + English), and notes on common mistakes.
Sindarin's Celtic origins, how Tolkien built place names from roots, and analysis of 25+ famous Middle-earth place names.
N1 grammar pattern 〜のみか: literary 'not only X but also Y,' escalating from X to the more surprising/larger Y. Formal and literary register.
HSK 5 | verb | to convey; to relay; to pass on (information, instructions, or feelings)
Complete JLPT N1 vocabulary reference: ~4,000 new words beyond N2 covering academic, literary, political, legal, medical, and financial Japanese with readings, meanings, and usage contexts.
HSK 5 | noun / verb | a bank loan; to borrow money from a financial institution
HSK 5 | noun | the central claim or argument in a piece of reasoning
Learn vocabulary for digital technology and media, and use 用 + tool + V and 关于 + topic constructions to discuss technology in natural Mandarin.
英 (ei/): England. JLPT N5 kanji.
HSK 5 | verb | to cross over or transcend a boundary, gap, or obstacle
HSK 3 | noun | reason, grounds, justification for an action or belief
HSK 5 | verb/noun | to overcome a barrier or achieve a significant advance
HSK 3 | adverb | introduces a result that is the opposite of what was expected or intended
HSK 2 | verb/adverb | please (polite request), to invite, to treat someone
HSK 3 | adjective | indicating many different varieties
HSK 5 | noun | chronic disease; long-term illness that persists over time
JLPT N4 kanji 練 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 4 | phrase | generally speaking; on the whole; overall
六 (roku/mut-tsu): six. JLPT N5 kanji.
We are given a list of tasks, each with an associated difficulty level, and we need to produce three different sequences in which all tasks are completed. Each sequence must respect the rule that tasks can only be reordered if their difficulties are equal.
HSK 3 | verb | to take part in, attend, or join an event or group
大 (big, large): 3 strokes, JLPT N5. On: ダイ、タイ. Kun: おお、おお-きい.
HSK 3 | noun | a public announcement promoting a product or service
HSK 3 | verb / noun | to explain, to interpret, to clarify; an explanation
HSK 5 | verb | to execute; to carry out; to implement a plan, order, or task
HSK 3 | noun | experience gained through practice or living, not academic knowledge
社会 (しゃかい / shakai): society. N4 level Japanese vocabulary.
HSK 4 | noun | a strategy, tactic, or planned approach to achieving a goal
HSK 5 | n | the art of effective language use; a rhetorical figure or device
HSK 3 | verb/noun | to make a request or an appeal
N1 grammar pattern 〜に至って(は): expressing 'now that it has come to ~, upon reaching this extreme point, at this point finally.'
HSK 3 | adjective | complex, complicated, involving many parts or factors
Sindarin conditional mood using aen and the subjunctive: expressing hypothetical situations, wishes, and uncertainty.
HSK 4 | noun | an aim or destination one works toward
HSK 3 | noun/adjective/adverb | the natural world; being natural or obvious
JLPT N4 kanji 路 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 5 | adjective/noun | acting with moral uprightness and impartiality in judgment
HSK 6 | n | a solemn duty or calling, especially one of great historical or moral significance
HSK 5 | noun | immersive computer-simulated environment
Dialectical reasoning, materialist philosophy, and the vocabulary of ethics — engaging with Chinese and Western philosophical discourse at C1 level
HSK 1 | pronoun | interrogative pronoun asking about things or identity
HSK 1 | conjunction | introduces a result or conclusion, paired with 因为
HSK 5 | noun | financial or physical assets; wealth owned by a person or entity
特別な (とくべつな / tokubetsu na): special. N4 level Japanese vocabulary.
HSK 3 | preposition | marks the object of a disposal action, placed before the verb
Narrate travel experiences using 经历 narrative patterns and vocabulary for memorable moments, local customs, and personal impressions at B1 level.
HSK 3 | verb / noun | to have an allergic reaction; an allergy to something
HSKK advanced patterns, spoken coherence strategies, and the pragmatics of formal spoken Chinese at C1 level
HSK 2 | adjective / noun | describes physical or mental well-being
HSK 3 | verb | to exceed, surpass, or go beyond a quantity, limit, or standard
Master Esperanto's 28-letter phonemic alphabet, including the six diacritic letters and the X-system alternative.
HSK 3 | adjective | slow-witted or physically awkward
Learn how direction complements combine with verbs to describe movement toward or away from the speaker, a key feature of Mandarin spatial expression.
JLPT N4 kanji 体 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
We have an $n times m$ chocolate bar made of unit squares. A cut is either horizontal or vertical, must follow grid lines, must lie strictly inside the chocolate, and cannot duplicate a previous cut.
HSK 4 | verb | to have thorough command of a skill or knowledge
一 (ichi/itsu/hito-tsu): one. JLPT N5 kanji.
HSK 3 | verb | to convey thoughts or feelings
HSK 5 | noun / verb | the process of embedding AI or smart technology into systems and products
Complete N3 exam preparation guide: test format, question type breakdowns, 90-day study plan from N4, most-confusable grammar pairs, kanji reading drills, and test-day strategies.
Complete N4 exam prep: format guide, 25 vocabulary questions, 25 grammar questions, reading passages with questions, listening scenarios, 60-day study plan, and N5→N4 pitfall guide.
駅 (eki/): station (train/bus). JLPT N5 kanji.
HSK 4 | verb | to guide; to lead; to direct
Learn three conditional structures: 如果...就 for general conditions, 只要...就 for sufficiency, and 只有...才 for necessity.
HSK 4 | verb/noun | to take part in an activity or event; active involvement
HSK 3 | verb | to flip, turn over, or browse through something
Codeforces 464B: Restore Cube
HSK 3 | noun | the title of a work, the topic of a discussion, or an exam question
HSK 4 | verb/noun | the act of choosing or a choice made
HSK 3 | adverb | once, at one time, formerly — always refers to past experience
HSK 1 | adjective | tired, fatigued, exhausted
〜と思う: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 5 | noun | a broadcast or online channel for content delivery
市 (city, market): 5 strokes, JLPT N5. On: シ. Kun: いち.
We are given a game with several players, each controlling an army composed of different types of soldiers. Each army is represented as a non-negative integer where the binary representation encodes the presence of soldier types: a bit set to 1 means the army includes that type.
Master Japanese reported speech patterns — 〜と言う, 〜と思う, 〜と聞く, 〜によると, 〜そうだ — and learn to convert direct speech to indirect speech with correct plain-form usage before と.
Sindarin i-affection plural formation: the a→ai, e→i, o→y vowel change patterns with 30+ worked examples.
HSK 4 | grammar pattern | expressing strong preference by choosing one thing over another
HSK 1 | adverb | negates past actions and the verb 有
HSK 1 | adverb/noun | the day before today
HSK 4 | verb | to refuse to admit or acknowledge something
HSK 3 | pronoun / adjective | refers to every member of a set without restriction
HSK 3 | noun | tour guide, someone who leads tourists
千 (thousand): 3 strokes, JLPT N5. On: セン. Kun: ち.
HSK 6 | verb | to express or articulate ideas clearly and systematically
Learn how to use 〜さえ〜ば (sae ~ ba) to express the minimum sufficient condition — 'if only ~, then everything is fine.' Includes structure, nuance, examples, and comparisons.
HSK 4 | phrase | it is necessary; there is a need
HSK 1 | adjective | short (in length or duration)
We are given a sequence of integers arranged in a line, and we want to split it into three contiguous segments. The cut points must produce three non-empty parts, and each part must have exactly the same sum.
HSK 6 | n | facts, examples, or reasoning used to support an argument
HSK 5 | discourse marker | in other words; to put it another way
HSK 3 | noun | the reasoning or principle behind something
HSK 4 | verb | to be in (a state, position, or situation)
Japanese time vocabulary: days of the week, months, seasons, clock time, time expressions, and relative time words.
HSK 5 | verb/noun | to spread information to promote a cause, idea, or policy
HSK 5 | verb/adjective | to streamline, trim, reduce to essentials; concise
Master the full range of concessive and adversative structures in Esperanto, from kvankam and malgraŭ to spite ke, eĉ se, and sen ke.
HSK 4 | verb/noun | to lead or a person in a leadership position
HSK 2 | verb + result complement | indicates work is done thoroughly and to a good standard
HSK 8 | noun | microprocessor; a single integrated circuit that functions as a CPU
HSK 5 | discourse marker | it is learned that; reportedly; used to introduce information from an unspecified source
HSK 4 | verb | to clearly express or demonstrate something
HSK 1 | noun | school, educational institution
Full guide to writing Sindarin in Tengwar: Standard Mode and Mode of Beleriand — with letter tables, vowel placement rules, and practice tips.
We are given an array of integers. The first number in the input is the array size n, and the next n integers are the array elements. The task is simply to output the same elements arranged in non-decreasing order.
Master the paired connectives 虽然...但是, 因为...所以, and 尽管...还是 to express contrast and cause-effect relationships in complex sentences.
HSK 5 | noun | the Internet; the World Wide Web
JLPT N4 kanji 頭 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 4 | noun | one's mental or emotional stance toward something
HSK 1 | noun | money, cash
HSK 3 | verb | indicating the origin or source of something
HSK 6 | verb | to look at carefully and critically; to examine with discernment
Esperanto vocabulary for body parts, health, illness, medical situations, and physical descriptions.
HSK 3 | noun | the center or core of something
HSK 4 | adjective | having a connection or relationship to the subject at hand
HSK 3 | verb + complement | verb-result compound meaning to write something wrong
HSK 1 | noun | water
Japanese work and business vocabulary: professions, office settings, keigo expressions, job hunting terms, and workplace culture.
HSK 1 | verb phrase | to sleep, to go to sleep
HSK 1 | verb | to be acquainted with someone; to recognize something
HSK 4 | verb / noun | to gradually accumulate or build up over time
HSK 4 | noun | physical facilities or infrastructure for a purpose
HSK 3 | adjective | remaining rational and unruffled in difficult situations
HSK 3 | verb / adjective / noun | putting in great effort toward a goal
HSK 4 | adverb | indicating an estimate rather than an exact figure
分 (minute, part): 4 strokes, JLPT N5. On: ブン、フン、ブ. Kun: わ-ける、わ-かる.
HSK 4 | verb | to face; to confront; to deal with
HSK 4 | noun | the underlying base upon which something is built or developed
HSK 5 | noun | the aggregate of people living together in a structured community
HSK 4 | noun | a place or system where goods and services are exchanged
HSK 4 | noun | the ability to accomplish something with minimal waste
HSK 1 | verb | to drink a liquid
Comprehensive N3 review: mock N3 reading passage with full analysis, 15 grammar transformation exercises, listening strategy guide, self-assessment checklist for all 10 lessons, and a Bridge to N2 preview.
Learn 〜に沿って and 〜に沿った to express acting in line with plans, policies, or guidelines in formal Japanese.
HSK 4 | verb/noun | to establish with certainty that something is true or agreed upon
HSK 4 | conjunction / adverb | introducing a mild contrast or limitation
HSK 3 | noun | guest, visitor — someone who visits your home or a business
HSK 1 | noun | the period from dusk to bedtime
Master the full range of Esperanto comic expression, from morphological puns and satirical register to the tradition of Esperanto folk humor.
HSK 1 | noun | a male child
HSK 6 | conj | used to extend a range to include an extreme or unexpected member
HSK 5 | verb / noun | to publish (a book or periodical)
Learn 〜がちだ (tend to, be prone to) — the N3 pattern for expressing undesirable tendencies or negative dispositions.
HSK 3 | verb | to feel or think something subjectively
HSK 3 | verb / adjective | to be suited to; to fit well with a person, purpose, or situation
HSK 6 | verb | to make clear, to illuminate the meaning of something complex
HSK 3 | noun | a paper container for sending letters
HSK 4 | verb/noun | a significant change or transformation
HSK 2 | verb | to become sick, to fall ill
HSK 3 | verb | to feel at ease; to set one's mind at rest
HSK 3 | noun | steamed stuffed bun, a classic Chinese food
Develop systematic strategies for reading complex Chinese texts efficiently, including contextual inference, identifying main ideas, and detailed comprehension.
Master the Japanese counting system, learn to tell time, and express dates accurately to navigate daily life in Japan.
HSK 5 | noun | background, backdrop, context, personal connections
HSK 5 | adjective | renewable (resources, energy)
HSK 3 | adjective | fluent, smooth — describes language production that flows without stopping
十 (juu/jit/too): ten. JLPT N5 kanji.
HSK 5 | noun | a medical institution providing inpatient and outpatient care
The document can be viewed as a vertical stack of rows, where each row has a fixed length and the cursor always sits at some coordinate inside this grid.
HSK 5 | noun | the supreme law of a country that defines government structure and rights
HSK 5 | noun | a formal agreement reached between parties, or a technical protocol
Mastering formal written and spoken Esperanto for business correspondence, academic writing, conference presentations, and professional contexts.
We have an elevator starting on the first floor. Every person is waiting on that floor and wants to reach a specific destination floor. The elevator can carry at most k people at once. Moving between floors costs time equal to the floor difference.
HSK 1 | noun | newspaper
HSK 5 | discourse marker | in summary, to summarise what has been said above
HSK 3 | noun | a piece of news or information about recent events
HSK 4 | verb/adjective | to coordinate; harmonious
Complete HSK 3 vocabulary reference: 499 new words introduced at A2 level. Organized by category with Chinese, pinyin, English, part of speech, and example sentences.
HSK 3 | noun | city, urban area — a large, populated urban settlement
HSK 3 | noun | a feeling of curiosity or enjoyment toward something
HSK 3 | verb / noun | to need, require; a need or requirement
HSK 4 | noun | a software application designed for a specific purpose
HSK 2 | adjective/adverb | early, in the morning; also used as a morning greeting
Learn 〜わけではない (it doesn't mean that ~, it's not that ~) — the N3 partial negation pattern for correcting over-generalizations and assumptions.
HSK 2 | adverb | negative imperative; commands someone not to do something
経験 (けいけん / keiken): experience. N4 level Japanese vocabulary.
HSK 3 | noun | quality, standard — how good or poor something is
HSK 1 | verb | seeing or viewing in multiple senses
Master formal written Chinese grammar patterns and register-appropriate vocabulary for academic and professional contexts.
HSK 3 | verb | to care about, show concern for someone or something
Side-by-side comparison of HSK old standard (2.0, 6 levels) and new standard (3.0, 9 levels), including vocabulary counts, CEFR alignment, and transition timeline.
JLPT N4 kanji 便 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 4 | verb/noun | to make a prediction or forecast about a future event
HSK 5 | noun | artificial intelligence; AI
HSK 5 | verb/noun | to govern or manage; systematic control and management
HSK 5 | noun | a system of interacting organisms and their environment
HSK 1 | noun | apple
HSK 4 | verb/noun | using words to portray something in detail
HSK 3 | adjective / adverb | the most important or primary aspect
HSK 4 | noun | a consumer or customer who purchases goods or services
HSK 1 | noun | public bus
JLPT N4 kanji 仕 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 5 | noun | cultural cultivation, personal literacy, core competency
HSK 4 | verb / noun | establishing or maintaining a link between people or things
We are given the beauty values of n flowers. We must choose exactly two flowers. Among all possible pairs, we are interested only in pairs whose beauty difference is as large as possible.
JLPT N4 kanji 顔 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 3 | noun / adverb | time reference word for events before a stated point
HSK 2 | noun | physical body, health condition
We are given exactly six stick lengths. To build an animal, four of the sticks must be used as legs, which means those four sticks must all have the same length. After choosing the four leg sticks, two sticks remain. These determine the animal type.
HSK 5 | noun | model example, exemplary case, paradigm
HSK 4 | verb / noun | creating or planning something with purpose and structure
HSK 3 | verb | to oppose, to object to, to be against
Roland wants to place n watch towers on a 2D integer grid around a rose at the origin so that the sum of squared distances between all pairs of towers is maximized. Each tower must lie within or on a circle of radius r centered at (0,0).
HSK 5 | discourse marker | based on this; accordingly; on the basis of the above
HSK 5 | adjective | resistant to change; preferring traditional values
Consolidation of all C1 patterns, high-frequency vocabulary review, and a comprehensive self-assessment framework for HSK 6 readiness
HSK 4 | noun | the physical or social surroundings of a place or situation
Apply all intermediate constructions to Caesar's De Bello Gallico, with a full analysis of BG 1.1.
HSK 1 | adverb | strong intensifier meaning extremely or exceptionally
HSK 2 | verb | to swim; also used as a noun for the sport of swimming
HSK 4 | noun | the specific function or capability of a device, system, or organ
HSK 3 | noun | knowledge; information learned through study or experience
聞 (bun/mon/ki-ku): hear. JLPT N5 kanji.
花 (カ/はな): flower; blossom. JLPT N5 essential kanji.
HSK 4 | noun | a consistent pattern or natural law governing events
HSK 3 | adjective/verb | fully developed in body, mind, or flavor
HSK 6 | n | the lowest acceptable limit; a non-negotiable minimum threshold
生活 (せいかつ / seikatsu): daily life, living. N4 level Japanese vocabulary.
HSK 4 | adverb/conjunction | introducing additional information
HSK 5 | adjective | on the brink of danger or extinction; endangered
Esperanto vocabulary for school, university, subjects, learning, and academic life.
HSK 4 | noun/adjective | the record of past events or something relating to the past
JLPT N4 kanji 習 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 4 | noun | a personal position or way of looking at something
Learn to describe your daily routine using time adverbs and the -iĝ- suffix for reflexive changes of state.
HSK 4 | noun/verb | an immune reaction to a normally harmless substance
Learn to recognize and translate result (consecutive) clauses and indirect questions in classical Latin.
HSK 6 | verb/noun | to put forward an initiative; a formal proposal or initiative
HSK 3 | noun | bread; a baked flour product
HSK 5 | noun | an advantage, gain, or benefit that serves one's goals or welfare
HSK 3 | preposition / noun | based on, according to; a basis or grounds
Learn to describe the weather and seasons, using impersonal verbs and the conjunction ĉar.
HSK 8 | noun | fiber optics; thin glass or plastic fibers transmitting data as light pulses
HSK 3 | verb / noun | to choose, select; a choice or selection
HSK 4 | noun | a defined section of space or territory
HSK 5 | noun | a biological preparation used to provide immunity against a disease
Perfect passive with PPP + esse, pluperfect active and passive, future perfect active, and Latin temporal sequence.
HSK 2 | pronoun | refers to all the people in a group
Essential Japanese vocabulary for illness, medicine, hospitals, exercise, nutrition. N1 level reference with readings, romaji, and examples.
HSK 5 | verb | to bear; to endure; to withstand pressure, pain, or hardship
HSK 4 | noun | a large populated urban settlement
HSK 6 | n | the principles of valid reasoning; the internal coherence of an argument or situation
〜と: JLPT N4 grammar pattern. Usage, structure, examples, and comparison with similar patterns.
HSK 3 | preposition | about, concerning, regarding — introduces the topic of a statement
〜が (ga): JLPT N5 grammar pattern. Usage, structure, examples, and comparison with は.
円 (yen, circle): 4 strokes, JLPT N5. On: エン. Kun: まる.
HSK 3 | verb/noun | an upward rise in quantity or degree
HSK 2 | adjective | morally bad, of poor quality, or physically broken/spoiled
HSK 3 | noun/verb | an activity, event, or to move around
HSK 1 | number | the digit two; used in isolation, ordinals, and compound numbers
Common Japanese sentence patterns organized by JLPT level N5 through N1, with structure, usage notes, and example sentences for each pattern.
HSK 4 | noun | money spent or charged for a service or activity
HSK 3 | noun | dish, course of food; vegetable broadly — general term covering both cooked dishes and raw produce
Discuss health and medical topics using advanced 被 constructions and 受 + emotional/physical state at B1 level.
HSK 3 | adverb | happening infrequently or from time to time
HSK 5 | verb | to complicate, to make more complex, to become complicated
HSK 4 | noun/verb | an official notice or the act of informing someone
HSK 4 | adverb | indicating something happened in the past but no longer continues
HSK 3 | verb / noun | to contact, to get in touch; connection, link, relationship
Side-by-side comparison of Sindarin and Quenya: phonology, grammar, vocabulary, and cultural context — for understanding both languages and avoiding confusion between them.
HSK 5 | verb/noun | growth and advancement toward a more complete state
HSK 2 | verb + result complement | indicates buying resulted in actually obtaining the item
Discuss technology and innovation using 利用 + tech + V and vocabulary for AI, big data, and the internet at B1 level.
HSK 2 | conjunction | presents alternatives in a statement; contrast with 还是 for questions
HSK 1 | verb, noun | to end, to finish, the end
HSK 1 | noun | chicken egg, egg
HSK 4 | verb / noun | to help; assistance
HSK 5 | noun | ability, capability, capacity to do something
HSK 5 | noun/verb | to deal with or respond to a difficult situation
Comprehensive N4 review: authentic dialogues using all N4 grammar, a reading passage with comprehension questions, a 20-point self-assessment checklist, 30 mixed-level practice questions, and a preview of what awaits at N3.
HSK 5 | noun phrase | the warming of Earth's atmosphere due to trapped heat
HSK 1 | proper noun | China
HSK 4 | verb | to reflect a situation or to report a problem to a higher authority
HSK 5 | verb / noun | to appeal, to call for, to urge; an appeal, a call
HSK 4 | noun | quality; standard; mass (physics)
HSK 3 | verb / adjective | to know well, be familiar with — through repeated exposure or experience
Learn 〜うる and 〜える to express logical possibility in formal academic, legal, and official Japanese writing.
HSK 5 | noun / verb | the point from which something originates; to originate from
新 (shin/atara-shii): new. JLPT N5 kanji.
Learn to form and use the Esperanto passive voice with esti + participles, agent phrases with de, and the impersonal oni construction.
HSK 5 | noun/verb | a period or process of change from one state to another
Learn 〜わけではない to express partial negation — it doesn't mean that X, clearing up misunderstandings while allowing for nuance.
HSK 5 | noun phrase | content recommendation driven by algorithms
Sindarin complex sentences: subordinate clauses, relative clauses with i, embedded questions, temporal clauses, and attested examples from Tolkien's texts.
Learn 〜に対して (toward, against, in contrast to) and its attributive form 〜に対する — key N3 patterns for expressing directed attitudes, responses, and contrasts.
HSK 3 | adjective | feeling joyful and pleased
HSK 4 | adjective / verb | having or providing plentiful variety and substance
新 (new): 13 strokes, JLPT N5. On: シン. Kun: あたら-しい、あら-た、にい.
HSK 3 | verb / preposition | to pass through or by; after going through a process
HSK 5 | verb | deliberate or observed reduction in scope, size, or gap
変える (かえる / kaeru): to change. N4 level Japanese vocabulary.
HSK 5 | verb/adjective | to contain something hidden within; not explicitly stated
Learn 〜とは限らない to express that something is not necessarily or universally true, countering general assumptions.
HSK 3 | noun / verb | a link between people or things; also in set phrases
A comprehensive review of all A1 grammar patterns and the top 50 most useful HSK 1 words, with practice exercises.
JLPT N4 kanji 局 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
HSK 5 | adjective | feeling relief and peace of mind after worry or tension is resolved
HSK 6 | n | the organized, regulated state of affairs in society, an institution, or nature
HSK 4 | verb | to initiate and carry out an activity or campaign
Mastering Chinese medical register: clinical documentation, patient communication, and biomedical discourse.
HSK 5 | adjective | having the qualities of democracy; based on popular participation
HSK 3 | adverb | approximately, probably, roughly — an estimate rather than exact figure
HSK 5 | noun | conclusion, final judgement reached through reasoning
HSK 4 | verb | to bring a goal or dream into reality
HSK 1 | number | the digit eight; the luckiest number in Chinese culture
HSK 1 | verb | to produce speech or language
HSK 5 | verb/noun | financial loss; to operate at a loss; deficit
HSK 5 | verb / noun | to call on, to appeal to; a call, an appeal
北 (ホク/きた): north. JLPT N5 essential kanji.
Complete reference for the 100 most important four-character idioms (四字熟語) for JLPT N1: readings, literal meanings, actual meanings, usage examples, and exam notes.
HSK 4 | noun | the effect, function, or role that something plays
HSK 5 | adjective | profound in meaning; deeply felt; incisive and penetrating
HSK 4 | adjective | feeling of anxiety, or a taut/scarce situation
HSK 5 | noun | political system based on concentrated authority and limited freedoms
HSK 4 | verb | to approve; to authorize; to sanction
HSK 1 | noun | one's elder sister
HSK 1 | verb phrase | to get up, to rise from bed
10 structured N2 lessons for upper-intermediate learners: formal Japanese, newspaper reading, grammar discrimination, academic patterns, business Japanese, and exam strategies.
HSK 1 | adjective | small, little, young
HSK 4 | noun | the fundamental nature or essence of something
HSK 5 | noun | one's sense of personal worth and dignity
Learn 〜こそ and 〜だからこそ to place emphatic focus on a specific element — 'it is precisely X' and 'precisely because of X.'
We are given a list of integers and a set of queries. Each query asks how many contiguous subarrays of the list have a greatest common divisor equal to the query value.
HSK 5 | noun/verb/adjective | the feeling of sorrow over something missed or lost
HSK 5 | noun/verb | a strategic game or contest involving competing interests and decisions
HSK 1 | pronoun | interrogative pronoun for location (standard/southern form)
全然 (ぜんぜん / zenzen): not at all (+ negative). N4 level Japanese vocabulary.
HSK 8 | noun | thermodynamics; the branch of physics dealing with heat and energy
HSK 1 | noun | bed
Complete the correlative table with the universal ĉi- (every/all) and negative neni- (no/nothing) series, plus -ajn and combining forms.
JLPT N4 kanji 公 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
Esperanto vocabulary for jobs, workplace, professional activities, and employment.
HSK 4 | adjective | having a quality or character that belongs to no other person or thing
HSK 4 | noun | a span of time or a period during which something occurs
HSK 1 | noun | a week; combined with numbers 1–6 or 日/天 to name weekdays
HSK 5 | verb | to set up, establish, or form an organisation or team
HSK 5 | noun | money paid for the use of borrowed capital, or earned on savings
Master six N3 scope and contrast patterns: 〜だけでなく〜も, 〜ばかりでなく, 〜どころか, 〜わけではない, 〜とは限らない, and 〜にほかならない — with full nuance analysis.
HSK 3 | preposition | expressing purpose or goal
HSK 4 | verb | to participate, to join, to attend
HSK 5 | noun | information cocoon; filter bubble; echo chamber
HSK 2 | adverb | marks the welcome or long-awaited end of a process
HSK 4 | adjective | having the pleasant taste of sugar; also used figuratively
HSK 5 | noun | intrinsic property or attribute belonging to something
HSK 3 | noun | a written character, word, or one's handwriting
Forming questions in Sindarin: question words, yes/no questions, and word order changes when asking who, what, where, when, why.
HSK 3 | noun | actor, actress, performer
HSK 6 | verb | to set forth in detail, to expound on a topic systematically
HSK 3 | verb + complement | verb-result compound meaning to finish a task
Mastering 之、乎、者、也 and the grammar of classical Chinese in modern scholarly and literary contexts.
JLPT N4 kanji 指 — readings, vocabulary, stroke order, mnemonic, and common confusion pairs.
Top 10 mistakes English speakers make in Esperanto — grammar errors, pronunciation issues, and common misconceptions.
HSK 3 | adverb / conjunction | in addition, besides, furthermore; another, other
HSK 4 | verb | to nurture and develop a skill, person, or quality
HSK 4 | verb/noun | a formal account of findings or events
HSK 4 | noun | knowledge, learning
Sindarin dual number: dual pronouns for pairs, dual verb forms (attested and reconstructed), and the dual possessive — with cultural context on Elvish pairing.
HSK 4 | adverb | expressing something that has always been true, or (negated) never
We are given an undirected weighted graph representing cities and roads. City 1 is the capital. In addition to normal roads, there are special train routes that connect the capital directly to some city. The government wants to close as many train routes as possible.
Learn how potential complements with 得 and 不 express ability or possibility, one of the most productive grammatical patterns in Mandarin.
HSK 1 | pronoun | first-person singular pronoun
We are given a binary string consisting only of 'a' and 'b'. For every substring, we repeatedly merge consecutive equal characters into a single character.
HSK 4 | noun | experience gained through practice or life
年 (nen/toshi): year. JLPT N5 kanji.
Learn 〜のもとで and 〜のもとに to express acting under conditions, authority, supervision, or established premises.
This problem asks us to find the length of the shortest substring in a string s that matches a pattern p, where p contains exactly two '' wildcards. Each '' can match zero or more characters.
The equation is
The previous solution failed at the structural point where it tried to characterize tetrahedra admitting a sphere tangent to all six edge-lines.
The problem gives us an integer array nums and asks us to count how many valid partition points produce an even difference between the sum of the left part and the sum of the right part.
The problem is a construction problem on a fixed circumcircle.
Consider the equation
We seek all real numbers $x$ satisfying
Let $n$ be a natural number ending in 6 such that moving the 6 to the front produces a number four times as large.
We are asked to construct a triangle $ABC$ when the side lengths
We are given an undirected tree with n nodes labeled from 0 to n - 1. The tree is initially completely unmarked. We then simulate a spreading process that behaves like a multi-source breadth-first expansion: at each second, every unmarked node that has at least one already…
We are given a binary array nums consisting only of 0s and 1s. We are allowed to perform an operation any number of times: choose any three consecutive elements and flip them, meaning 0 becomes 1 and 1 becomes 0.
The problem gives us a list of distinct points on a 2D plane. Each point is represented as [x, y], where x is the horizontal coordinate and y is the vertical coordinate. We must count how many ordered pairs (A, B) satisfy two conditions: 1.
The problem gives us a list of employee access records. Each record contains two strings: - The employee's name - A timestamp in 24-hour "HHMM" format We need to determine which employees accessed the system at least three times within a time window that is strictly less than…
The problem is asking us to modify an array nums such that the XOR of all contiguous subarrays (segments) of length k is equal to zero, while minimizing the number of changes. Each element of nums is a non-negative integer less than .
The problem is a pure proof problem.
The statement asks for a lower bound on $a^2+b^2+c^2$ in terms of the area $S$ of a triangle.
The problem asks for conditions on the parameters $a$ and $b$ under which the system admits three distinct positive numbers $x,y,z$.
The problem concerns a right circular cone inscribed in a sphere and a cylinder circumscribed about the same sphere.
Consider triangle $ABC$ with an interior point $P$.
The reviewer correctly identified a fatal flaw in the previous argument.
The previous solution failed at two genuinely important points.
The problem asks for the smallest integer $n$ such that every $n$-element subset of $S={1,2,\dots,280}$ contains five numbers that are pairwise relatively prime.
The reviewers identified two genuine problems in the previous proof.
This problem asks us to count how many arrays of length n can be formed using integers between 1 and m such that exactly k adjacent pairs are equal. In other words, we are looking for sequences where the number of times arr[i] equals arr[i-1] is precisely k.
This problem asks us to count the number of special subsequences of length four in a given array nums of positive integers. A special subsequence (p, q, r, s) must satisfy two requirements.
Let f be a function that satisfies the following conditions:
Let P be a set of 7 different prime numbers and C a set of
Let n \geq1 be an integer. A path from (0, 0) to (n, n) in the
Let P be a polynomial of degree n satisfying
Three equal circles touch the sides of a triangle and have
ABCD is a quadrilateral with BC parallel to AD. M is the
Let S = {1, 2, 3, . . ., 280}. Find the minimal natural num-
(ROM 1′) Let n be a composite natural number and p a proper divisor
Let … be real numbers such that
Let Tk = k −1 for k = 1, 2, 3, 4 and
Let a be a positive integer and let {an} be defined by a0 = 0
Let n > m \geq1 be natural numbers such that the groups of
Let a tetrahedron ABCD be inscribed in a sphere S. Find the
For a positive integer n, let f(n) denote the number of ways to
Let g : C \toC, w \inC, a \inC, w3 = 1 (w ̸= 1). Show that
Let a, b be natural numbers with 1 \leqa \leqb, and M =
C6 (FRA 2) Let O be a point of three-dimensional space and let l1, l2, l3
For any integer r \geq1, determine the smallest integer h(r) \geq1
6b.(CAN 5) Let x1, x2, . . . , xn be positive numbers. Prove that
A flock of 155 birds sit down on a circle C. Two birds Pi, Pj are
Let a > b > c > d be positive integers and suppose
Let V be a finite subset of Euclidean space consisting of
We consider three distinct half-lines Ox, Oy, Oz in a plane.
C5 (USS 5) The right triangles ABC and AB1C1 are similar and have
Let M and N be points inside triangle ABC such that
Let p \geq5 be a prime number. Prove that there exists an
Let x1, x2, . . . , xn be real numbers satisfying the conditions
Points A, B, C divide the circumcircle Ωof the triangle ABC
A point M is chosen on the side AC of the triangle ABC in
The prolongation of the bisector AL (L \inBC) in the acute-
Two students A and B are playing the following game: Each
Let f(x) = x2+1
Let A, B, and C be noncollinear points. Prove that there is
Let a be the greatest positive root of the equation x3−3x2+1 =
Let the sequence …, …, be generated as follows:
1a.(CZS 3) The positive integers x1, . . . , xn, n \geq3, satisfy x1 < x2 <
Given seven points in the plane, some of them are connected
Let f(n) be the least number of distinct points in the plane
Let a1, a2, . . . , an be positive real numbers, n > 1. Denote by
A circle of radius 1 is located in a right-angled trihedron and
From a bag containing 5 pairs of socks, each pair a different
Let R be the set of real numbers. Does there exist a function
Let n be an integer, n \geq3. Let a1, a2, . . . , an be real numbers
The nonnegative real numbers x1, x2, x3, x4, x5, a satisfy the
Around a circular table an even number of persons have a
Let p be an odd prime and n a positive integer. In the
Let triangle ABC be such that its circumradius R is equal to
Find all polynomials P(x) with real coefficients that
For any set S of five points in the plane, no three of which
Let r \geq2 be a fixed positive integer, and let F be an infinite
Find a set A of positive integers such that for any infinite
Prove that there exist infinitely many positive integers n such
Let U = {1, 2, . . ., n}, where n \geq3. A subset S of U is said to be
Determine all the triples (a, b, c) of positive real numbers such
An eccentric mathematician has a ladder with n rungs that he
II 4 (FIN 3)IMO2 Let riangleABC be a triangle. Prove that there exists a
Find all complex numbers m such that polynomial
Let A0, A1, . . . , An be points in a plane such that
Determine all pairs (m, n) of positive integers such that
A6 (VIE 1)IMO6 Let S be a square with sides of length 100 and let L be
Let … denote the set of nonnegative integers. Find a bijective function … from … into … such that for all …,
Given an integer n > 1, denote by Pn the product of all
We are given 3n points A1, A2, . . . , A3n in the plane, no three
Let dn be the last nonzero digit of the decimal representation
In a convex quadrilateral ABCD the diagonal BD does
II 2 (NET 3)IMO5 If a, b, c, d are arbitrary positive real numbers, find all
Let a, b, c be positive integers satisfying the conditions b > 2a
Let ABC be a triangle and P an exterior point in the plane
The circle \Gamma and the line ℓdo not intersect. Let AB be the
Several equal spherical planets are given in outer space. On the
Given a, \theta \inR, m \inN, and P(x) = x2m −2|a|mxm cos \theta+a2m,
Prove that for any integer n \geq2, there exists a set of 2n−1
Is it possible to plot 1975 points on a circle with radius 1 so
Let I be the incenter of triangle ABC. Let K, L, and M
On the sides of an arbitrary triangle ABC, triangles BPC,
Let n be an integer greater than 2. Define V = {1 + kn |
Let k be a positive integer. Prove that there are infinitely
Let S be a convex quadrilateral ABCD and O a point inside
Let a, b, c, d, m, n be positive integers such that a2+b2+c2+d2 =
On the sides of a square ABCD one constructs inwardly
5a.(FRA 3) Let K and K′ be two squares in the same plane, their sides
Let AX, BY, CZ be three cevians concurrent at an inte-
There are six ports on a lake. Is it possible to organize a series
The following operation is allowed on a finite graph: Choose
Prove that the sequence 2n −3 (n > 1) contains a subse-
The function \psi from the set N of positive integers into itself
Let m be a fixed integer greater than 1. The sequence
For all integers n, n \geq0, there exist uniquely determined
Let aij, i = 1, 2, 3, j = 1, 2, 3, be real numbers such that aij
Consider in a plane i the points O, A1, A2, A3, A4 such that
Four different points A, B, C, D are chosen on a circle \Gamma such
Let A, B be adjacent vertices of a regular n-gon in the
Let … be a convex quadrilateral, and let …, …, …, and … denote the circumradii of the triangles …, …, …, and ……
Let p and q be integers. Show that there exists an interval I of
Let a and b be two positive integers such that ab+1 divides
Find all bases of logarithms in which a real positive number
Given a point … and lengths …, prove that there exists an equilateral triangle … for which …, …, …, if and only if …,…
Let a1 = 1111, a2 = 1212, a3 = 1313, and
Let n \geq2 be a fixed integer. Find the least constant C
Let ABC be a triangle, H its orthocenter, O its circumcenter,
A positive integer is written in each square of an m imesn board.
Show that in the plane there exists a convex polygon of 1992
Find, with proof, all functions f defined on the nonnegative
Let n be an integer greater than 2. A positive integer is said to be
A finite set of (distinct) positive integers is called a “DS-set”
Let R1, R2, . . . be the family of finite sequences of positive inte-
Let … be a function from the set of real numbers … into itself such that for all …, we have … and
In the plane we have n rectangles with parallel sides. The
Find all the functions f : R oR that satisfy
A finite number of beans are placed on an infinite row of squares. A sequence of moves is performed as follows: at each…
S6 (IND) Let N denote the set of all positive integers. Prove that there
Let R be a set of exactly 6 elements. A set F of subsets of R
Let f(0) = f(1) = 0 and
The circle inscribed in a triangle ABC touches the sides
Prove that for all positive real numbers a, b, c,
Find all solutions of the following system of n equations in n
An … square is divided into … unit squares in the usual manner. Each of the … vertices of these squares is to be…
Ali Barber, the carpet merchant, has a rectangular piece of
There are 2n words of length n over the alphabet {0, 1}. Prove
To each vertex Pi (i = 1, . . . , 5) of a pentagon an integer
Let ABC be any triangle and P any point in its interior. Let
Prove that in the Euclidean plane every regular polygon having
Let T be the set of ordered triples (x, y, z), where x, y, z are
Let n \geq5 be a given integer. Determine the largest integer
In the tetrahedron SABC the angle BSC is a right angle,
Let p > 3 be a prime number. For each nonempty subset T of
A line in the plane of a triangle … intersects the sides … and … respectively at points … and … such that …. Find the…
Let p be the product of two consecutive integers greater than
Prove that for any positive integer m there exist an infinite
Find the real values of p for which the equation
Let real numbers x1, x2, . . . , xn satisfy 0 < x1 < x2 < \cdot \cdot \cdot <
Consider a regular 2n-gon and the n diagonals of it that
Let f(x) = xn where n is a fixed positive integer and x =
Let n \geq2 be a natural number and let the real numbers
The triangle ABC is acute-angled. Let L be any line in the
Let ABC be an acute triangle. Let DAC, EAB, and FBC
Let S be the set of real numbers greater than −1. Find
A biologist watches a chameleon. The chameleon catches
For which positive integers n does there exist a positive
Let {fn} be the Fibonacci sequence {1, 1, 2, 3, 5, . . .}.
Given a point M on the side AB of the triangle ABC, let
Circles S1 and S2 intersect at points P and Q. Distinct points
Let A be a set of N residues (mod N 2). Prove that there
Let … be an acute-angled triangle with …. Let … be the circumcenter, … its orthocenter, and … the foot of its altitude…
Solve the system of equations
Determine all pairs (a, b) of real numbers such that a\lfloorbn\rfloor= b\lflooran\rfloor
We say that a set E of points of the Euclidian plane is
Let n be an integer greater than 1. In a circular arrange-
Let p be a prime and A = {a1, . . . , ap−1} an arbitrary subset
Does there exist a function f : N oN, such that f(f(n)) =
Let … be a finite set and let …, … be bijective functions from … onto itself. Let
Each pair of opposite sides of a convex hexagon has the
For which digits a do exist integers n \geq4 such that each digit
(a) A plane \pi passes through the vertex O of the regular
A circle S is said to cut a circle \Sigma diametrally if their common
Let ABCD be a convex quadrilateral whose vertices do not
Consider a sequence of polynomials P0(x), P1(x), P2(x), . . . ,
Let ABC be a triangle with ∡BAC = 60◦. Let AP bisect
For a polynomial P of degree 2000 with distinct real co-
A1A2A3 is an acute-angled triangle. The foot of the
In a convex quadrangle with area 32 cm2, the sum of the
There are two circles in the plane. Let a point A be one
At a party attended by n married couples, each person talks
Denote by S the set of all primes p such that the decimal
Suppose x1, x2, . . . , xn are real numbers with x2
B2 (POL 4) A convex, closed figure lies inside a given circle. The figure
Determine all integers n > 3 such that there are n points
Let an be the last nonzero digit in the decimal representation
Let \alpha, \beta, \gamma be positive real numbers such that \alpha + \beta + \gamma < \pi,
Let A, B, C, and D be distinct points on a line, in that
In an acute-angled triangle ABC, let AD, BE be altitudes and
Let ABCD be a convex quadrilateral and O the intersection of
Inside triangle ABC there are three circles k1, k2, k3 each of
Determine all pairs (x, y) of positive integers such that x2y+
Is it possible to put 1987 points in the Euclidean plane
3b.(GBR 4) A sequence of polynomials Pm(x, y, z), m = 0, 1, 2, . . ., in
Let n be a positive integer. Find the number of odd coefficients
Let A, B, and C be three points on the edge of a circular
I 1 (USA 4)IMO1 Alice, Betty, and Carol took the same series of exam-
Let S1 and S2 be circles meeting at the points A and B. A
Let n be an even positive integer. Show that there is a
A circle O with center O on base BC of an isosceles triangle
Let n be a positive integer and let (x1, . . . , xn), (y1, . . . , yn)
Let x1, . . . , xn and y1, . . . , yn be real numbers. Let A =
In a certain city, age is reckoned in terms of real numbers
Let a, b, c be positive integers satisfying (a, b) = (b, c) =
Let ABC be an acute-angled triangle with AB ̸= AC.
For an n imes n matrix A, let Xi be the set of entries in row
Solve the following system of equations, in which a is a given
Find all positive integers x and y such that x+y2+z3 = xyz,
Given nine points in space, no four of which are coplanar,
For which integers n \geq3 does there exist a regular n-gon in the
Let ABCD be a convex quadrilateral such that AC =
Prove that
An infinite sequence a0, a1, a2, . . . of real numbers satisfies
Given a nonequilateral triangle ABC, the vertices listed coun-
Given a set S in the plane containing n points and satis-
An infinite square grid is colored in the chessboard pattern.
In acute triangle ABC with circumcenter O and altitude
Decide whether there exists a set M of natural numbers satis-
If … and … are arbitrary positive real numbers and … an integer, prove that
Let n \geq2 be a positive integer and \lambda a positive real
Let u1, u2, . . . , un, v1, v2, . . . , vn be real numbers. Prove that
The vertex A of the acute triangle ABC is equidistant from
Let O be the center of the circumsphere of a tetrahedron
Let a and b be natural numbers and let q and r be the
Let a and b be nonnegative integers such that ab \geqc2,
In the plane, let there be given a circle C, a line l tangent
Let a set of p equations be given,
The bisectors of angles A, B, C of a triangle ABC meet its cir-
B1 (CAN 2)
Let a0, a1, . . . , an, an+1 be a sequence of real numbers satisfying
Let A be the sum of the digits of the number 1616 and B
Find all pairs of functions f : R oR, g : R oR such that
Prove that for every natural number m \geq1 there exists a
The incenter of the triangle ABC is K. The midpoint of AB
In a plane two different points O and A are given. For
For a triangle T = ABC we take the point X on the side
Does there exist a positive integer n such that n has
Forty-nine students solve a set of three problems. The score for
Let a1 \geqa2 \geqa3 be given positive integers and let N(a1, a2, a3)
Suppose that every integer has been given one of the colors
A rectangular array of numbers is given. In each row and each
Let n1, n2 be positive integers. Consider in a plane E two dis-
Let D be the interior of the circle C and let A \inC. Show
C3 (CAN 5) Show that
B6 (FIN 3) Four distinct circles C, C1, C2, C3 and a line L are given in
The vertices D, E, F of an equilateral triangle lie on the sides
A circle \omega is tangent to two parallel lines l1 and l2. A second
(a) Decide whether the fields of the 8 imes 8 chessboard can be numbered
Prove that from x + y = 1 (x, y \inR) it follows that
On an infinite chessboard, a solitaire game is played as
Let … be an arbitrary triangle and … a point inside it. Let … be the distances from … to sides …; … the lengths of the…
S5 (FIN) For positive integers n, the numbers f(n) are defined induc-
At n distinct points of a circular race course there are n cars
Consider the two square matrices
Let R denote the set of all real numbers and R+ the subset
Let ABC be a triangle and L the line through C parallel to
For every integer n \geq2 determine the minimum value that the
Let Ox, Oy, Oz be three rays, and G a point inside the trihe-
A finite set of unit circles is given in a plane such that the area
Given a point P in a given plane \pi and also a given point
Prove that for all n \inN the following is true:
If … … are distinct non-zero real numbers, prove that the equation
Find all polynomials f(x) with real coefficients for which
Prove that every integer k greater than 1 has a multiple that is
Prove that the intersection of a plane and a regular tetrahedron
Every point with integer coordinates in the plane is the
A solitaire game is played on an m imes n rectangular board, using
Let n > 6 and a1 < a2 < \cdot \cdot \cdot < ak be all natural numbers
The sequence a0, a1, a2, . . . is defined as follows:
Given an oriented line … and a fixed point … on it, consider all trapezoids … one of whose bases … lies on …, in the…
Let S be a set of n points in the plane. No three points of
Show that if a, b, c are the lengths of the sides of a triangle
In a permutation (x1, x2, . . . , xn) of the set 1, 2, . . . , n we call
For a given triangle ABC, let X be a variable point on
In the triangle ABC let B′ and C′ be the midpoints of the sides
Let a, b, c be given integers a > 0, ac −b2 = P = P1 \cdot \cdot \cdot Pm
N is an arbitrary point on the bisector of ngleBAC.
Let A, B, C be fixed points in the plane. A man starts
Let n > 1 be an integer and let f(x) = xn + 5xn−1 + 3.
Ali Barber, the carpet merchant, has a rectangular piece of
Let …, …, and … be positive real numbers such that ….
Let ABC be an isosceles triangle with AC = BC, whose
The tangents at B and A to the circumcircle of an acute-
Prove that
Let a be a rational number with 0 < a < 1 and suppose that
Given … (…) points in space such that every three of them form a triangle with one angle greater than or equal to …,…
Let p be a prime number and let f(x) be a polynomial of degree
Let ABC be a triangle with centroid G. Determine, with
Prove that for any positive integers x, y, z with xy−z2 = 1 one
I 5 (GBR 3) Let Ar, Br, Cr be points on the circumference of a given
Find all functions f defined on the positive real numbers
The set M = {1, 2, . . ., 2n} is partitioned into k nonintersecting
Find all positive integer solutions x, y, z of the equation 3x +
Prove that it is possible to place 2n(2n + 1) parallelepipedic
Find the sphere of maximal radius that can be placed inside
A finite sequence of integers … is called quadratic if for each … we have the equality ….
On a circle, 2n −1 (n \geq3) different points are given. Find
An integer sequence is defined by
Let n \geq4 be a fixed positive integer. Given a set S =
A3 (USS 4)IMO3 Consider the infinite sequences {xn} of positive real
Determine the least possible value of f(1998), where f is a
A magician has one hundred cards numbered 1 to 100.
Peter has three accounts in a bank, each with an integral
Let a1, a2, a3, . . . be any infinite increasing sequence of pos-
I 3 (SWE 3)IMO6 Let P(x) be a polynomial with integer coefficients. If
Prove that on the coordinate plane it is impossible to draw a
Ten points such that no three of them lie on a line are marked in
The set {a0, a1, . . . , an} of real numbers satisfies the following
Find all functions f from the reals to the reals such that
A function f defined on the positive integers (and taking
Show that for any vectors a, b in Euclidean space,
A sequence of integers a1, a2, a3, . . . is defined as follows: a1 = 1,
In the plane we are given a set E of 1991 points, and certain
In a contest, there are m candidates and n judges, where
Let … denote the set of nonnegative integers. Find all functions … such that
A circle whose center is on the side ED of the cyclic
(IND 3′)IMO1 Given a circle with two chords AB, CD that meet at E, let
Let ABC be a triangle such that ngleA = 90◦and ngleB < ngleC. The
Let n be a positive integer. Let \sigma(n) be the sum of the natural
Prove for each triangle ABC the inequality
Let A1A2A3 be a nonisosceles triangle with incenter I. Let Ci,
Let n be an integer, n \geq3. Let x1, x2, . . . , xn be real numbers
Let O be the circumcenter and H the orthocenter of an acute
The positive integers … and … are such that the numbers
Find all solutions … of the equation
Let n be a positive integer. Each point (x, y) in the plane,
(a) If a 5 imes n rectangle can be tiled using n pieces like those
Find all finite sequences (x0, x1, . . . , xn) such that for every
Let Sn be the number of sequences (a1, a2, . . . , an), where ai \in
For three points A, B, C in the plane we define m(ABC)
A2 (YUG 1) Let K be a convex polygon in the plane and suppose that
Let f(x) be a polynomial with rational coefficients and lpha be
For a finite graph G, let f(G) be the number of triangles
Let ABCDEFGH be a parallelepiped with AE\parallelBF\parallelCG\parallelDH.
A brick staircase with three steps of width 2 is made of twelve
Two congruent equilateral triangles ABC and A′B′C′ in the
Determine the smallest integer n \geq4 for which one can choose
The numbers from 1 to n2 are randomly arranged in the cells
Let … be an acute-angled triangle with circumcenter … and circumradius …. Let … meet the circle … again in …, let ……
The circle S has center O, and BC is a diameter of S.
Let T1 be a triangle having a, b, c as lengths of its sides and let
Prove that there exists a unique triangle whose side
Real constants a, b, c are such that there is exactly one square
An acute triangle ABC is given. Points A1 and A2 are taken
Let F be a nonempty set of functions f : R oR of the
Prove that the set of positive integers cannot be partitioned
Let M be a point in the interior of triangle ABC. Let A′ lie
Let R be a rectangle that is the union of a finite number of
Consider the n imes n array of nonnegative integers
Let f(x) be a monic polynomial of degree 1991 with integer
A5 (NET 2)IMO5 Let A1A2A3A4A5A6 be a regular hexagon. Each of its
For any positive integer x0, three sequences {xn}, {yn}, and
A convex quadrilateral ABCD has perpendicular diagonals.
Let m and n be nonnegative integers. Prove that m!n!(m+
Let p be a prime number and let A be a set of positive integers
Let P be a polynomial with real coefficients such that P(x) > 0
Find all pairs of integers x, y \geq1 satisfying the equation
I 4 (USS 4) The sum of the squares of five real numbers a1, a2, a3, a4, a5
Let f(x) be a continuous function defined on the closed interval
Let A1A2 . . . An be a regular n-gon. The points B1, . . . , Bn−1
Among a group of 120 people, some pairs are friends. A weak
For a given positive integer k denote the square of the sum of
Let P be a cubic polynomial given by P(x) = ax3+bx2+cx+
Prove that the sum of the face angles at each vertex of a tetra-
Let …, …, … be real numbers. Prove that the system of equations
Let a, b, c be positive real numbers with product 1. Prove
On a 5 imes 5 board, two players alternately mark numbers on
Let a0, a1, a2, . . . be an arbitrary infinite sequence of positive
In a triangle ABC, let D and E be the intersections of the bisec-
II 6 (USS 1) In a certain language words are formed using an alphabet
Let … be three positive integers with ….
3a.(USS 3) Find a method by which one can compute the coefficients
A soldier has to investigate whether there are mines in an
Let n, k be positive integers such that n is not divisible by
Let b, m, n be positive integers such that b > 1 and m ̸= n. Prove
Consider the set of all strictly decreasing sequences of n natural
A cyclic quadrilateral ABCD is given. The lines AD and
The quadrilateral ABCD has the following properties:
Let ABCD be a convex quadrilateral with AB not parallel
Let n be an odd integer greater than 1 and let c1, c2, . . . ,
Let F(n) be the set of polynomials P(x) = a0+a1x+\cdot \cdot \cdot+anxn,
Let x1 \geqx2 \geq\cdot \cdot \cdot \geqxn and y1 \geqy2 \geq\cdot \cdot \cdot \geqyn be two
I 6 (ROM 4)IMO3 Does there exist a natural number n for which the
Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a…
Let A1A2 . . . An be a convex polygon, n \geq4. Prove that
Let … be a convex hexagon such that …, …, and …. Let …, …, … be the circumradii of triangles …, …, … respectively, and…
(a) For which values of n > 2 is there a set of n consecutive
Let P be a cubic polynomial with rational coefficients, and let
B5 (CAN 3)
A circle is called a separator for a set of five points in a plane
Let n be a positive integer that is not a perfect cube. Define
For what values of n does there exist an n imes n array of entries
Determine the maximum value of the sum
(FIN 2‘) Let E be a finite set of points such that E is not contained in
Let x, y, z be nonnegative real numbers with x+y +z = 1.
Determine the minimum of a2 + b2 if a and b are real
For all rational x satisfying 0 \leqx < 1, f is defined by
The set S = {2, 5, 13} has the property that for every
Let a, b, c be positive numbers with \sqrta+
Let r1, r2, . . . , rn be real numbers greater than or equal to 1.
A lattice point in the plane is a point both of whose coordinates
Find four positive integers each not exceeding 70000 and each
Let A1A2A3A4 be a tetrahedron, G its centroid, and
The triangular array (an,k) of numbers is given by an,1 = 1/n,
Let … be the real polynomial function
A4 (BUL 2) Determine all real values of the parameter a for which the
Let f, g, and a be polynomials with real coefficients, f and g
In a right-angled triangle ABC, let AD be the altitude
Given a tetrahedron ABCD whose all faces are acute-
Determine all m imes n rectangles that can be covered with
Let au(n) denote the number of positive divisors of the positive
If a, b, and c are sides of a triangle, prove that
For n \geq3 and a1 \leqa2 \leq\cdot \cdot \cdot \leqan given real numbers we
Let p be an odd prime. Determine positive integers x and
Let n \geq2 be a positive integer, with divisors 1 = d1 <
Let A1 be the center of the square inscribed in acute triangle
Prove that there exist two strictly increasing sequences (an)
A plane cuts a right circular cone into two parts. The plane is
Find the largest number obtainable as the product of pos-
Prove that the functional equations
For each positive integer n, denote by s(n) the greatest
A pile of n pebbles is placed in a vertical column. This
Given a real number a > 1, construct an infinite and
Let O be an interior point of acute triangle ABC. Let A1
If an acute-angled triangle ABC is given, construct an equilat-
On a semicircle with unit radius four consecutive chords AB, BC,
Find two positive integers a, b such that none of the num-
Prove that for every real number M there exists an infinite
In the coordinate plane a rectangle with vertices (0, 0), (m, 0),
Let Q be the center of the inscribed circle of a triangle ABC.
Each positive integer a undergoes the following procedure in
At a round table are 1994 girls, playing a game with a deck
C1 (NET 1)IMO2 A scalene triangle A1A2A3 is given with sides a1, a2, a3
On the sides of the triangle ABC, three similar isosceles tri-
Let n be an integer greater than 1. Define
Let n be an even positive integer. We say that two dif-
At a meeting of 12k people, each person exchanges greetings
A sequence (an) is defined by means of the recursion
We are given two mutually tangent circles in the plane, with
For an acute triangle ABC, M is the midpoint of the segment
A set of three nonnegative integers {x, y, z} with x < y < z
Consider a sequence of circles K1, K2, K3, K4, . . . of radii
For a positive integer n, let d(n) be the number of all positive
A, B, C, D are four points in the plane, with C, D on the
5b.(BEL 2)
Find all functions f : R oR satisfying
Prove:
We consider permutations (x1, . . . , x2n) of the set {1, . . . ,
Let a, b, c be integers and p an odd prime number. Prove that
Let ABC be a triangle and M an interior point. Prove that
Find all positive integers n having the property that 2n+1
Consider the polynomial p(x) = xn+nxn−1+a2xn−2 +\cdot \cdot \cdot+an
Let a1, a2, . . . , an, . . . be a sequence of real numbers such that
Prove that for every natural number k (k \geq2) there exists an
Prove that the product of five consecutive positive integers
S1 (UKR) Does there exist a sequence F(1), F(2), F(3), . . . of nonneg-
Let a, b, c > 0 and ab + bc + ca = 1. Prove the inequality
The incircle Ωof the acute-angled triangle ABC is tangent
Find all natural numbers n for which 28 + 211 + 2n is a perfect
Let … be integers such that …. Determine the maximum size of a subset … of the set … such that no … distinct elements…
A set of 10 positive integers is given such that the decimal
Let p1, p2, . . . , pn be distinct primes greater than 3. Show
An infinite arithmetic progression whose terms are positive in-
Two ships sail on the sea with constant speeds and fixed directions. It is known that at … the distance between them…
Let R+ be the set of all nonnegative real numbers. Given two
The point M inside the convex quadrilateral ABCD is such
The tangents at B and C to the circumcircle of the acute-angled
Let A be a 101-element subset of the set S = {1, 2, . . . ,
A function f : I oR, defined on an interval I, is called
Assume that the set of all positive integers is decomposed into
Let ABCD be a cyclic quadrilateral. Let P, Q, R be the
In a test, 3n students participate, who are located in three
4b.(IRE 4) A set of 1985 points is distributed around the circumference
Determine for which positive integers k the set
Let a, b, c, d be nonnegative real numbers such that ab + bc +
C8 (TUN 3) Let ABCD be a convex quadrilateral and draw regular tri-
For each finite set U of nonzero vectors in the plane we define
Let S be the set of all pairs (m, n) of relatively prime positive
2a.(USA 3) Determine the radius of a sphere S that passes through the
Let ABCD be a cyclic quadrilateral. Let E and F be variable
Find all integer triples (p, q, r) such that 1 < p < q < r
Prove that every positive rational number can be repre-
Inside an equilateral triangle ABC one constructs points P,
Find all positive integers k for which the following statement is
Let … have orthocenter …, and let … be a point on its circumcircle, distinct from …, …, …. Let … be the foot of the…
A line l does not meet a circle \omega with center O. E is the
Let ABCD be a tetrahedron having each sum of opposite sides
Let n \geq2 be an integer. Find the maximal cardinality of a set
Determine the maximum value of m2 + n2 where m and n
An international society has its members in 6 different
4a.(BUL 1)
Let n be a positive integer and let x1 \leqx2 \leq\cdot \cdot \cdot \leqxn be
A function f from the set of positive integers N into itself is
Given any point P in the interior of a sphere with ra-
Find the least natural number n such that if the set
Let N be the number of integral solutions of the equation
A rectangular box can be filled completely with unit cubes.
Determine all positive integers n for which there exists an integer
B4 (BRA 1) A box contains p white balls and q black balls. Beside the
Determine all positive integers n \geq2 that satisfy the following
Let E be the set of 19833 points of the space R3 all three
Let f : N oN be a function that satisfies the inequality
(a) Show that the set Q+ of all positive rational numbers can be par-
Assume that f(x, y) is defined for all positive integers x and
Show that any two points lying inside a regular n-gon E can
Determine whether there exist distinct real numbers a, b, c, t
In town A, there are n girls and n boys, and each girl knows each
Let n \inN, n \geq2, and A0 = (a01, a02, . . . , a0n) be any n-tuple
Prove that for each n \geq4 every cyclic quadrilateral can
Let P be a point inside a regular tetrahedron T of unit volume.
Prove that the product of the radii of three circles exscribed to a given triangle does not exceed … times the product…
Let a, b, n be positive integers, b > 1 and bn −1 | a. Show
Let ABCDEF be a convex hexagon with AB = BC =
Let k be a fixed integer greater than 1, and let m = 4k2 −5.
For what natural numbers n can the product of some of
II 3 (CUB 3) Let x, y, z be real numbers each of whose absolute value
We call a positive integer alternate if its decimal digits
Does there exist a second-degree polynomial p(x, y) in two
Let \Gamma1, \Gamma2, \Gamma3, \Gamma4 be distinct circles such that \Gamma1, \Gamma3 are
Let P be a convex polygon. Prove that there is a convex
A bicentric quadrilateral is one that is both inscribable in
Find, with proof, the point P in the interior of an acute-angled
Consider two segments of length … (…) and a segment of length ….
Let c be a positive integer. The sequence {fn} is defined as
Let ABC be a triangle, Ωits incircle and Ωa, Ωb, Ωc three
Define a k-clique to be a set of k people such that every pair
Given the integer n > 1 and the real number a > 0 determine
Show that for any n ̸quiv0 (mod 10) there exists a multiple of
Given a set M of 1985 positive integers, none of which
For any positive integer k, Ak is the subset of {k+1, k+
Let T denote the set of all ordered triples (p, q, r) of nonneg-
There are n + 1 fixed positions in a row, labeled 0 to n in
Let (Fn)n\geq1 be the Fibonacci sequence F1 = F2 = 1, Fn+2 =
Does there exist a set M in usual Euclidean space such that
Let n be a positive integer and let p be a prime number, p > 3.
A circle with center O passes through points A and C and
Does there exist an integer n > 1 that satisfies the following
Consider two concentric circles of radii R and r (R > r)
Let E be a set of n points in the plane (n \geq3) whose co-
Let xn =
A broken line A1A2 . . . An is drawn in a 50 imes50 square, so that
Let Z denote the set of all integers. Prove that for any integers
Let {f(n)} be a strictly increasing sequence of positive
For every integer d \geq1, let Md be the set of all positive
We are given a fixed point on the circle of radius …, and going from this point along the circumference in the positive…
Ten gangsters are standing on a flat surface, and the distances
Show that the set of positive integers that cannot be repre-
Cards numbered 1 to 9 are arranged at random in a row. In a
Given that 1 −1
Let b be an integer greater than 5. For each positive integer
For a triangle ABC, let k be its circumcircle with radius r. The
Let m, n \geq2 be positive integers, and let a1, a2, . . . , an
Let a, b, and c be positive real numbers such that abc = 1.
Let a, b, c, d be odd positive integers such that a < b < c <
For any positive integer x define
Let ABC be a triangle such that ngleACB = 2ngleABC. Let D be
For a positive integer n define a sequence of zeros and ones
Let n and k be positive integers. There are given n circles
Find all natural numbers n for which every natural number
Unit cubes are made into beads by drilling a hole through
Find all positive integers … for which …,
Let n be a positive integer and let a, b be given real numbers.
Let ABCD be a convex quadrilateral for which the circle
Determine whether or not there exist two disjoint infinite sets … and … of points in the plane satisfying the following…
The altitudes through the vertices A, B, C of an acute-angled
Let K be one of the two intersection points of the circles W1
Let a, b, A, B be given constant real numbers and
Determine all pairs (a, b) of positive real numbers with a ̸= 1
Does there exist a function s: Q … such that if x
Let ABC be a triangle. The bisector of angle A meets
Let p and q be relatively prime positive integers. A subset
Let x1, x2, . . . , xn be real numbers satisfying x1+x2+\cdot \cdot \cdot+xn =
Find the integer represented by
Let P(x) be a polynomial with real coefficients such that P(x) >
A natural number n is said to have the property P if whenever
Prove that there exists a four-coloring of the set M =
The triangle ABC is inscribed in a circle. The interior bi-
Let N = {1, 2, . . ., n}, n \geq2. A collection F = {A1, . . . , At}
Determine the smallest natural number n having the following
(POL 1b) Let I = (0, 1] be the unit interval of the real line. For a given
Suppose that {x1, x2, . . . , xn} are positive integers for which
Given a convex polyhedron P1 with 9 vertices A1, . . . , A9,
Let n be a natural number and a1, a2, . . . , a2n mutually distinct
O is a point inside a convex quadrilateral ABCD of area
The function f(x, y) is a homogeneous polynomial of the nth
The localities P1, P2, . . . , P1983 are served by ten international
Define a sequence ⟨f(n)⟩\infty
Let R+ be the set of all positive real numbers. Find all
Find all nondecreasing functions f : R oR such that
Let … denote the integer part of …, i.e., the greatest integer not exceeding …. If … is a positive integer, express as…
Let ABC be a triangle and let P be a point in its interior.
Find all solutions in positive real numbers xi (i =
A polynomial … with integer coefficients is said to be divisible by an integer … if … is divisible by … for all…
Let D1, . . . , Dn be closed disks in the plane. (A closed disk
Let [x] denote the greatest integer less than or equal to x.
Let B be a point on a circle S1, and let A be a point distinct
Find the point P inside the triangle ABC for which
Determine all triples of positive integers (a, m, n) such that
Let a0, a1, a2, . . . be an increasing sequence of nonnegative inte-
Let m be a positive odd integer, m \geq2. Find the smallest
Let n be a positive integer having at least two different prime
Two circles Ω1 and Ω2 touch internally the circle Ωin
Consider the system
Let A be a nonempty set of positive integers. Suppose that
Given riangleABC with no side equal to another side, let G, K,
A cube is assembled with 27 white cubes. The larger cube is then
Let f(n) be a function defined on the set of all positive integers
A sphere S is tangent to the edges AB, BC, CD, DA of a tetrahe-
There are 10001 students at a university. Some students join
We call a set S on the real line R superinvariant if for any
Let O be the circumcenter of an acute-angled triangle ABC
Show that the solution set of the inequality
Let … be a real number and … a real function defined on all of …, satisfying for all …,
C2 (AUS 4)
Given real numbers x1, x2, . . . , xn (n \geq2), with xi \geq1/n
For which positive integers n do there exist two polynomials f
Let B be a set of k sequences each having n terms equal to 1 or
A set S of points in space will be called completely sym-
Let S be any point on the circumscribed circle of rianglePQR. Then
In the triangle ABC, let D, E be points on the side BC such
Let S be the set of all the odd positive integers that are not
For n an odd positive integer, the unit squares of an n imes n
Does there exist a set M with the following properties?
Let {ak}\infty
Let D be an internal point on the side BC of a triangle ABC.
A wobbly number is a positive integer whose digits in base 10
Given a tetrahedron ABCD, let x = AB \cdot CD, y = AC \cdot BD,
Let a1, a2, . . . , an be positive real numbers such that a1 + a2 +
A number of signal lights are equally spaced along a one-way
Let f(k) be the number of integers n that satisfy the following
The length of a finite sequence is defined as the number of
Prove that every partition of 3-dimensional space into three
Let … be a point inside … such that
Consider the triangle ABC, its circumcircle k with center O
Prove that a convex pentagon (a five-sided polygon) ABCDE
Prove that there is no positive integer n such that for k =
Let u1, u2, . . . , um be m vectors in the plane, each of length
Let m and n be positive integers. The set A = {a1, a2, . . . ,
Let N = {1, 2, 3, . . .}. Determine whether there exists a
In a triangle ABC, choose any points K \inBC, L \inAC,
Two circles G1 and G2 intersect at M and N. Let AB
Let a, b, c, d, e be real numbers. Prove that the expression
Let n be a positive integer. How many integer solutions
Let n \geq3 be an integer and t1, t2, . . . , tn positive real
Let a, b, and c be given positive real numbers. Determine all
What is the smallest positive integer t such that there exist
A1 (GBR 3)IMO1 The function f(n) is defined for all positive integers
Let lpha(n) be the number of digits equal to one in the binary
A game is played by n girls (n \geq2), everybody having a ball.
Let N be a positive integer. Two players A and B, taking
Let … be nonnegative real numbers, not all zero.
Prove or disprove: Given a finite set of points with integer
The integer 9 can be written as a sum of two consecutive
Let a, b, c, d be four nonnegative numbers satisfying a+b+c+d =
An integer n is said to be good if |n| is not the square of
Let a0, a1, . . . , ak (k \geq1) be positive integers. Find all positive
Does there exist a finite set M of points in space, not all in
Prove that the set {1, 2, . . ., 1989} can be expressed as the
Let Q+ be the set of positive rational numbers. Construct
Find all functions f : R oR such that
Is it possible to find 100 positive integers not exceeding
Let f and g be two integer-valued functions defined on the set
Let a1, a2, . . . be an infinite sequence of real numbers for
II 1 (POL 2) Let ai, bi be coprime positive integers for i = 1, 2, . . . , k,
Let P1, P2, . . . , Pn be distinct points of the plane, n \geq2. Prove
In a triangle ABC we have AB = AC. A circle is tangent
Let d be the sum of the lengths of all diagonals of a convex
Given five real numbers u0, u1, u2, u3, u4, prove that it is always
C7 (CZS 3)
Let p be an odd prime. Find the number of p-element
Let ABC be an acute-angled triangle. Three lines LA, LB,
Given a convex polygon A1A2 . . . An with area S, and a point
Natural numbers from 1 to 99 (not necessarily distinct) are
Let a1, a2, . . . , an be positive numbers and q a given real
Let a, b be integers that are not perfect squares. Prove that if
Prove that the sum of an odd number of unit vectors passing
Circles G, G1, G2 are three circles related to each other as
Find all positive integers a1, a2, . . . , an such that
Given two congruent triangles … and … …, prove that there exists a plane such that the orthogonal projections of these…
Let ABCDEF be a convex hexagon such that AB = BC, CD =
Find all pairs of positive integers m, n \geq3 for which
Let a0 = 1994 and an+1 =
(a) Let (m, k) = 1. Prove that there exist integers a1, a2, . . . , am
Given n countries with three representatives each, m commit-
Find the number of all …-digit numbers for which some fixed digit stands only in the …th … place and the last … digits…
Let x1 and x2 be relatively prime positive integers. For n \geq2,
(GBR 1a)IMO6 For all positive integral n, un+1 = un(u2
Twenty-one girls and twenty-one boys took part in a
Prove that the volume of a tetrahedron inscribed in a right
We are given a positive integer … and a rectangular board … with dimensions …, …. The rectangle is divided into a grid…
Let ϕ : {1, 2, 3, . . .} … be injective. Prove that
Let ABC be a triangle. A circle passing through B and C in-
Let the sides of two rectangles be … and … with
Let f(x) = x2 + x + p, p \inN. Prove that if the numbers
Let n be a positive integer. A sequence of n positive integers
Let a1 \geq\cdot \cdot \cdot \geqan \geqan+1 = 0 be a sequence of real numbers.
Prove that N = 5125−1
The lock on a safe consists of three wheels, each of which may
C4 (GBR 2)IMO4 Prove that if n is a positive integer such that the
An n imes n matrix with entries from {1, 2, . . ., 2n −1} is called
Let x, y, and z be positive real numbers such that xyz = 1. Prove
A pentagonal prism A1A2 . . . A5B1B2 . . . B5 is given. The
(FIN 2′) Let f : [0, 1] oR be continuous and satisfy:
Let ABC be a triangle with semiperimeter s and inradius
Let ABCDEF be a convex hexagon such that ngleB +ngleD +ngleF =
Let A = (a1, a2, . . . , a2001) be a sequence of positive integers.
Let a and b be the bases of two number systems and let
Let ABC be a triangle with bisectors AA1, BB1, CC1 (A1 \in
In a given tetrahedron ABCD let K and L be the centers of
Find all functions f from the reals to the reals such that
Let f and ϕ be real functions defined on the set R satisfying
A semicircle \Gamma is drawn on one side of a straight line l. C
S4 (NZL) Suppose that x1, x2, x3, . . . are positive real numbers for which
Find all positive integers n such that
Let ABC be a triangle for which there exists an interior
A nonempty set A of real numbers is called a B3-set if the
Prove that for each positive integer n, there exists a positive
Find the number of partitions of the set {1, 2, . . ., n} into three
Let K denote the set {a, b, c, d, e}. F is a collection of 16 different
Let f(x) = x8 + 4x6 + 2x4 + 28x2 + 1. Let p > 3 be a prime
Let p be a prime number. Prove that there exists a prime
Let S1 and S2 be two spheres with distinct radii that touch
2b.(VIE 1)
Ten localities are served by two international airlines such
Prove the following assertion: The four altitudes of a tetrahe-
Let n be an even positive integer. Let A1, A2, . . . , An+1 be
Let n \geqk \geq0 be integers. The numbers c(n, k) are defined as
Find all functions f : R oR satisfying the equation
Let S and F be two opposite vertices of a regular octagon.
For points A1, . . . , A5 on the sphere of radius 1, what is the
Angles of a given triangle ABC are all smaller than 120◦.
For any positive integer n, let au(n) denote the number of its
A sequence of real numbers u1, u2, u3, . . . is determined by u1
Show that there exists a finite set A \subsetR2 such that for
Given any integer n \geq2, assume that the integers a1, a2, . . . , an
Is there a 1990-gon with the following properties (i) and
The matrix
Prove that for every integer n > 1 the equation
In the plane are given a point … and a polygon … (not necessarily convex). Let … denote the perimeter of …, … the sum…
II 5 (BUL 1)IMO4 Consider a partition of an 8 imes 8 chessboard into p
Let x0 = 5 and xn+1 = xn +
Determine all pairs (a, b) of positive integers such that
The incircle of ABC touches BC, CA, and AB at D, E, and
In the convex pentagon ABCDE, the sides BC, CD, DE have
Let n and k be positive integers such that n/2 < k \leq2n/3.
Show that 20
Let S be a set of n elements. We denote the number of all
In the plane we are given two circles intersecting at X and Y .
On an infinite square grid, two players alternately mark sym-
In the triangle ABC, with ∡A = 60◦, a parallel IF to AC
Suppose G is a connected graph with n edges. Prove that
6a.(SWE 3)IMO6 The sequence f1, f2, . . . , fn, . . . of functions is defined
Let n \geq3 be a positive integer. Let C1, C2, C3, . . . , Cn
Let ABC be a triangle and M an interior point in ABC.
Let … be the apothem (distance from the center to one of the sides) of a regular …-gon (…) inscribed in a circle of…
Two players A and B play a game in which they choose
Find the highest degree k of 1991 for which 1991k divides the
A polyhedron has 12 faces and is such that:
The function F is defined on the set of nonnegative integers
Consider a matrix of size n imesn whose entries are real numbers
Knowing that the system
Describe all closed bounded figures hi in the plane any two
Let … be an equilateral triangle and let … be a point in its interior. Let the lines …, …, … meet the sides …, …, … in…
Let lpha be the positive root of the equation x2 = 1991x + 1. For
Given a circle K, find the locus of vertices A of parallelograms
Suppose that n \geq2 and x1, x2, . . . , xn are real numbers between
1b.(TUR 5) Find the smallest positive integer n such that
A tetrahedron ABCD is given such that AD = BC = a;
Let … be given, and define recursively
Prove or disprove: From the interval [1, . . . , 30000] one
We are given 100 points in the plane, no three of which are
An odd integer n \geq3 is said to be “nice” if there is at least one
Find all pairs of positive integers (x, p) such that p is
S2 (POL)IMO4 The positive real numbers x0, x1, . . . , x1995 satisfy x0 =
How many words with n digits can be formed from the alphabet
Three persons A, B, C, are playing the following game: A k-
Let … be an even positive integer. Prove that there exists a positive integer … such that
An n \times n chessboard (n \geq2) is numbered by the numbers
Four integers are marked on a circle. At each step we simultaneously replace each number by the difference between this…
Let n, k be positive integers with k \leqn and let S be a set
Prove the inequality
Let M be the set of all positive integers that do not contain the
Consider pairs of sequences of positive real numbers a1 \geq
S3 (POL) For an integer x \geq1, let p(x) be the least prime that does not
Let x1, x2, . . . , xn be arbitrary real numbers. Prove the
Given k parallel lines and a few points on each of them, find
Let f(n, r) be the arithmetic mean of the minima of all r-
Let 1 = a0 \leqa1 \leqa2 \leq\cdot \cdot \cdot \leqan \leq\cdot \cdot \cdot be a sequence of
M is a subset of {1, 2, 3, . . ., 15} such that the product of
Find all positive integers … and … for which
Prove that there are exactly
Let ABCD be a regular tetrahedron and M, N distinct points
Consider the sequences (an), (bn) defined by
Prove that there exist infinitely many positive integers n
Consider on the first quadrant of the trigonometric circle the
Is there a positive integer m such that the equation
Find the minimum value of
A positive integer is called a double number if its decimal rep-
From a collection of n persons q distinct two-member teams
Let m positive integers a1, . . . , am be given. Prove that there
The diagonals of a quadrilateral ABCD are perpendicular:
Three distinct points A, B, C are fixed on a line in this order.
Let M be an interior point of the tetrahedron ABCD. Prove
Let P, Q, R be polynomials and let S(x) = P(x3) + xQ(x3) +
Let P(z) and Q(z) be complex-variable polynomials, with degree
B3 (GBR 1) Let ABC be a triangle, and let P be a point inside it such
Let P1(x) = x2 −2, Pj(x) = P1(Pj−1(x)), j = 2, 3, . . . .
Two identically oriented equilateral triangles, ABC with center
A particle moves from (0, 0) to (n, n) directed by a fair coin.
The sequence {an} of integers is defined by a1 = 2, a2 = 7,
(a) Do there exist functions f : R oR and g : R oR such that
The polynomial 1976(x+x2+\cdot \cdot \cdot+xn) is decomposed into a sum
Let n \geq2 be a natural number. Find a way to assign nat-
Each of the numbers in the set N = {1, 2, 3, . . ., n −1},
(a) Let n be a positive integer. Prove that there exist distinct
I 2 (POL 1) Prove that the squares with sides 1/1, 1/2, 1/3, . . . may be
The expression contains the denominator
We seek all three-digit integers whose quotient upon division by $11$ equals the sum of the squares of their digits.
We must prove the sharp two-sided inequality
The proposed solution captures the central idea of the exercise: avoid the trivial assignments in step E3 of Algorithm E by alternating the roles of variables and proceeding directly to the next divis…
The problem asks us to show that for any positive integer $d \neq 2,5,13$, there exist distinct $a,b \in {2,5,13,d}$ such that $ab-1$ is not a perfect square.
For $n=0$ the sum equals
The three equalities resemble moment conditions.
Define
The quantity to maximize is
The reviewers correctly identified two independent problems.
Let two circles intersect at points $A$ and $B$.
Consider a tetrahedron with vertices $A, B, C, D$ and edges $AB, AC, AD, BC, BD, CD$, all positive.
The previous proof failed at the point where it claimed that the local inclusions from color changes automatically combine into a cyclic chain forcing all sets $X_i$ to be equal.
We seek all real numbers $x$ satisfying
For small values, the alternating harmonic sums reduce to fractions whose numerators often contain large prime factors.
The previous construction failed because adding a small geometric perturbation to $a_k$ cannot repair arbitrarily bad ratios.
The soldier moves inside an equilateral triangle $ABC$ of side length $a$.
Let $G$ be a set of non-constant affine functions of the real variable $x$ of the form
Through a point P within a triangle ABC the lines l, m, and
Let
Let ABCD be a tetrahedron and O its incenter, and let the
Let E be the set of all bijective mappings from R to R satisfying
Given a ring G in the plane bounded by two concentric circles
Let S be an infinite set of integers containing zero and such
Find eight positive integers n1, n2, . . . , n8 with the follow-
Suppose that f is a real function defined for 0 \leqx \leq1 having
How many real solutions are there to the equation x =
Given a square ABCD, let P, Q, R, and S be four variable
Prove the inequalities
Find all positive numbers p for which the equation x2+px+3p = 0
One country has n cities and every two of them are linked by a
Let n be a positive integer, n \geq2, and consider the polynomial
Let f(x) = xm + a1xm−1 + \cdot \cdot \cdot + am−1x + am and g(x) =
Let k \geq2 and n1, n2, . . . , nk \geq1 natural numbers having the
Let n points be given on the surface of a sphere. Show that the
The polynomial P(x) = a0xk + a1xk−1 + \cdot \cdot \cdot + ak, where
Given integers a1, . . . , a10, prove that there exists a nonzero
Given an isosceles triangle ABC with a right angle at C,
For a matrix (pij) of the format m imes n with real entries, set
Let p, q and r be three lines in space such that there is no plane
Suppose we have a pack of 2n cards, in the order 1, 2, . . . , 2n. A
Let T be a triangle with inscribed circle C. A square with sides
Let P, Q, R be polynomials with real coefficients, satisfying
A circle of radius 1 rolls around a circle of radius
A set M is formed of
Prove that the equation
For a convex polygon P in the plane let P ′ denote the convex
Given a finite number of angular regions A1, . . . , Ak in a plane,
Construct a triangle ABC given the side AB and the distance
Prove that the two last digits of 999 and 9999
Prove that a < b implies that a3 −3a \leqb3 −3b + 4. When
Prove that the polynomial x4 + \lambdax3 + µx2 + \nux + 1 has no
Find the relations among the angles of the triangle ABC whose
In a given country, all inhabitants are knights or knaves. A
Let M be the set of all functions f with the following proper-
The solid S is defined as the intersection of the six spheres with
A convex quadrilateral ABCD with sides AB = a, BC = b,
For a positive real number p, find all real solutions to the equation
For every sequence (x1, x2, . . . , xn) of the numbers {1, 2, . . ., n}
Let x1, x2, . . . , xn (n \geq1) be real numbers such that 0 \leqxj \leq\pi,
A circle K centered at (0, 0) is given. Prove that for every vector
Prove that the sequence 5, 12, 19, 26, 33, . . . contains no term
Let g(k) be the number of partitions of a k-element set M, i.e.,
For which arrangements of two infinite circular cylinders does
Let Zm,n be the set of all ordered pairs (i, j) with i \in
Prove that if x, y, z are real numbers such that x2+y2+z2 = 2,
An (n2 +n+1) imes(n2 +n+1) matrix of zeros and ones is given.
The positive integer n has the property that in any set of n
Consider a finite set of vectors in space {a1, a2, . . . , an} and
In how many ways can 1, 2, . . . , 2n be arranged in a 2 imes n
Determine the volume of the body obtained by cutting the
Let S be a set of n2 + 1 closed intervals (n a positive integer).
Let a1, a2, a3, b1, b2, b3, c1, c2, c3 be nine strictly positive real
The square ABCD is to be decomposed into n triangles
Given a triangle ABC and external points X, Y , and Z such
Let c, s be real functions defined on R\{0} that are nonconstant
A convex quadrilateral is inscribed in a circle of radius 1. Prove
For each point X of a given polytope, denote by f(X) the sum
Given real numbers xi (i = 1, 2, . . . , 4x + 2) such that
In a chess tournament there are n \geq5 players, and they have
Suppose that 1985 points are given inside a unit cube. Show
Three of the roots of the equation x4 −px3 + qx2 −rx + s = 0
Prove that for any natural number n, the number
Denote by an the greatest number that is not divisible by 3
Let (ai)i\inN be a strictly increasing sequence of positive real
If Cp
Let l be tangent to the circle k at B. Let A be a point on k
In 3-dimensional space a point O is given and a finite set A
Given n real numbers 0 < t1 \leqt2 \leq\cdot \cdot \cdot \leqtn < 1, prove that
A square 2n imes 2n grid is given. Let us consider all possible
The decimal number 13101 is given. It is instead written as a
Suppose that n numbers x1, x2, . . . , xn are chosen randomly
A turtle runs away from an UFO with a speed of 0.2 m/s. The
Given a polynomial f(x) with integer coefficients whose value
Which natural numbers can be expressed as the difference of
The opposite sides of the reentrant hexagon AFBDCE in-
Let f1(x) = x3 +a1x2 +b1x+c1 = 0 be an equation with three
Let A1A2A3A4 be a quadrilateral inscribed in a circle C. Show
A ball K of radius r is touched from the outside by mutually
Without using any tables, find the exact value of the product
Let ABCD be a tetrahedron such that AB \perpCD,
Let M be a nonempty subset of Z+ such that for every element
Assume that the bisecting plane of the dihedral angle at edge
Let a and b be two natural numbers that have an equal number
Find the largest positive real number p (if it exists) such that
There are n points on a flat piece of paper, any two of them
A, B, C, D, E are points on a circle O with radius equal to r.
If x is a positive rational number, show that x can be uniquely
Prove the inequality
Let C be the circumcircle of the square with vertices (0, 0),
Find all pairs of integers a and b for which
Triangle ABC is given for which BC = AC + 1
Construct a triangle ABC given its side a = BC, its circum-
Show that the reciprocal of any number of the form 2(m2 +
Given n positive real numbers a1, a2, . . . , an such that a1a2 \cdot \cdot \cdot an
Show that the sequence {an}n\geq1 defined by an = [n
Let f : R oR be of the form f(x) = x + psilon sin x, where
Let C1, C2 be circles of radius 1/2 tangent to each other and
Three mutually nonparallel lines li (i = 1, 2, 3) are given
Prove the existence of a unique sequence {un} (n = 0, 1, 2 . . .)
Find all functions f defined for all x that satisfy the condition
For which natural numbers n do there exist n natural numbers
Consider a sequence of numbers (a1, a2, . . . , a2n). Define the
Let Li, i = 1, 2, 3, be line segments on the sides of an equilateral
Let n be a natural number, n \geq2, and let \varphi be Euler’s function;
Given m+n numbers ai (i = 1, 2, . . . , m), bj (j = 1, 2, . . ., n),
Suppose that a triangle whose sides are of integer lengths is
A curve determined by
Let ABC, AA1A2, BB1B2, CC1C2 be four equilateral triangles
Given a sequence (an), with a1 = 4 and an+1 = a2
Find the largest integer not exceeding $1992
Consider two quadrilaterals ABCD and A′B′C′D′ in an affine
Numbers d(n, m), with m, n integers, 0 \leqm \leqn, ae defined
Let [x] denote the greatest integer less than or equal to x. Let lpha
The distance between the centers of the circles k1 and k2 with
Given a quadrangle of sides a, b, c, d and area S, show that S \leq
Let k be a positive integer. Define u0 = 0, u1 = 1, and
Find the radius of the circle circumscribed about the isosceles
Prove that the equation 4x+6x = 9x has no rational solutions.
If p is a prime greater than 3, show that at least one of the
Every x, 0 \leqx \leq1, admits a unique representation x =
We wish to construct a matrix with 19 rows and 86 columns,
Let ABC be an equilateral triangle. Let D, E, F, M, N, and
Simplify
Given two numbers x0 and x1, let lpha and eta be coefficients
Find the maximum value of
Find for every value of n a set of numbers p for which the fol-
Let ABCD be a convex quadrilateral whose diagonals intersect
The equation x3 + ax2 + bx + c = 0 has three (not necessarily
On a line a set of segments is given of total length less than
(a) Solve the equation
Prove that n! cannot be the square of any natural number.
Let O be a point on the oriented Euclidean plane and (i, j)
A finite set of points P in the plane has the following prop-
Decompose into real factors the expression 1 −sin5 x−cos5 x.
Determine all positive roots of the equation xx = 1/
Let ABCD be an arbitrary quadrilateral. Let squares ABB1A2,
Prove that for any triangle with sides a, b, c and area P the
Prove that the equation in x
A “number triangle” (tnk) (0 \leqk \leqn) is defined by tn,0 =
Suppose that positive real numbers x1, x2, x3 satisfy
In a triangle, a symmedian is a line through a vertex that is
Find all x for which for all n,
Given a graph with n vertices and a positive integer m that is
Let G be the centroid of the triangle OAB.
Let f(x) = ax2 + bx + c and g(x) = cx2 + bx + a. If |f(0)| \leq1,
Let S be the unit circle with center O and let P1, P2, . . . , Pn
We are given n mass points of equal mass in space. We define
In a plane a circle with radius R and center w and a line ambda
Let ABCD be any quadrilateral. A square is constructed on
The ternary expansion x = 0.10101010 . . . is given. Give the
Let n \geq2 be an integer. Find the minimum k for which there
The area of a triangle is S and the sum of the lengths of its
The triangles A0B0C0 and A′B′C′ have all their angles
Let a real number \lambda > 1 be given and a sequence (nk) of positive
We define a binary operation ⋆in the plane as follows: Given
Let ABCD be a cyclic quadrilateral. Show that the centroids of
Let {un} be the Fibonacci sequence, i.e., u0 = 0, u1 = 1,
Let x1, x2, . . . , xn be n integers. Let n = p + q, where p and q
Prove that the equation
Let Fn be the nth Fibonacci number, defined by F1 = F2 = 1
Let P be a convex planar polygon with equal angles. Let
Determine the range of w(w + x)(w + y)(w + z), where x, y,
A hexahedron ABCDE is made of two regular congruent tetra-
Let ABC be an arbitrary triangle and let S1, S2, . . . , S7 be
From a point P exterior to a circle K, two rays are drawn
Let A be a set of positive integers such that for any two elements
Let x = \sqrta +
Given a positive integer n, find the greatest integer p with the
The function F is a one-to-one transformation of the plane into
Prove the identity
Given a set of 1988 points in the plane, no three points of the
Three faces of a tetrahedron are right triangles, while the fourth
Let A, B, C be points on the sides B1C1, C1A1, A1B1 of a
Integers a1, a2, . . . , an satisfy |ak| = 1 and
If A, B, C, and D are four distinct points in space, prove that
Let ABCD be a quadrilateral. Let A′BCD′ be the reflection
(a) Let g(x) = x5 + x4 + x3 + x2 + x + 1. What is the remainder when the
Let f : [0, 1] o[0, 1] satisfy f(0) = 0, f(1) = 1 and
Let I and J be the centers of the incircle and the excircle in
Let f be a real-valued function defined on I = (0, +\infty) and
Show that if n runs through all positive integers, f(n) =
The circles (R, r) and (P, ho), where r > ho, touch externally
Solve the system
Let P(x) be a polynomial such that the following inequalities
Given two segments AB and CD not in the same plane, find
A rectangle ABCD is given whose sides have lengths 3 and
Find all integer solutions of the equation
We consider a game on an infinite chessboard similar to that of
How many tangents to the curve y = x3 −3x (y = x3 + px)
Let n points be given arbitrarily in the plane, no three of
If a0 is a positive real number, consider the sequence {an}
Let P1(x, y) and P2(x, y) be two relatively prime polynomials
Given n real numbers a1 \leqa2 \leq\cdot \cdot \cdot \leqan, define
A subset S of the set of integers 0, . . . , 99 is said to have
A circle touches the sides AB, BC, CD, DA of a square at
The fraction
Given a segment AB of the length 1, define the set M of points
Prove that if a person a has infinitely many descendants (chil-
Let N be a point inside the triangle ABC. Through the mid-
Suppose medians ma and mb of a triangle are orthogonal.
Let a and b be two positive real numbers. If x is a real solution
The integers 1 through 1000 are located on the circumference
A directed graph (any two distinct vertices joined by at most
A sequence (an)N
Prove that there are infinitely many pairs (k, N) of positive
For each pair of positive integers k and n, let Sk(n) be the
Let \Gammai, i = 0, 1, 2, . . ., be a circle of radius ri inscribed in an
Father has left to his children several identical gold coins.
Given two sequences of positive numbers {ak} and {bk} (k \inN)
Which fraction p/q, where p, q are positive integers less than
Let AB and CD be two perpendicular chords of a circle with
For any angle lpha with 0 < lpha < 180◦, we call a closed convex
Let D be the point on the side BC of the triangle ABC such
Let a, b, c denote the lengths of the sides BC, CA, AB, respec-
Let a, b, x, y be positive integers such that a and b have no
Show that for every natural number n, n
Given a triangle ABC such that the circumcenter is in the
The altitude from a vertex of a given tetrahedron intersects
Let a, b, c be natural numbers such that a+b+c = 2pq(p30+q30),
Evaluate sec′′ \pi
Given two points A, B outside of a given plane P, find the
Let n and z be integers greater than 1 and (n, z) = 1. Prove:
An infinite increasing sequence of positive integers nj (j =
For any positive integer n we denote by F(n) the number of
Find all natural numbers n < 1978 with the following property:
Let two glasses, numbered 1 and 2, contain an equal quantity
There are six circles inside a fixed circle, each tangent to
Prove that if for a polynomial P(x, y) we have
Consider the integer d = ab−1
Find a function f(x) defined for all real values of x such that
Two persons, X and Y , play with a die. X wins a game if the
Let r and m (r \leqm) be natural numbers and Ak = 2k−1
A real-valued function f on Q satisfies the following conditions
It is well known that the binomial coefficients
Let E be a finite set, PE the family of its subsets, and f a
Simplify
Given the equation
Let F be the family of all k-element subsets of the set
Does there exist an infinite number of sets C consisting of 1983
Given that n elements a1, a2, . . . , an are organized into n pairs
Find all solutions (x, y) \inZ2 of the equation
If in a convex quadrilateral ABCD, E and F are the midpoints
A square hole of depth h whose base is of length a is given.
Solve the equation
Let a \inR and let z1, z2, . . . , zn be complex numbers of mod-
For each \lambda (0 < \lambda < 1 and \lambda ̸= 1/n for all n = 1, 2, 3, . . .)
Find the sum of the fiftieth powers of all sides and diagonals of
The numbers 1, 2, 3, . . ., 64 are placed on a chessboard, one
Find all the functions f : R+ oR satisfying the identity
Prove that in a Euclidean plane there are infinitely many
Let 0 \leqx1 \leqx2 \leq\cdot \cdot \cdot \leqxn \leq1. Prove that for all A \geq1
Let A be a set of polynomials with real coefficients and let
A regular tetrahedron A1B1C1D1 is inscribed in a regular
Find all pairs of natural numbers (m, n) for which 2m \cdot 3n + 1
Denote by xn(p) the multiplicity of the prime p in the canonical
Determine the sum of all positive integers whose digits (in base
(a) Calculate x = (11+6
We consider the infinite chessboard covering the whole plane.
Find the greatest integer A for which in any permutation of
Prove that a convex polyhedron all of whose faces are equilat-
Let {An | n = 1, 2, . . .} be a set of points in the plane such
A boy has a set of trains and pieces of railroad track. Each
Let f : (0, +\infty) \toR be a function having the property
Points A, B, C, D lie on a circle such that AB is a diameter and
Let p \geq2 be a natural number. Prove that there exists an
Let b1, b2, . . . , b1989 be positive real numbers such that the
Suppose that the sides AB and DC of a convex quadrilateral
The triangle ABC has a right angle at C. The point P is
Two cyclists leave simultaneously a point P in a circular run-
Prove that for all x1, x2, . . . , xn \inR the following inequality
Let ABCD and A′B′C′D′ be two squares in the same plane and
Consider h + 1 chessboards. Number the squares of each board
Let 1 \leqk < n. Consider all finite sequences of positive integers
Describe which natural numbers do not belong to the set
Let M be the point inside the right-angled triangle ABC
Let v1, v2, . . . , v1989 be a set of coplanar vectors with |vr| \leq1
The circle x2 + y2 = r2 meets the coordinate axes at A =
Let p be a prime. For which k can the set {1, 2, . . ., k} be
Show that the numbers tan(r\pi/15), where r is a positive integer
A one-person game with two possible outcomes is played as
Determine all pairs of positive integers (x, y) satisfying the
We are given an n imes n board, where n is an odd number. In
If r > s > 0 and a > b > c, prove that
The colonizers of a spherical planet have decided to build N
Five points in the plane are given, no three of which are collinear.
Let (an)n\inN be the sequence of integers defined recursively by
A is a 2m-digit positive integer each of whose digits is 1. B is
Given that
Find positive integers x1, x2, . . . , x29, at least one of which is
Let P and Q be distinct points in the plane of a triangle ABC
Given the vertex A and the centroid M of a triangle ABC,
Let a cube of side 1 be given. Prove that there exists a point
Consider all the sums of the form
If the inradius of a triangle is half of its circumradius, prove
In a plane we are given two distinct points A, B and two lines
Determine whether there exist 1976 nonsimilar triangles with
A regular octagon P is given whose incircle k has diameter 1.
The satellites A and B circle the Earth in the equatorial plane
Let (an)n\inN be the sequence of integers defined recursively by
Consider the square ABCD in which a segment is drawn
Given two distinct numbers b1 and b2, their product can be
The points A1, A2, . . . , A1983 are set on the circumference of a
Let Q be a square with side length 6. Find the smallest integer
Let \alpha + \beta + \gamma = \pi. Prove that
Are there integers m and n such that
Given n points A1, A2, . . . , An, no three collinear, show that
On a one-way street, an unending sequence of cars of width a,
Let a, b, and c be real numbers such that
Let C be a cube with edges of length 2. Construct a solid with
Given an acute triangle find a point inside the triangle such
Let A, B, and C denote the angles of a triangle. If sin2 A +
Let O be the center of a circle. Let OU, OV be perpendicular
Given f(x) \leqx for all real x and
Prove that for every triangle the following inequality holds:
It is proposed to partition the set of positive integers into two
Let a, b, c be positive real numbers, 0 < a \leqb \leqc. Prove that
Let n be a natural number not greater than 44. Prove that for
A circle of radius ho is tangent to the sides AB and AC of the
Find the number of positive integers n satisfying arphi(n) | n such
Let there be given two sequences of integers fi(1), fi(2), . . .
Let T be the set of all lattice points (i.e., all points with
Prove that if x, y, z > 1 and 1
Let A and B be points on the circle \gamma. A point C, different
Consider the equation x4 + ax3 + bx2 + ax + 1 = 0 with real
Let O be a point on a nondegenerate conic. A right angle with
Find all real values of the parameter a for which the system of
Let f(x) = a sin2 x+b sin x+c, where a, b, and c are real num-
A straight cone is given inside a rectangular parallelepiped
For k = 1, 2, . . . consider the k-tuples (a1, a2, . . . , ak) of positive
Let m and n be natural numbers with m > n. Prove that
Find, with proof, the smallest real number C with the following
Let mj > 0 for j = 1, 2, . . ., n and a1 \leq\cdot \cdot \cdot \leqan < b1 \leq\cdot \cdot \cdot \leq
The set X has 1983 members. There exists a family of subsets
Prove the inequality
(Alternative to GBR 2) Prove that there exists, for n \geq4, a
The points S(i, j) with integer Cartesian coordinates 0 < i \leqn,
We are given twelve coins, one of which is a fake with a different
Given an equilateral triangle ABC, let M be an arbitrary point
In the Martian language every finite sequence of letters of
Several segments, which we shall call white, are given, and
Prove that there exist distinct natural numbers m1, m2, . . . ,
The sides a, b, c of a triangle ABC form an arithmetic progression;
Consider the set of grid points (m, n) in the plane, m, n inte-
Let C be the curve determined by the equation y = x3 in the
For a finite set E of cardinality n \geq3, let f(n) denote the
Find all integers x, y, z that satisfy
Let n > 1 be a natural number, a \geq1 a real number, and
Let f(x) be a polynomial with integer coefficients. Prove that
A sequence a1, a2, a3, . . . is defined recursively by a1 = 1 and
We match sets M of points in the coordinate plane to sets M∗
Let a, b, c be integers. Prove that there are integers p1, q1, r1,
Show that for nonnegative real numbers a, b and integers n \geq2,
Prove that
From point P on arc BC of the circumcircle about triangle
Let ABC be an isosceles triangle with right angle at point A.
Define the functions f, F : N oN, by
A sequence of real numbers x0, x1, x2, . . . is defined as follows:
Two nonzero integers x, y (not necessarily positive) are such
Let M be a finite set and P = {M1, M2, . . . , Mk} a partition
Find the natural number n with the following properties:
Find all numbers N = a1a2 . . . an for which 9 imes a1a2 . . . an =
Let Z be a set of points in the plane. Suppose that there exists
Integers cm,n (m \geq0, n \geq0) are defined by cm,0 = 1 for all
Let x1, x2, x3, x4, and x5 be positive integers satisfying
One hundred red points and one hundred blue points are
Let ABC be a nonequilateral triangle. Prove that there exist
Let x, y, and z be real numbers satisfying x + y + z = xyz.
Let M be the circumcenter of a triangle ABC. The line through
A sequence (an)\infty
Let a and b be two nonnegative integers. Denote by H(a, b)
Let there be given a circle with center S and radius 1 in the plane,
Let O be a point outside a given circle. Two lines OAB, OCD
Let p be a prime number and a1, a2, . . . , a(p+1)/2 different nat-
Does there exist an integer z that can be written in two different
There are n \geq2 people in a room. Prove that there exist two
Find, with proof, all solutions of the equation 1
Let A and B be positions of two ships M and N, respectively,
There are 1979 equilateral triangles: T1, T2, . . . , T1979. A side of
Let g(n) be defined as follows:
Given a natural number n, prove that the number M(n) of
Set
The sum of all the face angles about all of the vertices except
On the three distinct lines a, b, and c three points A, B, and
In the coordinate system in the plane we consider a convex
We are given a finite collection of segments in the plane, of
Two circles touch each other from inside, and an equilateral
Show that for no integers a \geq1, n \geq1 is the sum
Let ABC and A′B′C′ be any two coplanar triangles. Let L be
Six points P1, . . . , P6 are given in 3-dimensional space such that
Let a1, a2, a3, b1, b2, b3 be positive real numbers. Prove that
Let M be a set, and A, B, C given subsets of M. Find a
Find all numbers x \inZ for which the number
A “large” circular disk is attached to a vertical wall. It rotates
Let ABC be a triangle, O its circumcenter, S its centroid, and
Let there be given an acute angle ngleAOB = 3lpha, where OA =
In a triangle ABC, the incircle touches the sides BC, CA, AB
We call a coloring f of the elements in the set M = {(x, y) |
Given a line p and a triangle rianglein the plane, construct an
Assume that two parallelograms P, P ′ of equal areas have sides
(1) Start with a white balls and b black balls.
Find the least integer n with the following property: For any
On the circle with center O and radius 1 the point A0 is
Find the least natural number k such that for any n \in[0, 1]
Let f be a strictly increasing function defined on the set of real
A tetrahedron is inscribed in a sphere of radius 1 such that the
In what case does the system
Let a, 0 < a < 1, be a real number and f a continuous function
Given a natural number n, find all polynomials P(x) of degree
We are given 2n natural numbers
Let ABC be an equilateral triangle and \Gamma the semicircle
A sequence of real numbers x1, x2, . . . , xn is given such that
The number 0 or 1 is to be assigned to each of the n vertices
Suppose p and q are two different positive integers and x is a
Find all plane triangles whose sides have integer length and
The sphere inscribed in tetrahedron ABCD touches the sides
The circle k and its diameter AB are given. Find the locus of
Prove that for any triangle ABC there exists a point P in the
Through the circumcenter O of an arbitrary acute-angled trian-
Suppose x0, x1, . . . , xn are integers and x0 > x1 > \cdot \cdot \cdot > xn.
In a school six different courses are taught: mathematics,
Construct a triangle given the three exradii.
Find the set of all a \inR for which there is no infinite sequence
In a group of n people each one knows exactly three others. They
Find all positive integers x such that the product of all digits
Does there exist an integer such that its cube is equal to
Let E be a finite set of points in space such that E is not
Let E be the set of all triangles whose only points with integer
Let a, b, c, d be a permutation of the numbers 1, 9, 8, 4 and let
A regular pentagon A1A2A3A4A5 with side length s is given.
Find all square numbers S1 and S2 such that S1 −S2 = 1989.
If n1, n2, . . . , nk are natural numbers and n1+n2+\cdot \cdot \cdot+nk = n,
The base of an inclined prism is a triangle ABC. The per-
Let N = {1, 2, . . ., n}, n \geq3. To each pair i, j of elements of N,
If lpha is the real root of the equation
Let M, N, P be the midpoints of the sides BC, CA, AB of a
Is it possible to cover a rectangle of dimensions m imes n with
The vertices of an (n + 1)-gon are placed on the edges of a
Find all triples (x, y, z) of integers such that
Let lpha(n) be the number of pairs (x, y) of integers such that
Let l denote the length of the smallest diagonal of all rectangles
Suppose that points X, Y, Z are located on sides BC, CA,
For positive numbers a, b, c define A = (a + b + c)/3, G =
Let a1, a2, . . . , an (n \geq2) be a sequence of integers. Show that
Diagonals of a convex quadrilateral ABCD intersect at a
Determine all positive integers n for which there exists a poly-
Prove the following statement: If a polynomial f(x) with
A parabola P1 with equation x2 −2py = 0 and parabola P2
Let a1, a2, . . . , an be positive numbers, mg = (a1a2 \cdot \cdot \cdot an)1/n
Suppose \alphai > 0, \betai > 0 for 1 \leqi \leqn (n > 1) and that
A regular n-gonal truncated pyramid is circumscribed around
Let S be a unit circle and K a subset of S consisting of several
Let A1A2A3A4A5A6 be a hexagon inscribed into a circle with
Compute the largest number of regions into which one can divide
Determine the least possible value of the natural number n
In a plane three points P, Q, R, not on a line, are given. Let
Let m and n denote integers greater than 1, and let u(n) be
All the irreducible positive rational numbers such that the prod-
Let n, k \geq1 be natural numbers. Find the number A(n, k) of
Two straight lines perpendicular to each other meet each side
Find the number of permutations a1, . . . , an of the set
Let A, B, C be angles of a triangle. Prove that
Find whether among all quadrilaterals whose interiors lie inside
Find the triples of positive integers x, y, z satisfying
Let a quadratic polynomial g(x) = ax2 + bx + c be given and
Exactly one side of a tetrahedron is of length greater than
Determine the sixth number after the decimal point in the
Do there exist two sequences of real numbers {ai}, {bi}, i \in
Three concentric circles with common center O are cut by a
Let P be the set of rectangular parallelepipeds that have at
Find the integer solutions of the equation
A polygon (not necessarily convex) with vertices in the lattice
Prove that if P(x) = (x−a)kQ(x), where k is a positive integer,
In a group of interpreters each one speaks one or several foreign
The bisectors of the angles B, C of a triangle ABC intersect
Prove that for a, b \inN, a!b! divides (a + b)!.
A cylindrical container has height 6 cm and radius 4 cm. It
Consider the set E consisting of pairs of integers (a, b), with a \geq
ABCD is a parallelogram; AB = a, AD = 1, lpha is the size
Find, with argument, the integer solutions of the equation
The expressions a + b + c, ab + ac + bc, and abc are called the
Let S be a k-element set.
Let E = {1, 2, . . ., 16} and let M be the collection of all
From a square board 11 squares long and 11 squares wide, the
A desert expedition camps at the border of the desert, and
Find values of the parameter u for which the expression
Let M be the set of the lengths of an octahedron whose sides
Prove that for arbitrary positive numbers the following in-
Consider the set S of all the different odd positive integers
For each nonzero complex number z, let arg z be the unique
Suppose tan lpha = p/q, where p and q are integers and q ̸= 0.
In making Euclidean constructions in geometry it is permit-
Prove that the volume V and the lateral area S of a right circular
Given a sphere K, determine the set of all points A that are
Let P be a set of n points and S a set of l segments. It is
A regular triangular prism has height h and a base of side length
For each nonnegative integer n, Fn(x) is a polynomial in x of
Let n > 1 be a fixed integer. Define functions f0(x) = 0,
Prove that the product of two sides of a triangle is always
Let Pn = (19 + 92)(192 + 922) \cdot \cdot \cdot (19n + 92n) for each positive
Let z be an integer > 1 and let M be the set of all numbers
(Variant of GDR 4) Given a nonconstant function f : R+ oR
Let n and k be natural numbers and a1, a2, . . . , an positive real
Let ABC be a triangle with interior angle bisectors AA1,
In the set of 20 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C,
The circle inscribed in the triangle A1A2A3 is tangent to
Let N = B1 \cup\cdot \cdot \cdot\cupBq be a partition of the set N of all positive
A circle k = (S, r) is given and a hexagon AA′BB′CC′ inscribed
Let a0, a1, a2 be determined with a0 = 0, an+1 = 2an + 2n.
Let P be a prime number and n a natural number. Prove that
In the triangle ABC, let B1 be on AC, E on AB, G on BC,
Let a, b, c, r, and s be real numbers. Show that if r is a root of
Two perpendicular chords are drawn through a given interior
Let E1, E2, and E3 be three mutually intersecting ellipses, all
Given the side a and the corresponding altitude ha of a triangle
In a triangle P1P2P3 let PiQi be the altitude from Pi for
Let ABCD be a rhombus with angle ngleA = 60◦. Let E be a
Let a and b be arbitrary integers. Prove that if k is an integer
Let P(x) be a polynomial with integer coefficients such that
If a, b, c are side lengths of a triangle, prove that
Let n be a natural number. Solve in integers the equation
Let M be a set of points in a plane with at least two elements.
Let A and B be fixed distinct points on the X axis, none of
Denote by [x] the greatest integer not exceeding x. For all
Determine the maximum value of x2y2z2w when x, y, z, w \geq0
If p is a prime number greater than 2 and a, b, c integers not
Consider the sequence (cn):
The perpendicular line issued from the center of the circum-
Solve in the set of real numbers the equation 3x3 −[x] = 3,
Prove that there are infinitely many positive integers that
Let the circles C1, C2, and C3 be orthogonal to the circle C
In a multiple choice test there were 4 questions and 3 possible
Let p be a prime odd number. Is it possible to find p−1 natural
A triangle ABC with ngleA = 30◦and ngleC = 54◦is given. On
Let ak be positive numbers such that a1 \geq1 and ak+1 −ak \geq1
Let A1, A2, . . . , An+1 be positive integers such that (Ai, An+1)
For a real number x, let [x] denote the greatest integer not
Let ABC be a triangle. Denote by a, b, and c the lengths of
If n is a natural number, prove that
(a) The polynomial x2k + 1 + (x+ 1)2k is not divisible by x2 +x+ 1. Find
Prove that a pyramid A1A2 . . . A2k+1S with equal lateral edges
Let a regular 7-gon A0A1A2A3A4A5A6 be inscribed in a circle.
Prove the identity
A regular 14-gon with side length a is inscribed in a circle of
The circles c1 and c2 are tangent at the point A. A straight
An urn contains balls of k different colors; there are ni balls
Let F be the correspondence associating with every point P =
A line parallel to the side BC of a triangle ABC meets AB
Prove the following statement: If r1 and r2 are real numbers
Given a triangle ABC, three equilateral triangles AEB, BFC,
Prove the following statement: There does not exist a pyramid
Let t(n), for n = 3, 4, 5, . . ., represent the number of distinct,
Let Ax, By be two noncoplanar rays with AB as a common per-
T is a given triangle with vertices P1, P2, P3. Consider an arbi-
Let a, b, c be integers different from zero. It is known that the
Find x such that
An infinite set of rectangles in the Cartesian coordinate
It is given that x = −2272, y = 103 +102c+10b+a, and z = 1
If
Into every lateral face of a quadrangular pyramid a circle is
Given natural numbers k and n, k \leqn, n \geq3, find the set
Let n + 1 (n \geq1) positive integers be given such that for each
In an urn there are n balls numbered 1, 2, . . . , n. They are
Consider infinite diagrams
Given a circle, construct a chord that is trisected by two given
Let n = 2k −1, where k \geq6 is an integer. Let T be the set
For every integer r > 1 find the smallest integer h(r) > 1
(a) Prove that (a1 +a2 +\cdot \cdot \cdot+ak)2 \leqk(a2
Prove that the system of equations
Let {an}\infty
A wheel consists of a fixed circular disk and a mobile circular
Given an equilateral triangle ABC of side a in a plane, let
Find digits x, y, z such that the equality
There are a board with 2n\cdot2n (= 4n2) squares and 4n2−1 cards
Find necessary and sufficient conditions on given positive num-
The integers 1, 2, . . ., n2 are placed on the fields of an n imes n
In the plane a point O and a sequence of points P1, P2, P3, . . .
Let n be a positive integer. Find the maximal number of non-
Prove that a triangle with angles \alpha, \beta, \gamma, circumradius R, and
Prove that there are infinitely many positive integers n for
Let ABCD be a regular tetrahedron. To an arbitrary point
Two equilateral triangles are inscribed in a circle with radius
The quadrilateral A1A2A3A4 is cyclic and its sides are a1 =
A circle \gamma is drawn and let AB be a diameter. The point C
If a1, a2, . . . , an are real constants, and if
The set {1, 2, . . ., 49} is divided into three subsets. Prove that
Find the last eight digits of the binary development of 271986.
Let ABCD be a convex quadrilateral whose diagonals AC and
In the plane of a given triangle A1A2A3 determine (with proof)
The lengths of the sides of a rectangle are given to be odd
Solve the equation 28x = 19y + 87z, where x, y, z are integers.
Five points in a plane are given, no three of which are collinear.
The vertices of a given square are clockwise lettered A, B, C, D.
The function ϕ(x, y, z), defined for all triples (x, y, z) of real
Let ABC be an isosceles triangle, AB = AC, ngleA = 20◦. Let
Let c be the inscribed circle of the triangle ABC, d a line tan-
Prove the trigonometric inequality cos x < 1 −x2
Determine an equation of third degree with integral coefficients
Prove that there is a positive integer n such that the decimal
Show that for any natural number n there exist two prime
Let (an)\infty
Let p and q be two prime numbers greater than 3. Prove that
Congruent rectangles with sides m (cm) and n (cm) are
For every a \inN denote by M(a) the number of elements of
In how many different ways can three knights be placed on a
Let S be the point of intersection of the two lines l1 : 7x−5y +
Each of the numbers x1, x2, . . . , xn equals 1 or −1 and
Let S be a subset of the real numbers with the following
Prove that in every convex hexagon of area S one can draw
Let L denote the set of all lattice points of the plane (points
Let ABC be a triangle with angles \alpha, \beta, \gamma commensurable with
A sequence of numbers an, n = 1, 2, . . ., is defined as follows:
Numbers un,k (1 \leqk \leqn) are defined as follows:
We have p players participating in a tournament, each player
Four faces of tetrahedron ABCD are congruent triangles whose
For hi : N oZ let us define M hi = {f : N oZ; f(x) >
The n points P1, P2, . . . , Pn are placed inside or on the bound-
Let ABCD be a regular tetrahedron and Z an isometry map-
Determine whether there exist 100 distinct lines in the plane
We call a tetrahedron right-faced if each of its faces is a right-
An Egyptian number is a positive integer that can be expressed
Let S be a set of positive integers n1, n2, . . . , n6 and let n(f)
Let f be a function defined on the set of pairs of nonzero
The sequence (an) of real numbers is defined as follows:
(a) Find the equations of regular hyperbolas passing through
Given five points in the plane, no three of which are collinear,
Prove that the perpendiculars drawn from the midpoints of the
Consider the expansion
Let b \geq2 be a positive integer.
Let A, B denote two distinct fixed points in space. Let X, P
Given that x1+x2+x3 = y1+y2+y3 = x1y1+x2y2+x3y3 = 0,
Let ABC be a triangle. Prove that there is a unique point U
Let P be a fixed point and T a given triangle that contains the
If a1, a2, . . . , an denote the lengths of the sides of an arbitrary
On the sides AB and AC of triangle ABC two points K and
Let S be a circle, and lpha = {A1, . . . , An} a family of open arcs
In riangleABC with ngleC = 60o, prove that c
Solve the equation
Find all integer solutions of the equation
Let S \subset[0, 1] be a set of 5 points with {0, 1} \subsetS. The graph
Prove that 2147 −1 is divisible by 343.
A convex planar polygon M with perimeter l and area S is given.
A regular n-gon A1A2A3 . . . Ak . . . An inscribed in a circle of
Fibonacci numbers are defined as follows: F1 = F2 = 1, Fn+2 =
Let Sn = {1, . . . , n} and let f be a function that maps every
Let ABC be a triangle with inradius r and circumradius R.
The plane is divided into equal squares by parallel lines; i.e.,
Let f1 = (a1, a2, . . . , an), n > 2, be a sequence of integers.
Let a regular (2n + 1)-gon be inscribed in a circle of radius r.
A solid right circular cylinder with height h and base-radius
In riangleABC, the inscribed circle is tangent to side BC at X.
A square ABCD is divided into (n −1)2 congruent squares,
One hundred convex polygons are placed on a square with edge
In a plane, a circle with center O and radius R and two points
Given positive integers k, m, n with km \leqn and nonnegative
Seventeen cities are served by four airlines. It is noted that
Show that if the sides a, b, c of a triangle satisfy the equation
A fair coin is tossed repeatedly until there is a run of an odd
A game consists in pushing a flat stone along a sequence of
Let S be the set of all sequences {ai | 1 \leqi \leq7, ai = 0 or 1}.
Five distinct numbers are drawn successively and at random
In a triangle ABC, let H be its orthocenter, O its circumcenter,
Given the equation
Let C be a class of functions f : N oN that contains the
A function f has the following property: If k > 1, j > 1,
Find the positions of three points A, B, C on the boundary of
Find the total number of different integers that the function
Let a be a real number such that 0 < a < 1, and let n be a
Let k be one of the integers 2, 3, 4 and let n = 2k −1. Prove
A sequence (un) of integers is defined for n \geq0 by u0 = 0,
Let f(x) = (x −a1)(x −a2) \cdot \cdot \cdot (x −an) −2, where n \geq3
If T and T1 are two triangles with angles x, y, z and x1, y1, z1,
A lampshade is part of the surface of a right circular cone
Let P, Q, R be the polynomials with real or complex coefficients
Given an integer n \geq2, determine all n-digit numbers
A sequence {an} of positive integers is defined by
Let n and p be two integers such that 2p \leqn. Prove the
A 2 imes 2 imes 12 box fixed in space is to be filled with twenty-four
Let a and b be integers. Prove that 2a2−1
Let a and b be integers. Is it possible to find integers p and q
Prove that the product of two natural numbers with their sum
Let Ak (1 \leqk \leqh) be n-element sets such that each two
Prove that the numbers A, B, and C are equal, where we
Let a1, a2, . . . , an be n real numbers such that 0 < a \leqak \leqb
In connection with a convex pentagon ABCDE we consider
There are n \geq3 job openings at a factory, ranked 1 to n in
Let P be a convex 1986-gon in the plane. Let A, D be interior
Call a four-digit number (xyzt)B in the number system with
Poldavia is a strange kingdom. Its currency unit is the bourbaki
An arithmetic function is a real-valued function whose do-
Solve the system of simultaneous equations
A triangle ABC is given. Each side of ABC is divided into equal
Prove that if a diagonal is drawn in a quadrilateral inscribed
Let log2
Let p be a prime number greater than 5. Let V be the collection
Notice that in the fraction 16
Let p be a prime number. A rational number x, with 0 < x < 1,
Given a triangle ABC and a plane \pi having no common points
Prove that the number 191976 + 761976:
In a triangle ABC, ngleBAC = 100◦, AB = AC. A point
A square ABCD is given. A line passing through A intersects
Let k be a positive integer and Mk the set of all the integers
Let A, B, C be three points with integer coordinates in the
Let a, b, c be positive real numbers and let [x] denote the
For a \geq0, b \geq0, c \geq0, d \geq0, prove the inequality
Let (an)n\geq0 and (bn)n\geq0 be two sequences of natural numbers.
Let C1 and C2 be circles in the same plane, P1 and P2 arbitrary
Let A and B be two finite disjoint sets of points in the plane
Let squares be constructed on the sides BC, CA, AB of a trian-
Let X be a bounded, nonempty set of points in the Cartesian
Let x1, x2, x3, x4, x5, x6 be given integers, not divisible by 7.
Find the maximum value that the quantity 2m + 7n can have
Express the number 1988 as the sum of some positive integers
You are given an algebraic system admitting addition and
A boy at point A wants to get water at a circular lake and
Determine positive integers p, q, and r such that the diagonal
A variable tetrahedron ABCD has the following properties:
For n \inN, let f(n) be the number of positive integers k \leqn
Let a polynomial p(x) with integer coefficients take the value
An accurate 12-hour analog clock has an hour hand, a minute
If a, b, c, d are integers such that ad is odd and bc is even, prove
To each pair (x, y) of distinct elements of a finite set X a number
Let (an), n = 0, 1, . . ., be a sequence of real numbers such that
Let n and k be positive integers such that 1 \leqn \leqN + 1,
Three disks of diameter d are touching a sphere at their centers.
The points A, B, C are in this order on line D, and AB = 4BC.
Prove that there exist 78 lines in the plane such that they have
Let f be a function from the real numbers to the real numbers
What is the greatest number of balls of radius 1/2 that can be
Thirty-four countries participated in a jury session of the IMO,
Decompose the number 51985−1 into a product of three integers,
Given a regular convex 2m-sided polygon P, show that there is
The runs of a decimal number are its increasing or decreasing
Consider the binomial coefficients
Determine all functions f : R oR satisfying the following two
Prove that there exist infinitely many natural numbers a
Let x = p, y = q, z = r, w = s be the unique solution of the
Let ABC be an arbitrary scalene triangle. Define \Sigma to be the
Two families of parallel lines are given in the plane, consisting
In a company of n persons, each person has no more than d
In a room there are nine men. Among every three of them there
Which of the numbers 1, 2, . . ., 1983 has the largest number of
Determine all real functions f(x) that are defined and contin-
The function f(n) is defined on the nonnegative integers n by:
Consider a polynomial P(x) = ax2 + bx + c with a > 0 that
Find all pairs of integers (p, q) for which all roots of the trino-
Solve the following system of linear equations with unknown
Is it possible to partition 3-dimensional Euclidean space into
Let n be a positive integer, X = {1, 2, . . ., n}, and k a positive
Consider the number lpha obtained by writing one after another
Let g(x) be a fixed polynomial and define f(x) by f(x) =
In the triangle ABC, let D, E, and F be the midpoints of the
Let a1, a2, . . . , an be positive real numbers. Prove the inequality
Let four points Ai (i = 1, 2, 3, 4) in the plane determine four
Prove that
If 0 < k \leq1 and ai are positive real numbers, i = 1, 2, . . . , n,
(a) Let ABC be a triangle with AB = 12 and AC = 16. Suppose M is the
Find a necessary and sufficient condition on the natural num-
Let A be a set of positive integers such that no positive integer
Prove that if the equation x4 + ax3 + bx + c = 0 has all its
Let M be an interior point of tetrahedron V ABC. Denote
(a) Prove that for a, b, c, d \inR, m \in[1, +\infty) with am + b =
In the plane a circle C of unit radius is given. For any line l
Let O be an interior point of a tetrahedron A1A2A3A4. Let
Let 2n + 3 points be given in the plane in such a way that
Four swallows are catching a fly. At first, the swallows are
A fox stands in the center of the field which has the form of an
Let there be given three circles K1, K2, K3 with centers
The bisectors of the exterior angles of a pentagon B1B2B3B4B5
Let a, b, c, d be the lengths of the sides of a quadrilateral
If A1A2 . . . An is a regular n-gon (n \geq3), how many different
Let ABCD be a convex quadrilateral whose diagonals AC and
Let n be an integer that is not divisible by any square greater
Find all possible finite sequences {n0, n1, n2, . . . , nk} of integers
(a) Four balls of radius 1 are mutually tangent, three resting an the floor
In space, n points (n \geq3) are given. Every pair of points
Outside an arbitrary triangle ABC, triangles ADB and BCE
Does there exist a 2n-digit number a2na2n−1 . . . a1 (for an
A pack of 2n cards contains n different pairs of cards. Each
Find all cubic polynomials x3 + ax2 + bx + c admitting the
Connecting the vertices of a regular n-gon we obtain a closed
The faces of a convex polyhedron are six squares and eight
Let ABCD be a rectangle, AB = a, BC = b. Consider the
Given n points X1, X2, . . . , Xn in the interval 0 \leqXi \leq1,
Let p, q, and r be the angles of a triangle, and let a = sin 2p,
There are some boys and girls sitting in an n imes n quadratic
Let a and b be integers and n a positive integer. Prove that
Let (u1, . . . , un) be an ordered ntuple. For each k, 1 \leqk \leqn,
Let a1 \leqa2 \leq\cdot \cdot \cdot \leqan and b1 \leqb2 \leq\cdot \cdot \cdot \leqbn be two
Does there exist a number lpha (0 < lpha < 1) such that there is an
How many permutations a1, a2, . . . , an of {1, 2, . . ., n} are
Solve the equation |x2 −1| + |x2 −4| = mx as a function of the
Find the last two digits of a sum of eighth powers of 100
The sequence an,k, k = 1, 2, 3, . . ., 2n, n = 0, 1, 2, . . ., is defined
Let d and p be two real numbers. Find the first term of an arith-
Solve the system of equations
Determine the smallest positive integer m such that 529n +m\cdot
Let A1, A2, . . . , A29 be 29 different sequences of positive integers.
The incenter of a triangle is the midpoint of the line seg-
Superchess is played on on a 12 imes 12 board, and it uses su-
For given positive integers r, v, n let S(r, v, n) denote the num-
Prove that in any parallelepiped the sum of the lengths of the
Given a triangle ABC, let R be the radius of its circumcir-
Let ABC be a triangle. For every point M belonging to segment
Show that if 994 integers are chosen from 1, 2, . . . , 1992 and
Let n be a natural number. If 4n + 2n + 1 is a prime, prove
Let n be a positive integer. Prove that the number of ways
Let r > 1 be a real number, and let n be the largest integer
Let xn = 22n + 1 and let m be the least common multiple of
Given a polynomial
Show that the set S of natural numbers n for which 3/n
One Martian, one Venusian, and one Human reside on Pluton.
Evaluate
Five points lie on the surface of a ball of unit radius. Find the
Given a pyramid whose base is an n-gon inscribable in a circle,
Let ABCDS be a pyramid with four faces and with ABCD
Prove that for a natural number n > 2,
Let a, b, c be nonnegative integers such that a \leqb \leqc, 2b ̸=
Let −1 < x < 1. Show that
Let a, b, c be positive real numbers and p, q, r complex numbers.
Let k and s be positive integers. For sets of real numbers
Prove the identity
Prove that a regular polygon with an odd number of edges
Find the number of five-digit numbers with the following
Let PQ be a line segment of constant length \lambda taken on the
Consider the set Q2 of points in R2, both of whose coordinates
Let k, m, and n be positive integers such that m+k + 1 is
Find a natural number n such that for all prime numbers p, n
Suppose ABCD and A′B′C′D′ are two parallelograms arbi-
Prove that 1
The intersection of a plane with a regular tetrahedron with
Alice has two urns. Each urn contains four balls and on each
Given any integer m > 1 prove that there exist infinitely
Evaluate (cos(\pi/4) + i sin(\pi/4))10 in two different ways and
A smooth solid consists of a right circular cylinder of height
The real numbers lpha1, lpha2, lpha3, . . . , lphan are positive. Let us denote
Consider a cube C and two planes \sigma, \tau, which divide Euclidean
Prove that for a > b2,
Show that the triangle whose angles satisfy the equality
A town has a road network that consists entirely of one-way
In a plane a finite number of equal circles are given. These circles
Let an =
A finite number of parallel segments in the plane are given with
A two-person game is played with nine boxes arranged in a
Let O be the midpoint of the axis of a right circular cylinder.
Prove the inequality
Let n numbers x1, x2, . . . , xn be chosen in such a way that
Consider the set A = {0, 1, 2, . . ., 9} and let (B1, B2, . . . , Bk)
Find all rational solutions of
Let A, B, C, D be four arbitrary distinct points in space.
Prove that the sequence (an)n\geq0, an = [n
Given a unit cube, find the locus of the centroids of all tetra-
Let f and g be functions from the set A to the same set A.
Let ABCD be a quadrilateral inscribed in a circle with diam-
Let A be an n imesn matrix whose elements are nonnegative real
Two half-lines a and b, with the common endpoint O, make an
Find the conditions on the positive real number a such that
For a positive integer n, let 6(n) be the natural number whose
Prove the inequality
Construct the circle that is tangent to three given circles.
Find the average of the quantity
Let X and Y be two sets of points in the plane and M be a set
Solve the set of simultaneous equations
For each P inside the triangle ABC, let A(P), B(P), and
(a) Find the number of ways 500 can be represented as a sum of
Two moving bodies M1, M2 are displaced uniformly on two
Let Sn = {1, 2, . . ., n} and fn : Sn oSn be defined inductively
Prove that for every positive integer n coprime to 10 there
Assuming that the roots of x3+px2+qx+r = 0 are all real and
Consider 37 distinct points in space, all with integer coordi-
A plane rectangular grid is given and a “rational point” is
All edges and all diagonals of regular hexagon A1A2A3A4A5A6
Let K be a convex set in the xy-plane, symmetric with respect
Under the conditions x1, x2 > 0, x1y1 > z2
Construct a scalene triangle such that
Suppose that n > m \geq1 are integers such that the string of
Let r1, . . . , rn be the radii of n spheres. Call S1, S2, . . . , Sn the
A collection of 2n letters contains 2 each of n different letters.
In a plane, three pairwise intersecting circles C1, C2, C3 with
(a) Consider a circle K with diameter AB, a circle L tangent to AB and
Two mirror walls are placed to form an angle of measure lpha. There
Solve the system
Numbers 1, 2, . . . , 16 are written in a 4 imes4 square matrix so that
Let d \geq1 be an integer that is not the square of an integer.
From each of the vertices of a regular n-gon a car starts to
If a, b, c, d are real numbers such that a2 + b2 + c2 + d2 \leq1,
Points D and E are chosen on the sides AB and AC of the
A set of n standard dice are shaken and randomly placed in a
A linear binomial l(z) = Az + B with complex coefficients A
Let n be a positive integer. Show that (
Prove that for all X > 1 there exists a triangle whose sides
If 0 \leqa \leqb \leqc \leqd, prove that
Let ABC and DEF be acute-angled triangles. Write d = EF,
Through a point O on the diagonal BD of a parallelogram
An alphabet consists of n letters. What is the maximal length
A circle K with radius r, a point D on K, and a convex
For two given triangles A1A2A3 and B1B2B3 with areas ∆A
We are given n > 3 points in the plane, no three of which lie on
Let n be an integer > 1. In a circular arrangement of n lamps
Find all (real) solutions of the system
(a) Given a tetrahedron ABCD and its four altitudes (i.e.,
We are given a circle K and a point P lying on a line g. Construct
Prove that for an arbitrary pair of vectors f and g in the
Given n points in the plane such that no three of them
The Fibonacci sequence is defined by
Two concentric circles have radii R and r respectively. Determine
In the plane 4000 points are given such that each line passes
Let S =
Let N = {1, 2, 3, . . .}. For real x, y, set S(x, y) = {s | s =
Is it possible to choose a set of 100 (or 200) points on the
The equation
Show that there exists a convex polyhedron with all its vertices
Let P1(x), P2(x), . . . , Pn(x) be polynomials with real coefficients.
Let I, H, O be the incenter, centroid, and circumcenter of the
Let a and b be coprime integers, greater than or equal to 1.
Let A1A2, B1B2, C1C2 be three equal segments on the three
If n is even, prove that
Let Ka, Kb, Kc with centers Oa, Ob, Oc be the excircles of a
M = (ai,j), i, j = 1, 2, 3, 4, is a square matrix of order four.
Find values of n \inN for which the fraction 3n−2
Given a positive integer k, find the least integer nk for which
Let a1, . . . , an be distinct positive integers that do not contain
Let arphi(n, m), m ̸= 1, be the number of positive integers less
A balance has a left pan, a right pan, and a pointer that moves
In an urn there are one ball marked 1, two balls marked 2, and
(a) Show that the set N of all natural numbers can be parti-
Let K1, . . . , Kn be nonnegative integers. Prove that
In a school, n children numbered 1 to n are initially arranged in
Let ABCD be a convex quadrilateral. DA and CB meet at
Let AA1, BB1, CC1 be the altitudes in an acute-angled triangle
Prove that
A polynomial P(x) has degree at most 2k, where k = 0, 1,
The sequences a0, a1, . . . and b0, b1, . . . are defined by the equal-
A be an infinite set of positive integers such that every n \inA is
If p and q are distinct prime numbers, then there are integers
Let m be a positive integer and define f(m) to be the number
A park has the shape of a convex pentagon of area 5
Prove that the set {1, 2, . . ., 1986} can be partitioned into 27
Show that if n runs through all positive integers, f(n) =
In the set Sn = {1, 2, . . ., n} a new multiplication a∗b is defined
Let the numbers 1, 2, . . . , n2 be written in the cells of an n imes n
Let f(x) be a periodic function of period T > 0 defined over R.
Determine all continuous functions f such that
Let n be an integer greater than 1. In the Cartesian coordinate
The segment AB perpendicularly bisects CD at X. Show that,
We are given a circle K with center S and radius 1 and a square
On a chessboard (8 imes 8 squares with sides of length 1) two
Let ABC be an equilateral triangle with side length equal to a
Let the polynomials
Let us define u0 = 0, u1 = 1 and for n \geq0, un+2 = aun+1+bun,
(a) What is the maximal number of acute angles in a convex
Let l, l′ be two lines in 3-space and let A, B, C be three points
Find the number of lines dividing a given triangle into two parts
Let a, b, c, d be positive integers such that ab = cd and a + b =
A figure of area 1 is cut out from a sheet of paper and divided
A rectangular pool table has a hole at each of three of its
A pentagon ABCDE inscribed in a circle for which BC < CD
Prove the following assertion: If c1, c2, . . . , cn (n \geq2) are real
Show that the equation
Let {un} be the sequence defined by its first two terms u0, u1
The four circumcircles of the four faces of a tetrahedron have
Prove that
Solve the system of equations
Determine all pairs of positive integers (x, y) satisfying the equa-
A family of sets A1, A2, . . . , An has the following properties:
Show that tan 7◦30′ =
Prove that all numbers in the sequence
Find all numbers lpha for which the equation
Let a be a number different from zero. For all integers n define
A set G with elements u, v, w, . . . is a group if the following
Let (an)n\geq1 and (bn)n\geq1 be two sequences of natural numbers
To every natural number k, k \geq2, there corresponds a sequence
A rhombus with its incircle is given. At each vertex of the
Let f : R oR be a continuous function. Suppose that the
The diagonals of a convex 18-gon are colored in 5 different
Which regular polygons can be obtained (and how) by cutting
Is it possible to put 100 (or 200) points on a wooden cube such
Note that 83 −73 = 169 = 132 and 13 = 22 + 32. Prove that
Prove that 1
Given any triangle ABC and any positive integer n, we say
By \omega(n), where n is an integer greater than 1, let us denote
Given the expression
A system of n numbers x1, x2, . . . , xn is given such that
Let CD be a diameter of circle K. Let AB be a chord that is
We are given n points in space. Some pairs of these points
It is given that a11, a22 are real numbers, that x1, x2, a12, b1, b2
In a plane are given n points Pi (i = 1, 2, . . . , n) and two
Let x, y, z be nonnegative real numbers satisfying
Find the greatest number c such that for all natural numbers
Let \theta1, \theta2, . . . , \thetan be real numbers such that sin \theta1 + \cdot \cdot \cdot +
Let AA′, BB′, CC′ be the bisectors of the angles of a triangle
A fixed point A inside a circle is given. Consider all chords
Let n > 1 and xi \inR for i = 1, . . . , n. Set Sk = xk
In a triangle ABC for which 6(a + b + c)r2 = abc, we consider
Given two natural numbers w and n, the tower of n w’s is the
We are given a triangle ABC and three rectangles R1, R2, R3
A line l is drawn through the intersection point H of the
Let a, b, and c denote the three sides of a billiard table in the
By h(n), where n is an integer greater than 1, let us denote the
We are given three equal rectangles with the same center in
Let X be an arbitrary nonempty set contained in the plane and
Establish the maximum and minimum values that the sum
Given a triangle, prove that the points of intersection of three
Given a finite sequence of complex numbers c1, c2, . . . , cn, show
Decide whether it is possible to color the 1984 natural numbers
Given 5 points in the plane, no three of which are collinear, prove
We are given a bag of sugar, a two-pan balance, and a weight of
Show that there exists a set S of 15 distinct circles on the
Let us consider a variable polygon with 2n sides (n \inN) in a
Let a, b, c be positive integers satisfying (a, b) = (b, c) = (c, a) =
We are given n (n \geq5) circles in a plane. Suppose that every
A right-angled triangle OAB has its right angle at the point B.
Consider the regular 1987-gon A1A2 . . . A1987 with center O.
In the system of base n2 + 1 find a number N with n different
Prove that the center of the sphere circumscribed around a
Let Q+ denote the set of nonnegative rational numbers. Show
Let p(x, y) and q(x, y) be polynomials in two variables such
If x, y, z are real numbers satisfying the relations x+y+z = 1
Find the maximal number of regions into which a sphere can
(a) Prove that if 0 \leqa0 \leqa1 \leqa2, then
Prove that there exists a natural number k0 such that for
For a point O inside a triangle ABC, denote by A1, B1, C1
Let AB be a segment of unit length and let C, D be variable
Let a, b be coprime integers. Show that the equation ax2 +
Let two circles A and B with unequal radii r and R, respec-
In a Cartesian coordinate system, the circle C1 has center
With the medians of an acute-angled triangle another triangle is
Let f : N oN be such that
Let there be 3399 numbers arbitrarily chosen among the first
Show that there do not exist more than 27 half-lines (or rays)
Find all real solutions of the system of equations
Among all triangles with a given perimeter, find the one with
Let A, B, C, D be points in space. If for every point M on the
For any positive integer n consider all representations n =
Find all real numbers \lambda such that the equation
We consider n real variables xi (1 \leqi \leqn), where n is an
The polynomial
The reviewers correctly identified that the previous proof failed at the planar lemma.
The reviewer identified a false claim in the previous proof:
The problem asks for the existence of a regular tetrahedron with one vertex on each of four given distinct parallel planes in $\mathbb{R}^3$.
The functional equation is
The recurrence is
The system involves five positive real numbers $(x_1, x_2, x_3, x_4, x_5)$ linked cyclically by inequalities of the form $(x_i^2 - x_{i+2}x_{i+4})(x_{i+1}^2 - x_{i+2}x_{i+4}) \le 0$, where indices are…
Consider small values of $m$ and $n$ to examine the expression
A cyclic quadrilateral is given.
Consider a set of ten distinct two-digit numbers, $S = {a_1, a_2, \dots, a_{10}}$, and examine the sums of all its non-empty subsets.
The system consists of $p$ homogeneous linear equations in $q=2p$ variables, with coefficients in ${-1,0,1}$.
The reviewer identified only one critical flaw, namely the final deduction from
For small values of $m$, explicit examples suggest a graph-theoretic interpretation.
The path $XYZTX$ lies on the four faces adjacent cyclically around the tetrahedron.
Place the figure in the coordinate plane with
The sequence begins
Let $P_1$ be a convex polyhedron with nine vertices $A_1,\dots,A_9$ in $\mathbb{R}^3$, and let $P_i = P_1 + (A_i - A_1)$ for $i=2,\dots,9$.
Define
The statement concerns an arbitrary increasing sequence of positive integers.
The statement concerns all $\binom{100}{3}$ triangles determined by a set of points in general position.
Consider small examples with $n = 2$ and $n = 3$ to gain intuition.
The problem requires constructing an isosceles trapezoid $ABCD$ with $AB$ parallel to $DC$, vertices $A$ and $C$ in distinct planes $P$ and $Q$, and vertices $B$ and $D$ lying in the respective planes…
The problem concerns the integers $k$ for which $P(k)=\pm1$.
We are given a positive length $c$, and we must construct a right triangle whose hypotenuse has length $c$.
The quantity attached to a rectangle is not its area but half its area, because a rectangle containing equally many white and black squares has even area and the number of white squares equals the num…
Let $t=\cos x$ and $u=\cos 2x$.
We are asked to determine all real numbers $x$ such that
The fraction
The previous solution correctly derived
We are asked to consider a set of linear functions, each defined by a slope $ki$ and intercept $bi$, and to analyze their sum.
We have an undirected graph representing intersections and roads inside the park. Some intersections contain volunteers. PMP starts at intersection s, and the bus station is at intersection t. PMP has weak memory.
The problem asks us to generate a sequence of dates starting from a given start date and ending at a given end date, incrementing by a fixed number of days defined by step.
We are given an n × m grid and two cells: - source = [sx, sy] - dest = [dx, dy] A move consists of jumping from one cell to another cell that shares either the same row or the same column. The destination cell must be different from the current cell.
We are given an undirected tree with n nodes. A tree is a connected graph with exactly n - 1 edges and no cycles. Each node has a special propagation delay determined entirely by its parity: - Odd-numbered nodes become marked 1 time unit after one of their neighbors is marked.
The problem is asking us to count the number of distinct ways to build a house of cards using exactly n cards. Each house consists of one or more rows of triangles formed by leaning two cards together, with horizontal cards placed between adjacent triangles.
We are asked to repair a road represented as a linear sequence of n holes. There are m construction companies, each offering to fix a contiguous segment of the road at a given cost.
This problem asks us to design a mutable data structure that supports two operations efficiently over two arrays, nums1 and nums2. The first array, nums1, is fixed after initialization and never changes.
The problem gives us two integer arrays, nums1 and nums2, which have the same length. We are told that every element in nums1 was modified by adding the same integer x, and after this transformation the resulting array became equal to nums2.
The problem is asking us to split a given array of non-negative integers into contiguous subarrays in a way that maximizes the number of subarrays, while minimizing the sum of their bitwise AND scores.
We have a fixed set of surnames. Each surname can be either active or inactive. Initially every surname is active. The system must process three kinds of operations. A query +i activates the i-th surname. A query -i deactivates the i-th surname. A query ?
The problem asks us to find a subset of rows in a binary matrix where each column’s sum in that subset is at most half of the number of rows chosen, rounded down.
We are given a set of points in a 2D plane, and we want to measure how quickly a specific closest-pair algorithm behaves on a worst-case input.
We are given a chronologically ordered stream of log entries, where each entry has an exact timestamp down to the second and an associated message describing a program warning.
The problem asks us to identify the top-performing driver for each fuel type based on the trips data. We are given three tables: Drivers, Vehicles, and Trips. Each driver may operate one or more vehicles, and each vehicle may have multiple trips.
We are given a row of domino tiles. Each tile has two numbers: one on the top half and one on the bottom half. We are allowed to flip a tile, which swaps its top and bottom numbers. Each flip costs one unit of time.
We are asked to divide an orange consisting of nk segments among k children so that each child receives exactly n segments, each child definitely receives the segment they chose, and no segment is given to more than one child.
The original HQ9+ language has several commands with special behavior. This problem asks about a reduced version called HQ..., and the destroyed statement leaves only enough clues to reconstruct the intended task. We are given a single string representing a program.
The problem asks us to determine the minimum number of characters to take from either end of a string s consisting only of the letters 'a', 'b', and 'c' so that we collect at least k of each character.
The problem gives us a database table named Tweets, where each row represents a tweet posted during February 2024. Every tweet contains plain text in the tweet column, and that text may contain one or more hashtags.
We are given a list of strings, and for each string, we need to count the number of its substrings that appear in at least k strings from the list. A substring is any contiguous segment of a string.
The floor is made from unit hexagonal tiles, and the whole hall itself forms a larger hexagon. The six sides of the hall contain a, b, c, a, b, c tiles respectively as we walk around the boundary.
The problem asks us to find the length of the longest subsequence in an integer array nums such that the subsequence is strictly increasing and the difference between consecutive elements does not exceed a given integer k.
The problem presents a list of intervals derived from the start array and a fixed d. Specifically, each interval is [start[i], start[i] + d].
We are given the six pairwise Hamming distances between four unknown binary strings. Every string contains only 'a' and 'b', and all four strings must have the same length. The task is not to recover the original strings.
The problem gives us an integer array nums, an integer k, and an integer multiplier. We must perform exactly k operations on the array. In each operation, we locate the minimum value currently present in the array.
We are given a sequence of positive integers, all between 1 and 7, whose length is divisible by three. The task is to split this sequence into triplets so that within each triplet the numbers are strictly increasing and each number divides the next.
The problem requires us to count k-subsequences of a string s that have the maximum beauty, where beauty is defined as the sum of the frequency of each character in the subsequence. A k-subsequence is a subsequence of length k where all characters are unique.
The problem asks us to compute the maximum frequency score among all contiguous subarrays of length k from a given integer array nums. A frequency score is defined as the sum of each distinct element raised to the power of its frequency within the subarray, taken modulo 10^9 + 7.
We have an array and a fixed window length len. For every subarray of length len, we compute its sum and then take the absolute value. The "optimal sum" of the whole array is the maximum absolute subarray sum among all windows of that fixed length.
The problem provides an array of positive integers nums and asks us to compute a sum based on an encryption transformation. The transformation encrypt(x) replaces every digit in x with the largest digit in x.
Roma has a collection of positive integers, and he is fascinated with numbers whose decimal digits consist only of 4 and 7. These are called lucky numbers. The task is to determine, from his collection, how many numbers have at most k lucky digits.
The problem gives us an integer array nums, and we must determine whether it is possible to split the array into contiguous groups such that every group satisfies one of three valid patterns. A valid group can be: 1. Exactly two equal numbers, such as [5,5] 2.
The problem requires finding a peak element in a 2D matrix. A peak element is defined as an element that is strictly greater than its adjacent neighbors to the top, bottom, left, and right.
The problem asks us to find the top 3 trending hashtags from a table of tweets for a specific month, February 2024. Each tweet contains exactly one hashtag. The input is represented by a Tweets table with columns userid, tweetid, tweet, and tweetdate.
This problem describes a scenario where a group of friends attends a party with an infinite number of chairs labeled from 0 upwards. Each friend has a specific arrival and leaving time. When a friend arrives, they must occupy the smallest-numbered unoccupied chair.
The problem asks whether it is possible to make all strings in the input array words equal by redistributing characters between them. Specifically, in one operation, you can pick any character from one string and move it to another string at any position.
This problem asks us to classify bank accounts into three salary categories based on their monthly income, then count how many accounts belong to each category. The input is a database table named Accounts.
We are given an array of integers a with n elements. The task is to select two distinct indices i and j and compute a function f(i, j) that combines both the squared distance between the indices and the squared sum of the elements strictly between them.
We are asked to construct a "beautiful permutation" of the integers from 1 to n, where n can be as large as one million. The permutation is built by repeatedly applying a block-cyclic left shift operation on increasing block sizes.
This problem describes a two-player game played on a line of colored pieces. The input colors is a string representing these pieces, where each character is either 'A' or 'B'. Alice always moves first and can only remove 'A' pieces that are surrounded on both sides by 'A'.
The problem involves simulating operations on an infinite number line starting at 0 and extending towards the positive x-axis.
The problem asks us to count the number of ways to select exactly 3 buildings from a street represented as a binary string s, such that no two consecutive buildings among the selected ones have the same type. Here, 0 represents an office, and 1 represents a restaurant.
We are given a string consisting solely of the characters "x" and "y". Two operations can be applied repeatedly in a specific order. The first operation swaps a consecutive "y" followed by "x" into "x" then "y". The second operation removes a consecutive "x" followed by "y".
We are given a sequence of integers, and for each distinct value we want to understand how its occurrences are spaced across the array. For any value $x$, we look at all indices where $x$ appears.
We are asked to color a set of points on a number line using two colors, red and blue, so that for each given segment, the number of red and blue points inside that segment differ by at most one.
Each trolleybus starts from the depot at a fixed departure time. It begins with speed 0, can accelerate at most a, and can never exceed its own speed limit v[i]. The destination is d meters away.
The problem gives us a binary string s consisting only of characters '0' and '1'. Our goal is to make every character in the string equal, meaning the final string must become either all '0' characters or all '1' characters. We are allowed to perform two types of operations: 1.
We need to simulate a tiny shell that supports only two commands. The command cd path changes the current directory. The path may be absolute, meaning it starts from the root /, or relative, meaning it starts from the current directory. Inside a path, the token ..
The problem asks us to determine whether two JSON values, o1 and o2, are deeply equal. Deep equality goes beyond simple reference or shallow equality. For primitive values such as numbers, strings, booleans, or null, equality is straightforward using strict comparison (===).
The spiral starts at the origin and grows outward by alternating directions: right, up, left, down. Each new segment is longer than the previous pair. The first few moves are: - right 1 - up 1 - left 2 - down 2 - right 3 - up 3 and so on forever.
The problem asks us to determine whether we can select two or more elements from a given array of positive integers such that their bitwise OR produces a number whose binary representation ends with at least one zero.
The problem asks us to find two indices i and j in the array such that two conditions are satisfied simultaneously: 1. The indices must be far enough apart: 1. The values at those indices must differ enough: We may return any valid pair if multiple answers exist.
We are looking for bases in which a very special cyclic behavior exists. Take a number with exactly n digits in base b. Leading zeroes are allowed, so 0011 in base 2 is a valid length-4 number.
We are given the chronological contest scores of one programmer. A performance is called "amazing" when the current score is strictly greater than every previous score, or strictly smaller than every previous score.
The problem asks us to analyze customer purchasing behavior using two relational tables: Transactions and Products. The Transactions table contains every purchase made by customers, including the transaction ID, customer ID, product ID, transaction date, and amount spent.
We are given a row of positions from 1 to n, and each position currently holds a glass. A marble is initially hidden under the glass at position s. The only way the configuration changes is by applying a fixed rearrangement rule multiple times.
The problem asks us to form the largest palindromic integer using the digits from a given string num. A palindrome is a number that reads the same forwards and backwards, like 121 or 7449447. The input string num consists only of digits (0 to 9) and may include repeated digits.
This problem asks us to count the number of valid words in a sentence. The sentence consists of lowercase letters, digits, hyphens, punctuation (!, ., ,), and spaces. A word is defined as a token separated by spaces, and it is considered valid if it satisfies three conditions: 1.
The problem asks us to process an array of lowercase strings words and a single lowercase character x, and return all indices of the words that contain this character. In simpler terms, for each string in the array, we must check whether x appears anywhere in that string.
We are given a lowercase English string s and a positive integer k. We must count how many non-empty substrings satisfy the definition of a beautiful substring.
The problem asks us to identify hills and valleys in an integer array nums. A hill is a position where the closest non-equal neighbors on both sides are smaller than the current value, and a valley is where those neighbors are larger.
This problem asks us to verify whether a string follows a very specific spacing rule between repeated characters. We are given a string s that contains only lowercase English letters. Every character that appears in the string appears exactly twice.
The Friends table represents an undirected friendship graph. Each row (userid1, userid2) means the two users are directly connected as friends. The problem asks us to find every friendship pair where the two users do not share any common friend.
We are given only a single integer $n$, and we must construct an array of length $n$ that either breaks a very specific sorting procedure or prove that no such array exists.
The problem provides a database table named Data with two integer columns: firstcol and secondcol. Each row represents a pair of numbers, and duplicate rows are allowed.
The problem asks us to process a database table, CoffeeShop, which contains two columns: id and drink. Each row represents a drink order. Some drink values may be NULL.
This problem asks us to find the last day we can cross a grid from the top row to the bottom row, walking only on land. The grid is initially all land (0), and each day, specific cells are flooded with water (1) according to the cells array.
This problem asks us to reshape a dataset from a wide format to a long format, also known as “melting” in data analysis. In the input DataFrame, each row represents a product, and each column beyond the first represents quarterly sales (quarter1 to quarter4).
The problem gives us an integer array nums and a value k. We are allowed to repeatedly merge adjacent elements under one condition: - If two adjacent values x and y satisfy x y <= k, then we may replace them with a single value equal to x y.
The problem asks us to construct a string from the characters of a given string s under a specific constraint: no character can appear more than repeatLimit times consecutively.
The problem presents an integer array nums and a positive integer k. You can perform a bitwise operation on any two distinct elements of the array any number of times.
We are given four players split into two fixed teams of two players each. Each player has two independent strengths: one for defence and one for attack. Before the match, each team must assign one of its players to attack and the other to defend.
The problem asks us to divide a given string s into the minimum number of substrings such that every substring contains only unique characters. In other words, within a single substring, no character may appear more than once.
The problem asks us to design a data structure, FrequencyTracker, that can efficiently track the frequency of numbers and answer queries about whether any number exists with a specific frequency.
We are asked to select the largest possible subset of integers from a given set such that any two numbers in the subset can be summed by Vasya. Vasya’s summing rule is unusual: for each decimal place, at least one of the two numbers must have a zero in that place.
The problem asks us to determine whether a given string qualifies as a "valid word" according to four specific rules. First, the word must contain at least 3 characters. Any string shorter than 3 is automatically invalid.
We are given a list of segments on a number line, each defined by a left and right endpoint. The task is to find if there exists a single segment among them that fully contains every other segment.
The problem is asking us to simulate a process where we distribute elements from a 1-indexed array of distinct integers nums into two separate arrays, arr1 and arr2, following a set of deterministic rules.
That is a long, structured reference document with multiple required sections, detailed prose, two full implementations, worked examples, test cases, and edge-case analysis. To keep quality high and avoid truncation, I will provide it in a complete guide format.
This problem is essentially asking how many distinct ways we can select questions from multiple types to reach exactly a target score. Each type of question has a fixed number of questions (counti) and a fixed number of points per question (marksi).
We are given a row of balls, each colored with one of k colors. No color initially appears three times in a row. Iahub holds a single extra ball of a given color x and can insert it anywhere in the row, including before the first ball or after the last.
The problem is asking us to compute a special score for each string in a list of strings. Specifically, for a string term, its score is defined as the number of strings in the array that have term as a prefix.
The problem asks us to transform an input integer array nums into a beautiful array by performing a minimum number of increment operations. An increment operation increases a single element of nums by 1.
We are given a string consisting only of four bracket characters: (, ), [ and ]. The string itself is not guaranteed to be balanced. Our task is to find a contiguous substring that forms a valid bracket sequence.
We are given an array of daily temperatures. We want the sequence to look like this: First, several consecutive negative values. Then, several consecutive positive values. Both parts must be non-empty, and zero is forbidden anywhere in the final sequence.
The problem gives an n x n integer matrix and allows an operation where we pick any two adjacent cells (sharing a side) and multiply both values by -1.
The problem gives us a 0-indexed array nums of length n containing distinct positive integers. The goal is to determine the minimum number of right shifts required to sort the array in strictly increasing order.
The problem requires determining the maximum possible length x of ribbons such that, after cutting or keeping the given ribbons in the array ribbons, you can obtain at least k ribbons of length x.
We are given an array maximumHeight where maximumHeight[i] represents the largest height that tower i is allowed to have. Our goal is to assign an actual height to every tower such that: - Every assigned height is a positive integer.
The problem asks us to find the minimum number of operations needed to ensure that every element in an array is at least k. Each operation consists of removing the smallest element from the array.
That is a long, multi-section technical guide. To make sure I target the correct problem and avoid producing the wrong reference document, can you confirm the exact LeetCode problem number/title? You mentioned LeetCode 2577 - Minimum Time to Visit a Cell In a Grid.
We are given an $n times n$ grid of lowercase letters. A valid string is formed by walking from the top-left cell to any reachable cell by moving only right or down, always starting at $(1,1)$.
The problem asks us to determine the number of distinct ways to spend a given amount of money, total, on pens and pencils, each with fixed costs, cost1 for pens and cost2 for pencils.
The problem asks us to count how many integers in the range [1, n] contain only distinct digits. A number is considered special if no digit appears more than once in its decimal representation.
The problem asks us to take a positive integer num and split its digits into two non-negative integers num1 and num2 such that the sum num1 + num2 is minimized.
The problem asks us to count the number of valid pairs of arrays (arr1, arr2) derived from a given array nums. Both arr1 and arr2 have the same length as nums. The constraints on these arrays are: 1. arr1 is non-decreasing. 2. arr2 is non-increasing. 3.
We are given a set of $n$ points on a 2D plane, each with integer coordinates, and all points are distinct. The task is to count how many axis-aligned squares exist whose four corners are all points from this set.
We are given a decimal number as a string. We may independently permute its digits twice, producing two new numbers that contain exactly the same multiset of digits as the original number. Leading zeroes are allowed after permutation.
We have a collection of shoes, where every shoe has a unique size and a price. We also have customers, where each customer has a budget and a foot size. A customer can buy a shoe only if two conditions hold.
We are simulating a single-threaded printer that receives tasks over time. Each task arrives at a given time, has a known number of pages, and a priority that determines the order in which it is served when multiple tasks are waiting.
We are given an integer array nums. In one operation, we may choose any subsequence of the array that is strictly increasing and remove all of its elements simultaneously. A subsequence does not need to be contiguous. We only need to preserve the relative order of elements.
The problem describes multiple virus variants spreading across an infinite 2D grid. Each virus starts from its own origin point on day 0.
The problem asks us to determine how many integer points on a number line are covered by at least one car. Each car is represented as a range [starti, endi] of integers, inclusive. For example, a car [3,6] covers the points 3, 4, 5, and 6.
We are given a lowercase string and an integer k. We may rearrange the letters however we want. The goal is to build a new string that consists of exactly k identical blocks placed one after another.
This problem is asking us to take an undirected tree where each node has a numerical value and determine how many edges we can remove such that every resulting connected component has the same total value.
The problem gives us two collections of items, items1 and items2, where every item is represented as a pair: The value acts like a unique identifier for an item, while weight represents that item's associated weight.
The problem gives us an undirected weighted graph with n cities and a list of highways. Each highway connects two cities and has an associated toll cost. We must find the maximum possible total toll for a trip that uses exactly k highways. There are two important restrictions: 1.
This problem asks us to find the maximum possible variance among all substrings of a given string. The string contains only lowercase English letters, and the variance of a substring is defined as the largest difference between the counts of any two characters that both appear…
The problem is asking us to maximize the total price of items placed in a bag with a fixed capacity. Each item is defined by a price and a weight, but unlike traditional knapsack problems, items can be divided proportionally, meaning we can take fractions of an item.
We are given a collection of distinct points on a 2D plane. Each point is represented as (x, y). An increasing path is a sequence of points where both coordinates strictly increase from one point to the next.
This problem asks us to take an input array nums of length n and produce a new array ans of length 2n where the first half of ans is identical to nums and the second half is also identical to nums. In other words, ans is formed by concatenating nums with itself.
We are asked to partition a club of members into groups based on responsibility and age constraints. Each member has a responsibility value and an age.
This problem asks us to divide a message into multiple parts while respecting a strict maximum length constraint for every part.
This problem asks us to compute the sum of all good subsequences in a given integer array nums. A subsequence is any sequence derived from nums by deleting zero or more elements without changing the order of the remaining elements.
We are asked to determine the maximum expected total listening time for Manao's playlist, given that each song has a certain probability of being liked and a fixed length.
This problem asks us to maximize the number of rows covered in a binary matrix after selecting exactly numSelect columns. Each row is covered if all 1s in that row are located in the selected columns, or if the row contains only 0s.
We are looking for the smallest positive integer that has exactly n digits and is divisible by every one of the numbers 2, 3, 5, and 7 at the same time. In other words, we want the minimal n-digit number that is a multiple of the least common multiple of those four integers.
This problem asks us to schedule a collection of tasks into the minimum number of work sessions. Each task has a fixed duration, and every work session has a maximum allowed length, sessionTime. A task cannot be split across multiple sessions.
The problem asks us to find the maximum length of a contiguous sequence of cities that appears in every friend's travel path. Each friend has a path represented as an array of city IDs. A subpath is simply a contiguous segment of that array.
This is a Type B - “Prove that” problem. The statement to prove is: If $$a+b=tanfrac{gamma}{2}(atanalpha+btanbeta),$$ then the triangle is isosceles. The proposed proof attempts exactly this implication. It does not merely prove a weaker statement.
This problem asks us to determine how long we can safely wait before starting to move from the top-left corner of a grid to the bottom-right corner while a fire spreads across the grid over time.
The problem provides a table of phone call records where each row contains a caller, a recipient, and a timestamp. Each call is bidirectional in the sense that both participants are considered to have made and received the call simultaneously.
The problem asks us to determine the maximum number of jumps one can make from the first element of an array nums to the last element, subject to a jumping constraint.
This problem asks us to determine how the global rankings of national teams change after their points are updated. We are given two database tables: The TeamPoints table contains the current ranking information for each team.
We maintain a set of currently active colliders. Each collider is identified by an integer from 1 to n. The system is safe only if every pair of active colliders is coprime. In other words, no prime factor may appear in two different active numbers at the same time.
This problem asks us to calculate the total amount of money spent by Premium and VIP members on Fridays of each week in November 2023.
This problem requires us to classify a document into one of three subjects based on its textual content. Each document consists of an identifier, a name, and a body of text.
This problem asks us to determine which recipes can be created when we start with a set of available supplies and are allowed to recursively create additional recipes. Each recipe has a list of required ingredients.
This problem is asking us to implement an immutability helper for JSON-like objects in JavaScript. Specifically, we need to create a class ImmutableHelper that allows users to "mutate" a proxy version of the object without affecting the original object.
The problem asks us to implement a specialized string compression algorithm. Given an input string word, we are required to build a compressed version by repeatedly taking prefixes consisting of consecutive repeating characters, limited to a maximum length of 9, and appending…
The problem gives us a compressed string where every character is immediately followed by its frequency. For example, the string "a3b2" represents the original expanded string "aaabb". However, the input compression is not guaranteed to be optimal.
This problem models airline flight bookings with limited seating capacity. The Flights table contains one row per flight. Each flight has a unique flightid and a capacity, which represents the maximum number of passengers that can be seated on that flight.
The problem asks us to find three indices (i, j, k) such that the middle element forms the peak of a mountain. More specifically, the indices must satisfy: - i < j < k - nums[i] < nums[j] - nums[k] < nums[j] Among all valid mountain triplets, we must return the minimum…
The problem gives us an array nums containing positive integers. Each integer represents a potential side length that we may use when constructing a polygon. A polygon must have at least three sides.
This problem asks us to compute the score of a string, where the score is defined as the sum of the absolute differences of ASCII values of consecutive characters in the string.
In this problem, we are given a database table named Orders. Each row represents a single order and contains three columns: - orderid, the unique identifier for the order - customerid, the customer who placed the order - ordertype, which is either 0 or 1 The task is to return…
We are given a single non-negative integer, called the magic number. At each step, we can subtract from it any of its digits to produce a new number. We repeat this operation until the number reaches zero.
The problem asks us to calculate the maximum score achievable by hopping through an array from the first element to the last. You start at index 0, and at each step, you can jump to any subsequent index j i. When you jump, you accumulate a score of (j - i) nums[j].
We are asked to determine whether a given company name is symmetric with respect to a vertical mirror. In practical terms, we are given a single string consisting of uppercase English letters, and we need to check whether the string would appear identical if reflected in a…
The shape of the table is fixed. Row lengths are non-increasing, so every row is no longer than the row above it. The cells contain all integers from 1 to s exactly once, where s is the total number of cells. We may swap the contents of any two cells.
The problem gives us three parallel arrays: - creators[i] represents the creator of the ith video - ids[i] represents the video ID of the ith video - views[i] represents the number of views for the ith video Each index corresponds to one video.
The problem asks us to determine the minimum number of bit flips required to transform an integer start into another integer goal. A bit flip is defined as changing a single bit in the binary representation of a number from 0 to 1 or from 1 to 0.
We are given a line of numbered chests, each containing some number of coins. On each move, a player chooses an integer position $x$, and that move simultaneously affects three specific chests: $x$, $2x$, and $2x+1$. From each of these chests, one coin is removed if it exists.
The problem gives us an array of positive integers called nums. We need to compute two different values from this array. The first value is the element sum, which is simply the sum of every number in the array.
This problem asks us to identify symmetric coordinate pairs from a database table called Coordinates. Each row in the table represents a coordinate (X, Y), and duplicate rows are allowed.
We are given an integer array nums. We may repeatedly choose any element and replace it with any positive integer x. Replacing a value nums[i] with x costs |nums[i] - x|.
We are given two positive integers, n and target. We need to construct an array of exactly n distinct positive integers such that no two different elements add up to target. Among all arrays satisfying these conditions, we want the one with the smallest possible sum.
This problem takes place on a two-dimensional grid where each cell represents a type of terrain. The grid contains: - "S": your starting position. - "D": the destination you want to reach. - ".": an empty cell that can be walked on. - "X": a stone cell that cannot be entered.
This problem asks us to determine the number of possible original strings Alice intended to type based on the final string displayed on her screen.
We are asked to simulate a pursuit scenario between a princess and a dragon. The princess runs at a constant speed, and the dragon flies faster but only begins chasing after a delay. Each time the dragon catches up, the princess can drop a bijou to distract him.
We are given a company where employees participate in several independent chat groups. Each chat has a fixed membership defined in advance. Over time, a log records messages: each event says that a particular employee posts in a particular chat.
The problem gives us a sequence of processor temperatures recorded over n steps. The chief engineer has reliable notes of the minimum and maximum temperature observed, but the assistant only recorded m of the n temperatures.
The problem gives us an undirected graph where each node has an associated integer value. We are asked to form a star graph and compute the maximum possible star sum. A star graph is defined by choosing one node as the center and selecting up to k of its neighbors.
The problem asks us to determine how many times a user changes keys while typing a string s. A change of key occurs when the current character typed differs in its lowercase form from the previous character typed.
This problem asks us to take an array of positive integers, nums, and transform it into another array, answer, where each element of nums is broken down into its constituent digits in order.
The problem gives us three integer arrays, a, b, and c. We must count how many triplets (a[i], b[j], c[k]) produce a bitwise XOR result with an even number of set bits. A set bit is a bit equal to 1 in the binary representation of a number.
The problem is asking for the maximum length of a substring in a given string s such that no character occurs more than twice within that substring. In other words, for any valid substring, each character can appear at most two times.
In this problem, we are given an integer num, which may be positive, negative, or zero. Our goal is to rearrange its digits so that the resulting number is as small as possible while preserving the original sign. The important detail is that the sign cannot change.
We are given an integer array nums, and we must count how many subsequences of length 5 satisfy a very specific condition: - The subsequence must have exactly 5 elements.
We are tasked with reorganizing a hard disk so that files occupy contiguous clusters from the beginning of the disk, and free clusters are pushed to the end. The disk has n clusters, numbered from 1 to n, and m files.
The problem gives us a 0-indexed integer array nums. We must find the smallest index i such that: The expression i mod 10 means the remainder when i is divided by 10. We are asked to scan the array and determine whether any index satisfies this condition.
We are given an array of integers, and some of these integers are "lucky numbers," meaning they consist only of the digits 4 and 7.
The problem gives us an integer array nums and asks us to count how many elements satisfy a very specific condition.
The problem gives us two positive integers, a and b, and asks us to count how many integers divide both numbers evenly. A number x is considered a common factor if: - a % x == 0 - b % x == 0 This means x divides both integers without leaving a remainder.
In this problem, we are given a row of plants where each plant requires a certain amount of water. Two people, Alice and Bob, water the plants simultaneously from opposite ends of the array. Alice starts from the left side at index 0 and moves toward the right.
The problem provides a Pandas DataFrame named employees with two columns: | Column | Type | | --- | --- | | name | object | | salary | int | Each row represents one employee and their current salary.
The problem asks us to distribute weights.length marbles into k contiguous bags, where the cost of a bag is defined as the sum of the first and last marble in that bag.
We are given two integers, l and r. We may choose any two numbers a and b such that both lie inside the interval [l, r] and a ≤ b. Among all such pairs, we must compute the maximum possible value of a XOR b. The XOR operation compares bits position by position.
The problem asks us to minimize the number of connected groups in a set of intervals by adding exactly one interval of length at most k. Each interval [start, end] represents a continuous range on the number line.
We maintain two rooted trees. Initially each tree contains only the root vertex 1. Every operation adds one new vertex to one of the trees. The new vertex becomes a child of an existing vertex, and the connecting edge stores one lowercase letter.
We are given a directed graph where each node represents a city junction, and each directed edge represents a one-way road. Each junction has a cost to build a police checkpost.
The problem asks us to sort an array of integers based on a custom digit mapping rather than their natural numeric value.
We are given a matrix of integers and a number d. The penguin can either add or subtract d from any matrix element in a single move. The goal is to make all elements equal using the fewest moves possible.
The problem gives us an integer array nums and asks us to find the length of the longest contiguous subarray that is either strictly increasing or strictly decreasing. A subarray must consist of consecutive elements from the original array.
Here is the complete, detailed technical solution guide for LeetCode 2643 - Row With Maximum Ones following your exact formatting requirements. The problem presents a binary matrix mat of size m x n, where each element is either 0 or 1.
We start with a segment representing Liss’s current safe region, initially the interval from 0 to 1. Stones fall one after another, and each stone always lands exactly at the midpoint of Liss’s current interval.
The problem gives us a numeric string s, where every character is a digit between '1' and '9', and an integer k. We must split the string into contiguous substrings such that every substring, when interpreted as an integer, has a value less than or equal to k.
The problem gives us a string s and two smaller strings, a and b. We need to find every index i where substring a appears in s, but only if there exists at least one occurrence of substring b close enough to it. More formally, an index i is considered beautiful when: 1.
The problem asks us to identify words in the queries list that can be transformed into a word in the dictionary list with at most two character edits. Each edit consists of changing a single character to another lowercase English letter.
This problem asks us to compute the average number of items per order from a compressed representation of order data. Instead of storing every individual order as a separate row, the table groups together orders that contain the same number of items.
We are asked to compute the minimum time needed for a player to open all portals in a country modeled as a graph. The country consists of n cities connected by m bidirectional roads with positive travel times. Some subset of cities, k of them, have portals.
We are given a collection of snowballs, each with a specific radius, and our goal is to assemble as many snowmen as possible using these snowballs. Each snowman must be made of exactly three snowballs, and each of those three must have a distinct radius.
In this problem, we are given the positions of n rooks placed on an n x n chessboard. Each rook is represented as a pair [xi, yi], where xi is the row index and yi is the column index.
We are given a group of $n$ people standing on one river bank. Each person has a weight of either 50 or 100, and a boat that can carry a limited total weight $k$.
The problem gives us a string word and a list of forbidden strings forbidden. A substring is considered valid if none of its internal substrings appear in the forbidden list.
We are given an undirected weighted graph where each node has an associated value. A path is considered valid if: 1. It starts at node 0 2. It ends at node 0 3.
That is a very large, comprehensive request for a Hard problem, with full sections, worked examples, Python and Go implementations, detailed prose, test cases, and edge case analysis.
The problem asks us to count how many distinct anagram sentences can be formed from a given string s. The input string contains one or more words separated by single spaces. An anagram sentence must preserve the structure of the original sentence.
We are given a sequence of movie genres scheduled over n days, with exactly one movie per day. There are k genres, and each genre appears at least once. Valentine, a critic, experiences stress whenever the genre of consecutive movies he watches changes.
This problem asks us to implement a custom version of the forEach method for arrays in JavaScript. The goal is to extend the Array.prototype so that any array can call forEach(callback, context) and execute the callback on each element.
The problem gives us three types of fixed two-character strings: - "AA" appears x times - "BB" appears y times - "AB" appears z times We may choose any subset of these strings and concatenate them in any order.
The problem asks us to transform an array of tower heights into a mountain-shaped arrangement while removing as few bricks as necessary.
The problem asks us to count how many contiguous subarrays of nums have a bitwise AND equal to k. A subarray is any contiguous segment of the array. For every possible subarray, we compute the bitwise AND of all elements inside it.
The problem asks us to construct an array of n distinct positive integers such that no two elements in the array sum to a given integer k. This type of array is called k-avoiding.
We are given a square matrix of size n × n, where n is guaranteed to be an odd number. Each element of the matrix is a non-negative integer.
The problem requires categorizing a box based on its dimensions and mass according to given criteria. We are given four integers: length, width, height, and mass. The output is a string that represents the category of the box: "Bulky", "Heavy", "Both", or "Neither".
The problem is asking us to find the minimum number of swaps required to group all 1s in a binary circular array together at any contiguous segment. The array is circular, which means the end of the array wraps around to the beginning.
The problem asks us to transform a one-dimensional integer array nums into a two-dimensional array (a list of lists) while satisfying three specific conditions. First, every element from nums must appear in the 2D array exactly as many times as it appears in the original array.
The problem asks us to find indices of "stable" mountains from a given array of mountain heights. Each element in the array represents the height of a mountain in sequence.
We have trees placed on a number line. Each tree may fall left, fall right, or remain standing. The probabilities for these three outcomes are given independently for every tree. A falling tree destroys mushrooms in a half-open interval determined by the tree position and height.
I can do that, but the complete guide with all requested sections and the level of detail you specified will exceed a single message limit. I will split it cleanly into multiple parts while preserving your exact formatting requirements. Planned structure: 1.
We are given a tree with n nodes rooted at node 0. Each node contains a certain number of coins. We must collect coins from every node while respecting the tree hierarchy, meaning a node can only be processed after all of its ancestors have already been processed.
We are asked to count the number of ways an ant can start at vertex D of a tetrahedron and return to D after exactly n steps, moving along edges at every step. The tetrahedron has four vertices labeled A, B, C, D, and each vertex is connected to the other three.
You are given a 0-indexed integer array arr. For every index i, you must compute the sum of distances between i and every other index j where arr[i] == arr[j].
This guide will use the weighted median optimization approach, which is the key insight behind solving this problem efficiently within the constraints. The problem gives us two arrays, nums and cost, each of length n.
We are given a row of n lights, some of which are initially on. Each light can only be turned on if it has at least one neighbor that is already on. The task is to count how many sequences of switches exist that will eventually turn all the lights on.
This problem asks us to find a contiguous subarray whose elements are all sufficiently large relative to the subarray length. More specifically, for a subarray of length k, every element inside that subarray must satisfy: We are allowed to return the size k of any valid subarray.
The problem asks us to determine the minimum number of swap operations required to sort the values of a binary tree level by level in strictly increasing order. Specifically, at each level of the tree, we can only swap values of nodes that exist on that level.
We are asked to maximize Vasya's score in a game where destroying geometric figures earns points based on a per-figure cost and a global factor. Each figure type has a quantity and a point value.
The kingdom of gnomes consists of castles connected by a very specific graph structure. There is a cycle of m castles called the Good Path, connected sequentially with roads and forming a closed loop.
The problem asks us to find the k-th smallest positive palindrome of a fixed length for multiple queries. Specifically, we are given an array queries where each element indicates the position of a palindrome we need to find, and an integer intLength which specifies the number…
This problem asks us to count how many different ways we can represent a given integer n as the sum of distinct positive integers raised to the power x.
We are given a list of travel times from the town Rozdil to each of n other towns. Each town has a positive integer time, and the towns are numbered from 1 to n. The goal is to find which town has the smallest travel time.
The problem asks us to count the number of bad pairs in an array. We are given a 0-indexed integer array nums, and a pair of indices (i, j) is considered bad if: - i < j - j - i !
Gennady spends the first 10 minutes of the contest only reading the statements. After that, he has exactly 710 minutes left for writing solutions. Each problem requires a fixed amount of writing time, and he may pause and resume problems whenever he wants.
We are given a non-negative integer s and need to determine the largest possible dimension n of a three-dimensional array A. The array has dimensions n × n × n, and every element is defined as: where | denotes the bitwise OR operation.
We are given a rooted forest representing family relationships, where each node corresponds to a person, and each edge points from a child to their parent. A person may have no parent, in which case they are a root of one of the trees in the forest.
We are given an array of positive integers and a set of "bad" prime numbers. Every other prime not in the bad set is implicitly "good." Each number in the array contributes to a total "beauty" score determined by its prime factorization.
The problem asks us to compute the triangular sum of an array of digits. The triangular sum is obtained by repeatedly reducing the array according to a simple rule: for every pair of adjacent elements, sum them modulo 10 to form a new array of length one less than the current…
Valera wants to buy exactly one antique item from a set of sellers. Each seller offers multiple items with a current auction price. Valera can only secure a deal if he offers strictly more than the current price of an item, and he has a fixed budget v.
We are asked to count substrings of a given string that satisfy a complicated set of constraints. Each constraint specifies a letter and a minimum and maximum number of times that letter can appear.
The problem asks us to count the number of positive integers less than n (where n is given in binary as a string s) that are k-reducible.
The problem gives us n sticks with unique lengths from 1 to n. We must arrange these sticks in some order so that exactly k sticks are visible when looking from the left side. A stick is visible if every stick before it is shorter.
The problem asks us to determine the minimum initial money required to complete all transactions in any order. Each transaction is defined by [costi, cashbacki], meaning that performing the transaction requires at least costi money, and after completing it, we receive…
The problem gives us a lowercase string s and defines a repeated operation. During one operation, we scan through all letters from 'a' to 'z'. For each letter, if that letter appears in the current string, we remove its first occurrence.
We are given a permutation of length n, which means an array of integers where each number from 0 to n - 1 appears exactly once. In this array, a "fixed point" is an index where the value equals the index itself.
We have a large circular table with radius R and want to place n identical circular plates, each with radius r. Every plate must satisfy three conditions simultaneously. First, the entire plate must stay inside the table. Second, every plate must touch the boundary of the table.
You included two different problems in one message, and the second prompt overrides the first at the end. I will answer for LeetCode 2988 - Manager of the Largest Department.
This problem is intentionally unusual for Codeforces. There is no hidden graph, no combinatorics trick, and no numeric constraint to optimize around. Instead, we are asked to classify a text document into one of three categories using a fixed training corpus.
This problem asks us to select exactly one element from each row of a given m x n integer matrix mat such that the sum of the selected elements is as close as possible to a given integer target.
This problem asks us to calculate how much money each user has spent across all of their purchases. We are given two database tables: The Sales table stores purchase records.
The problem presents a binary tree where leaf nodes represent boolean values 0 (false) or 1 (true), and internal nodes represent boolean operations OR, AND, XOR, and NOT, encoded as integers 2, 3, 4, and 5. You are also given a target boolean result.
The problem describes a snake moving inside an n x n grid. Every cell in the grid is assigned a numeric position using the formula: This means the grid is numbered row by row, starting from the top-left corner.
The problem asks us to calculate the k-beauty of a given integer num. To do this, we treat num as a string and examine every possible contiguous substring of length k. For each substring, we interpret it as an integer and check whether it divides the original number num evenly.
We are given an array usageLimits where usageLimits[i] tells us how many times the number i can be used across all groups. The numbers available are exactly 0 through n - 1, where n is the length of the array.
This problem asks us to implement a class ArrayWrapper that encapsulates an array of integers and supports two operations in a custom way.
We are given a sequence of non-negative integers and we want to pick a contiguous block of elements such that when we take the bitwise XOR of everything inside that block, the result is as large as possible.
The problem gives us the head of a singly linked list and asks us to remove every node that has a strictly greater value somewhere to its right. In other words, for each node, we must determine whether there exists another node later in the list whose value is larger.
We are given a tree with $2n$ nodes, and the goal is to place each node into a 2-row by $n$-column table so that each edge of the tree connects two cells sharing a side. Each node occupies exactly one cell, and each cell contains exactly one node.
We are asked to analyze a two-player string game. Players alternate moves. On a turn, a player can select any string fragment available (initially the whole string) and cut a character that is the center of a palindrome of odd length.
The problem asks us to determine if an even-length array can be split into two equal-sized subarrays, each containing only distinct elements.
The problem gives us an array nums, and we are allowed to repeatedly perform a special operation on pairs of elements.
This guide is long enough that a complete, high quality response will exceed a practical single-message length if I include all requested sections in full detail, including detailed worked examples, Python and Go implementations, test coverage, and edge case analysis.
We are given a sequence of events happening in time order in a city. Each event is either the arrival of one or more new police recruits, or the occurrence of a crime. When recruits arrive, they increase the number of available officers.
We have a sequence of trees, each with a strictly increasing height, and each tree has an associated cost that represents the price of recharging the chainsaw after completely cutting the highest-indexed tree so far. Kalila and Dimna need to reduce all tree heights to zero.
The problem gives us an integer array candies, where each value represents the flavor of a candy. We must give exactly k consecutive candies to our sister. After removing those k candies, we keep the remaining candies for ourselves.
The problem asks us to analyze a given string word where each lowercase English letter is mapped to a digit according to a classic phone keypad scheme.
The input is a long string made only of digits and hyphens. Somewhere inside this string, there may be many substrings that look like dates written in the exact format dd-mm-yyyy. We must find which valid date appears most often as a substring.
Codeforces 350A: TL
We are given a set of snow drifts on a 2D grid. Bajtek can move from one snow drift to another by sliding along the rows or columns until he reaches another snow drift, moving strictly in the north, south, east, or west directions.
We work in a positional numeral system with base b. For a divisor d, we must determine which kind of divisibility rule exists in this base. The problem defines several categories.
The problem gives us an integer array nums of length n. We must create and return a new array called answer of length n - 1.
In this problem, we are given a lowercase English string s, and we must split it into one or more contiguous substrings such that every substring is balanced. A substring is considered balanced when every distinct character inside it appears the same number of times.
This problem asks us to verify a self-descriptive property of a string of digits. You are given a string num of length n, where each character is a digit between '0' and '9'.
This problem asks us to count how many nodes in a binary tree satisfy a specific condition: the node’s value must equal the average value of all nodes in its subtree. A subtree consists of the current node and every descendant below it.
The problem gives us a three digit integer n. We must determine whether n is a fascinating number. A number is considered fascinating when we concatenate three values together: 1. n 2. 2 n 3.
We are given two valid Gregorian calendar dates in the format yyyy:mm:dd. The task is to compute how many days lie between them.
This problem gives us a weighted, undirected graph. Each intersection is a node, and each road is an edge with a travel time. We start at intersection 0 and want to reach intersection n - 1. The important detail is that we are not simply looking for the shortest distance.
The problem gives us exactly five playing cards. Each card is represented by two pieces of information: - ranks[i] represents the numerical rank of the card, from 1 to 13 - suits[i] represents the suit of the card, using characters from 'a' to 'd' We need to determine the…
The problem requires calculating the sum of all integers from 1 up to a given positive integer n that are divisible by 3, 5, or 7. In simpler terms, we need to consider each number in the range [1, n] and check if it is a multiple of any of these three numbers.
We have a collection of items sold in a supermarket. Every item has a price and a type. Type 1 means the item is a stool, type 2 means it is a pencil. Polycarpus owns exactly k shopping carts, and every cart must contain at least one item.
We are given two sequences of integers, a and b, and Sereja has some initial energy s and a fixed energy cost e for one type of operation. He can perform two operations repeatedly until both sequences are empty.
The problem gives us two starting values, initialEnergy and initialExperience, which represent the player's stats before entering a sequence of competitions. We are also given two arrays, energy and experience, where each index corresponds to an opponent.
The problem gives an array of positive integers and asks for the length of the longest contiguous subarray that is considered "nice". A subarray is nice when every pair of distinct elements inside it has a bitwise AND equal to 0.
The problem asks us to compute the absolute difference between the sum of elements to the left and the sum of elements to the right for each element in a given array nums. In other words, for every index i, we want to know how different the sums on either side of that index are.
The problem gives us an array nums, where nums[i] represents the cost of buying a chocolate currently located at index i. Initially, the chocolate at index i is also considered to be of type i. Since there are n positions, there are exactly n chocolate types.
The problem involves two circular sequences of unique hieroglyphs. We are asked to "cut" these circles at some point to turn them into linear arrays, then find the longest contiguous segment from the first array that appears as a subsequence in the second array.
This problem asks us to maximize the number of tasks that can be completed using a group of workers, where each worker can perform at most one task. Each task has a required strength value, and each worker has a current strength value.
The problem gives us a Transactions table that stores three pieces of information for every transaction: - userid, which identifies the user - spend, which represents the transaction amount - transactiondate, which represents when the transaction occurred The pair (userid…
We have a matrix of integers and three kinds of operations applied to it repeatedly. One operation swaps two rows. Another swaps two columns. The third asks for the value currently visible at a specific row and column.
This problem describes a weighted, undirected graph where each city is a node and each highway is an edge with an associated toll cost. The goal is to travel from city 0 to city n - 1 while minimizing the total travel cost. The special twist is the presence of discounts.
We are given one long string that originally consisted of several file names written back to back with no separators. Every valid file name must look like name.ext. The rules are strict. The part before the dot contains only lowercase letters and has length from 1 to 8.
We are given a single string containing printable ASCII characters. Some of those characters may be digits from '0' to '9'. The task is to remove every digit and print the remaining characters in their original order.
The problem describes a Martian with a row of eyes, each covered by a patch marked with an uppercase letter. The string of letters visible when all eyes are opened represents a sequence s of length n.
The problem gives us a list of inclusive intervals, where each interval is represented as [left, right]. We must divide all intervals into groups such that no two intervals inside the same group intersect. The important detail is that the intervals are inclusive.
This problem gives us two arrays of strings, words1 and words2. Our task is to count how many strings appear exactly once in both arrays. The key detail is that a word only qualifies if: 1. It appears exactly one time in words1 2.
The problem provides a database table named transactions, where each row represents a transaction with three fields: - transactionid, a unique identifier - amount, the transaction amount - transactiondate, the date on which the transaction occurred For every distinct…
The problem gives us a database table named Users. Each row represents a purchase made by a user. The columns include: - userid, the identifier of the user - item, the purchased product - createdat, the purchase timestamp - amount, the purchase value The table may contain…
The problem asks us to find the smallest rectangle that covers all the 1's in a given 2D binary grid. The grid consists of rows and columns where each cell is either 0 or 1.
We have n ice sculptures placed evenly on a circle. Each sculpture has a value, which may be positive or negative. We may remove some sculptures, but the remaining ones must still form a regular polygon.
This problem asks us to determine whether a given message should be classified as spam. We are given two arrays of strings: - message, which contains the words appearing in the message. - bannedWords, which contains words that are considered banned.
This problem asks us to compute the sum of squares of certain elements in an array, specifically the special elements. The array nums is 1-indexed, meaning the first element is at index 1 (not 0).
The problem asks us to transform a string num representing a non-negative integer into a special number by deleting as few digits as possible. A special number is defined as an integer divisible by 25.
We are given a sequence of hills placed along a line. Each hill has a fixed horizontal position and a height, and we imagine it as a vertical segment rising from the ground.
This problem asks us to implement a very small testing utility that mimics the behavior of assertion libraries used in real software development. We need to create a function named expect that accepts any value and returns an object containing two methods: toBe and notToBe.
The problem asks us to calculate the final price for each product in a database, taking into account any category-specific discounts. We are given two tables: Products and Discounts. The Products table contains each product's unique ID, its category, and its original price.
We are given a string made of lowercase letters and a way to assign a numerical weight to each letter. The value of a full string is computed by summing, over all positions, the product of the position index (starting from 1) and the weight of the character at that position.
This problem asks us to implement a custom version of JavaScript’s Function.prototype.call method, called callPolyfill. The purpose is to execute a function with an explicit this context. Normally, in JavaScript, this depends on how a function is called.
We have a set of reading heads positioned on an infinitely long tape of tracks. Each head starts at a distinct track, and it can move left, right, or stay put once per second.
We are given a column of tanks, numbered from 1 to n, each with a message receiving radius. The goal is to transfer n messages from the front of the column to the end under specific rules.
We have a rectangular board where every cell contains one of three symbols. A move selects an active cell and disables cells along diagonals passing through it. The exact diagonals depend on the symbol. A cell marked L attacks the two diagonals with constant i + j.
The problem defines an infinite string-building process that starts with the string "a". At every operation, we take the current string and create a transformed version where every character is replaced by the next character in the alphabet.
The problem gives us two integer arrays, nums1 and nums2, both of the same length n, along with an integer k. We must choose exactly k indices from the arrays.
The problem gives us the root of a binary tree and asks us to replace every node’s value with the sum of all of its cousins’ values.
The problem gives us a 0-indexed array of distinct integers. Among these integers, exactly one value is the smallest element in the array, and exactly one value is the largest element in the array. The task is to remove both of these elements using the fewest deletions possible.
The problem gives us an integer array nums, where each position represents a possible location we can stand on. We always begin at index 0, and we must eventually reach the last index of the array. From any index i, we are allowed to jump to any later index j where j i.
The problem gives us a ladder represented by a strictly increasing array called rungs. Each value in the array represents the height of a rung above the floor. You begin standing on the floor at height 0, and your goal is to reach the final rung.
The problem gives us a numeric string word and an integer m. For every prefix of the string, we must determine whether that prefix represents a number divisible by m. A prefix word[0...i] means the substring starting at index 0 and ending at index i, inclusive.
We are given a set of cities on the plane. Every city has a unique x coordinate and a unique y coordinate. The first k cities are given explicitly, and the remaining cities are generated by two linear recurrences.
The problem gives us a database table named Store. Each row in the table represents a single bill issued to a customer.
The problem gives us a 0-indexed integer array nums, and we must count how many triplets (i, j, k) satisfy two conditions.
The problem asks us to process a string representing a sentence containing words and prices, where prices are words that start with a dollar sign 1e5 or 5$) should remain unchanged.
The problem asks us to find the length of the longest contiguous subarray in a given integer array nums that satisfies three specific conditions. First, the subarray must start with an even number.
For any array, its imbalance number is determined after sorting the elements. Suppose we have a subarray and sort it into sarr. We examine every adjacent pair in the sorted order.
We are given a rectangular table of integers with $n$ rows and $m$ columns. Each cell contains a number that could be positive, negative, or zero. Harry can perform two types of operations: flip the sign of all numbers in a row or flip the sign of all numbers in a column.
The problem asks us to determine the minimum number of operations to convert a given n x n grid into a representation of the letter Y, under specific conditions. The grid contains integers 0, 1, or 2.
The problem gives us two database tables, Keywords and Posts. The Keywords table maps words to topic IDs. A single topic can have multiple words associated with it, and a single word may belong to multiple topics.
We build strings using the Fibonacci recurrence, except concatenation replaces addition. The first two strings are: Every later string is formed by concatenating the previous string with the one before it: The beginning looks like this: For each query string s, we must count…
The problem asks us to find the length of the longest substring in a given string s where the characters appear in consecutive alphabetical order. An alphabetical continuous substring means that every adjacent pair of characters differs by exactly one in the alphabet.
The problem asks us to take a 1D array of integers and convert it into a 2D matrix of specified dimensions, rowsCount and colsCount, following a snail traversal order by columns.
The problem gives us an integer array nums and asks us to count how many contiguous subarrays have an even product. A subarray is a continuous segment of the array. For each possible subarray, we compute the product of all its elements.
We are given an array nums, and we must consider all non-empty subsequences of that array. A subsequence preserves the original order of elements, but elements do not need to be contiguous.
The problem describes a simulation of passengers arriving at a bus station and buses arriving to pick them up. Each bus has a unique ID, an arrival time, and a limited capacity. Each passenger has a unique ID and an arrival time.
Codeforces 409H: A + B Strikes Back
The problem gives us two integer arrays, nums1 and nums2, where some elements may be 0. Every 0 must be replaced with a strictly positive integer, meaning every replacement must be at least 1.
The problem describes an array that continuously goes through a repeating two-phase process. At minute 0, the array is unchanged. Every following minute, the leftmost element is removed until the array becomes empty.
We have a collection of cards. Every card gives two things when played. The first number adds to the score, and the second number adds extra opportunities to play more cards. The game starts with exactly one available move.
We are given a square grid of size $n times n$, where $n$ is an odd integer. Each cell contains a lowercase English letter. The task is to determine whether the pattern of letters forms an “X” shape under a strict rule.
Codeforces 424B: Megacity
Codeforces 402B: Trees in a Row
The problem asks us to analyze three relational tables: Users, Items, and Orders. Each user (seller) has a favorite brand, each item has a brand, and orders record which seller sold which item on which date.
We are given an undirected graph where every edge already has a flow amount attached to it. The graph is supposed to represent a valid maximum flow from vertex 1 to vertex n, but the directions of the edges were lost.
We must construct a small string rewriting program. The program consists of ordered commands. Each command searches for a substring and replaces its first occurrence with another string. Some commands continue execution after replacement, while others terminate immediately.
The problem gives us a list of events, where each event is represented as: Each event occupies an inclusive time interval from startTime to endTime. If we attend that event, we earn value points. We are allowed to attend at most two events, but the chosen events must not overlap.
The problem asks us to identify the most frequent even number in an integer array nums. If multiple even numbers share the highest frequency, we should return the smallest among them. If the array contains no even numbers, the function should return -1.
The problem asks us to count the number of ways to split a given string s of digits into exactly k non-overlapping substrings, where each substring satisfies specific rules: it must start with a prime digit (2, 3, 5, 7), end with a non-prime digit (1, 4, 6, 8, 9), and have…
The problem asks us to compute the sum of all consecutive subarrays in a given integer array nums. A consecutive subarray is defined as one where each element differs from the previous by exactly 1 or -1.
The problem asks us to analyze shift overlaps for employees. We are given a table EmployeeShifts with columns employeeid, starttime, and endtime. Each row represents one work shift for an employee.
We are given a binary string s, meaning the string contains only the characters '0' and '1', along with a positive integer k. The goal is to find a substring of s that is considered beautiful, where a beautiful substring contains exactly k occurrences of '1'.
The problem asks us to determine the minimum number of coins we need to add to an existing list of coins so that every integer from 1 to a given target can be formed as the sum of some subsequence of the coins.
We are given a multiset of digits and asked to construct the largest integer from some or all of them such that the resulting number is divisible by 2, 3, and 5 simultaneously.
We have a rectangular grid and two chips placed on different cells. The first player moves first. On every turn, the first player chooses one chip that is still movable and shifts it by one square in one of the four cardinal directions.
The problem asks us to find the smallest integer that satisfies three specific conditions. Given an integer k and two digits digit1 and digit2, we need an integer that is strictly larger than k, is a multiple of k, and consists only of the two given digits.
This problem asks us to analyze sales data to determine which products each user spent the most money on. We are given two tables: Sales and Product. The Sales table contains individual transactions, showing which user bought which product and in what quantity.
The problem gives us an array nums and an integer k. We begin with a score of 1, and we are allowed to perform at most k operations. In each operation, we choose a subarray that has not been chosen before. From that subarray, we select the element with the highest prime score.
The problem asks us to determine the minimum possible length of a string after repeatedly removing the substrings "AB" or "CD". We are given a string s consisting solely of uppercase English letters.
The problem asks us to calculate the total number of ways to distribute n candies among exactly three children, with the constraint that no child can receive more than limit candies.
We are asked to determine how many generals need warm blankets at a rectangular table placed on an infinite Cartesian plane. The table corners are given by two points with integer coordinates, and each integer point along the perimeter of the rectangle hosts a general.
We have a sequence of trees along a straight street. Each tree has a certain height, and on top of each tree is a nut that Squirrel Liss wants to eat. Liss starts at the base of the first tree.
The problem requires splitting a given array of positive integers into contiguous subarrays such that the greatest common divisor (GCD) of each subarray is strictly greater than 1. The goal is to minimize the number of subarrays after splitting.
We are asked to count how many ways a simple polygon with $n$ vertices can be triangulated. Triangulation here means splitting the polygon into triangles that exactly cover the polygon without overlap and without introducing new points.
We have a hash table with h cells numbered from 0 to h - 1. Every inserted object has a fixed hash value t. If cell t is occupied, we try the next candidate cell by repeatedly adding m modulo h: $$t, (t+m)bmod h, (t+2m)bmod h,dots$$ The first empty cell receives the object.
This problem provides a database table named Tasks, where each row represents a submitted task. Every task contains three fields: - taskid, the unique identifier for the task - assigneeid, the user assigned to the task - submitdate, the date the task was submitted The goal is…
We are asked to simulate a game between two players arranging colored cubes in a line. Petya wants to maximize the number of consecutive cubes of the same color, while Vasya wants to maximize the number of consecutive cubes of different colors.
The problem asks us to calculate the maximum number of points that can be collected in a grid for a series of queries. The grid is represented by an m x n matrix of integers, where each cell has a value.
We are given a stripe consisting of $n$ square cells, each painted in one of $k$ colors. The stripe is represented as a string of uppercase letters, where each letter corresponds to a color. The goal is to repaint some cells so that no two adjacent cells share the same color.
The problem is asking us to implement a simple counter function with a closure-like behavior. Given an integer n, we need to return a function counter() that, when called the first time, returns n, and then increments the returned value by one for every subsequent call.
We are given all possible integer intervals on a line from 1 to n, and we form a subset S by choosing some of these intervals. For any chosen set S, we look at how many intervals we can pick from S such that no two overlap.
The problem defines a special array called the uniqueness array. For every possible subarray of nums, we compute how many distinct values appear inside that subarray. We then collect all of those distinct counts into a single array and sort it in non decreasing order.
This problem requires us to reorder the vowels in a string while keeping all consonants in their original positions.
We are given a rooted tree with n nodes. Node 0 is the root, and the parent-child relationships are described by the parent array. Each node i also has an associated character s[i]. For any node x, a DFS traversal is defined as follows: 1.
The problem asks us to find the maximum length of a special substring that appears at least three times in the given string. A substring is considered special if it consists entirely of a single repeated character.
The problem gives us a lowercase English string s. We must perform exactly one operation: 1. Choose any non-empty substring. 2. Replace every character in that substring with the previous character in the alphabet.
This problem models a bottle exchange process where the exchange rate increases after every successful trade. You start with numBottles full water bottles. Every time you drink a full bottle, it becomes an empty bottle.
The problem gives us an m x n rectangular cake and asks us to cut it into individual 1 x 1 pieces. We are allowed to cut along predefined horizontal and vertical lines. Each line has a fixed cost associated with it.
We repeatedly apply the same operation to two positive integers. At every step, we subtract the smaller value from the larger one. The process stops as soon as one number becomes zero. The task is to count how many subtraction operations are performed for each pair.
This problem is asking us to count all "smooth descent periods" in a stock price array. A smooth descent period is defined as a contiguous sequence of days where the stock price decreases by exactly 1 each day, except the first day which is always counted as a valid period of…
We are given a rooted tree with n nodes, numbered from 1 to n, with node 1 as the root. Each node has an initially empty list of numbers.
The problem describes a two-player turn-based game between Alice and Bob with a row of stones, each having an integer value.
We are given an array of positive integers and multiple queries, each specifying a contiguous subarray. For each query, we are asked to count how many numbers appear in the subarray exactly as many times as their own value.
The problem gives us a square matrix grid of size n x n, where every element is an integer. We need to count how many (row, column) pairs are exactly equal.
The problem gives us two database tables, NewYork and California. Each table contains the exam scores of students from a university. Every row represents a single student, identified by a unique studentid, along with their score. The competition rule is straightforward.
The problem gives us a weighted tree representing servers connected by edges with weights. The servers are numbered from 0 to n-1. Each edge has a weight representing distance or cost.
This problem asks us to construct a complete knight’s tour on an m x n chessboard. A knight’s tour is a sequence of knight moves such that every cell on the board is visited exactly once.
The problem asks us to compute the total amount spent on Fridays in each week of November 2023. We are given a database table named Purchases, where each row represents a purchase made by a user on a specific date, along with the amount spent.
We are given a sequence of integers, and we are allowed to repeatedly swap adjacent elements. Because adjacent swaps can generate any permutation of the array, the real freedom we have is complete reordering of the elements.
The problem gives us an array order that represents the exact sequence in which values are inserted into a binary search tree, usually abbreviated as BST. The array is a permutation of integers from 1 to n, which means every value appears exactly once and there are no duplicates.
We are given a connected weighted undirected graph with no loops or multiple edges. Each edge has a positive weight.
The problem gives us an integer array nums, and asks us to compute the sum of the ranges of every possible non-empty contiguous subarray. The range of a subarray is defined as: For every possible subarray, we calculate its range, then add all of those ranges together.
The problem gives us limit + 1 balls labeled from 0 to limit. Initially, none of the balls have a color assigned. We then process a sequence of queries, where each query is of the form [x, y]. This means ball x should now be painted with color y.
We are given a list of eggs, each of which can be painted by either of two children, A or G. Each child quotes a price for painting each egg, and these prices for a single egg always sum to exactly 1000. Uncle J.
We are given a linear banner split into n fixed positions, and a cursor-like ladder that starts at position k. Each position corresponds to exactly one character of a target string, and we must eventually print that string left to right, one character per position.
The problem gives us an integer array nums and an integer k. We are allowed to repeatedly choose any contiguous subarray of length k and decrease every element inside that subarray by 1.
The problem gives us an integer array nums and asks whether it is possible to transform the array into a strictly increasing sequence by performing a special operation.
We are given a sequence of distances from a single unknown root vertex in an unknown undirected simple graph. Each vertex has a known shortest-path distance to that root, and we are also told that in the original graph every vertex had degree at most k.
In this problem, we are given a string s containing only lowercase English letters and an integer k. We need to find the length of the longest subsequence such that the difference between every pair of adjacent characters in that subsequence is at most k.
We have two arrays, a and b, both of length n. Queries arrive online. One type of query copies a contiguous segment from a into a segment of b. If the query is (x, y, k), then: The second type asks for the current value at a single position in b.
This is a SQL database problem involving two tables: | productid | price | | --- | --- | | Unique product identifier | Unit price of the product | | invoiceid | productid | quantity | | --- | --- | --- | | Invoice identifier | Product purchased | Number of units purchased |…
The problem asks us to transform a one-dimensional array, original, into a two-dimensional array with m rows and n columns.
The problem asks us to find the smallest integer strictly greater than n such that the number is numerically balanced. A number is numerically balanced when every digit that appears in the number appears exactly as many times as its value.
The problem asks us to count pairs of elements in an integer array hours such that the sum of the two elements is a multiple of 24, which we call a "complete day." Each pair (i, j) must satisfy i < j.
Problem Understanding
This problem requires counting the number of distinct subarrays of a given integer array nums that satisfy a specific constraint: each subarray can contain at most k elements divisible by p.
The problem is asking us to compute the minimum number of days required to defeat all monsters in an array power, where power[i] represents the strength of the i-th monster. You start with zero mana and gain mana daily, with the initial daily gain of 1.
Here’s a automatically by
Here is the complete, detailed technical solution guide for LeetCode 2250 - Count Number of Rectangles Containing Each Point in a single, comprehensive response. The problem provides two arrays: rectangles and points.
We are given a one-dimensional street made of n consecutive sections. Each section is either a house that must receive exactly one kilogram of sweets, a shop that can provide at most one kilogram of sweets, or empty space that only matters for movement.
The problem asks us to count submatrices within a given 2D character matrix grid that satisfy three conditions. A submatrix is defined by a contiguous rectangle within the grid, and the submatrix must include the top-left cell grid[0][0].
We have n people sitting at a table, each of whom is either always honest or always lies. Honest people always answer truthfully about how many honest people are at the table. Liars can pick any number from 1 to n except the true number.
We are given a list of strings called words and another string called target. The task is to construct target by concatenating several smaller strings, where each smaller string must be a prefix of at least one word in words. A prefix means the beginning portion of a word.
The problem gives an array obstacles, where each value represents the height of an obstacle. For every position i, we must determine the length of the longest valid obstacle course that ends exactly at index i.
The problem gives us a binary string s that contains only '0' and '1'. We are allowed to swap any two characters in the string, not necessarily adjacent ones. Our goal is to transform the string into an alternating binary string using the minimum number of swaps.
This problem asks us to repeatedly modify an array according to a very specific rule. We are given: - An integer array nums - An integer k, representing how many operations to perform - An integer multiplier For each of the k operations, we must find the smallest value…
The problem asks us to traverse a m x n integer matrix and generate numbers by moving in straight lines along eight possible directions: east, south-east, south, south-west, west, north-west, north, and north-east.
The problem asks us to compute the number of ways to build a wall of given height and width using bricks of specified widths, such that the wall is sturdy.
We are given an array nums where every element is a prime number. For each value nums[i], we must find the smallest non-negative integer ans[i] such that: where | denotes the bitwise OR operation. If no such value exists, we must place -1 in the answer array at that position.
The problem asks us to count all positive integers less than or equal to a given number num such that the sum of their digits is even. In other words, we are asked to evaluate each integer 1 through num, compute the sum of its digits, and check whether that sum is divisible by 2.
We are given a connected weighted undirected graph representing a road network. Junctions are graph vertices, roads are weighted edges, and the restaurant may be placed anywhere, not only at vertices but also at any point along an edge.
We are given an array and several range queries. Each query asks for the sum of all elements between two indices.
The problem gives us an integer array nums and an integer k. We need to count how many contiguous subarrays have a greatest common divisor (GCD) exactly equal to k. A subarray is any non-empty contiguous segment of the array.
The problem asks us to analyze the binary representation of a positive integer n and count how many 1 bits appear at even indices and how many appear at odd indices. Bit positions are counted from right to left, starting at index 0.
We are given an undirected simple graph and two small integers $h$ and $t$. We are asked to determine whether inside this graph there exists a very specific structure consisting of two special vertices connected by an edge.
The problem asks us to find the largest possible even sum of a subsequence of length k from a given integer array nums. A subsequence is a sequence that can be obtained by deleting some elements from the array without changing the relative order of the remaining elements.
The problem gives us an array of integers nums. For every number, we can determine its largest digit. For example: - 2536 has digits 2, 5, 3, 6, so its largest digit is 6 - 112 has digits 1, 1, 2, so its largest digit is 2 - 71 has digits 7, 1, so its largest digit is 7 We…
The problem asks us to transform data in a SQL table named Person. Each row represents a person, with three columns: personid, name, and profession.
We are given a lowercase string and asked to extract a palindromic subsequence from it. A subsequence keeps the original order of characters but may skip positions. The output requirement has a twist.
We are given a set of points on a two-dimensional plane, each with real-valued coordinates. The task is to find the largest possible angle formed at the origin by any three of these points, where the origin serves as the vertex and the rays extend to the selected points.
We have two versions of a mobile application. The paid version costs p rubles and contains no ads. The free version contains c banners. Each user has two limits. The value a[i] is the maximum amount this user is willing to pay for the paid version.
This will be very large if done comprehensively and correctly under your formatting rules, likely exceeding a practical single-message limit and risking truncation or reduced quality.
The problem gives us two integer arrays, nums1 and nums2, along with a positive integer k. We need to count how many index pairs (i, j) satisfy the following condition: In mathematical terms, a pair is considered good if: The task is not asking us to return the pairs themselves.
The problem gives us a 0-indexed integer array nums and an integer k. We need to count how many pairs of indices (i, j) satisfy all of the following conditions: 1. 0 <= i < j < n 2. nums[i] == nums[j] 3.
This problem provides a database table named Spotify that contains information about songs appearing in a Spotify ranking list.
The problem asks us to find the largest magic square inside a given m x n grid. A magic square is a k x k subgrid where the sum of each row, the sum of each column, and the sums of the two main diagonals are all equal.
Here’s a fully detailed technical solution guide for LeetCode 3080 following your requested format. The problem gives a zero-indexed array nums of size n consisting of positive integers and a 2D array queries of size m where each query is [indexi, ki].
The problem gives us a singly linked list where the values are grouped between 0 nodes. The list always starts with 0 and ends with 0, and there are no two consecutive zeros. Every sequence of non-zero nodes between two zeros represents one group.
This problem asks us to maximize the total value of coins collected from multiple piles while following a strict removal rule. Each pile is ordered from top to bottom, meaning we cannot arbitrarily pick any coin in a pile. We may only remove coins from the top, one at a time.
This problem gives us a weighted directed graph with n nodes and a list of directed edges. Each edge has a positive weight. We are also given two source nodes, src1 and src2, along with a destination node dest.
We are given a sorted array of exponents. Instead of the exponents themselves, Ivan writes the corresponding powers of two on paper: $$2^{a1}, 2^{a2}, dots, 2^{an}$$ We may add more numbers, but every added number must also be a power of two.
The game consists of n questions, each with some base value a[i]. Some questions are marked as auctions. Team R2 starts by choosing the first question. After that, whoever answered the previous question correctly gets to choose the next question.
We are given a square image represented as an $n times n$ grid of pixels, where each pixel is either black (1) or white (0). Within this image are a few geometric shapes: circles and squares. Our goal is to count how many circles and squares appear.
We are given a row on size, showing the algorithm handles boundary constraints uniformly.
Please provide the specific problem number/title you want the detailed solution guide for (for example, “LeetCode 1861 - Rotating the Box”), and I will format it exactly according to your requirements.
We are given a single long string made of lowercase letters, digits, and the characters '.', '', and '@'. We need to consider every contiguous substring and decide whether it can be interpreted as a valid email address under a strict format, then count how many such substrings…
We are asked to take a positive integer and express it as a product of prime numbers, showing each prime the number of times it appears in the factorization. For example, the number 245 can be expressed as 5 multiplied by 7 twice, so the output would be 577.
We are given three integers: - n, the length of the array. - m, the maximum value allowed in the array. - k, the exact number of special adjacent positions we want. Every element of the array must be chosen from the range [1, m].
We are given a row of vertical stacks of cubes, where each position holds a certain number of cubes. You can think of this as an array where each index represents a column and the value is its height.
The statement is written as a LOLCODE program. The actual task is to understand what this program computes. The input is a sequence of digits between 0 and 9, one per line. The program repeatedly reads numbers until it encounters 0.
We are given a string s representing Alice’s moves across n rounds of a game. Each character corresponds to one creature: - 'F' = Fire Dragon - 'W' = Water Serpent - 'E' = Earth Golem In every round, Alice and Bob each choose one creature simultaneously.
The problem asks us to find a sequence of indices from word1 such that the characters at those indices, when concatenated in order, form a string that is almost equal to word2.
We start with a line of students, each holding a uniquely numbered ball from 1 to n in order. The system evolves through operations where a teacher selects two students and swaps the balls they currently hold.
The problem asks us to identify all good indices in an array nums based on a window size k. A good index i satisfies two conditions: the k elements immediately before it form a non-increasing sequence, and the k elements immediately after it form a non-decreasing sequence.
We are given an array nums containing positive integers. We need to count how many pairs of indices (i, j) with i < j satisfy a special condition called almost equal.
We start with a string start and want to reach another string end after exactly k operations. One operation chooses a non-empty prefix and a non-empty suffix. If the current word is written as xy, the operation transforms it into yx.
This problem asks us to implement a cancellable delayed function execution mechanism. Essentially, you are given a function fn, an array of arguments args, and a timeout t in milliseconds.
The problem asks us to maximize a running total reward by selecting elements from an integer array rewardValues. Each element in the array represents a reward at that index. Initially, the total reward x is 0, and no indices are marked.
This problem asks us to extend the behavior of strings so that every string supports a replicate(x) method. The method should return a new string where the original string is repeated exactly x times.
The problem asks us to count how many digit strings of length n satisfy a specific positional rule. A digit string is considered "good" when: - Every digit placed at an even index, meaning indices 0, 2, 4, ..., must itself be an even digit.
This problem gives us a Directed Acyclic Graph (DAG) with n nodes labeled from 0 to n - 1, along with a list of directed edges. Each edge [u, v] means there is a one way connection from node u to node v. The goal is to compute, for every node i, all of its ancestors.
We have a rectangular grid where every cell has a cost, the number of flowers destroyed if we pave that cell with concrete. Among all cells, there are up to seven special cells containing important buildings.
This problem asks us to simulate watering a row of plants with a watering can that has a fixed capacity. The plants are arranged along a straight line with indices representing their positions.
This problem gives us a connected undirected graph with n nodes and exactly n edges. A connected graph with n nodes normally forms a tree when it has n - 1 edges. Since this graph has one extra edge, it contains exactly one cycle.
The problem requires calculating the total spending by users on each Friday in November 2023, broken down by week of the month. The input is a Purchases table containing userid, purchasedate, and amountspend.
The problem describes an online chess tournament where a new round begins every 15 minutes. A player can only be credited for a full round if they are present for the entire duration of that round.
We are counting colorings of an $n times m$ grid with black and white cells that form a very specific geometric structure. Most rows are completely white. There is one contiguous block of rows, say from row $l$ to row $r$, where every row contains exactly two black cells.
The problem gives us an array tasks, where each value represents a task type. The tasks must be completed strictly in the given order, which means we cannot rearrange them to optimize the schedule. We are also given an integer space.
This problem asks us to repeatedly combine the two smallest elements of an array nums until all elements are greater than or equal to a threshold value k. The combination operation is not a simple sum, but a formula: min(x, y) 2 + max(x, y).
The problem gives us an integer array nums and an integer k. We must count how many index pairs (i, j) satisfy two conditions: 1. i < j 2.
We have a cup with capacity s and several mugs containing water. Players take turns choosing one non-empty mug and pouring all of its water into the cup. The cup starts empty and water is never removed.
The input is a sequence of integers. The first number tells us how many additional integers follow. If the sequence is: then there are four values: 1, 2, 3, 4. The task itself is intentionally disguised by the strange statement.
We are looking for the number of positive integer triples $(a,b,c)$ such that three smaller cubes of sizes $a^3$, $b^3$, and $c^3$ together are short of exactly $n$ unit cubes when trying to build one large cube of side length $a+b+c$.
This problem asks us to determine whether a given string is a pangram. A pangram is a sentence that contains every lowercase English letter, from 'a' through 'z', at least once. The input is a single string named sentence.
This problem asks us to implement a function cancellable that repeatedly calls a given function fn with a set of arguments args at a fixed interval t milliseconds and allows this repeated execution to be stopped by a cancel function cancelFn.
The problem asks us to design a ticket reservation system for a concert hall. The hall contains n rows, and each row contains exactly m seats. Seats in every row are numbered from left to right starting at 0.
The problem asks us to determine if two events on the same day overlap in time. Each event is given as a pair of strings in HH:MM format representing the start and end times. The output should be a boolean: true if there is any overlap between the two events, and false otherwise.
This problem asks us to design a class named Calculator that supports basic arithmetic operations while enabling method chaining. Method chaining means that after calling one method, we can immediately call another method on the same object.
We are asked to analyze a turn-based game played by two players, Yaroslav and Andrey, each starting with a binary string of length 2·n.
We maintain a hash table with h cells numbered from 0 to h - 1. Every insertion starts from the object's hash value and probes cells by repeatedly adding m modulo h until it finds an empty slot.
We are given the final state of a row of boxes after a very specific operation was applied exactly once in reverse history. Originally, each box contained some number of balls.
The problem asks us to pair elements in an array of even length such that the largest sum among all pairs is minimized. In simpler terms, imagine we have a collection of numbers and we want to form pairs of two numbers each. Each number can only belong to one pair.
The problem asks us to compute the minimum number of adjacent swaps needed to transform a given number num into the kth smallest wonderful integer. A wonderful integer is any permutation of num's digits that is strictly greater than num.
The problem provides a Pandas DataFrame named products with three columns: | Column | Type | | --- | --- | | name | object | | quantity | int | | price | int | The task is to replace all missing values in the quantity column with 0.
The problem models a faulty program that continuously consumes memory over time. There are two memory sticks, represented by the integers memory1 and memory2. At each second, the program allocates an increasing number of bits: - At second 1, it allocates 1 bit.
The problem asks us to transform a given string into a "fancy string". A fancy string is defined as a string that does not contain three consecutive identical characters anywhere in the string.
This problem asks us to count the number of unique pairs of indices (i, j) in a given integer array nums such that the sum of the two numbers at those indices is strictly less than a given target.
We are given a rectangular grid where each cell has a height. A biathlon track must be the boundary of a sub-rectangle, and athletes run clockwise along this boundary.
We need to fill a 2 x 2 grid with four distinct digits from 1 to 9. The grid looks like this: $$begin{matrix} a & b c & d end{matrix}$$ The input gives us six sums.
Each day Mashmokh receives a pile of tokens, and at the end of that day he can exchange some of them for money using a fixed conversion rule.
This problem describes a group of employees sitting around a circular table. Every employee has exactly one favorite coworker, represented by the array favorite, where favorite[i] is the person employee i wants to sit next to.
We are given a string that represents a partially specified code. Each position in the string contributes constraints on what digit can appear there.
The problem asks us to identify users who demonstrate "bursty behavior" in their posting patterns during February 2024.
The problem requires us to identify the last time each bike was used based on a table of rides. Each row of the table corresponds to a unique ride and contains the rideid, bikenumber, starttime, and endtime.
We are given an array of integers and a number k. The task is to find a contiguous subarray, or segment, such that it contains exactly k distinct integers.
The problem asks us to repeatedly remove characters from a string based on a specific rule. Specifically, for any character s[i] in the string, we can remove the nearest occurrence of the same character to the left of i and the nearest occurrence to the right of i.
We are given a directed structure on n objects representing a ski resort. Each object is either a mountain or a hotel. Every object has at most one outgoing ski track leading to another object, and a hotel never has outgoing tracks at all.
The problem gives us an integer array nums and an integer k. We need to count how many pairs of indices (i, j) satisfy two conditions: 1. i < j 2.
This problem asks us to find the longest contiguous subarray that follows a very specific alternating pattern. Given an integer array nums, we need to identify subarrays where: 1. The subarray length is at least 2. 2.
The problem presents a 0-indexed array nums of non-negative powers of 2 and an integer target. Each element in nums is guaranteed to be a power of two, which is important because it allows us to reason about sums and splits in terms of binary representation.
The problem asks us to count all palindromic subsequences of length 5 within a given string of digits s. A palindromic subsequence is a sequence that reads the same forward and backward, and a subsequence can be formed by deleting zero or more characters without changing the…
We are given a very large range of possible values for a parameter m, and for each such value a deterministic procedure produces a matrix filled using bitwise XOR rules. The matrix has size (m + 1) by (m + 1), and the filling follows a fixed recursive or constructive pattern.
We are given a sequence of hours on a train journey, each with an associated light level between 0 and 100. Vasya wants to read for exactly k of these hours.
We are given an array of n integers. From this array, we form every ordered pair (a[i], a[j]), including pairs where i = j. Since both positions are chosen independently, there are exactly n² pairs. All these pairs are sorted lexicographically.
We are given a ticket represented as a string of 2n digits, where n is between 1 and 100. The first half of the ticket contains the first n digits and the second half contains the remaining n digits. We are asked to determine whether the ticket is “definitely unlucky.
The problem gives us a 0-indexed integer array nums of length n and asks us to compute the maximum sum score among all indices of the array.
The problem asks us to compute the total "power" across every possible non-empty subset of heroes. For any selected group of heroes, its power is defined as: We are given an array nums, where each value represents the strength of one hero.
We are working with a set of cities and we are allowed to build undirected roads between some pairs of them, except for a set of forbidden pairs that are explicitly given.
The problem asks us to count how many strings of length n made of lowercase English letters can be rearranged to contain the substring "leet". In other words, a string is "good" if, after any permutation of its characters, "leet" appears as a contiguous sequence.
The problem asks us to count the number of contiguous subarrays in an integer array nums that consist entirely of zeros. A subarray is defined as any consecutive sequence of elements from the original array.
The problem gives us a string s containing lowercase English letters, vertical bars '|', and asterisks ''. The key rule is that every two consecutive vertical bars form a pair. Any asterisk located between the two bars of a pair must be ignored when counting.
This problem is about simulating the motion of pistons in a car engine and computing the maximum total area under the pistons over time. Each piston moves either up or down by 1 unit every second, reversing direction when it hits the top (height) or bottom (0).
We are given a game split into n parts. Each part must be completed on a specific computer, and some parts depend on others, forming a dependency graph without cycles. Rubik can start at any computer and spends exactly one hour to complete a part.
The problem asks us to determine whether a path exists between two nodes in an undirected graph. The graph is defined by n vertices labeled from 0 to n - 1 and a list of edges, where each edge connects two distinct vertices.
The problem asks us to find the largest square that can fit inside the overlapping region of at least two rectangles on a 2D plane. Each rectangle is axis-aligned, meaning its sides are parallel to the x-axis and y-axis.
This problem asks us to format a sentence into multiple rows such that each row has length at most k. The sentence consists of words separated by spaces, and words cannot be split across rows. We may only insert line breaks between words.
We are given a table with n rows and m columns. We can place a single point in any cell, and the goal is to fill the table so that every contiguous square subtable of size n × n contains exactly k points.
The problem asks us to find the largest positive integer k in a given array nums such that its negative counterpart -k also exists in the array. In other words, for each positive integer in the array, we need to check whether its negative exists.
We have a directed graph of debts. An edge a - b means employee a owes money to employee b. We must arrange all employees in a sequence such that for every pair of consecutive employees (x, y) in the sequence, there is no edge x - y.
The problem asks us to take an array of integers, nums, and repeatedly form pairs of equal numbers until no more pairs can be formed. Each operation removes exactly two identical numbers from the array, and we continue doing this until it is no longer possible.
The problem presents a rooted tree with n nodes numbered from 0 to n-1, where node 0 is the root. Each node has a gate that can either cost money to open (negative value) or give a reward (positive value). Alice starts at the root, and Bob starts at a specified node.
This problem asks us to simulate a task scheduling system with two priority rules. We are given a list of servers and a list of tasks. Each server has a weight, and each task has a processing duration. At second j, task j becomes available and enters a queue.
The problem gives us an m x n matrix called grid, where every cell contains a positive integer. From any cell (r1, c1), we are allowed to move to another cell (r2, c2) as long as the destination is either strictly below or strictly to the right.
The problem asks us to construct the lexicographically smallest number from a string pattern consisting of 'I' and 'D'.
We are given a connected weighted undirected graph representing cities and roads. City s is the capital. A missile silo may be located either exactly on a city or at some interior point of a road. A position is valid if its shortest-path distance to the capital is exactly l.
That prompt contains two different LeetCode problems merged together, but the second problem, LeetCode 2461 - Maximum Sum of Distinct Subarrays With Length K, appears to be the target because the detailed sections and code stubs at the end correspond to it.
In this problem, we are given a 0-indexed integer array nums, and for every index i, we must compute something called the "average difference".
We are given a vertical stack of levels, each level containing three colored blocks arranged left, middle, right.
This problem gives us a finite array nums, but asks us to imagine an infinite array called infinitenums created by repeating nums forever. For example: We must find the shortest contiguous subarray in this infinite sequence whose sum is exactly equal to target.
We are asked to schedule a day of phone calls for Mr. Jackson in a way that maximizes the longest uninterrupted segment of sleep. Each call has a specific start time and duration.
The problem asks us to repeatedly replace a number n with the sum of its prime factors until it reaches the smallest value it can take.
We are given a finite maze of size n × m. Some cells are walls, some are open, and one cell contains the starting position S. The maze is not used only once.
We are given a directed graph of shops and streets. Every directed edge has an attached sequence of shop numbers called a vision. Whenever Neo traverses that edge, he sees exactly that sequence.
We are asked to count the number of valid sequences of nucleotides of length n for a Martian DNA strand, given a total of m nucleotides and a list of forbidden consecutive pairs. Each nucleotide is represented by a letter from 'a' to 'z' and 'A' to 'Z' depending on its index.
You included two different LeetCode problems with conflicting templates. The second problem appears to be the one you want covered: LeetCode 3051 - Find Candidates for Data Scientist Position (Database) However, the required sections still reference Python and Go solutions…
We are asked to work with a contiguous range of integers from l to r, and for every possible subset of size k within this range, we consider the Fibonacci numbers at positions given by the subset elements.
=== 1996-N5 === Origin: ROM Let denote the set of nonnegative integers. Find a bijective function from into such that for all , We first observe that the given functional equation is equivalent to This gives us the idea of introducing a function defined as By the above…
The problem describes a line of students, each holding a uniquely numbered ball. Each student can participate in at most one or two throws, and in each throw, two students exchange the balls they are holding.
We are given a set of levels in a game, each with two possible completion times. Completing a level for one star takes a[i] time, and completing it for two stars takes b[i] time, with a[i] < b[i].
We are given two lowercase strings, word1 and word2. A substring of word1 is considered valid if its characters can be rearranged so that word2 becomes a prefix of the rearranged string. To understand what this means, suppose word2 = "abc".
We are given a set of soda bottles, each labeled with a brand and a "can open" brand. Each bottle can be used to open other bottles of the brand it can open. A bottle can be used to open itself or others.
We start with a complete undirected graph on n vertices. Every pair of vertices has exactly one edge between them. Alice keeps m of those edges, and Bob receives all remaining edges.
We have a group of boys, each with a skill level in football. The task is to split them into two teams such that the teams are nearly equal in size and the total skill levels of the teams are roughly balanced.
The problem asks us to determine the year with the highest population given a list of birth and death years. Each element in the input logs is a pair [birthi, deathi] representing the inclusive start of life at birthi and exclusive end at deathi.
We are given a permutation of numbers from 1 to n, and we repeatedly modify it by swapping positions. On top of that, we are asked queries about intervals of values, not positions.
The problem gives us an integer array nums. In one operation, we are allowed to either increase or decrease any single element by exactly 1. Our goal is to make every number in the array divisible by 3, while using the smallest possible number of operations.
The problem gives us a string s that was formed by concatenating several strings together, where every piece is an anagram of the same unknown string t. Our task is to determine the minimum possible length of t.
We are given a permutation of integers from 1 to n and a list of queries asking for the positions of specific elements. The task is to compare two linear search strategies.
The problem asks us to compute the total number of vowels that appear across every possible substring of a given string. A substring is any contiguous sequence of characters. For a string of length n, there are n (n + 1) / 2 total substrings.
The problem asks us to find the largest-valued odd integer substring from a given string num that represents a large integer. Here, a substring is any contiguous sequence of characters in num.
We have a line of people waiting at a bus stop, but they are organized into groups. Each group has a fixed number of people and stands consecutively in the queue. A bus comes that can carry at most m people, and the people enter in the order of their groups.
The problem asks us to distribute a set of cookie bags among k children such that the unfairness of the distribution is minimized. Here, unfairness is defined as the maximum total number of cookies any single child receives.
We maintain an array under two operations. The first operation assigns a constant value to an entire subarray. If we receive = l r x, every position from l through r becomes x. The second operation asks for a weighted sum over a subarray. For a query ?
The problem gives us a permutation nums of length n. A permutation means the array contains every integer from 1 to n exactly once. A permutation is considered semi-ordered when: - The first element is 1 - The last element is n We are allowed to repeatedly swap adjacent elements.
The problem asks us to count all strictly increasing subarrays in a given array nums of positive integers. A strictly increasing subarray is a contiguous sequence of numbers where each element is strictly larger than the previous one.
We are given a fixed 5 by 5 grid that contains mostly zeros and exactly one cell containing a one. In one move, we are allowed to swap adjacent rows or swap adjacent columns. Each such swap moves the entire row or column by exactly one position.
Each input line represents a sentence from a chat log, and the task is to classify who likely said it based on simple pattern clues at the beginning and end of the string. A sentence is attributed to Freda if it ends with the substring "lala.".
We have a line of soldiers, each with a height. The general only cares about two positions in the lineup. The tallest soldier must stand at the very front, and the shortest soldier must stand at the very end. The order of everyone else is irrelevant.
The problem presents a 0-indexed 2D array items of length n, where each element represents an item with two values: a profit and a category. You are asked to select exactly k items to form a subsequence. A subsequence preserves the original order of items but can skip elements.
You are given two arrays, nums1 and nums2, both of the same length n. For every index i, you may optionally swap the pair (nums1[i], nums2[i]). Each swap counts as one operation.
We are given an array of distinct positive integers. For every subarray, we look at its largest value and its second largest value, then compute their XOR. Among all possible subarrays, we need the maximum such XOR value.
The problem asks us to analyze a sentence represented as a string s and determine whether all numbers embedded in the sentence are strictly increasing from left to right.
The problem asks whether there is a clear, unobstructed path from the bottom-left corner (0, 0) to the top-right corner (xCorner, yCorner) of a rectangle, such that the path does not touch or go inside any given circles.
The problem is asking to select the maximum number of integers from the range [1, n] under several constraints. First, integers in the banned array cannot be chosen. Second, each integer can be chosen at most once. Third, the sum of all chosen integers must not exceed maxSum.
We are given an integer array nums and a list of interval queries. Each query [l, r] allows us to reduce every element inside that range by at most 1. The important detail is that the decrement is optional and independent for every index in the range.
We are asked to determine whether a stone thrown onto a hopscotch court lands strictly inside a square, and if so, to identify which square. The court is constructed from squares of side length a, arranged in rows with a repeating 1-1-2-1-2 pattern.
The game consists of two flower lanes containing x and y flowers respectively. Alice moves first, and on every turn a player removes exactly one flower from either lane. A very important observation is that the players do not have any meaningful strategic choice.
We are asked to calculate probabilities for Anton picking up chocolates from a moving conveyor belt. The belt has a straight visible part of length l and loops back under the floor, making the total belt length 2l.
This problem asks us to maximize the total number of fruits collected by three children traversing a square grid of rooms, each with a certain number of fruits. The grid has dimensions n x n, and the fruits are given in a 2D array fruits[i][j].
The problem asks us to determine the minimal set of initial triangles to color red in an equilateral triangle of side length n, such that by repeatedly applying a propagation rule, all triangles eventually become red.
The problem requires determining whether there exists a path from the top-left corner (0, 0) to the bottom-right corner (m - 1, n - 1) in a binary m x n matrix grid, while maintaining a health value greater than or equal to 1.
The problem gives us two integers, num1 and num2. In one operation, we may choose any integer i between 0 and 60, inclusive, and subtract the value 2^i + num2 from num1. The goal is to determine the minimum number of operations required to make num1 become exactly 0.
The problem gives us a string s and a character c. We must count how many substrings of s both start and end with the character c. A substring is any contiguous portion of the string. For every possible substring, we check two conditions: 1. The first character must equal c 2.
The problem asks us to design a data structure that continuously processes a stream of integers and determines whether the most recent k integers are all equal to a specific target value.
The problem presents a collection of binary strings of a fixed length, but rather than giving the strings explicitly, each string is described compactly: the first $fi$ bits are the same, $si$, and the remaining $n-fi$ bits are the opposite of $si$.
The problem asks us to identify all valid chemical bonds that can form between elements in a given table. The Elements table contains three columns: symbol, type, and electrons. The type column is an enumeration of 'Metal', 'Nonmetal', and 'Noble'.
The problem requires generating the lexicographically smallest beautiful string that is strictly larger than a given beautiful string s. A string is beautiful if it uses only the first k letters of the English alphabet and contains no palindromic substring of length 2 or more.
The problem asks us to transform a given array of positive integers nums into its lexicographically smallest form by performing a specific swap operation any number of times.
We are given a single string representing a command line in a simplified Pindows operating system. Each "lexeme" in this command line is either a contiguous sequence of non-space characters or a sequence of characters enclosed in double quotes.
We start with a number q written on paper. On each turn, a player must replace the current number with one of its non-trivial divisors, meaning a divisor strictly between 1 and the number itself. If a player cannot make a move, that player wins immediately.
We are given a weighted tree. Every node represents a village, and every edge represents a road with a length. Villages can appear and disappear over time through online queries. At any moment, only a subset of villages currently exists.
The problem asks us to count how many non-empty substrings of a given string are considered "wonderful". A substring is wonderful if at most one character appears an odd number of times inside that substring.
The problem asks us to make three strings s1, s2, and s3 identical by repeatedly deleting their rightmost characters. Each deletion is counted as one operation. The key restriction is that we cannot completely delete a string, so at least one character must remain in each string.
We are given a chronological log of a single night in a club. Every character in the input string represents one event: a “+” means someone entered the club, and a “-” means someone left.
The problem asks us to find the smallest positive sum among all subarrays whose lengths fall within a given range [l, r]. A subarray is a contiguous section of the array. We are allowed to choose any non-empty contiguous segment as long as: 1.
We are asked to determine the winner of a two-player game involving numbers from 1 to n. The players, Vasya and Petya, take turns picking numbers. Once a number x is chosen, neither x nor any of its higher integer powers (x², x³, …) can be chosen in future turns.
This problem asks us to construct the lexicographically largest possible numeric string after optionally applying a single transformation operation on one contiguous substring of the input string num.
We are given a row of matchboxes, each containing some number of matches. The total number of matches is divisible by the number of boxes, so there exists a target configuration where every box ends up holding exactly the same number.
We are given two binary strings, s1 and s2, of the same length n, along with a positive integer x. Our goal is to transform s1 into s2 using the minimum possible cost. We are allowed to perform two kinds of operations: 1. Choose any two positions i and j, then flip both bits.
The problem gives us an array nums of positive integers and asks us to count how many contiguous non-empty subarrays can be removed so that the remaining elements form a strictly increasing array.
A ball is thrown from the origin along some chosen direction, and it repeatedly appears at equally spaced points along that ray: first at distance $d$, then at $2d$, then $3d$, and so on. The direction is fixed once chosen, but it can be any real direction in the plane.
We are given an array of positive integers. A pair (x, y) is called cool if x can be written as the sum of y consecutive integers, where the sequence may contain negative numbers and zero. The sequence itself is cool when every adjacent pair satisfies this condition.
The problem asks us to count how many substrings of a given string are considered "complete". A substring is complete if it satisfies two conditions simultaneously: 1. Every distinct character in the substring appears exactly k times. 2.
We are moving on a one-dimensional track from position 0 to position L. Walking is simple, one meter costs exactly one second, and we may move in either direction as long as we never go below 0. A ramp gives a shortcut, but using it has a strict structure.
We are asked to compute the sum of two integers, but the emphasis is on careful output formatting. Specifically, the input consists of two numbers, each on its own line, and we need to print their sum without any leading zeros.
The problem asks us to compute the number of distinct integers in an array after performing a specific operation on each element: reversing its digits and adding it to the array. In other words, for every integer in nums, we generate its reverse and append it.
We are given an array of integers and a number k. We need to count how many contiguous subarrays contain some value at least k times. The condition is not about having k distinct equal pairs or anything global across the subarray.
Vasya has a set of electrical devices and a limited number of wall sockets in his flat. Additionally, he owns multiple supply-line filters, each of which provides extra sockets but occupies one itself when plugged in.
We have several stone piles, each with some initial size. One operation chooses a source pile and merges it into another pile. The source pile disappears, the destination pile grows, and the operation cost equals the current size of the source pile.
We are given the final standings of a programming contest. The scores are already sorted in non-increasing order, meaning each participant has a score greater than or equal to the next participant. A participant advances to the next round if two conditions are true.
Here is a complete technical solution guide for LeetCode 3015 - Count the Number of Houses at a Certain Distance I, formatted according to your specifications. The problem describes a simple linear city with n houses numbered from 1 to n.
The problem describes a process of coloring cells on an infinite two-dimensional grid over n minutes. At minute 1, we start by coloring exactly one arbitrary cell blue.
The problem gives us an integer n and an array sums of length 2^n, which contains every possible subset sum of an unknown array of length n.
The problem asks us to implement a counter object that maintains a mutable integer value, initialized to a given number init. This counter object must provide three operations: increment, decrement, and reset.
This problem asks us to design a lightweight in memory database system that supports four core operations on multiple tables: 1. Insert a row into a table 2. Remove a row by id 3. Select a single cell value 4.
We are given a square table that was originally generated from a hidden array of non-negative integers. Each off-diagonal entry is the bitwise AND of the corresponding pair of hidden values, while diagonal entries are artificially replaced by -1.
Certainly. Here is a complete, detailed technical solution guide for LeetCode 2254 - Design Video Sharing Platform, following your requested formatting and style. The problem asks us to design a class to simulate a video sharing platform.
The problem gives us a string s consisting only of digits from 0 to 9. We need to find the length of the longest substring that is considered semi-repetitive. A substring is semi-repetitive if it contains at most one pair of equal adjacent digits.
We are given a rooted tree with n vertices, where each vertex explicitly lists its children in a defined order. The tree allows three types of queries. The first type asks for the distance between two nodes in terms of the number of edges along the shortest path.
The problem asks us to count how many integers in the inclusive range [low, high] are stepping numbers. A stepping number is defined as a number where every pair of adjacent digits differs by exactly 1.
This problem asks us to find the length of the longest subsequence in an array nums such that for every consecutive pair of elements in the subsequence, the sum of the pair modulo k is the same. In other words, if we denote a valid subsequence as sub = [a1, a2, ...
We are asked to construct a walk on a directed functional graph defined by a very specific transition rule. Each state is a node from 0 to n − 1. From a current node i, the next state is not arbitrary: it must be either 2i mod n or (2i + 1) mod n.
The problem asks us to determine whether there exist two contiguous subarrays of length 2 in a given array nums that have the same sum. A subarray of length 2 is simply any consecutive pair of elements in the array.
The problem requires calculating the number of distinct averages generated from a sequence of numbers using a specific process. The input is an integer array nums of even length.
The problem asks us to evaluate every contiguous subarray of length k in the given array nums. For each subarray, we must determine whether it satisfies two conditions simultaneously: 1. The elements are sorted in strictly ascending order. 2.
The problem asks us to count all substrings of a given string s such that at least one character in the substring appears at least k times. In simpler terms, for every substring we consider, we check whether there is any character that has frequency greater than or equal to k.
We are given a string word consisting of lowercase English letters. Two adjacent characters are considered almost-equal if either: - They are exactly the same, such as 'a' and 'a' - Their positions in the alphabet differ by exactly one, such as 'a' and 'b', 'c' and 'b', 'x'…
The problem presents an array nums of length n that is a permutation of integers from 0 to n - 1. We are asked to find a permutation perm of the same range [0, 1, 2, ...
You are given a tree with n nodes. Every node has a value stored in the array nums, and every edge connects two nodes in an undirected manner.
The problem asks whether it is possible to create a specific target triplet [x, y, z] as an element of a given list of triplets by repeatedly merging pairs of triplets using a component-wise maximum operation.
We are working on an enormous infinite grid of unit square cells, and we can think of each cell as a “bed” in a potato field. We start in the center cell and walk according to a sequence of axis-aligned moves.
We are given a rooted tree where each vertex represents a water tank. Initially all tanks are empty, but we are allowed to place up to $k$ liters of water, each liter placed into a distinct non-root node.
The problem asks us to find the minimum cost to travel from city 0 to city n-1 within a given time limit, maxTime, where the cost is defined by passing fees associated with each city visited. The country has n cities connected by bi-directional roads with varying travel times.
We are given a circular arrangement of cows, each cow facing exactly one of its two neighbors. Represent the circle as a binary string where each character describes direction: one symbol means “point to clockwise neighbor” and the other means “point to counterclockwise…
We are given a directed graph representing a city with junctions and streets. Urpal starts at junction a and wants to reach junction b using buses.
This problem asks us to model the spread of an infection across a binary tree. Each node in the tree has a unique integer value, and the infection starts from a specific node, identified by start.
The problem gives us a string s containing only lowercase English letters. We are allowed to perform operations where we change any single character into any other lowercase character.
The problem asks us to examine every contiguous subarray of length k in the input array nums and determine its "power". A subarray has valid power only if two conditions are simultaneously true: 1. The elements are sorted in strictly ascending order. 2.
The problem gives us a final string s and describes a process where the string is built by repeatedly prepending characters to the front. This means that every intermediate string si is actually a suffix of the final string s.
We are given a single positive integer and must print its representation in base 2. In other words, instead of writing the number using decimal digits from 0 to 9, we must write it using only binary digits, 0 and 1.
This problem asks us to repeatedly transform a binary string according to a very specific rule. During each second, every occurrence of the substring "01" is replaced simultaneously with "10". The important detail is that all replacements happen at the same time.
We are given an undirected connected graph describing a subway system. Every edge is a tunnel, every vertex is a station. The graph is guaranteed to be a vertex cactus, meaning each vertex belongs to at most one simple cycle. The subway is composed of lines of two possible types.
We are given a sequence of distinct integers. We may keep any subsequence, preserving the original order, and remove the rest. The remaining sequence must satisfy two conditions simultaneously. First, the xor of all remaining numbers must be zero.
The problem gives us a date string in the format yyyy-mm-dd. The year always has four digits, while the month and day always have two digits because of zero padding. We need to convert each of the three components, year, month, and day, into their binary representations.
The problem requires us to find the number of arithmetic triplets in a strictly increasing array of integers. An arithmetic triplet (i, j, k) satisfies the conditions i < j < k, nums[j] - nums[i] == diff, and nums[k] - nums[j] == diff.
The problem is asking us to determine the optimal way for Bob to distribute his numArrows across 12 scoring sections (from 0 to 11) in an archery competition in order to maximize his total points against Alice. Alice's arrow distribution is given in aliceArrows.
We are given a single positive integer and must print its representation in base 2. In other words, we want to express the number as powers of two and output the corresponding sequence of bits.
We are asked to count the numbers in a given interval [l, r] where the first digit equals the last digit. The input consists of two integers l and r, defining the interval. The output is a single integer, the count of numbers satisfying this property.
We are given an $n times m$ grid where each cell contains an integer representing a “shoe pair ID”. Every ID appears exactly twice in the grid, so each number defines exactly two positions.
In this problem, we are given several gardens, where flowers[i] represents how many flowers are already planted in the i-th garden. We are also given a limited number of additional flowers, newFlowers, that we may distribute among the gardens.
I can do that, but the guide will be very long for a single chat response because your required format includes detailed prose, brute force and optimal approaches, proof sketch, Python and Go implementations, worked traces for every example, complexity analysis, comprehensive…
We start with exactly one upward-pointing triangle. Every year, each triangle splits into four smaller triangles. Three of them keep the same orientation as the parent, while one flips direction. The input gives the number of years n.
The problem gives us the head of a singly linked list and asks us to delete the middle node. The definition of the middle node is based on 0-based indexing. If the list has n nodes, then the middle node is the node at index ⌊n / 2⌋.
We are given a list of 2D points, where each point is represented as [x, y], and an integer k. The distance between two points is not the usual Euclidean or Manhattan distance. Instead, it is defined as: where ⊕ denotes the bitwise XOR operation.
This problem is not a traditional algorithmic task. Instead of processing arrays or graphs, we are given a text document and must determine which of three categories it belongs to. A training dataset is provided externally, containing labeled examples for all three subjects.
We are given a rooted tree with n nodes, each node labeled with a 0 or 1. We are also given a target configuration of 0s and 1s for each node. The only operation allowed is to "pick" a node, which flips its value and every second-level descendant down the tree.
The problem asks us to reverse the values of nodes at odd levels of a perfect binary tree. A perfect binary tree is a tree in which every parent node has exactly two children and all leaves appear at the same depth.
The problem presents an m x n matrix grid of positive integers and asks us to simulate a repeated deletion process on it. In each operation, for every row in the matrix, we remove the greatest element.
We are given two integer arrays, one belonging to Devu and the other to his brother. We are allowed to repeatedly increase or decrease any single element of either array by 1 in one operation.
The problem asks us to determine whether a given sentence is circular. A sentence is defined as a string of words separated by a single space, with no leading or trailing spaces. Words contain only uppercase and lowercase English letters.
We need to construct a convex polygon with exactly n vertices such that every interior angle is equal, but every side length is different. A polygon where all angles are equal is called equiangular. For a convex equiangular polygon, the direction of each edge is fixed.
The problem asks us to construct a binary tree from a list of relationships, where each relationship is given as [parent, child, isLeft].
You are given two integer arrays, nums1 and nums2, and both arrays are sorted in non-increasing order. That means the values either stay the same or decrease as we move from left to right. A pair of indices (i, j) is considered valid if two conditions are satisfied: 1. i <= j 2.
The problem asks us to compute a Premier League-style ranking table from a TeamStats table, which contains each team's matches played, wins, draws, and losses. We are required to calculate three additional columns for each team: points, position, and tier.
The problem asks us to determine the top three wineries in each country based on the total points accumulated by each winery. We are given a table called Wineries that contains columns id, country, points, and winery.
We are given a multiset of cards, each card showing either the digit 0 or the digit 5. From these cards we may choose any subset and arrange the chosen digits into a number written in a single line. Our goal is to build the largest possible number that is divisible by 90.
The problem requires calculating a Premier League-style ranking for teams in each season based on their match results.
The problem gives us two empty arrays, arr1 and arr2, and asks us to fill them with positive integers under several constraints. The first array, arr1, must contain exactly uniqueCnt1 distinct positive integers, and none of those integers can be divisible by divisor1.
The problem asks us to calculate the total number of valid move combinations for a small set of chess pieces (up to four) on an 8 x 8 chessboard. Each piece-rook, bishop, or queen-can move according to standard chess rules, but only along paths defined by the piece's movement.
The problem describes a row of stones, each colored red, green, or blue, represented by a string s of characters 'R', 'G', and 'B'. Squirrel Liss starts on the first stone and executes a sequence of color-based instructions, given by a second string t.
The problem asks us to count the number of valid ways to sequentially build all rooms in an ant colony, given a dependency tree that dictates which rooms must be built before others. Each room i has a predecessor prevRoom[i] that must be built before i.
We are asked to simulate a peculiar kind of calendar. Vasya has a clock that shows days from 1 to d. Each month has a certain number of days, and the clock does not know which month it is.
The problem is asking us to select exactly k disjoint subarrays from a given array nums such that the last element of each subarray comes before the first element of the next subarray.
This problem gives us an array words containing distinct strings. Every string has length exactly 2, and all strings consist of lowercase English letters.
The problem asks us to determine the minimum number of characters that need to be appended to the end of string s so that string t becomes a subsequence of s. A subsequence is a sequence that appears in the same order as in another string, but not necessarily consecutively.
We have a sequence of skyscrapers, each with a height chosen randomly and independently according to a geometric distribution where floor i exists with probability 2⁻ⁱ. The number of skyscrapers is unknown, but uniformly distributed between 2 and 314!.
The problem gives us three inputs: - An integer array nums - An integer array queries - An integer x We need to answer each query independently. A query asks for the index of the kth occurrence of the value x inside the array nums.
We are given 2 n real numbers representing Jeff's birthday gifts. Jeff dislikes fractional numbers, so he performs n pairwise operations to "adjust" the numbers.
The problem asks us to find a subsequence of length k from a given integer array nums such that the sum of the elements in the subsequence is maximized.
The problem gives us an array nums and an integer target. We must find a subsequence whose elements add up exactly to target, while maximizing the number of elements included in that subsequence.
The problem gives us a collection of 2D points, where each point is represented as (xi, yi). We must cover every point using a set of rectangles. Each rectangle has a very special form: - Its bottom edge always lies on the x-axis, meaning the rectangle starts at y = 0.
The problem gives us an integer array nums and two integers, low and high. We need to count how many index pairs (i, j) satisfy two conditions: - i < j - low <= nums[i] XOR nums[j] <= high The XOR operation compares bits between two numbers.
We are given a simple polygon whose vertices are listed in counterclockwise order. A spider starts at vertex s and wants to reach vertex t. The spider has two kinds of movement. The first move is walking along the polygon boundary.
In this problem, we need to count how many binary arrays can be formed using exactly zero copies of 0 and exactly one copies of 1, while satisfying an additional stability condition.
The problem describes a lunch scenario where three Rabbits have a fixed break of k time units and a list of n restaurants. Each restaurant is defined by two numbers: the joy fᵢ the Rabbits gain if they finish on time, and the time tᵢ it takes to eat there.
You are given a binary string s whose length is guaranteed to be even. A binary string contains only the characters '0' and '1'. A string is considered beautiful if it can be divided into one or more contiguous substrings such that: - Every substring has even length.
The problem describes a two-player game between Alice and Bob played on a string s. Alice always moves first. The rules for each player are based on vowel counts in substrings: Alice can remove any non-empty substring containing an odd number of vowels, while Bob can remove…
We have several wires stacked vertically. Each wire contains some birds sitting in a row from left to right. Every shot targets exactly one bird on one wire. When a bird on wire x is shot at position y, three things happen immediately: 1. The bird itself disappears. 2.
The problem asks us to determine the number of valid times that can be formed from a partially known digital clock string of the format "hh:mm", where unknown digits are represented by '?'. Each '?
The problem asks us to take a four-digit integer num and split its digits into two new integers such that the sum of these two integers is minimized. We are allowed to use all digits exactly once, and the new integers can have leading zeros.
This problem asks us to determine the minimum number of insertions needed to transform a given string word into a valid string. A valid string is defined as one that can be formed by concatenating the sequence "abc" one or more times.
We are given an array of integers and must count how many subsequences of length exactly k satisfy a special restriction on lucky numbers. A number is lucky if every decimal digit is either 4 or 7. Examples are 4, 47, and 744. Numbers like 5, 17, and 467 are not lucky.
We are given an 8×8 grid representing a chessboard, but the colors of the squares may not be correct. Each square is either black, represented by "B", or white, represented by "W".
The problem gives us an integer array nums, and we need to return the number whose value is closest to 0. The phrase "closest to zero" means we compare numbers using their absolute values. For example: - |-2| = 2 - |5| = 5 Since 2 < 5, the number -2 is closer to zero than 5.
This problem asks us to maximize the total tastiness of fruits we purchase while staying within a fixed budget. Each fruit has two attributes: a price and a tastiness value. We may either skip a fruit or buy it, but each fruit can only be purchased once.
We are given a circular array called colors, where each element is either 0 or 1. These represent red and blue tiles arranged in a ring, which means index 0 and index n - 1 are adjacent.
This problem provides a database table named Listings that contains information about home listings. Each row represents a single home listing and contains three fields: - listingid, a unique identifier for the listing - city, the city where the home is located - price, the…
We are given a graph of counters, where each counter starts at zero and is connected by undirected wires. Pressing a button on a counter increases its own value by one, and also increases the value of every directly connected neighbor by one.
The problem is asking us to find the maximum distance between two houses that have different colors. You are given a 0-indexed list colors where each element represents the color of a house.
Polycarpus already knows the best regular price for his product, p. He is willing to reduce it by at most d bourles if that gives the final price a more attractive ending, specifically as many trailing nines as possible. A trailing nine means a digit 9 at the end of the number.
This problem asks us to design a simplified text editor that supports four operations: 1. Inserting text at the current cursor position 2. Deleting characters to the left of the cursor 3. Moving the cursor left 4.
This problem asks us to implement a generator that produces the factorial sequence up to a given integer n. Recall that the factorial of a positive integer is defined as: Additionally, by definition: The generator should not return only the final factorial value.
We are given a group of boys and girls who will participate in a sequence of dances. Each dance pairs exactly one boy with one girl. The restriction is about novelty: in any valid pair, at least one participant must be dancing for the first time in the entire sequence.
The problem asks us to construct a strictly increasing array nums of length n such that the bitwise AND of all elements equals x. Among all valid arrays, we want to minimize the last element, nums[n - 1]. The constraints are important: - Every element must be a positive integer.
Each year in Berland has a certain number of days. The first year has a days, the next has a + 1, and so on for n consecutive years. For a year with x days, the government chooses a month length that satisfies two conditions. First, the month length must be a perfect square.
We are asked to select k lemmings from a group of n and assign them to k ledges of increasing heights so that heavier lemmings occupy higher ledges, while minimizing the time t needed for all selected lemmings to reach their assigned ledges.
The problem asks us to implement a generator function that yields numbers from the Fibonacci sequence. The Fibonacci sequence is defined recursively as Xn = Xn-1 + Xn-2 with the starting values X0 = 0 and X1 = 1.
This problem gives us a directed graph represented by an array called edges. The graph contains n nodes labeled from 0 to n - 1. Each node has at most one outgoing edge.
The reviewer identified a sign error in the swap argument, so the first task is to determine the correct monotonicity direction before rebuilding the proof. For , there is only one permutation, so the statement is trivial.
We are given a weighted tree with n vertices. Every edge contributes two different quantities to a path. The first quantity is the path length, meaning the number of edges on the path. The second quantity is the path weight, meaning the sum of edge weights on the path.
The problem gives us an integer array nums and another array queries. For every query value q, we must calculate the minimum number of operations required to make every element in nums equal to q. One operation consists of increasing or decreasing a single element by exactly 1.
We have a list of chore difficulties. Vasya must receive exactly b chores whose difficulty is at most x, while Petya must receive exactly a chores whose difficulty is strictly greater than x. The value x must be an integer. We need to count how many integers satisfy the split.
We have several mugs that already contain different amounts of Ber-Cola. There is also some drink left in the bottle. We must pour the entire remaining amount into the mugs so that every mug ends up with exactly the same volume. If the initial amounts are a1, a2, ...
The problem asks us to compute the beauty of every contiguous subarray of size k within a given integer array nums. The beauty of a subarray is defined as the xth smallest negative number in that subarray. If a subarray contains fewer than x negative numbers, the beauty is 0.
The problem requires transforming a string of digits into its lexicographically smallest form by performing at most one swap between adjacent digits of the same parity. Here, parity refers to whether a digit is even or odd.
The problem asks us to implement a page recommendation system based on users’ friendships and their page likes. We are given two database tables: Friendship and Likes.
We are given a network of coders, and each coder makes a claim during a meeting: they point to two other coders and assert that the culprit is one of those two people. We then have to choose exactly two coders to bring in as suspects.
We are given an integer array nums, and we want to count how many ways we can split it into exactly three non-empty contiguous subarrays: - nums1 - nums2 - nums3 such that: where + means concatenation. A split is considered beautiful if at least one of these conditions holds: 1.
This problem asks us to divide a given list of player skill levels into teams of exactly two players such that every team has the same total skill.
We are given a decimal number as a string and another string containing extra digits that we may use for replacements. Every digit from the second string can be used at most once. For each chosen digit, we may replace any single position in the original number.
The problem gives us a numeric string s and asks whether it can be split into at least two non-empty substrings such that their integer values form a strictly descending sequence where every adjacent pair differs by exactly 1.
We are given n cities labeled from 0 to n - 1. Initially, the graph forms a simple directed chain: - 0 - 1 - 1 - 2 - 2 - 3 - ...
We are given a circle with $2n$ labeled points arranged in order. These points are connected pairwise by $n$ chords, and each point is used exactly once, so the chords form a perfect matching on the circle. From these endpoints, we must choose exactly three of the given chords.
This problem gives us an undirected tree with n nodes and a list of directed parent-child guesses. Each guess claims that one node is the parent of another node when the tree is rooted in some way. The important detail is that the tree itself is undirected.
The problem gives us a 0-indexed integer array nums and asks us to find the maximum possible value of: subject to the constraint: In other words, we must choose three indices in increasing order.
We are given a recursive procedure that operates on a permutation of numbers from 1 to n. The function behaves like this: For f(x), it first recursively processes the first x - 1 elements, then swaps positions x - 1 and x.
The problem requires designing a memory allocator that manages a fixed-size memory array. Initially, all memory units are free, and the allocator supports two operations: allocating a block of consecutive free memory units for a given ID (mID) and freeing all memory units…
We are asked to assign k deputies to n cities, with the condition that each deputy manages exactly three cities. The cities are placed on two sides of a river, and some pairs of cities are connected by bridges that span the river.
We are given a fixed string consisting of lowercase letters, and many independent queries. Each query specifies a contiguous segment of this string, and for that segment we must count how many substrings are palindromes.
The problem asks us to transform an integer array so that its median becomes exactly k, using the minimum number of operations. In one operation, we may either increase or decrease any single element by 1. The median is defined after sorting the array in non-decreasing order.
We are given two separate undirected trees. The first tree contains n nodes and is represented by edges1, while the second tree contains m nodes and is represented by edges2. A tree is a connected graph with no cycles.
The problem asks us to enumerate all possible ways to pair up a set of n men with n women using a given list of k possible marriages. Each marriage has a happiness score, and each man or woman can appear in at most one marriage in a set.
The problem gives us an undirected graph with n nodes labeled from 0 to n - 1. We must place every node into a 2D grid so that adjacency in the grid matches adjacency in the graph exactly.
You asked for a complete guide, but the previous message included two different LeetCode problems with full requirements: - LeetCode 2692 - Make Object Immutable - LeetCode 2699 - Modify Graph Edge Weights A full reference-quality guide for either one is quite large and should…
We are given a single integer a, and we need to compute a number that depends only on this value. The input is small enough that the answer is expected to be derived from a direct mathematical pattern rather than any simulation or search.
The problem is asking us to find a valid split point in an array nums where the product of elements to the left of the split and the product of elements to the right are coprime. Formally, if we split at index i, the left product is nums[0] nums[1] ...
The problem asks us to determine the kth smallest amount that can be formed using coins of given denominations under a strict limitation: we can only use a single type of coin at a time. In other words, combinations of different denominations are not allowed.
The problem requires us to partition an array of positive integers nums into two ordered groups such that the sum of elements in each group is at least k. A partition is considered great if this condition is satisfied.
This problem provides a database table named usercontent with two columns: | Column | Description | | --- | --- | | contentid | Unique identifier for each row | | contenttext | A text string containing words and spaces | The task is to produce a result table that contains: 1.
The problem gives us two arrays, nums1 and nums2, where each array contains unique digits from 1 to 9. We need to construct the smallest possible number such that the number contains at least one digit from each array.
We are given a string consisting of digits, <, and characters. Every query selects a substring and treats it as a standalone program in a tiny tape language. The interpreter keeps two pieces of state.
The problem asks us to compute the number of distinct prime factors in the product of an array of positive integers, nums.
Kirito starts with some initial strength and must defeat every dragon on the level. Each dragon has two values: the minimum strength needed to beat it, and the bonus strength Kirito gains afterward.
The problem is asking us to transform a JSON-like array of objects or arrays into a matrix representation. Each row of the resulting matrix corresponds to one object (or array) in the input array, and the first row contains the column names.
This problem is unusual compared to standard competitive programming tasks because the original contest expected participants to build a document classifier from a training dataset.
We are given a large binary image represented as an n × n grid. A value of 1 means a black pixel, and 0 means a white pixel. The black regions correspond to geometric figures. Every figure is either a circle or a square.
We are asked to count the number of ways to build a fence of exact length l using boards of n types, where each type is a rectangle of dimensions ai by bi.
The problem gives us a sorted array stones, where each value represents the position of a stone in a river. The frog starts on the first stone, which is always at position 0, and must travel to the last stone and then eventually return to the first stone.
The problem asks us to determine the minimum cost to buy exactly one apple starting from each city in a network of cities connected by bidirectional roads. Each city has a specific cost for buying an apple, and each road has a travel cost.
The problem gives us a list of mechanics, where each mechanic has a specific rank. A mechanic with rank r takes r n^2 minutes to repair n cars. This means the repair time grows quadratically as the number of assigned cars increases.
The field is represented as a large rectangle whose outer boundaries are fixed fences. Inside the field, additional horizontal and vertical fences divide the area into smaller sections.
We need to construct an array of groom fortunes so that the princess reacts in exactly the required way. For every groom after the first one, two special situations are possible. If the current fortune is larger than every previous fortune, the princess says Oh....
The problem asks us to determine whether a given list of nodes represents the preorder traversal of a binary tree. Each node is represented as a pair [id, parentId], where id is the unique identifier of the node, and parentId identifies its parent in the tree.
The problem gives us a binary string s, along with two coprime integers num1 and num2. We need to count how many non-empty substrings contain 0s and 1s in the exact ratio num1 : num2.
This problem is asking us to determine the minimum amount of time that a computer must be turned on to execute a set of tasks, given that each task has a specified time window [starti, endi] and requires a total duration durationi that does not need to be continuous.
The problem asks us to compute the sum of elements in an array, but not based on the values themselves. Instead, we decide whether to include an element by examining the binary representation of its index. You are given a 0-indexed integer array nums and an integer k.
The problem asks us to find the smallest positive integer that cannot be represented as a bitwise OR of any subsequence of a given array nums. The input array contains positive integers, and the subsequences can have any length, including length 1.
We are given a short binary array, where each position is either 0 or 1. We are allowed to choose exactly one contiguous segment and invert every value inside it, turning 0 into 1 and 1 into 0.
We are given a tree whose structure has been erased, but two pieces of information remain attached to each vertex: its color and a number that equals the total weight of all edges incident to it in the original tree.
We have a sequence of balls, each with a color and a value. We may pick any subsequence while preserving order. The score of the chosen subsequence depends on how consecutive chosen balls relate by color.
We have several shelves of porcelain items. Inside one shelf, the items form a line, and at any moment we may only remove the current leftmost item or the current rightmost item. After removing one item, the next item on that side becomes accessible.
We are given a directed graph where every node represents a program state. Each state does one of three things to a variable: 0 means the variable is ignored. 1 means the variable is assigned a new value. 2 means the variable is used.
This problem asks us to construct the longest possible palindrome using a list of two-letter words. Each word can be used at most once, and the order of concatenation can be chosen freely.
Codeforces 409D: Big Data
The problem asks us to transform an integer x into another integer y using the minimum number of operations. At every step, we are allowed to perform exactly one of four actions: 1. Divide x by 11 if it is divisible by 11. 2. Divide x by 5 if it is divisible by 5. 3.
The problem asks us to determine whether a given integer is a Harshad number. A Harshad number is an integer that is divisible by the sum of its digits. We are given a single integer x, and we must perform two operations: 1. Compute the sum of all digits in x 2.
In this problem, we have a regular polygon with n vertices, and each vertex initially contains exactly one monkey. Every monkey must move simultaneously to one of its two neighboring vertices.
The problem asks us to count how many distinct subsequences of a binary string are considered “good.” A subsequence is any sequence formed by deleting zero or more characters without changing the relative order of remaining characters.
We are given two lowercase strings. From each string we can look at every contiguous segment, and we care about those segments that behave unusually: a segment is considered special if it appears exactly once in its own string.
The problem requires us to determine how many strings in targetWords can be formed from strings in startWords through a specific transformation.
The problem gives us an array happiness representing the happiness values of n children standing in a queue and asks us to select exactly k children to maximize the sum of their happiness.
You are given an undirected weighted tree with n nodes. Since the graph is a tree, there is exactly one simple path between any two nodes. Each edge has a weight between 1 and 26. For every query [a, b], we look at the unique path from node a to node b.
The problem gives us a list of robots positioned on an infinite number line. Each robot starts at a unique coordinate from the array nums, and each robot has an associated movement direction from the string s. A robot moves exactly one unit per second.
This problem asks us to implement a function similar to JavaScript's Promise.allSettled(), but without actually using the built-in method. We are given an array of functions functions, where each function, when invoked, returns a promise.
The problem asks us to find the minimum cost to move from a starting point start = [startX, startY] to a target point target = [targetX, targetY] in a 2D plane.
The problem requires counting overlapping shifts for each employee from a database table called EmployeeShifts. Each row in this table represents a single shift worked by an employee, with a starttime and an endtime.
We are asked to find the largest subset of a given set of positive integers such that no element in the subset is exactly k times another element in the subset.
The problem asks us to find the longest subsequence in an integer array nums where each adjacent pair of elements in the subsequence has the same parity sum, modulo 2.
The problem is asking for the maximum integer value that can be created by concatenating the binary representations of all elements in a list of three integers. Each number should be converted to its binary representation without leading zeros.
This problem requires building a hierarchical view of a company's employee structure starting from the CEO. The Employees table contains employee records, including their unique ID, name, manager ID, and salary. The CEO is identified as the employee with a NULL managerid.
The problem gives us n players and a list called pick, where each entry is of the form [xi, yi]. This means player xi picked a ball with color yi. A player wins if they have picked strictly more than their player index number of balls of the same color.
We are asked to choose three positive integers, each at most n, to maximize their least common multiple (LCM). The input is a single integer n, and the output is a single integer representing the largest possible LCM achievable using three integers not greater than n.
That is a long-form reference document with multiple substantial sections, complete Python and Go implementations, detailed walkthroughs, and worked examples. To keep formatting quality high and avoid truncation, I will provide it in a structured document format.
The problem asks us to find the minimum number of operations required to reduce a given positive integer n to 0, where each operation consists of adding or subtracting a power of two from the current value of n. A power of two is defined as any number of the form where .
We are given a chronological log of private messages between users in a social network. Each record contains the sender, the receiver, and the timestamp of the message. Two users become friends if one of them replies to the other's message quickly enough.
We are given a tree with n cities connected by n-1 roads. Each city must be assigned an officer with a rank from 'A' (highest) to 'Z' (lowest).
We have a two-dimensional gym represented as a grid of size n × m. Each cell contains a positive integer representing the calories burned by doing the workout at that location.
The problem asks whether an integer remains unchanged after performing the digit reversal operation twice. A reversal operation takes the decimal representation of a number and reverses the order of its digits.
The problem asks us to count how many substrings of a given string satisfy two conditions simultaneously: 1. The substring contains all five vowels, 'a', 'e', 'i', 'o', and 'u', at least once. 2. The substring contains exactly k consonants.
The problem asks us to simulate a process of hiring k workers from a list costs, where each worker has an associated cost. We can hire workers only from the first candidates workers or the last candidates workers in each hiring session.
Here is the complete technical solution guide for LeetCode 2755 following your requested format. The problem asks us to deep merge two JSON values, obj1 and obj2. The merging rules depend on the type of the values at each position in the objects or arrays.
The problem asks us to split a given array nums into exactly three disjoint contiguous subarrays and minimize the sum of their costs. The important detail is how cost is defined. The cost of a subarray is simply its first element.
The problem gives us a binary matrix grid, where each cell contains either 0 or 1. We need to count how many valid right triangles can be formed using cells whose value is 1.
We are asked to bring all roads in Berland to an asphalted state using a special operation. Each road connects two cities and has an initial state: asphalted or not.
We are given a valid parenthesis sequence. Every opening bracket has a unique matching closing bracket, and the pairs are properly nested. We want to assign colors to brackets under two rules.
We have a rooted tree with root 1. Every edge from a parent to a child contains a lowercase string. If we walk downward through the tree and read characters along the edges, we obtain a long text embedded into the tree. A position is not a vertex.
Polycarpus has a sequence of tasks, each with a start window, an end window, and a duration. Formally, task i is represented by (li, ri, ti).
The tournament is organized as a sequence of elimination rounds. In every round, players are paired symmetrically from the two ends of the current lineup. The first player faces the last player, the second player faces the second-to-last player, and so on.
We have a ribbon of total length n. Every cut piece must have one of exactly three allowed lengths: a, b, or c. The goal is not just to split the ribbon successfully, but to maximize how many total pieces we obtain.
We are given an array nums, where each element represents the value assigned to a node in a graph. The graph contains exactly n nodes, one node for each array element.
We are given a collection of booking requests, where each request represents a group of guests who either all get seated together or do not come at all.
We are given a rectangular grid of integers. In one operation we may pick any single cell and increase its value by exactly one. We can repeat this as many times as we want on any cells. The goal is not to make the whole matrix prime.
This problem asks us to determine the maximum number of operations that can be performed on an array of integers, where every operation removes two elements from specific positions (either the first two elements, the last two elements, or the first and last elements), and…
The problem gives us two strings, start and target, both having the same length. Each position in the strings contains one of three characters: - 'L', representing a piece that can only move left - 'R', representing a piece that can only move right - '', representing an empty…
We are asked to count divisors of a given positive integer x that share at least one digit with x. In other words, we examine every number d that divides x evenly and check whether there exists a digit appearing both in x and in d.
We are working on a very narrow grid: only two rows, but a large number of columns. Each cell in this 2 by n strip is either free or blocked.
We are given a list of abbreviated king names in chronological order. Each name is a lowercase string. We may choose some of these names to form a dynasty, while preserving their original order. A valid dynasty must satisfy two conditions.
We are given two base strings, call them a and c, and two integers b and d. From these, two longer strings are conceptually constructed: the first string is a repeated b times, and the second is c repeated p times for some unknown positive integer p.
The problem gives us a binary array nums, meaning every element is either 0 or 1. We need to count how many subarrays are alternating. A subarray is considered alternating if no two adjacent elements inside that subarray are equal.
This problem asks us to simulate how people pass through a single door over time while following a specific priority policy. Every person arrives at a certain second and wants to either enter or exit.
The problem gives us an array of positive integers, nums. In one operation, we may choose two adjacent elements, nums[i] and nums[i+1], and replace either one of them with the greatest common divisor, gcd, of the pair.
The problem gives us a database table named cities, where each row contains a state name and a city name. The pair (state, city) is guaranteed to be unique, which means the same city will not appear twice for the same state. We must generate a report for qualifying states.
The problem asks us to generate every binary string of length n such that no substring of length 2 contains two consecutive zeros. A binary string contains only the characters "0" and "1". A substring of length 2 means every adjacent pair of characters in the string.
The problem asks us to repeatedly remove occurrences of a substring part from a string s. The important detail is that on every operation, we must remove the leftmost occurrence of part. We continue performing removals until part no longer appears anywhere inside s.
This problem asks us to design a very simple banking system that supports three operations: 1. Transfer money between two accounts 2. Deposit money into an account 3. Withdraw money from an account The bank contains n accounts numbered from 1 to n.
We are asked to simulate the spread of a signal across an n × n grid. Initially, a single cell at row x and column y is turned on, and in each second, any cell that is side-adjacent to a turned-on cell also turns on.
We are asked to arrange a sequence consisting of two kinds of objects, boys and girls, into a single line. The only freedom we have is the order.
We must fill an $n times n times n$ cube with the integers from $1$ to $n^3$. Consecutive numbers must always occupy cubes that share a face, so the numbering forms a Hamiltonian path through the 3D grid. The second requirement is the unusual one.
We are given a line of apples numbered from 1 to n. Each apple can be given to exactly one of two hamsters, Arthur or Alexander. The only restriction is that Arthur must receive only apples he likes, and Alexander must receive only apples he likes.
The problem gives us an array of strings words. We start with words[0] and then process the remaining strings one by one. For each new word words[i], we have exactly two choices: 1. Append it to the right side of the current string. 2.
The problem asks us to compute the greatest common divisor (GCD) of two specific values derived from the input array nums: the smallest element in the array and the largest element in the array.
The problem gives us the root of a binary tree and an integer k. For every level in the tree, we calculate the sum of all node values that appear on that level. After computing all level sums, we must return the kth largest level sum.
This problem asks us to identify employees whose assigned project workload is greater than the average workload of employees within their own team. We are given two database tables: The Project table stores information about project assignments.
We are given a set of $n$ cities, each represented as a point on a 2D Cartesian plane. The distance between any two cities is measured using the Manhattan metric, which is the sum of the absolute differences of their coordinates.
This is a SQL database problem where we need to group employees into teams based on salary. The important detail is that a team is defined entirely by salary, meaning every employee in a team must have exactly the same salary, and all employees with the same salary must belong…
The airport has multiple planes, each with a certain number of empty seats. Every passenger in a queue can choose any plane to buy a ticket from, and the cost of the ticket equals the number of currently empty seats in that plane at the time of purchase.
The problem asks us to select a number from an array of distinct positive integers such that the chosen number is neither the minimum nor the maximum in the array. In other words, we must return a number that lies strictly between the smallest and largest elements.
We are given a collection of gifts, each with a name and a set of distinct prices. Some names may appear multiple times, each with different prices. The old man can request exactly n gifts, specifying only names.
The problem asks for the minimum number of straight cuts required to divide a circle into n equal slices. A valid cut can either pass through the center and touch two points on the circle, or touch one point on the circle and pass through the center.
This problem asks us to transform a starting integer into a target integer using the minimum number of operations.
The problem asks us to take a JSON-formatted string str and convert it into the equivalent native data structure.
The problem describes a turn-based game between Alice and Bob, where both players have a lexicographically sorted list of words. Alice always starts by playing her lexicographically smallest word, and the players then alternate turns.
Vasily the Bear is creating a geometric structure on the plane using a sequence of squares and line segments, all centered at the origin. The parameter n controls the number of nested squares he draws, each forming vertices at multiples of 2i and 2i+1 along the axes.
The problem asks us to count the number of ways we can split a given binary array nums into contiguous subarrays such that each subarray contains exactly one 1. The input is a binary array, meaning it only contains 0s and 1s.
The problem gives us a binary string s, which means the string contains only the characters '0' and '1'. We need to find the length of the longest substring that is considered balanced. A substring is balanced when two conditions are satisfied: 1.
The problem asks us to find the smallest missing integer in the array that is greater than or equal to the sum of the longest sequential prefix. A sequential prefix means the beginning part of the array where every number increases by exactly 1 compared to the previous number.
This problem asks us to count the number of special subsequences in an array nums that consists only of integers 0, 1, and 2. A subsequence is special if it follows the pattern of one or more 0s, followed by one or more 1s, followed by one or more 2s.
The problem describes a competition where players stand in a queue and repeatedly compete against each other. Every player has a unique skill value, and whenever two players play, the one with the larger skill always wins. At each step: 1.
We are given an undirected graph where each vertex carries a positive weight and each edge also carries a positive weight.
The problem gives us a positive integer n and asks us to find a special integer x, called the pivot integer, such that: - The sum of all integers from 1 to x is equal to - The sum of all integers from x to n The important detail is that x belongs to both sums.
The problem asks us to maximize a score after performing exactly k operations on an array of integers. Each operation allows us to choose any element nums[i], add its value to the score, and then replace it with its ceiling division by 3.
The problem gives us a single positive integer k. We start with an array containing exactly one element: We are allowed to perform two kinds of operations any number of times: 1. Increase the value of any existing element by 1 2.
The problem presents a scenario where we have n flower seeds, each of which requires two phases to bloom: a planting phase and a growth phase. Each seed i takes plantTime[i] full days to plant, which can be spread across non-consecutive days.
This problem asks us to analyze phone call records stored across two database tables and return the three longest calls for each call type, incoming and outgoing. The Contacts table stores information about people.
The problem is a pure inequality with a geometric constraint. The condition that gives by the Pythagorean theorem in . The second condition, that the foot of the perpendicular from to plane is the orthocenter of , is much more restrictive than it first appears.
This problem asks us to calculate the confirmation rate for each user in a system where users can request confirmation messages after signing up. We are given two tables: Signups and Confirmations.
We are asked to count rhombi inside a rectangle of width w and height h, where each rhombus has its vertices on integer coordinates and diagonals aligned with the axes. Each rhombus must have positive area and be fully contained in the rectangle.
We have a binary matrix where each row can be rotated cyclically. A left rotation moves every element one position left and wraps the first element to the end. A right rotation does the opposite.
The problem gives us an integer array nums containing exactly 2 n elements. Our task is to determine whether it is possible to split the array into n valid pairs, where each pair contains two identical numbers.
This problem asks us to calculate the total sales value generated by the customers assigned to each salesperson. We are given three database tables: The Salesperson table contains the list of salespeople. Each salesperson has a unique salespersonid and a name.
The problem asks us to convert a doubly linked list into a standard integer array while preserving the order of elements. In a doubly linked list, each node contains a value (val) and two pointers: next (pointing to the next node) and prev (pointing to the previous node).
We are given a queue made of boys and girls, represented as a string containing only B and G. Every second, each adjacent pair "BG" swaps positions and becomes "GB". The swaps happen simultaneously. That detail matters a lot.
We are asked to calculate the minimum time required for a group of n aliens to move from the third section of a three-section spacecraft to the first section.
The playlist is built from blocks of repeated songs. Song i has duration t[i], and Eugeny listens to it c[i] times consecutively before moving to the next song. If a song lasts 4 minutes and is repeated 3 times, that block contributes 12 minutes to the playlist timeline.
This problem asks whether a matrix returns to its original form after repeatedly applying cyclic row shifts. The matrix has m rows and n columns. Every operation affects all rows simultaneously, but the direction depends on the row index.
Each friend has a phone book containing numbers written in the format XX-XX-XX. Every phone number belongs to exactly one of three categories. A taxi number uses the same digit everywhere. Examples are 11-11-11 or 55-55-55.
The problem asks us to implement an asynchronous function sleep that pauses execution for a given number of milliseconds, specified by the input millis.
The problem asks us to maximize the points a tourist can earn over a fixed number of days, k, while visiting n cities.
This problem gives us two arrays of equal length: - names[i] represents the name of a person - heights[i] represents the height of that same person The two arrays are aligned by index, meaning the person at index i has both the name names[i] and the height heights[i].
The problem asks us to count the number of triplets (a[i], b[j], c[k]) from three integer arrays a, b, and c such that the bitwise XOR of the three numbers has an even number of set bits (bits equal to 1 in binary representation).
This problem asks us to merge two arrays of JSON objects based on a shared id field. Every object in both arrays contains a unique integer id, and our goal is to produce a single merged array where objects with the same id are combined together.
The problem presents a classroom scenario where students are given problems in a round-robin order, consuming a certain number of chalk pieces for each problem.
We are given a multiset of strings and must choose exactly k of them. The score of the chosen group is based on longest common prefixes between every unordered pair of chosen strings. If two strings share a prefix of length t, then that pair contributes t to the total score.
We are given a database table called Sessions, where each row represents one user session on a platform. Every session belongs to a specific userid, has a start and end timestamp, a unique sessionid, and a sessiontype that can be either "Viewer" or "Streamer".
This problem asks us to implement a cache that stores key-value pairs, but unlike a normal hash map, each entry has an expiration time. A stored value remains accessible only for a limited duration measured in milliseconds.
We are asked to find the longest chain of stars the Donkey can select under a geometric rule. The stars are points on a plane, and we begin at the origin. From any star, we imagine two rays at fixed angles relative to the horizontal axis.
This problem asks us to find the smallest non-empty contiguous subarray whose bitwise OR is at least k. The input consists of: - An integer array nums - An integer k For any subarray nums[left:right+1], we compute the bitwise OR of all elements inside that range.
The problem describes an event with n performers and x available stages. Each performer must be assigned to exactly one stage. Multiple performers may share the same stage, which means they form a band together. Some stages may remain unused.
The problem gives us a tree with n nodes labeled from 0 to n - 1. The tree is undirected and rooted at node 0. The tree structure is described by the edges array, where each edge [a, b] connects nodes a and b. Each node also has an associated cost, given in the cost array.
We are given two lowercase strings, s and t. The task is not simply to check whether t is a subsequence of s. We must answer a stronger question. Consider every possible way to obtain t as a subsequence of s.
We are tasked with equipping as many soldiers as possible with bulletproof vests. Each soldier has a preferred vest size, but they are willing to tolerate deviations within a given range. Specifically, the i-th soldier can wear any vest with a size between a[i] - x and a[i] + y.
We are given a string of lowercase letters and need to count the number of pairs of non-overlapping palindromic substrings. Formally, we are looking for tuples (a, b, x, y) such that the substring from a to b and the substring from x to y are both palindromes and b < x.
We have a square campus with coordinates from (0, 0) to (b, b). Inside the campus there are several directed axis-aligned segments, called arrows. Each arrow is either horizontal or vertical, and the arrows never intersect or touch.
We are given two integers. The first number, a, is arbitrary. The second number, b, is guaranteed to be lucky, meaning every digit of b is either 4 or 7. For any positive integer, its "mask" is formed by taking only the digits 4 and 7 from left to right and concatenating them.
This problem is implemented in JavaScript on LeetCode, but you requested Python and Go reference solutions. Since the original stub uses JavaScript-style nested arrays with mixed integer/array values, I will model the structure idiomatically in Python and Go while keeping the…
We need to construct an undirected graph that contains exactly k triangles. A triangle means three different vertices where every pair is connected by an edge. The graph may contain any number of other structures, but the total number of triangles must be exactly k.
We have an n × n grid that starts completely white. Cells are painted black one by one, and every move paints a different cell. After each move, we want to know whether the board already contains a completely black 3 × 3 square.
We are given an integer array nums, and we must find three indices (i, j, k) such that: - i < j < k - nums[i] < nums[j] < nums[k] Among all valid increasing triplets, we want to maximize the expression: The task is not to maximize the sum of the triplet.
We are given a production line where multiple paintings move through a fixed sequence of painters. Every painting must pass through all painters in order, from the first to the last.
We are given an array a of n non-negative integers, and a number m specified as a binary string. Each integer in a represents a weight associated with a position, and for any number x between 0 and m inclusive, we can select positions where the bits of x are set and sum the…
This problem gives us two collections of geometric objects on a 2D plane. The first collection is an array called points, where each element is a coordinate pair [x, y]. Each pair represents a point on the plane.
The problem gives us two lowercase strings, str1 and str2. We are allowed to perform at most one global operation on str1. During this operation, we may choose any subset of indices in str1, and increment the character at each chosen index by one alphabetically.
The problem asks us to find, for each index in a given array nums, the length of the shortest contiguous subarray starting at that index whose bitwise OR is equal to the maximum possible bitwise OR obtainable from that index onward.
The problem gives us a permutation perm of the integers [1, 2, ..., n]. A permutation is simply an arrangement of all numbers from 1 to n where each number appears exactly once.
This problem asks us to determine how many monsters we can eliminate before any one of them reaches the city. Each monster starts at some distance from the city and moves toward it at a constant speed.
After the theft, the barn contains a smaller rectangular box of hay blocks. If the original dimensions were $A times B times C$, then the remaining pile has dimensions $(A-1) times (B-2) times (C-2)$.
This problem models a garbage collection system with three separate garbage trucks: - One truck collects metal garbage, represented by 'M' - One truck collects paper garbage, represented by 'P' - One truck collects glass garbage, represented by 'G' The input array garbage…
This problem gives us two positive integers, n and m. We need to examine every integer in the inclusive range [1, n] and separate the numbers into two groups based on divisibility by m. The first group contains all numbers that are not divisible by m.
We are given a tree $s$ with $n$ vertices. We must construct another tree $t$, whose nodes are subsets of vertices of $s$. Each subset is usually called a bag. The decomposition must satisfy three conditions. First, every original vertex must appear in at least one bag.
The problem gives us a 0-indexed array of integers, nums, which can include negative numbers, zero, and positive numbers. We are allowed to reorder the elements in any way we choose.
This problem asks us to take a string of lowercase English letters and convert it into a numeric value, then repeatedly transform that number by summing its digits.
We have a triangular pyramid of scales. The top row has n scales, the second row has n-1, the third has n-2, down to the last row with a single scale. Each scale has a maximum weight it can hold without breaking. Initially, each top-row scale receives a certain amount of cereal.
The problem asks us to construct an array ans from a given array nums of prime integers. For each element nums[i], we need to find the smallest integer ans[i] such that the bitwise OR of ans[i] and ans[i] + 1 equals nums[i]. Formally, ans[i] | (ans[i] + 1) == nums[i].
This problem gives us three arrays: - heroes, where heroes[i] represents the power of the ith hero. - monsters, where monsters[j] represents the power of the jth monster. - coins, where coins[j] represents the reward obtained for defeating the jth monster.
The problem asks us to design a simulation of an ATM machine that can handle deposits and withdrawals with strict rules about how money is dispensed. The ATM stores five types of banknotes: 50, 200, and $500.
Xenia has a cupboard shaped like a semicircular arch on top of two vertical walls. The semicircle has radius r and the walls have height h, with the cupboard’s depth also equal to r. Inside, she wants to store spherical balloons of radius 1.
This problem is asking us to split a given undirected tree into as many connected components as possible, under the condition that the sum of values of nodes in each component is divisible by a given integer k.
We have an array of soldier reliabilities. Every contiguous subarray represents one possible police group. The general always chooses the strongest unused group first, meaning all subarray sums are sorted in descending order and picked one by one.
The problem asks us to determine whether two strings s1 and s2 of equal length can be made identical using a specific type of swap operation.
We are asked to embed a tree onto a set of points on a plane in such a way that tree edges correspond to straight line segments connecting the points.
The problem asks us to manipulate an array of integers, nums, so that its prefix sums are never negative. A prefix sum at index i is simply the sum of all elements from the start of the array up to i.
The tree in this problem is not given explicitly through an edge list. Instead, the structure is defined mathematically. Every node v has parent floor(v / 2), which creates the same structure as a binary heap.
The problem gives us a sequence of keyboard button press events. Each event is represented as: Here: - index is the identifier of the button that was pressed. - time is the moment when the press happened.
This problem requires implementing a SQL-style pivot operation programmatically. The input is a table named Products with columns productid, store, and price, where each row represents the price of a product in a specific store.
The problem gives us an array amount of length 3. Each index represents the number of cups that need to be filled for a specific water type: - amount[0] represents cold water cups - amount[1] represents warm water cups - amount[2] represents hot water cups Every second, the…
We are asked to work with an infinite matrix that is filled in a very specific order. Each cell contains a positive integer, starting from 1, and the ordering rule is based on the maximum of the row and column indices.
The problem requires counting the number of vowel substrings in a given string word. A substring is any contiguous sequence of characters within the string.
This problem gives us an undirected weighted graph with n nodes and a list of weighted edges. Each edge connects two nodes and has a positive weight. We are also given a source node s, a destination node d, and an integer k.
The problem gives us a string word and a character ch. We must locate the first occurrence of ch inside the string. Once we find it, we reverse the substring starting at index 0 and ending at that occurrence, inclusive.
The problem asks us to count the number of "good" strings that can be constructed by repeatedly appending either '0' exactly zero times or '1' exactly one times, starting from an empty string. A string is good if its length lies between low and high inclusive.
The problem asks us to identify the longest square streak in an array of integers nums. A square streak is a subsequence of at least length 2 where, after sorting, every element is the square of the previous element. In other words, if the sorted subsequence is [x1, x2, ...
Codeforces 313E: Ilya and Two Numbers
This problem asks us to determine whether a given integer num can be represented as the sum of three consecutive integers. If such a representation exists, we must return those three integers in sorted order. Otherwise, we return an empty array.
This is a Type A, “Find all X” problem. A complete solution must do two things: First, it must show that every listed family of solutions actually satisfies the system. Second, it must prove that no other solutions exist. The proposed proof does both parts.
The problem describes a street represented by positions from 0 to n - 1. There are multiple street lamps, each defined by its position and range. A lamp at position p with range r illuminates all positions from max(0, p - r) to min(n - 1, p + r), inclusive.
The problem gives us an array price where each element represents the price of a candy. We must choose exactly k distinct candies and maximize the basket's "tastiness".
The problem asks us to break a string s into non-overlapping substrings such that each substring exists in a given dictionary. Characters in s that cannot be matched with any dictionary word are considered extra characters.
We are given a chessboard represented as an n by m grid where each cell is either good or bad. A good cell is indicated by a "." and a bad cell by a "-".
The problem asks us to repeatedly perform an operation on an integer array nums. In each operation, we remove the first two elements of the array and compute a score, which is simply the sum of those two removed values.
We are asked to simulate the operation of a single bus that repeatedly transports students from a bus stop at coordinate 0 to their respective destinations along the positive axis. Each student arrives at the stop at a distinct time and has a fixed destination coordinate.
The problem asks us to determine the minimum number of cells we need to visit in order to reach the bottom-right corner of a given m x n grid. Each cell (i, j) contains a number grid[i][j] representing the maximum number of steps we can move right or down from that cell.
Each mountain is represented by a peak point (x, y). Because the mountain is a right-angled isosceles triangle with slopes +1 and -1, its shape is completely determined by its peak.
This problem asks us to find the kth largest value among a list of integers that are represented as strings. Each element in nums is a non-negative integer encoded as a string with no leading zeros, and we are required to return the kth largest value according to numeric order…
Each Olympiad happens on a fixed calendar date in 2013. Before that date, a group of jury members must work continuously for several days. If an Olympiad needs p people and t preparation days, then exactly p people are busy on each of the t days immediately before the Olympiad.
We are given an m x n rectangular cake and need to divide it completely into 1 x 1 pieces. The cake can be cut along predefined horizontal and vertical lines.
We are asked to count the number of obtuse triangles that can be formed on a triangular tennis racket with evenly spaced holes along its sides. Each side has n holes dividing it into n+1 equal segments.
The problem gives us two integers, red and blue, representing the number of red and blue balls available. We want to build a triangle where: - Row 1 contains exactly 1 ball - Row 2 contains exactly 2 balls - Row 3 contains exactly 3 balls - And so on There are two additional…
The problem asks us to count how many integers in the inclusive range [l, r] are not special. A number is considered special if it has exactly two proper divisors. A proper divisor of a number x is any positive divisor of x other than x itself.
This problem asks us to determine the maximum possible distance from the origin after performing a sequence of moves on a number line. We start at position 0, and we are given a string moves, where each character represents one movement instruction.
The problem gives us two square binary matrices, mat and target, both of size n x n. Each cell contains either 0 or 1. Our task is to determine whether mat can be transformed into target by rotating it clockwise in 90 degree increments. A rotation can be performed multiple times.
The problem gives us a list of match results. Each match is represented as a pair: This means the player winner defeated the player loser. Our goal is to return two separate lists: 1. All players who never lost any match. 2. All players who lost exactly one match.
We have two collections of objects. Markers are described by (color, diameter) and caps are also described by (color, diameter). A cap can be attached to a marker only if the diameters are equal.
The problem asks us to find the number of index pairs (i, j) in an array nums such that i < j and the integers nums[i] and nums[j] are almost equal.
The problem asks us to design a throttled batching system for asynchronous queries. Instead of immediately sending every incoming request individually, we want to intelligently combine multiple requests together whenever possible.
This problem asks us to count how many contiguous subarrays satisfy two exact boundary conditions at the same time: 1. The minimum element in the subarray must be exactly minK. 2. The maximum element in the subarray must be exactly maxK.
We are asked to calculate the probability of performing well in a sequence of wizard contests, given both your chances of winning each contest and the types of prizes you may receive.
This problem asks us to analyze a cinema seating table and identify the longest continuous block of available seats.
This problem asks us to simulate a sequence of device tests based on battery percentages. We are given a 0-indexed integer array batteryPercentages, where each element represents the current battery level of a device.
The problem requires determining how many possible text messages Alice could have sent given a sequence of digit key presses received by Bob. Each digit from '2' to '9' maps to a set of letters on a phone keypad.
The problem asks us to compute, for every element in the array, its "second greater element" to the right. For an index i, we are interested in elements positioned after i. Among those elements, we only care about values strictly greater than nums[i].
The problem asks us to count all substrings in a given string s that start and end with the same character. A substring is a contiguous sequence of characters, so the order and adjacency of characters matter.
We are given a set of n points positioned on a one-dimensional number line. Each point has a unique coordinate, and these coordinates can be very large in absolute value, up to 10^9.
The problem gives us an integer array nums and an integer k. We need to determine whether the array contains two adjacent subarrays, each of length k, such that both subarrays are strictly increasing.
The problem asks us to group a list of unique lowercase strings based on a special notion of connectivity. Each string can be represented as a set of letters, and two strings are connected if one can be transformed into the other with exactly one operation: adding a letter…
In this problem, each bomb is represented by three integers: its x-coordinate, y-coordinate, and explosion radius. If a bomb explodes, every other bomb whose center lies within or on the boundary of that radius will also explode.
We have an array of fence heights. A query gives one contiguous segment of the fence, and we must count how many other segments match it. Two segments match if they have the same length, do not overlap, and their heights differ by the same constant at every position.
We are asked to simulate a painter robot moving diagonally across a rectangular floor made of tiles. Each tile starts white, and the robot paints a tile black whenever it enters it. The robot can face one of four diagonal directions: up-left, up-right, down-left, or down-right.
Here is the complete technical solution guide for LeetCode 2285 following your requested format and level of detail: The problem gives you n cities, numbered from 0 to n - 1, and a list of bidirectional roads connecting pairs of cities.
The problem asks us to determine the maximum possible sum when picking exactly k items from a bag containing items labeled 1, 0, or -1.
The problem asks us to transform one integer n into another integer m by repeatedly modifying individual digits of n. At each step, we can increase a digit that is not 9 or decrease a digit that is not 0.
This problem provides a Pandas DataFrame named animals with four columns: | Column | Type | | --- | --- | | name | object | | species | object | | age | int | | weight | int | The task is to return a new DataFrame containing only the names of animals whose weight is strictly…
The problem gives us two integers, num and t. We are allowed to perform a specific operation at most t times. In one operation, we must simultaneously modify both numbers involved: - Increase or decrease x by 1 - At the same time, increase or decrease num by 1 The goal is to…
The problem is asking us to identify all the peaks in a given array mountain. A peak is an element that is strictly greater than its immediate neighbors. Importantly, the first and last elements of the array cannot be peaks because they do not have two neighbors.
We are given a rectangular grid of integers. Inside this grid, we may place any odd-sized square, for example a 3×3, 5×5, or 7×7 subgrid. Inside that square we draw the standard spiral that starts at the top-left corner and winds inward. The spiral does not visit every cell.
The problem gives us an integer array nums, where some elements may already be zero and others are positive integers. We must choose a starting index curr such that nums[curr] == 0, and also choose an initial movement direction, either left or right.
The problem asks us to find the minimum total time to complete a race of numLaps laps using a collection of tires, where each tire has two parameters: a base lap time fi and a multiplier ri.
We are given a map of a city as an n × m grid, where each cell represents a district. Three of the districts have been robbed, marked by , and all other districts are empty (.).
The problem asks us to find two prime numbers inside a given inclusive range [left, right] such that the difference between them is as small as possible.
The problem presents a directed graph with n nodes, where each node has exactly one outgoing edge defined by the array edges. Specifically, edges[i] indicates that there is a directed edge from node i to node edges[i].
The problem is asking us to maximize the number of unique elements we can have in a set after removing exactly half of the elements from two arrays nums1 and nums2.
We are given a string representing a Martian time in the format "a:b", where a is the hour component and b is the minute component. Unlike Earth time in base 10, these strings could represent numbers in any numeral system with a base greater than 1.
That is a long, structured technical guide with multiple required sections and two full implementations. To keep the quality high and follow your formatting rules exactly, I will provide it in a single comprehensive response.
This problem asks us to determine the maximum length of a non-decreasing array that can be obtained from a given integer array nums by performing a sequence of subarray sum operations.
The problem presents a scenario where Igor wants to paint a number on a fence using a limited amount of paint. Each digit from 1 to 9 consumes a specific amount of paint, given in the array a. Igor cannot use zero.
We are asked to model a simple randomized iterative process. We start with a number $x$ and perform $k$ steps. In each step, a random number between 1 and 100 is drawn. With probability $p/100$, the current number is doubled; otherwise, it is incremented by 1.
The problem asks us to transform a binary string s into an alternating string using the minimum number of flip operations (Type-2), while optionally performing any number of rotations (Type-1).
This problem asks us to determine the number of possible sets of branches that can be closed such that the maximum distance between any pair of remaining open branches does not exceed a given maxDistance.
We are tracking a runner moving along the perimeter of a square of side length a, starting from the bottom-left corner (0, 0) and moving counter-clockwise. After every fixed distance d, we want to know the runner’s exact coordinates on the square boundary.
The problem gives a queue of n people standing from left to right, where each person has a distinct height. For each person i, we need to determine how many people to their right they can see.
The previous solution fails immediately under a counterexample check. The proposed path was the union of the segments from to the midpoints of and . If the side length is , then the detector radius is The distance from to that path is , while Hence is not detected at all.
The problem asks us to analyze a directed graph where each node has exactly one outgoing edge, represented by the array edges. Each index i represents a node, and edges[i] represents a directed edge from node i to node edges[i].
We are asked to find the smallest angle with its vertex at the origin that contains all given points (mannequins) on a 2D plane. Each point is defined by integer coordinates, and no mannequin is located at the origin itself.
The problem asks us to enhance arrays such that any array can call a groupBy method with a callback function fn. This function will determine the key for grouping each element.
We are given an array of integers and a limited number of increment operations. Each operation increases a single element by one, and the same element can be increased multiple times.
The problem asks us to count how many substrings of a given string contain only unique characters. A substring is considered special if no character appears more than once inside that substring. We are given a string s containing only lowercase English letters.
We are given two lowercase strings. A string d is called a divisor of another string s if s can be formed by concatenating d several times in a row. For example, "ab" divides "ababab" because repeating "ab" three times gives the full string.
We are given an array of integers of length n and asked to determine if we can swap any two elements at distinct positions such that the resulting array is no longer sorted. Sorting here is defined broadly: the array is sorted if it is either non-decreasing or non-increasing.
This problem asks us to count how many connected components in an undirected graph are complete graphs. We are given an integer n, representing the number of vertices labeled from 0 to n - 1, and a list of undirected edges.
This problem gives us an m x n grid containing three types of cells: 1. Guard cells 2. Wall cells 3. Empty cells Each guard can observe cells in the four cardinal directions: - Up - Down - Left - Right A guard continues seeing cells in a direction until the view is blocked by…
We are given a group of n friends who want to play multiple rounds of Mafia, but in each round only one person acts as the supervisor while the remaining n−1 players participate as regular players.
We are given several drinks, and each drink already contains some percentage of orange juice. Vasya mixes the same amount from every drink into one large cocktail. The task is to compute the final percentage of orange juice in the mixture.
The problem gives us an integer array nums, where every value is positive. We need to determine which elements appear most frequently, then return the total number of occurrences contributed by all such elements.
This problem is asking us to simulate a progressive tax system. We are given a list of tax brackets where each bracket specifies an upper bound of income and a tax percentage.
This problem asks us to determine the maximum "beauty" of an array after performing a set of allowed operations. Specifically, the beauty is defined as the length of the longest subsequence consisting of equal elements.
This problem asks us to partition an array nums into exactly m contiguous subarrays such that the bitwise AND of elements in the i-th subarray equals andValues[i].
This problem asks us to find the airport(s) with the most traffic using a table of flights. Each row of the Flights table represents a direct connection from departureairport to arrivalairport along with the number of flights flightscount for that route.
That is a long, structured technical guide. To make sure I target the correct problem and avoid wasting your time, can you confirm the exact LeetCode problem title for 2762?
In this problem, we are given a document with three components: an identifier, a title, and the main text. Each document belongs to one of three subjects, labeled 1, 2, or 3. The task is to determine the correct subject for any given document.
This problem asks us to count all ordered triples (a, b, c) such that: - 1 <= a, b, c <= n - a² + b² = c² This is the classic Pythagorean theorem relationship. Any triple satisfying this condition is called a square triple in the problem statement.
The Votes table records every voter-candidate selection. A voter may vote for multiple candidates, vote for exactly one candidate, or choose not to vote at all. The important detail is that every voter contributes a total voting weight of exactly 1.
Edit This problem gives us a perfect binary tree with n nodes, where each node has an associated cost. The tree follows a strict indexing rule: node i has a left child at 2 i and a right child at 2 i + 1.
We are given a binary matrix grid with m rows and n columns. Every cell contains either 0 or 1. We may flip any cell, meaning we can change 0 to 1 or 1 to 0. The goal is to perform the minimum number of flips such that two conditions become true simultaneously: 1.
We are given an undirected weighted tree with n nodes and n - 1 edges. Since the graph is a tree, there is exactly one simple path between any pair of nodes, and there are no cycles. Each edge has a weight, and we may remove any number of edges.
We are asked to count the number of distinct substrings of a string s that satisfy a set of rules. Each rule consists of a string p and a range [l, r]. A substring t of s is good if, for every rule (p, l, r), the number of times t occurs in p lies within the given range.
We need to construct the lexicographically smallest lucky number consisting only of digits 4 and 7 such that four substring counts match given values.
The problem asks whether it is possible to transform a given integer array nums into a Zero Array, where all elements are zero, using a series of decrement operations defined by queries.
We are tasked with transporting robots across a fixed distance using a limited fuel supply. Each robot has three characteristics: the number of other robots it can carry, the amount of fuel it consumes to move on its own, and the maximum distance it can travel.
The problem gives us an m x n grid where each cell contains a small integer value. We start at the top left corner (0, 0) and must reach the bottom right corner (m - 1, n - 1) by moving only right or down. Along every path, we compute the XOR of all visited cell values.
This problem asks us to determine the minimum cost to make all elements in an array equal. You are allowed two types of operations: increment a single element at a cost of cost1, or increment any two distinct elements simultaneously at a cost of cost2.
We are given a string and an integer k. We may change any characters we want, and the goal is to transform the string into a concatenation of at most k palindromes while minimizing the number of modified characters. The partition boundaries are not fixed.
The problem presents a scenario where we have a list of positive integers nums representing targets placed on a number line. We also have an integer space representing the step interval of a machine.
We are given two polynomials, $P(x)$ and $Q(x)$, written in descending powers of $x$. The task is to compute: $$lim{x to infty} frac{P(x)}{Q(x)}$$ The input gives the degrees of the two polynomials and all coefficients from the highest-degree term down to the constant term.
The problem asks us to transform a given integer array nums into a beautiful array using the minimum number of deletions. A beautiful array satisfies two conditions: its length is even, and no two consecutive elements at even indices are equal (nums[i] !
We have a square campus with coordinates from (0, 0) to (b, b). Inside the square there are several directed segments, called arrows. Every arrow is either horizontal or vertical, and no two arrows intersect or even touch.
This problem gives us a list of directed pairs, where each pair [starti, endi] represents a directed edge from starti to endi. We must rearrange all pairs so that adjacent pairs connect correctly.
The problem gives us a string s consisting only of lowercase English letters. We want to completely delete the string using the maximum possible number of operations. In a single operation, we have two possible actions: 1. Delete the entire remaining string immediately. 2.
The problem asks us to determine the minimum number of keypresses required to type a given string s using a keypad with 9 buttons. Each button can map to at most 3 letters, and each letter must be mapped to exactly one button.
The problem provides a database table named Students, where each row represents a student and contains three fields: - studentid, the unique identifier for the student - departmentid, the department the student belongs to - mark, the student's exam score The goal is to compute…
We have a rectangular grid representing the park. Some cells initially contain spiders, and each spider permanently moves in one fixed direction: left, right, up, or down. Every second, a spider moves to the adjacent cell in that direction.
The problem provides a table Rides that records rides between drivers and passengers. Each row has a unique rideid, a driverid, and a passengerid. The task is to determine, for each driver, how many times that driver has appeared as a passenger in the table.
The factory has a row of lights indexed from left to right. Every light starts in the “on” state. Mashmokh performs a sequence of button presses, and each button has an index that determines how far to the right its effect extends.
The problem gives us two positive integers, n and k. We are allowed to perform one specific operation on n: choose any bit that is currently 1 in the binary representation of n and change it to 0.
We are asked to determine how many problems a team of three friends will attempt during a programming contest. Each friend independently has confidence about each problem, represented as a binary value: 1 if the friend is confident they know the solution, 0 otherwise.
We maintain a dynamic array of integers where three kinds of updates and queries are mixed together in arbitrary order.
This problem is asking us to determine the longest sequence of consecutive wins, also called a winning streak, for each player in a match history. The input is a Matches table with three columns: playerid, matchday, and result.
This problem asks us to determine how many distinct categories exist among n elements. The elements are labeled from 0 to n - 1, but we are not given the category values directly.
We are asked to simulate a waterfall that flows from the top of a wall to the bottom across horizontal panels. Each panel is a horizontal segment at some height, spanning a range along the x-axis.
We are asked to determine the number of days the opposition can walk along a boulevard of tiles before it becomes impossible due to the tiles being destroyed by rain. Each tile has a durability measured in days.
This problem introduces a generalized version of the bitwise OR operation called the K-or. In a normal bitwise OR, a bit in the result becomes 1 if at least one number has that bit set. The K-or operation changes this rule.
We are dealing with a very large directed walk-counting problem on a graph that is heavily structured but too large to ever build explicitly. There are $2m$ islands, and DZY starts from a home node. From home, he can move to island $i$ in $ai$ different ways.
We are given a tree with n junctions and n - 1 roads. Every road has an integer beauty value. We must choose two junctions so that the path between them contains between l and r edges inclusive.
The problem asks us to evaluate a full binary tree where each node represents either a boolean value or a boolean operation. Leaf nodes have values 0 (False) or 1 (True). Non-leaf nodes have values 2 (OR) or 3 (AND).
The infinite string in this problem is built recursively. Start with: In general: The alphabet contains 36 symbols: The full string after 30 steps has length: $ For k = 30, the length is about 10^9, which matches the input bounds.
The problem gives us an integer array nums and asks for the length of the longest contiguous subarray whose first element is strictly greater than its last element. A subarray is any contiguous segment of the array. For a subarray nums[i...
The problem gives us a list of items, where each item has two properties: - price - beauty Each entry looks like: We are also given a list of queries.
This problem asks us to count how many subsets of the input array have a product that can be written as a multiplication of distinct prime numbers. The phrase "distinct prime numbers" is the key restriction.
We have a meeting log for a team of developers where each log entry records either a user logging in or logging out. The log may start or end in the middle of the meeting, so we do not know who was online before the first recorded message.
This problem asks us to determine whether each passenger's ticket is confirmed or placed on the waitlist, based on the booking order and the capacity of the flight they booked.
We are given two 1-indexed arrays: - nums, where nums[i] represents how many decrement operations index i still needs before it becomes zero. - changeIndices, where changeIndices[s] tells us which index is eligible to be marked at second s. Initially, every index is unmarked.
This problem requires us to maximize the number of matchings between players and trainers under the condition that a player's ability cannot exceed the trainer's training capacity. In other words, a player i can only be assigned to trainer j if players[i] <= trainers[j].
The problem asks us to count how many permutations of the numbers [0, 1, 2, ..., n - 1] satisfy a collection of inversion constraints on prefixes.
We have several jobs, each with a start time, a duration, and a profit. A machine that starts a job stays occupied for the entire interval from s through s + t - 1. At most k jobs may run simultaneously because we only own k machines. The goal is not to schedule all jobs.
The problem asks us to calculate the minimum amount of time required for a team of workers to reduce the height of a mountain to zero, given that each worker has a specific time cost for performing work and the total work time grows linearly for consecutive units of height.
Each employee follows a rigid repeating schedule. Once hired on day x, the employee works for n consecutive days, then rests for m consecutive days, then repeats forever. So the cycle length is n + m, and inside each cycle the employee is active for the first n days.
The problem asks us to examine every possible non-empty subarray of the given array nums. For each subarray, we compute how many distinct values appear inside it. After finding this distinct count, we square it and add it to the final answer.
The problem asks us to work with a conceptual infinite array called bignums, which is generated by taking every positive integer i, converting it to its powerful array (the sorted array of powers of two that sum to i), and concatenating all these arrays sequentially.
This problem gives us a 0-indexed array nums containing non-negative integers. We must perform a sequence of operations on adjacent elements, then rearrange the resulting array by moving all zeros to the end. The first phase consists of processing the array from left to right.
The graph family is built recursively. D(0) is a single vertex. D(1) is a single edge. For every larger order, D(n) is formed by taking a copy of D(n-1) and a shifted copy of D(n-2), then connecting them with two extra edges.
The problem requires transforming a given integer array nums into an alternating array with the minimum number of operations.
This problem asks us to determine the largest price that cannot be formed using an unlimited number of coins of two given prime denominations. We are given two distinct prime numbers, primeOne and primeTwo.
This problem asks us to determine if there exists a path in a binary matrix from the top-left corner (0, 0) to the bottom-right corner (m - 1, n - 1) such that the number of 0s visited along the path is equal to the number of 1s.
The problem gives us an Orders table where each row represents a purchase made by a customer. Every order has an orderid, a customerid, an orderdate, and a price.
We have two uppercase strings of equal length. We may replace characters in the first string, and after all replacements the final string only needs to be an anagram of the second string.
The problem asks us to determine the minimum time required for a fleet of buses to collectively complete a given number of trips. Each bus in the fleet has its own fixed trip duration, denoted by the array time.
The problem asks us to remove all trailing zeros from a string representing a positive integer. The input num is a string consisting only of digits with no leading zeros, and the length of the string is between 1 and 1000.
This problem gives us an undirected graph with n nodes and a list of edges. A connected trio is a group of exactly three distinct nodes where every pair of nodes has an edge between them. In graph theory terms, this is simply a triangle.
We are given two strings, s and t, and we are asked to count how many distinct pairs (x, y) exist such that x is a substring of s, y is a subsequence of t, and x and y are equal as strings. The key distinction is in how “distinct” is defined.
In this problem, we are asked to simulate Vasya's prize redemption strategy. Vasya collects points from chocolate bar wrappings over time. Each wrapping contributes a certain number of points, and the points accumulate sequentially.
We are given an integer that represents a bank balance. This balance can be positive or negative, and we are allowed to perform at most one modification operation that consists of removing a single digit from the number.
The problem asks us to compute a special value called the x-sum for every contiguous subarray of length k. For each window of size k, we first count how many times each number appears. After that, we only keep the contributions of the top x most frequent distinct values.
The problem asks us to find the maximum area rectangle that can be formed from a given set of points on a 2D plane, under strict constraints.
The problem asks us to compute an alternating sum of the digits of a positive integer. The alternation starts from the most significant digit, which always has a positive sign. Every following digit flips the sign from the previous one.
The problem gives us an n x n matrix called grid, where every number from 0 to n n - 1 appears exactly once. Each value represents the order in which a knight visited that cell during a tour of the chessboard. A knight in chess moves in an L-shape.
This problem asks us to determine the minimum number of moves required to identify a critical floor f in a building with n floors, using exactly two eggs. The critical floor f has the following meaning: - Any egg dropped from a floor higher than f will break.
That is a long, multi-section technical guide with code, worked examples, test cases, and detailed explanations. I can provide the full reference document, but it will be quite large.
This problem asks us to compute, for every possible subarray size, the best possible minimum value among all subarrays of that size. Given an array nums of length n, we must evaluate every window size from 1 to n. For each size k, we consider all contiguous subarrays of length k.
The problem gives us an integer primeFactors, representing the maximum total number of prime factors we are allowed to use when constructing some positive integer n. The goal is not to maximize n itself. Instead, we want to maximize the number of "nice divisors" of n.
In this problem, we are given a list of stock prices over different days. Each element in stockPrices is a pair: This represents a point on a 2D graph where: - The X-axis is the day - The Y-axis is the stock price on that day The line chart is formed by connecting consecutive…
We are given an array of length (2·n - 1). Yaroslav can perform an operation any number of times where he selects exactly n elements and multiplies each by -1. Our task is to determine the maximum sum achievable by applying this operation optimally.
The problem is asking us to implement a function partial that takes a target function fn and a list of arguments args. Some of these arguments may be placeholders represented by the string "".
This problem gives us two arrays: - processorTime, where each value represents the time when a processor becomes available - tasks, where each value represents how long a task takes to execute Each processor has exactly 4 cores, and each core can execute exactly one task.
We are given a list of submissions made by participants in a programming contest. Each submission is described by two numbers: x, which counts how many unique solutions this participant had already submitted before this one, and k, the participant's ID.
The problem asks us to determine, given a positive integer num, how many of its digits evenly divide num. In other words, for each digit d in the number, we check if num % d == 0.
The problem asks us to make a given integer array nums non-decreasing by performing a specific type of operation any number of times. Each operation consists of selecting an element and dividing it by its greatest proper divisor.
We are maintaining an initially empty square grid of size $n times n$, where every cell starts as zero. Two kinds of operations are performed over this grid. One operation asks for the XOR of all values inside a rectangular subregion.
The problem gives us a 50 x 50 chessboard containing exactly one knight and up to 15 pawns. The knight starts at position (kx, ky), and every pawn is placed at one of the coordinates in positions. Two players, Alice and Bob, alternate turns. Alice moves first.
The input describes a rooted binary tree using a parent array. Every node is identified by an integer from 0 to n - 1, and parents[i] tells us which node is the parent of node i. The root node is always 0, so its parent is -1. The goal is to compute a score for every node.
We are asked to construct a square matrix of size n × n, filled with zeros and ones, that satisfies two properties: it must be clear, meaning that no two ones are adjacent horizontally or vertically, and it must be symmetrical along both the horizontal and vertical axes.
This problem asks us to identify the top 5 percentile of insurance claims by fraud score for each state in a Fraud table. The table has three columns: policyid, state, and fraudscore.
The problem gives us two arrays, nums and removeQueries, both of length n. The nums array contains positive integers. Initially, all elements are present, forming one contiguous segment. Then, elements are removed one by one according to the order defined in removeQueries.
We are given an array of integers and need to support two types of queries efficiently. The first type requires adding Fibonacci numbers to a contiguous subarray: for indices from l to r, we add F₁ to the element at l, F₂ to the element at l+1, up to F{r-l+1} at position r.
You are given a binary matrix grid with m rows and n columns. Every cell contains either 0 or 1. A row is considered palindromic if reading it from left to right gives the same sequence as reading it from right to left.
We are given a 1-indexed array prices, where prices[i] represents the stock price on day i. We want to choose a subsequence of indices, not necessarily contiguous, such that the selected indices satisfy a special linearity condition.
We are given an undirected tree with n nodes labeled from 0 to n - 1. The tree is rooted at node 0, which means every node except the root has exactly one parent, and each node may have zero or more children. The task is to count how many nodes are considered "good".
We are given a scenario where Qwerty's ship has an initial magnetic gripper and is surrounded by n scattered grippers from two crashed ships. Each gripper has a location, a mass, a power, and a radius.
We start with an array of length n. One operation replaces every element with its prefix sum. After one operation: $$ai leftarrow sum{j=1}^{i} aj$$ After repeating this process k times, we must print the resulting array. The operation is linear and highly structured.
We start with an array of positive integers. For every element, we are allowed to decrease it by at most k, and the value must stay positive.
The problem provides a database table named Salaries with three columns: | Column | Meaning | | --- | --- | | empname | Employee name | | department | Employee department | | salary | Employee salary | The combination of (empname, department) is unique, meaning there are no…
In this problem, we are given two arrays, nums1 and nums2, each containing every integer from 0 to n - 1 exactly once. In other words, both arrays are permutations of the same set of values.
The problem asks us to determine whether every integer within a given inclusive range [left, right] is covered by at least one of the intervals specified in the ranges array.
Vanya has discovered a subset of cards from his original infinite collection, where each card carries an integer between -x and x. His goal is to balance the sum of these found cards to zero by potentially adding additional cards from the infinite supply.
The problem gives us a single floating point value named celsius, representing a temperature measured in degrees Celsius. Our task is to convert this temperature into two other temperature scales, Kelvin and Fahrenheit, and return both converted values in an array.
We have a person walking around the border of a square whose side length is n. He starts at the lower-left corner and places a cross there immediately. After that, he keeps moving clockwise along the perimeter, placing another cross every n + 1 meters.
We are given two integers, n and limit. There are exactly three children, and we want to distribute all n candies among them. If we let the number of candies received by the three children be: then the distribution must satisfy: with the additional restriction: for every child.
In this problem, every character in the game has two attributes: - attack - defense The input is a 2D array called properties, where: represents the stats of the i-th character.
We are given a set of domino tiles, each with two numbers on its halves. The task is to arrange all dominoes in a sequence so that the touching halves of adjacent dominoes have the same number. A domino can be flipped, which swaps its two numbers.
This problem asks us to count how many contiguous subarrays have a median equal to a given value k. The array nums contains every integer from 1 to n exactly once, which means all elements are distinct.
The problem presents n bags, each with a defined maximum capacity and a current number of rocks. The input consists of two arrays: capacity and rocks, where capacity[i] is the maximum number of rocks bag i can hold, and rocks[i] is how many rocks are currently in that bag.
The problem gives us an integer array nums and an integer d. We must count how many triplets of indices (i, j, k) satisfy two conditions: 1. The indices are strictly increasing, meaning i < j < k. 2. The sum of the corresponding elements is divisible by d.
We have a collection of apples where every apple weighs either 100 grams or 200 grams. The task is to split all apples into two groups so that both groups have exactly the same total weight. The input gives the number of apples and then the weight of each apple.
This problem asks us to invert the relationship between keys and values in a JSON object or array. The input can be either: - A JSON object, where keys are strings and values are strings. - A JSON array, where indices act as keys and elements are strings.
This problem asks us to split an input array into smaller subarrays, commonly called chunks, where each chunk has a fixed maximum size.
This problem asks us to determine the maximum number of consecutive robots that can be run without exceeding a given budget. We are given two arrays of length n: chargeTimes and runningCosts.
We are given a strictly increasing array of integers. We want to build the longest subsequence such that every pair of neighboring numbers shares at least one common prime factor.
Codeforces 251B: Playing with Permutations
This problem asks whether a given parentheses string can be transformed into a valid parentheses sequence under certain constraints. We are given two strings: - s, containing only '(' and ')' - locked, containing only '0' and '1' Both strings have the same length n.
We are given a single string representing a program written in the joke language HQ9+. Most characters do nothing, but four specific uppercase characters are meaningful. The instruction H prints "Hello, World!
The problem asks us to examine every possible non-empty subset of the given array nums and compute the bitwise OR value of each subset. Among all these OR values, we need to determine the maximum possible value, then count how many different subsets produce that maximum.
We are given a graph with n cities connected by m bidirectional roads, each with a positive length. The goal is to determine, for every pair of cities (s, t) with s < t, how many roads can appear on at least one shortest path from s to t.
The problem asks for the k-th smallest product that can be formed by multiplying one element from the sorted array nums1 with one element from the sorted array nums2. Both arrays may contain negative numbers, zeros, and positive numbers.
This problem asks us to implement a custom version of the array transformation operation, similar to JavaScript’s Array.map, but without using the built in Array.map method. We are given two inputs: 1. An integer array arr 2.
The problem asks us to analyze a database table named Files that contains two columns: filename and content. Each row corresponds to a unique file and its textual content.
We are given many queries. In each query, we take all numbers of the form $$k^{2l}+1, k^{2l+1}+1, dots, k^{2r}+1$$ and must compute their least common multiple modulo a prime $p$. The interval length can be enormous because $r$ may reach $10^{18}$.
The problem gives us a list of package sizes and several suppliers. Each supplier offers an unlimited number of boxes, but only in certain fixed sizes. We must choose exactly one supplier and pack every package using only the box sizes that supplier provides.
The problem describes a simulation of a robot moving on a rectangular grid defined by width x height. The robot starts at the bottom-left corner (0, 0) facing East and moves in discrete steps.
This problem presents a ball-passing game among n players, represented by an array receiver of length n. Each element receiver[i] indicates which player receives the ball when player i passes it. The game starts by selecting a player i as the first to hold the ball.
We have n people placed on a line at strictly increasing coordinates. PMP starts at person s, which immediately counts as visited. Every later move must go either strictly left or strictly right, depending on the ticket used for that move.
The problem gives us a string s made of two characters: - '' represents a plate - '|' represents a candle We are also given multiple queries, where each query specifies a substring of s using indices [left, right].
This problem asks us to transform a given string into a palindrome using the minimum number of adjacent swaps. A palindrome is a string that reads the same forward and backward. For example, "abba" and "racecar" are palindromes.
We are given a sequence of length $n$, where each position is either a fixed positive integer or unknown (marked as $-1$). The task is to interpret this sequence as being formed by concatenating several arithmetic progressions, one after another.
We are given an array nums of positive integers. We need to count how many contiguous subarrays satisfy a very specific condition: - The first element of the subarray and the last element of the subarray must both be equal to the maximum value inside that subarray.
We are asked to compute the sum of a number a and the reverse of another number b. Reversing a number means reading its digits from right to left and treating the result as an integer.
The problem is asking us to determine the minimum number of operations required to make all elements of an array equal, given a very specific operation. The operation allows choosing a prefix of the array, selecting any integer k, and adding k to every element in that prefix.
This problem asks us to repeatedly transform an array until only one number remains. At every round, we reduce the size of the array by half using alternating min and max operations on adjacent pairs.
This problem asks us to transform an array so that the bitwise XOR of all elements becomes exactly k, while minimizing the number of operations. An operation consists of selecting any element in the array and flipping exactly one bit in its binary representation.
We are asked to determine whether a ticket number is lucky. A lucky ticket number satisfies two conditions simultaneously. First, every digit of the number must be either 4 or 7.
The problem gives us a binary string s consisting of only '0' and '1' characters and an integer numOps representing the maximum number of bit flips we can perform.
The problem asks us to construct a string that contains three given strings a, b, and c as substrings while minimizing its length. If multiple strings satisfy the minimum length condition, the lexicographically smallest one must be returned.
The problem gives us n retail stores and an array quantities, where each element represents how many products exist for a particular product type. The important restriction is that a single store may contain products from only one product type.
Codeforces 403D: Beautiful Pairs of Numbers
This problem gives us n houses arranged on a number line and a collection of purchase offers. Each offer is represented as [start, end, gold], meaning a buyer wants to purchase every house in the inclusive range [start, end] and is willing to pay gold units of gold.
The problem asks us to figure out how many dragons get affected by a princess who has a unique way of defending herself. She targets every k-th, l-th, m-th, and n-th dragon with different actions. The total number of dragons is d.
That is a long, structured reference document with multiple sections, full prose explanations, worked examples, two complete implementations, test suites, and edge case analysis.
The problem gives us an integer array nums and asks us to count how many contiguous subarrays of length exactly 3 satisfy a specific mathematical condition.
In this problem, we are given a 2D binary matrix called room. Each cell represents a position in the room: - 0 means the space is empty and can be cleaned. - 1 means the space contains an object and cannot be entered.
The problem gives us an array nums representing the positions of marbles on a number line. Multiple marbles may exist at the same position, so nums can contain duplicates. We are also given two arrays, moveFrom and moveTo, which describe a sequence of move operations.
The problem gives us two integer arrays, nums1 and nums2, and both arrays are already sorted in non-decreasing order. Our task is to find the smallest integer that appears in both arrays. If there is no value shared between the two arrays, we must return -1.
The problem describes a line of n people, numbered sequentially from 1 to n. Initially, person 1 holds a pillow. Every second, the pillow is passed to the adjacent person. The direction of passing changes whenever the pillow reaches either end of the line.
The problem gives us an array called piles, where each element represents the number of stones in a pile. We are also given an integer k, representing the exact number of operations we must perform. In one operation, we choose a pile and remove floor(pile / 2) stones from it.
This problem asks us to transform a given string s into another string t such that the total transformation cost does not exceed k. Among all possible valid strings, we must return the lexicographically smallest one. The key detail is how the transformation cost is defined.
We start with an array of length n filled with zeros. One operation chooses a segment [l, r] and adds 1 to every element inside that segment. The final array is given.
We are given a polyline representing mountain peaks. The polyline has 2n + 1 vertices, with even-indexed vertices (2, 4, 6, ..., 2n) representing peaks. In the initial picture, each peak is strictly higher than its neighbors, i.e., for every even i, y[i-1] < y[i] y[i+1].
The problem is asking us to determine how many words in a given string can be fully typed on a malfunctioning keyboard. The keyboard has some broken letter keys that cannot be used.
We are given the title words of Lesha's new problem and several archive problems from Torcoder. The words in Lesha's title are all distinct. An archive title may repeat words.
We are given two strings representing the genomes of two dwarves. The goal is to determine whether these two genomes could belong to the same race under a very specific definition: the first genome can be transformed into the second genome by swapping exactly two characters in…
We have a directed graph where every edge enforces a strict relationship between the colors of its endpoints. If a vertex has color c, then every outgoing neighbor must have color c + 1, wrapping around modulo k.
This problem is long and deserves a detailed explanation to do it properly, especially with all requested sections, worked examples, Python and Go implementations, proof sketch, comprehensive tests, and edge cases.
The problem gives us an integer array nums and an integer target. We must sort the array in non-decreasing order and then return every index where the value equals target.
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The problem asks us to construct a polynomial equation from a database table called Terms. Each row in the table represents a single term in the equation, where power is the exponent of X and factor is the coefficient.
In this problem, we are asked to calculate the total number of cells in a star-shaped pattern drawn on a grid, given a number of layers. The input is a single integer a, representing the number of concentric layers in the star.
We are given an undirected graph where each vertex represents a horse and each edge represents mutual enmity. We must color every vertex with one of two colors, corresponding to the two parties, such that every horse has at most one enemy inside its own party.
We are given two integers, a and b. The task is deceptively small because the entire original statement is represented only by a picture.
The problem asks us to determine the minimum number of months required to complete all courses given prerequisite relationships and the time each course takes. Each course is labeled from 1 to n, and relations describes which courses must be completed before others.
The problem gives us a string s consisting only of lowercase English letters. We are allowed to replace characters in the string with other lowercase letters. Our goal is to transform the string into a palindrome. A palindrome is a string that reads the same forward and backward.
We are given a long string s and a list of pairs of small strings (ai, bi). For each pair, we need to find the shortest substring of s that contains both ai and bi as substrings. If no substring contains both, we report -1.
The problem gives us a target string and a very restricted keyboard with only two operations. The first key always appends the character "a" to the end of the current string. The second key changes only the last character of the current string to the next letter in the alphabet.
This guide will be quite long because you requested a comprehensive reference document with multiple sections, detailed walkthroughs, complete Python and Go implementations, worked examples, test cases, edge cases, and complexity analysis.
We are asked to distribute a certain number of nuts into boxes, and we are given a limited number of divisors that can split a box into multiple sections. Each box can have at most k sections, and each section can hold at most v nuts. We have a total nuts and b divisors.
This is a Type B - “Prove that / determine a value” problem. The task is to deduce a uniquely determined numerical quantity from the hypotheses.
The problem asks us to find the largest integer num such that the accumulated price of all numbers from 1 to num does not exceed a given threshold k.
We are given a set of students at a university, each with a record of how many times they have already participated in the ACM ICPC world championship.
The problem gives an integer array nums of length 3, where each value represents the length of a side of a possible triangle. Our task is to determine what kind of triangle these three sides can form.
We have an array of friend attractiveness values. Every unordered pair of distinct friends produces one possible picture, and the value of that picture is the xor of the two attractiveness values.
The problem gives us two arrays, reward1 and reward2, where each index represents a specific type of cheese. Every cheese must be eaten by exactly one of the two mice. If cheese i is eaten by the first mouse, we gain reward1[i] points.
We are asked to minimize the number of steps required to reduce a given integer n to zero. Each step consists of choosing a digit from the current number and subtracting it from that number.
This problem asks us to create an immutable wrapper around a JSON object or array. The returned structure must behave exactly like the original data for read operations, but any attempt to mutate it must immediately throw a string error with a very specific format.
We are given a string consisting of uppercase and lowercase English letters, and an integer k. The task is to modify the string so that exactly k characters become uppercase. The relative order of characters must stay unchanged, only the letter cases may change.
We start with an array a and must perform exactly u operations. Every operation is one of two types. The first type applies a bitwise xor independently to every position: $$ai leftarrow ai oplus bi$$ The second type simultaneously permutes the array using permutation p and…
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This problem asks us to design a custom Bitset data structure that supports several operations efficiently. A bitset is simply a sequence of binary values, where each position stores either 0 or 1.
The problem is asking us to reverse-engineer an array arr from its prefix XOR array pref. Specifically, pref[i] represents the XOR of all elements in arr from index 0 to i. Our goal is to reconstruct arr given only pref.
This problem asks us to compute the number of valid infection sequences in a line of n people, where some people are initially infected. The array sick represents the indices of people who are already infected at the start.
The problem gives us a string word and an integer k. Every second, we are forced to perform two operations in sequence: 1. Remove the first k characters from the string. 2. Append any k characters to the end of the string. The appended characters are completely under our control.
This problem gives us a Pandas DataFrame named weather with three columns: - city, the name of a city - month, the name of a month - temperature, the recorded temperature for that city during that month The task is to reshape the table using a pivot operation.
The problem asks us to determine the number of houses on a circular street where we can only interact with the street through the provided Street interface. Each house has a door that can either be open or closed.
The problem asks us to determine the latest time we can arrive at a bus station to catch a bus, given the departure times of buses, arrival times of other passengers, and the maximum capacity of each bus.
The array contains every integer from 0 to n - 1 exactly once. The value 0 represents the empty space, while every other number represents an item that should eventually appear in sorted order. The operation is unusual compared to normal array sorting problems.
We are given a word consisting of distinct lowercase English letters. We are allowed to completely redesign the mapping of letters onto the telephone keypad keys 2 through 9. There are 8 available keys (2 to 9).
The problem gives us a string s that represents events happening in a waiting room over time. Each character corresponds to one second. If the character is 'E', one person enters the room and occupies a chair. If the character is 'L', one person leaves the room and frees a chair.
We are given an array of integers that is supposed to be sorted in non-decreasing order. The Little Elephant suspects that at most one swap operation may have disturbed the array.
This problem asks us to count servers that did not receive any requests within a certain time window for multiple queries. You are given n servers, each with a unique ID from 1 to n.
The problem gives us a square matrix grid of size n x n. We must determine whether this matrix satisfies the definition of an X-Matrix. An X-Matrix has a very specific structure.
The problem asks us to find the maximum possible sum of a contiguous subarray where the absolute difference between the first and last element of that subarray is exactly k. More formally, for a subarray nums[i..
The problem describes a two player game played on a numeric string of even length. The string contains digits and possibly some '?' characters. Alice and Bob alternate turns replacing one '?' with a digit from '0' to '9'. Alice moves first. At the end of the game, every '?
The problem asks us to count the number of distinct quadruplets (a, b, c, d) in a given array nums such that the sum of the first three elements equals the fourth, i.e., nums[a] + nums[b] + nums[c] == nums[d], and the indices satisfy a < b < c < d.
The problem asks us to split the array nums into one or more contiguous non-empty subarrays such that the total cost is minimized. For every subarray, we define a special quantity called its importance value.
We are given the final results of a programming contest. Each team has two values attached to it: how many problems it solved and its total penalty time. The ranking rule is the standard ICPC-style ordering. A team ranks higher if it solved more problems.
In this problem, two players, Nikephoros and Polycarpus, play multiple rounds of rock-paper-scissors. Each player has a fixed sequence of moves that they cycle through as the rounds progress.
The problem asks us to find the smallest integer x such that: 1. x = n 2. Every bit in the binary representation of x is set to 1 A number whose binary representation contains only set bits looks like this: - 1 → binary "1" - 3 → binary "11" - 7 → binary "111" - 15 →…
We are given two people, Alice and Bob, each of whom has caught a collection of fish. Every fish belongs to one of k species, and species numbers are ordered so that species with a larger index are guaranteed to be at least as heavy as those with a smaller index.
The problem provides two arrays, A and B, each a permutation of integers from 1 to n. A permutation means each number from 1 to n appears exactly once in the array.
The problem is asking us to analyze call records from a Calls table and determine the peak calling hour for each city. Each row in the table contains a callerid, recipientid, a timestamp (calltime), and the city where the call originated.
This problem asks us to find the k-th lucky number, where a lucky number is defined as an integer consisting only of the digits 4 and 7. For example, the sequence of lucky numbers in increasing order starts as 4, 7, 44, 47, 74, 77, 444, and so on.
The task gives a tree of hotels, where every road connects two hotels and moving along a road costs one unit of time. For each query, two hotels are chosen as main event locations.
The problem asks whether it is possible to split an input array nums into exactly n arrays of size one using a series of valid splits. Each split must take an existing array of length at least two and divide it into two good arrays.
We are given an undirected tree with n nodes. Every node has a price value. Since the graph is a tree, there is exactly one simple path between any two nodes. The problem allows us to choose any node as the root of the tree.
The problem gives us an integer array nums, and we must divide the array into one or more contiguous subarrays. Every element must belong to exactly one subarray.
We are given an integer array nums and several queries. Each query specifies a subarray using two indices [fromi, toi]. For every query, we must determine whether the subarray nums[fromi..toi] is a special array.
We are given a string num consisting only of digits. We may rearrange the digits in any possible way, but we only count distinct permutations. A permutation is considered balanced if the sum of the digits placed at even indices equals the sum of the digits placed at odd indices.
We are given a rooted forest describing family relations. Every person has a name and at most one parent. Multiple roots are allowed because some people may have no ancestor at all.
LeetCode 2537: Count the Number of Good Subarrays (Medium)
We are
We are asked to simulate the movement of a snake on a small grid and determine the minimal number of moves required for the snake to reach an apple. The grid contains walls, empty squares, the snake itself, and one apple.
We are given two integers, s and k. The task is to express s as a sum of distinct numbers taken from the k-bonacci sequence. The sequence behaves like Fibonacci, but instead of summing the previous two values, each term is the sum of the previous k terms.
The input is a binary image, essentially a grid where each cell is either background or part of a drawn object. Inside this grid, there are multiple disjoint shapes called suns.
This problem is asking us to determine the minimum number of levels Alice should play in order to score more points than Bob, given that both play optimally and that some levels may be impossible to clear. The input is a binary array possible of length n.
In this problem, Bob must break a collection of locks, where each lock requires a certain minimum amount of energy before it can be destroyed. The input array strength represents these requirements.
The problem asks us to append k unique positive integers to an existing array nums such that none of the integers we add are already in nums, and the sum of the numbers we append is minimized.
The problem asks us to count subarrays in an integer array nums where the maximum element of the subarray occurs at least k times.
The problem requires decoding a string that was encoded using a slanted transposition cipher. In this cipher, the original text is written diagonally in a grid with a fixed number of rows.
We have a row of balloons placed at increasing positions on a line. Each balloon has a pressure endurance, which limits how large its radius can grow. We inflate balloons sequentially from left to right.
This problem gives us a fixed 3 x 3 grid containing only two possible characters, 'B' for black and 'W' for white. We are allowed to change the color of at most one cell.
We are given a list of measurement results from a physics experiment. Vasya wants to keep as many measurements as possible, but the remaining set must satisfy one condition: the largest remaining value cannot be more than twice the smallest remaining value.
This problem asks us to search for a rectangular submatrix inside a larger integer matrix, where the submatrix follows a pattern described by digits and lowercase letters.
The orbit contains n stations placed uniformly on a circle. Adjacent stations are exactly m miles apart, so the whole circumference equals n m.
Here is a comprehensive, reference-quality solution guide for LeetCode 2463 - Minimum Total Distance Traveled following your formatting requirements.
We are given a main string $s$. We want to count how many different substrings of $s$ are “valid” under a set of constraints. Each constraint gives us another string $p$ and two integers $l$ and $r$.
The problem gives us two arrays, present and future, where each index represents a stock. The value present[i] is the price of buying the i-th stock today, while future[i] is the price at which the same stock can be sold one year later.
We have several independent piles of coins. A move picks one pile with size x and replaces it with some smaller value y such that $$x^{1/4} le y le x^{1/2}$$ and y < x. The player who cannot make a move loses.
The problem requires identifying the largest "good" integer from a given string of digits. A "good" integer is defined as a substring of length 3 where all digits are identical.
We are given a collection of strings words and a target string target. A string is considered valid if it is a prefix of at least one word in words. This means that for every word, all of its prefixes are available for use.
This problem asks whether it is possible to reach a specific target cell (fx, fy) from a starting cell (sx, sy) in an infinite 2D grid in exactly t seconds. Each second, movement must occur to one of the eight adjacent cells, including diagonals.
A bus travels through a fixed sequence of road segments. Between consecutive segments there are traffic lights, and every light follows the same synchronized cycle. Each cycle lasts g + r seconds.
We are given a permutation of numbers from 1 to n. We want to count all sets of positions that can be represented as two non-overlapping segments and whose values form a consecutive range of integers.
We have two binary grids. A shift (x, y) means that cell (i, j) in the first grid is compared with cell (i + x, j + y) in the second grid. The score of a shift is the number of positions where both cells exist and both contain 1.
We are given a directed acyclic graph with vertices numbered from 1 to n, where every edge goes from a smaller-numbered vertex to a larger-numbered vertex. Some edges already exist, and we are allowed to add more edges under certain constraints.
The problem is asking us to count the number of ways to paint an m x n grid using three colors: red, green, and blue, with the constraint that no two adjacent cells can have the same color. Adjacent cells include both vertically and horizontally neighboring cells.
The problem gives us a square matrix nums of size n x n. We need to examine the values that appear on the two diagonals of the matrix and return the largest value among them that is prime.
We are given two strings, word1 and word2. We need to count how many substrings of word1 are considered valid. A substring is valid if its characters can be rearranged so that word2 becomes a prefix of the rearranged string.
This is a Type B, “Prove that” problem. The statement to prove is: Among all triangles determined by 100 points in general position, at most 70% are acute. The proposed proof attempts to establish a universal upper bound on the number of acute triangles.
Codeforces 261D: Maxim and Increasing Subsequence
We have a line of prisoners, and each prisoner has a crime severity value. We must choose exactly c consecutive prisoners for transfer. Every prisoner inside the chosen segment must have severity at most t.
This is a Type C - Optimization problem. The task is to determine the maximum number of intersections among perpendiculars drawn from each of five points to lines formed by the other four points.
This problem asks us to write an SQL query that calculates a special bonus for every employee in the Employees table.
This problem asks us to recursively filter a JSON-like structure that may contain nested objects and arrays. The input consists of two parts: - obj, which can be either: - a primitive value, - an array, - or an object containing nested arrays and objects - fn, a predicate…
We are asked to reconstruct the possible positions of a specific beaver in a queue based on partial ordering information. Each beaver either knows who should be directly in front of them in the line, or does not know, represented by zero.
This problem describes a circular passing game among n friends numbered from 1 to n. The ball always starts with friend 1, and each turn increases the number of clockwise steps by a multiple of k.
We are asked to model a journey along a straight line of cities connected by one-way roads, where a car travels at a constant speed of one kilometer per hour and consumes one liter of fuel per kilometer.
We are working with two dynamic rooted trees that start with a single node each. Operations are sequential and extend the trees by attaching a new child to an existing node with an edge labeled by a lowercase English letter.
The problem asks us to transform the integer 1 into the integer target using the minimum number of moves. At every move, we are allowed to perform one of two operations: 1. Increment the current number by 1 2.
We are given a string consisting of three possible characters: fixed black cells, fixed white cells, and unknown cells that we are free to assign either color.
The problem asks us to enhance JavaScript arrays with a method called upperBound(). Given a sorted array of numbers and a target value, the method should return the last index where the target appears. If the target does not exist in the array, the method should return -1.
We are given a container string p and a target message string s. The container is arbitrary, while the message contains the sequence we wish to extract. The extraction mechanism is governed by a binary key q of length k.
We are given a sequence of integers, a[1…n], and two parameters k and d. The goal is to find the longest contiguous subsegment of a that can be extended into an arithmetic progression with difference d by adding at most k elements.
We are given a collection of “bag capacities” $a1 < a2 < dots < an$, each capacity describing a special requirement: the bag can exactly accommodate some multiset of item weights whose sum is exactly that capacity, but it is forbidden to use a configuration whose total…
We have a two-player game played with two numbers on a blackboard, call them a and b. Each player can, on their turn, either replace the larger number with the remainder of dividing it by the smaller number, or subtract a positive multiple of the smaller number from the larger…
We work on the integer lattice inside a rectangle. Every point $(x,y)$ with $0 le x le w$ and $0 le y le h$ is available. We must count ordered triples of distinct lattice points that form a nondegenerate triangle whose area is an integer.
We need to count how many non-negative integer pairs (a, b) satisfy two equations at the same time: - a² + b = n - a + b² = m The input gives the two target values n and m.
The problem describes a simplified scoring system for a bowling game between two players. Each player has an array representing the number of pins hit in each turn, and there are exactly n turns.
The problem provides two integer arrays nums1 and nums2 of equal length, along with a target integer x. Each second, every element in nums1 increases by the corresponding element in nums2. After the increment, you can choose one index and set its value in nums1 to zero.
The problem asks us to find the largest possible group of numbers from the candidates array such that the bitwise AND of every number in that group is greater than 0. A bitwise AND operation only keeps bits that are set to 1 in every participating number.
Each pattern is a string made of lowercase English letters and the wildcard character ?. A wildcard can represent any lowercase letter. Two patterns intersect if there exists at least one concrete string that matches both of them. For example, a?
This problem asks us to maximize the number of times a two-character string pattern occurs as a subsequence in a given string text, after inserting exactly one character.
We are given a list of integers representing a sequence where each element is at least as large as the previous one but no more than double the previous one.
This problem asks us to analyze a binary tree and count how many nodes satisfy a specific property: the value of the node equals the sum of all its descendants. A descendant of a node is any node that lies in the subtree rooted at that node excluding the node itself.
This problem gives us the positions of three chess pieces on a standard 8 x 8 chessboard: - A white rook at (a, b) - A white bishop at (c, d) - A black queen at (e, f) We are allowed to move only the white pieces. The queen remains stationary.
This problem asks us to determine the minimum number of straight lines needed to cover a given set of points on an X-Y plane. Each point is defined by its (x, y) coordinates, and a straight line can pass through any number of points as long as they are collinear.
We are given two strings of equal length, both consisting only of the characters 4 and 7. The goal is to transform the first string into the second using the fewest operations. There are two allowed operations. We may flip a single digit, changing 4 to 7 or 7 to 4.
The problem asks us to build a series of towers along a coordinate line, where each tower has a height constrained by a given maxHeights array.
We are given an undirected connected graph representing a maze of rooms. DZY starts at room 1 with a fixed number of lives. Each time he is in a room, he randomly chooses one of its outgoing corridors uniformly and moves to the adjacent room.
This problem asks us to construct a set of positive integers such that: 1. Every number in the set has a units digit equal to k. 2. The sum of all numbers equals num. 3. We want the smallest possible number of integers in the set.
The problem gives us an m x n matrix called grid, where every cell contains a positive integer. We may begin from any row in the first column, meaning any cell (row, 0) is a valid starting point.
The problem requires tracking obstacles on an infinite 2D plane and answering, after each obstacle is placed, the distance to the k-th nearest obstacle from the origin (0, 0) based on Manhattan distance, defined as |x| + |y|.
We are given a line of points fixed at coordinates $(1,0), (2,0), dots, (n,0)$. Each point initially has a lowercase letter label, except that some of these labels were erased and replaced with question marks. All uppercase letters were also erased.
The problem asks us to count symmetric integers within a given range [low, high]. A symmetric integer is defined as an integer with an even number of digits, where the sum of the first half of the digits is equal to the sum of the second half.
We have n coin types, where coin type i has value a[i]. We want to count how many different multisets of coins sum to exactly t. The twist is that we are also given inequalities between coin counts.
This problem asks us to transform text stored in a database table while preserving the original formatting structure. The table usercontent contains two columns: a unique contentid and a contenttext string.
We are maintaining a mutable array of integers where two types of operations are performed repeatedly: range sum queries and range updates where every element in a subarray is XORed with a fixed value.
This problem asks us to implement a concurrency limiter for asynchronous operations. We are given an array called functions, where each element is itself a function. When one of these functions is called, it returns a Promise.
We are given a set of points on the plane representing the crystals that survived after part of a symmetric snowflake melted.
We are given a sequence of people standing in a line, indexed from 1 to n. We must split this line into k consecutive groups.
This problem asks us to determine how many indices in an array satisfy a very specific condition related to smaller values on both sides. For an index i to be considered k-big, two separate requirements must both hold: 1.
The problem asks us to find the minimum number of steps required to make two strings, s and t, anagrams of each other. An anagram means that both strings must contain the exact same characters with the same frequency, though their order can differ.
The problem gives us a zero-based permutation array called nums. A permutation means that every integer from 0 to n - 1 appears exactly once, where n is the length of the array.
This problem describes a painting scenario represented as a one-dimensional number line. Each element in the input array paint[i] = [starti, endi] represents the section that needs to be painted on the ith day.
This problem asks us to return the rows of the Olympic table in a very specific sorted order. Each row represents a country and the number of gold, silver, and bronze medals that country won in the Olympic games.
In this problem, we are given an integer array nums and an integer k. We may perform at most k operations, where each operation selects one element and multiplies it by 2. Multiplying by 2 in binary is equivalent to shifting all bits one position to the left.
We are asked to select soldiers from a battalion to participate in a beauty pageant over several days. Each soldier has a unique beauty value. On each day, we must send a group of soldiers whose combined beauty is unique compared to the other days.
The problem gives us a string called number, which represents a positive integer, and a character digit, which is guaranteed to appear at least once inside number.
We are given a number a and a target divisor b, along with a count n representing how many digits we want to append to a. Each operation consists of appending a single digit to the right of the current number such that the new number is divisible by b.
The problem asks us to take a circular linked list of positive integers and split it into two separate circular linked lists. The first list should contain the first half of the nodes, rounded up (ceil(length / 2)), and the second list should contain the remaining nodes.
The problem gives us a two dimensional matrix called grid with m rows and n columns. We must verify whether every cell satisfies two separate rules. The first rule applies vertically. For every cell, if there is a cell directly below it, both values must be equal.
We have a robot starting at the origin on a 2D grid. Several bombs are placed at distinct coordinates, and the robot must destroy all of them.
The problem is asking us to compute a special sum over a given array nums by repeatedly concatenating the first and last elements of the array until it is empty.
We are asked to determine how a buyer, constrained to coins whose values are powers of three (1, 3, 9, 27, …), could pay an amount n marks in such a way that he cannot pay n exactly and must overpay using the minimum number of coins possible.
We are given a list of phone numbers from friends in a city. Each phone number is a string of digits, and all numbers have the same length. The task is to find the city phone code, which Polycarpus defines as the longest common prefix shared by all these numbers.
The problem presents a long corridor represented as a string, where 'S' denotes a seat and 'P' denotes a plant. The goal is to partition this corridor into sections, such that each section contains exactly two seats and any number of plants.
Codeforces 263B: Squares
The problem is asking us to implement currying for a given function. Currying is a functional programming technique where a function with multiple parameters is transformed into a sequence of functions, each accepting a subset of the original parameters.
This problem asks us to take a binary array nums, which contains only 0s and 1s, and transform it so that all elements become 1 using the minimum number of allowed operations.
We are given an n×n grid where each cell contains an integer, which can be positive or negative. Furik starts at the top-left corner (1,1) and moves only right or down, while Rubik starts at the bottom-right corner (n,n) and moves only left or up.
The problem gives us a singly linked list and asks us to identify all of its critical points. A critical point is a node that is either a local maximum or a local minimum.
We are given a set of stations placed on a 2D grid. Moving between any two stations takes time proportional to their Manhattan distance multiplied by a constant factor $d$.
The problem asks us to calculate the difference between the highest and lowest total scores among students in a class. Each student has three assignment scores, and the total score is simply the sum of these three values.
The problem asks us to determine whether a given string s is good, meaning that all characters that appear in the string occur the same number of times.
The problem asks us to process an Orders table where each row contains a minute and the number of orders received during that specific minute. We are tasked with calculating the total number of orders in intervals of six consecutive minutes.
Polycarpus has a row of n marbles, each either red or blue, and he wants to count how many subsequences of these marbles form a zebroid, which is a sequence where the colors strictly alternate. A zebroid can be as short as one marble.
The problem presents a scenario in which Alice has an original array arr of length n consisting of positive integers. She chooses a positive integer k and generates two new arrays: lower and higher.
Each person remembers a single number, how many taller people stood before them in the queue. We no longer know either the original order or the actual heights. The task is to reconstruct any valid queue order together with heights that satisfy every person's remembered value.
The problem presents a simple two-player game between Alice and Bob using an array of positive integers. Each number in the array is either a single-digit number (1 to 9) or a double-digit number (10 to 99).
The problem presents a scenario where we have multiple piles of gifts, represented as an integer array gifts, with each element indicating the number of gifts in that pile.
The problem gives us a string s containing only uppercase and lowercase English letters. Our task is to find the greatest English letter that appears in both lowercase and uppercase forms somewhere in the string. The answer must be returned as an uppercase letter.
The problem describes an infinite two dimensional grid where every integer coordinate (i, j) contains an apple tree.
We are given up to one hundred thousand axis-aligned rectangles on a plane. Rectangles never overlap in their interiors, although touching at borders is allowed.
The problem asks us to determine the maximum possible minimum power that any city can have after optimally adding k new power stations to an existing configuration.
We are given a directed graph with n focus points, numbered from 0 to n - 1. Some focus points already contain magic crystals. These are the starting sources of magic energy. Directed runes represent one-way magic flow between focus points.
The problem requires displaying the first three rows of a given DataFrame named employees. The DataFrame contains four columns: employeeid, name, department, and salary.
The problem asks us to determine the number of ways a string of digits, num, can be split into a sequence of positive integers that are non-decreasing and have no leading zeros.
The problem provides a 2D grid representing a map of land and water cells. Each cell can either be land (value 0) or water containing a positive number of fish.
The problem provides two binary strings, s and target, of equal length n. You are allowed to perform a specific bitwise operation on s any number of times, which involves picking two distinct indices i and j and updating s[i] to s[i] OR s[j] and s[j] to s[i] XOR s[j].
We are given several independent sequences of integers, one sequence per scientist. Each sequence must be kept in its original internal order, but we are allowed to interleave these sequences arbitrarily when forming one global list.
The problem asks us to implement a function named createHelloWorld that returns another function. The returned function must always produce the string "Hello World" whenever it is called.
We are given an array nums and two different operations that can reduce the values of elements in the array. The first operation divides a number by 2 and rounds the result upward.
We have a string consisting only of digits 4 and 7. Two kinds of operations are performed on it. The first operation flips every digit in a segment. Every 4 becomes 7, and every 7 becomes 4.
In "Black Square," Jury must press on one of four vertical strips whenever a black square appears. Each strip has a fixed energy cost, measured in calories, for pressing it.
This problem asks us to determine the minimum number of coins required to acquire all fruits in a market where buying a fruit grants a special reward. You are given a 0-indexed array prices, where prices[i] represents the cost of purchasing the (i + 1)th fruit.
We are given a number as a string and must print it in a banking-style money format. The formatting rules combine several independent transformations. The integer part must contain commas every three digits, counting from the right.
You are given an array heights, where each index represents a building and the value represents that building's height.
We are given a sequence of recollections of Fridays the 13th. Each day is represented by a character: "0" for a normal day, "1" for a bad day, and "?" for an unknown day. Along with this sequence, we are provided a probability p that a "?" corresponds to a bad day.
We are asked to manipulate dates by adding or subtracting a number of days. The input consists of a date string in the "DD.MM.YYYY" format and an integer representing a shift in days, which can be positive or negative.
The problem asks us to find the substring of a given string s that maximizes a custom cost function. Each character in the string has a value: if it exists in the string chars, its value is taken from the corresponding index in the array vals; if it does not exist in chars…
The forum keeps a list of threads ordered by the time of their latest message. Whenever someone posts in a thread, that thread immediately moves to the front of the list. No other reordering happens. Initially the threads are ordered as 1, 2, 3, ..., n.
The problem asks us to select exactly p disjoint pairs from the array nums such that the largest difference among all chosen pairs is as small as possible.
The problem asks us to count how many ordered triplets of distinct indices (i, j, k) satisfy a very specific divisibility condition.
The input consists of a 2D array dimensions, where each element represents a rectangle. For a rectangle dimensions[i], the value dimensions[i][0] is its length and dimensions[i][1] is its width.
We are tasked with optimizing the gain of a biologist, SmallR, who can change the sex of each of her n dogs at a cost. Each dog has an initial sex, either female (0) or male (1), and a cost to change sex.
This problem asks us to redesign a telephone keypad so that typing a given word requires the fewest total key presses possible. A traditional telephone keypad contains keys 2 through 9, giving us exactly 8 available keys.
The problem asks us to partition a list of student grades into multiple ordered groups under two strict conditions. First, the total sum of grades in the i-th group must be less than that of the (i+1)-th group.
The problem asks us to count the number of pairs of non-empty disjoint subsequences from a given array nums such that the GCD of the elements in each subsequence of the pair is equal.
The graph in this problem is almost a simple path. Every vertex has degree at most 2, except possibly one special vertex that may have larger degree. A tree with this shape looks like several chains glued together at one center. Initially every edge is black.
The problem requires identifying all indices in an array nums that are "k-distant" from at least one occurrence of a given value key.
The problem is asking us to implement a food rating system that supports dynamic updates to the ratings of individual food items and allows querying for the highest-rated food for a given cuisine.
The problem is asking us to compute a new matrix answer based on a given m x n grid, where each cell in answer represents the absolute difference between the number of distinct elements on the diagonal above and to the left of the current cell, and the number of distinct…
We are working with a 3D grid of size $k times k times k$, where every cell is a unit cube in a larger cube. Each unit cube has up to six face-adjacent neighbors in the 3D grid.
I can provide the full guide, but I want to avoid giving you a technically incorrect reference document. For LeetCode 3009, there are two subtly different interpretations that materially change the optimal algorithm and correctness proof: - whether intersections at a chart…
We are given a hidden cyclic arrangement of the integers from 1 to n placed around a circle. From this arrangement, someone constructed a set of ordered pairs describing connections between values.
We are given a string s of even length, and we may swap any two characters any number of times. Since unrestricted swapping allows us to rearrange the string arbitrarily, the real task is to determine whether some permutation of the characters can form an anti-palindrome.
The problem gives us the head of a singly linked list whose length is always even. The nodes are grouped into pairs based on their indices: - Nodes at indices (0, 1) form the first pair - Nodes at indices (2, 3) form the second pair - Nodes at indices (4, 5) form the third…
This problem asks us to compute the bitwise OR of every possible subsequence sum of a given array. A subsequence is formed by choosing any subset of elements while preserving their original order.
The problem asks us to maintain a collection of IDs that is updated incrementally based on two arrays, nums and freq. Each index i represents a step.
This problem asks us to design a simplified event system similar to the one used in environments like Node.js or browser DOM events. The goal is to implement an EventEmitter class that supports two operations: 1. Subscribing callback functions to named events 2.
The problem gives us the root of a binary tree, where every node contains a positive integer value. Our task is to determine which level of the tree has the smallest sum of node values.
We have two permutations of the numbers from 1 to n. For every cyclic rotation of permutation b, we must compute how close the rotated permutation can align with permutation a. For a fixed pair of permutations, the distance is defined using matching values.
We are given a square grid of size $n times n$, where every cell contains either 0 or 1. The operation allowed is to permute the columns arbitrarily.
The problem requires simulating a repetitive transformation on a string word where, at each second, the first k characters are removed, and then any k characters are appended to the end.
The problem asks us to take an array of functions, each returning a promise, and a delay time ms. We are to return a new array of functions where invoking any function in this array returns a promise that behaves like the original promise but resolves or rejects only after an…
We are given a string consisting solely of opening and closing round brackets. The task is to determine whether this sequence is balanced, meaning it could represent a correct arrangement of parentheses in a mathematical expression.
The hero starts at point A, rides to point B, and then must continue toward point C. At point B, he is facing in the direction from A to B. We need to determine whether reaching C requires turning left, turning right, or continuing straight.
We are given a sequence of non-negative integers. For any contiguous subarray of this sequence, we can compute its bitwise OR. The problem asks for the number of distinct values obtained from all such subarrays.
This problem gives us a boolean expression containing only: - '0' and '1' - binary operators '&' and '|' - parentheses The expression is guaranteed to be valid, which means every operator has valid operands and every parenthesis is properly matched.
The problem asks us to manually implement the behavior of JavaScript's JSON.stringify for valid JSON values, without using the built in function itself.
This problem asks us to repeatedly reduce numbers in an array until the total sum of the array has been reduced by at least half, while using the minimum number of operations.
This problem asks us to create a memoized version of a function. Memoization is a technique where you store the results of expensive function calls and return the cached result when the same inputs occur again.
This problem asks us to find the earliest substring of length k whose polynomial rolling hash equals a given target value. The hash function is defined as: where: - val('a') = 1 - val('b') = 2 - ...
Polycarpus has a certain number of candies and a number of friends. He wants to distribute all the candies among his friends in such a way that every friend gets a positive number of candies, and the difference between the friend who receives the most candies and the friend…
We know how many times each player flipped a coin. Vasya flipped x times and Petya flipped y times. Every head gives one point, every tail gives nothing. Valera does not remember the exact final scores, but he remembers three facts.
We are given a directed acyclic graph of cities and one-way flights. Every flight initially takes 1 hour. We may independently change any flight duration to either 1 or 2 hours.
Vasya has to write at least n lines of code during one night. He starts with productivity v, meaning he writes v lines before the first tea break. After every break, his productivity drops by a factor of k, using integer division.
We are given a collection of coins, each with a positive integer value. The task is to choose a subset of these coins such that the total value of our chosen coins is strictly greater than the total value of the coins left for the other person.
The problem asks us to implement a function that checks whether a given value is an instance of a specified class or any of its superclasses.
The problem asks us to count how many arrays of length n satisfy a divisibility condition while keeping every value within the range [1, maxValue]. An array arr is considered ideal if: 1. Every element is between 1 and maxValue. 2.
We start with an array of integers and a target value x. We may append any number of extra integers to the array, and we want the median of the final array to become exactly x. The task is to compute the smallest number of added elements needed to make that happen.
This problem asks us to maximize the number of indices in an array that can be "marked" using a specific operation. You are given an integer array nums and can repeatedly pick two different unmarked indices i and j such that 2 nums[i] <= nums[j].
The problem asks us to determine how many strings in a given array words are prefixes of a target string s. A prefix of a string is defined as any substring that starts at the first character and continues for any length up to the length of the string itself.
This problem asks us to analyze email addresses stored in a database table and determine how many people belong to each unique email domain, but only for domains ending in .com.
We are given up to five axis-aligned rectangles on a plane. Each rectangle is defined by its bottom-left and top-right coordinates. No two rectangles overlap, although they may touch at edges or corners.
The problem gives us a matrix called grid with m rows and n columns. Every cell contains a unique integer from 0 to m n - 1. We may start from any cell in the first row and move downward one row at a time until we reach the last row. The movement rule is very flexible.
The task is to find the number of ways two turtles can move from the top-left corner of a grid to the bottom-right corner without meeting along the way, except at the start and the end. The grid has cells that are either free or blocked.
The problem asks us to construct a new matrix p from a given matrix grid. For every position (i, j), the value p[i][j] must equal the product of every element in the matrix except grid[i][j], and the result must be taken modulo 12345.
This problem asks us to maximize the frequency of the most common value in an array after performing at most k operations. Each operation allows us to choose any element and either increase or decrease it by exactly 1.
The problem asks us to count the number of complete subarrays in a given array nums. A subarray is complete if it contains all distinct elements that exist in the entire array.
The problem gives us two integers, n and t. We must find the smallest integer greater than or equal to n whose digit product is divisible by t. The digit product of a number is obtained by multiplying all of its digits together.
All criminals stand on the x-axis, and we must choose one integer coordinate for the police station. The patrol car starts from the station, visits some criminals, brings them back to the station, then repeats until everyone is arrested. Each trip can carry at most m criminals.
We are asked to maximize a geometric metric called "beauty" within a rectangle. The rectangle is axis-aligned and has width w and height h. Inside it, there are n given points with coordinates (xi, yi).
The problem gives us an array rewardValues, where each element represents a reward we may choose exactly once. We begin with a total reward x = 0, and we are allowed to repeatedly pick an unmarked element only if its value is strictly greater than the current total reward.
We are given an array prices, where each value represents the price of a chocolate. For every query [k, m], Bob wants to choose exactly m chocolates. The payment rule depends on the threshold k: - If a chocolate costs at most k, Bob pays the entire price.
The problem asks us to partition a given binary string s into the minimum number of substrings such that each substring is beautiful. A substring is beautiful if it represents a power of 5 in decimal and does not have leading zeros.
We are given an integer array nums, where each position represents a possible jump starting point. We begin at index 0 and must eventually reach index n - 1. From any index i, we may jump to any later index j where j i.
The problem gives us two inputs: - An array of strings called words - A string called pref We must count how many strings inside words start with the string pref. A prefix means the beginning portion of a string.
We are asked to count the number of distinct ways to perform exactly k painting moves on an n×n table with a black border.
This problem asks us to implement a custom version of Promise.all, but without using the built in Promise.all API. We are given an array called functions, where every element is an asynchronous function. Each function takes no arguments and returns a promise when executed.
We are given a multiset of characters in string s. We may rearrange these characters in any order, but we must use every character exactly once. Among all such permutations, we want the lexicographically smallest string that is still strictly larger than another string t.
This problem asks us to generate a league standings table from a database table named TeamStats. Each row in the table represents a football team and stores how many matches the team has played, won, drawn, and lost.
This problem provides a database table named cities, where each row represents a relationship between a state and one of its cities.
There are two independent tasks available during a contest that lasts from minute 0 up to minute t minus 1. Each task has a score that decreases linearly with time.
This problem asks us to determine whether a move in a board game is legal based on specific line rules. The board is an 8 x 8 matrix where cells can be empty '.', white 'W', or black 'B'.
We are given a rooted tree with n nodes numbered from 0 to n - 1. The tree structure is represented by the parent array, where parent[i] is the parent of node i. Since node 0 is the root, its parent is -1.
This problem models a scenario where you are traversing a train line with two parallel routes: a regular route and an express route. Each route has a series of consecutive stops, and the cost to move from one stop to the next is provided in the arrays regular and express.
The problem is asking us to purchase exactly two chocolates from a store given their individual prices, while ensuring that after buying them we do not end up in debt.
The problem asks us to find the smallest positive integer that is a multiple of both 2 and a given positive integer n. In other words, we are looking for the least common multiple (LCM) of 2 and n. The input n is guaranteed to be between 1 and 150, which is a very small range.
The problem is asking us to optimally pair students with mentors such that the sum of their compatibility scores is maximized. Each student and each mentor has answered n yes/no questions represented by 0s and 1s.
We are given a set of points on a 2D Cartesian plane. A point is called supercentral if there exists at least one other point directly to its left, one directly to its right, one directly above it, and one directly below it.
This problem asks us to modify a singly linked list by removing all nodes whose values appear in a given array nums. The input consists of two elements: an array of integers nums and the head of a linked list.
The problem presents a 0-indexed array nums of integers and a positive integer x. You start at the first element, nums[0], and can move to any subsequent position j where j i. For each element you visit, you accumulate its value as part of your score.
We are given a set of points in the plane, each with distinct coordinates, and we need to construct a rooted binary tree such that each node has at most two children. The arcs, which connect parents to children, must be directed strictly downward in terms of y-coordinates.
The problem asks us to find a subarray within a given array nums such that the absolute difference between the integer k and the bitwise OR of the subarray elements is minimized.
We are asked to model the growth of bacteria under a specific rule. In the first experiment, each bacterium multiplies by a factor of k every second and then b extra bacteria appear due to some abnormal effect.
We are given a permutation of numbers from 1 to n. For every query interval [l, r], we must count how many ordered pairs of positions (q, w) inside that interval satisfy: $$p[q] mid p[w]$$ Since all values are distinct and form a permutation, every number appears exactly once.
The problem presents an array nums of non-negative integers and an integer k. You are allowed to perform at most k operations, where each operation merges two adjacent elements using the bitwise AND operator.
The problem gives us a string s containing lowercase English letters and the character '?'. Every '?' must be replaced with a lowercase letter so that the resulting string has the minimum possible value.
We are given an m x n matrix called board, where each cell contains an integer value. We must place exactly three rooks on the board. A rook attacks every square in the same row and the same column.
We are given three integers. The length of the array cannot exceed n, every value must lie between 1 and m, and the number of distinct cyclic arrangements that satisfy a special adjacency rule must be between 1 and k. The adjacency rule defines a "good" array.
The problem asks us to count how many non-empty subsets of the array nums are considered "beautiful". A subset is beautiful if there are no two numbers inside the subset whose absolute difference equals k.
The problem asks us to find a beautiful pair of indices (i, j) in two integer arrays nums1 and nums2 of equal length. A pair (i, j) is beautiful if it minimizes the sum of the absolute differences between the corresponding elements: among all possible pairs where i < j.
We have a rectangular table with dimensions a × b and an unlimited supply of identical circular plates with radius r. Two players alternate placing plates on the table. Every plate must lie completely inside the rectangle, and plates may touch but cannot overlap.
The problem presents a rectangular basement of a store as an n × m grid, where some cells are walls, some are empty, and some contain sleeping rats.
The problem asks us to perform a series of replacement operations on an array of distinct integers. Specifically, we have an initial array nums containing n distinct positive integers. We are also given m operations, each consisting of a pair [oldValue, newValue].
This problem provides a database table named books. Each row in the table represents a single book and contains information such as the book's ID, title, author, publication year, and rating. The important detail is that the rating column can contain NULL values.
This problem provides a pandas DataFrame named students with three columns: | Column | Description | | --- | --- | | studentid | Unique identifier for a student | | name | Student name | | age | Student age | The task is to return only the name and age columns for the student…
We are given a binary string s of length n and an integer k. An operation consists of selecting any contiguous substring of length k and flipping every bit in that substring. A 0 becomes 1, and a 1 becomes 0. The operation may be applied any number of times, including zero times.
This problem asks us to calculate the percentage of immediate food delivery orders for each unique orderdate. The Delivery table contains one row per order. Each row includes: - deliveryid: A unique identifier for the delivery. - customerid: The customer who placed the order.
The Chamber of Secrets is represented as a grid of size n by m, where each cell is either empty or contains a column. A basilisk is stationed in the bottom-right corner and looks left, while a person trying to enter starts at the top-left corner and looks right.
In this problem, we are given a matrix named score with dimensions m x n. Each row represents a student, and each column represents an exam. The value score[i][j] is the score obtained by the ith student on the jth exam.
Each planet receives exactly a[i] suitcases, and every suitcase contains the same number x of presents. That means planet i starts with a[i] x presents. The Beaver always spends the first day on a planet without giving anything away.
The problem asks us to select a non-empty subset of integers from a given array nums such that the product of the numbers in the subset is maximized. Each number represents a student’s exam score, and the subset represents a group of students.
We are given a collection of segments on a number line. Each segment spans from a left endpoint to a right endpoint, and we must assign each segment one of two colors.
We are asked to count the number of ways to assign ballot numbers to 7 political parties in a zoo election so that the Little Elephant Political Party (LEPP) ends up with a “luckier” number than the sum of the lucky digits in the other six parties’ numbers.
The problem asks us to find the widest subarray (continuous segment) in two binary arrays nums1 and nums2 such that the sum of elements in that segment is equal for both arrays. Formally, we need to find indices (i, j) with i <= j such that: and maximize the distance j - i + 1.
The problem describes a line of people represented by a binary array team, where each index corresponds to a person. A value of 1 indicates a person who is “it”, and a value of 0 indicates a person who is not “it”.
The corridor is represented as a string where each character is either 'S' for a seat or 'P' for a plant. We already have fixed dividers at both ends of the corridor, and we may optionally place additional dividers between adjacent positions.
We are given a 2D integer matrix with n rows and m columns. The task is to find a rectangular submatrix where all elements are distinct and whose area (number of elements) is maximized.
The problem gives us an integer array nums, along with two integers, k and numOperations. We are allowed to perform exactly numOperations operations.
This problem asks us to double a number represented as a singly linked list. Each node of the list contains a single digit, and the digits are stored in the order from most significant to least significant. For example, the list [1,8,9] represents the number 189.
We are given an n×n grid representing a city built from unit cubes stacked in towers. Each cell of the grid contains an integer indicating the height of the tower at that location.
The problem gives us an array of lowercase strings called words. We need to find the longest word such that every prefix of that word also exists in the array. A prefix means the string formed by taking characters from the beginning of the word.
We can think of the company as a graph of employees. Two employees are directly connected if they share at least one language. Communication is allowed through intermediates, so if employee A can talk to B, and B can talk to C, then A and C are effectively connected as well.
We are tasked with filling a warehouse grid of size n by m with the maximum number of T-shaped turboplows. Each turboplow occupies five cells in a specific T pattern, but it can be rotated in any of four orientations.
This problem asks us to compute the maximum value of strings in an array according to a specific definition of value. Each string can either be entirely numeric or alphanumeric. If a string consists only of digits, its value is the integer it represents.
A group of friends wants to make identical toasts using three resources: soft drink, lime slices, and salt. Every toast consumes a fixed amount of each resource. The task is to determine how many complete toasts each friend can make before at least one resource runs out.
The problem presents a circular street where each house has a door that can be either open or closed. You start at an arbitrary house and can perform three actions: check if the door is open, close the door, or move to the next house in the circular street.
The problem gives us a string s, a target string sub, and a list of character replacement rules called mappings. Each mapping [old, new] means that a character old inside sub may be replaced with new.
The problem is about collecting coins located on the nodes of a tree with the minimum number of edge traversals. We are given a tree with n nodes, represented by an edge list, and an array coins indicating whether a coin is present at each node.
The problem gives us a two dimensional integer matrix called grid with m rows and n columns. Our task is to compute the width of every column independently. The width of a column is defined as the maximum string length among all integers appearing in that column.
We are given an integer array nums and an integer x. We need to find two elements whose indices are separated by at least x, and among all such valid pairs, return the minimum possible absolute difference between their values.
This problem asks us to insert a single digit x into a very large integer n, represented as a string, in a position that maximizes the resulting numerical value. The integer n can be either positive or negative.
The problem asks us to determine how many possible sequences of integers exist that match a given array of differences between consecutive elements, while staying within a specified inclusive range.
The problem asks us to determine whether two strings are "almost equivalent" based on the frequency of each lowercase English letter. We are given two strings, word1 and word2, both of the same length.
The input describes a tree of cities. Since there is exactly one simple path between every pair of cities, the road network forms a connected acyclic graph. Then we are given several pairs of cities.
We are given a rectangular harp represented as a grid of size n × m, where pins are placed along the edges: n pins on the left and right sides, and m pins on the top and bottom sides. Each string connects exactly two pins located on different sides of the rectangle.
The problem asks us to count the number of ordered pairs (i, j) in a list of strings words such that i < j and words[i] is both a prefix and a suffix of words[j].
The problem provides a database table named Days with a single column called day. Each value in this column is a valid SQL DATE, and every value is unique.
We have 2n cards laid out in a sequence. Every card has an integer written on it, and each card also has an index from 1 to 2n. The task is to divide all cards into exactly n pairs such that both cards inside every pair contain the same number.
We are given a main string s. From this string, we consider every possible substring and want to count how many distinct substrings are considered “good”. Whether a candidate substring t is good is not decided by properties of s itself, but by a small set of constraints.
We are asked to construct a complete record of matches between n teams, where every match has a winner and a loser, and no pair of teams plays more than once.
The problem asks us to find the maximum number of non-overlapping palindrome substrings in a given string s such that each substring has a length of at least k.
We have a straight bus route with fixed stop coordinates. A passenger normally pays the full distance between their boarding and exit stops. The conductor is allowed to "hide" at most one continuous segment of that trip from the ticket.
The problem gives us an array called milestones, where each value represents how many milestones belong to a particular project. Every week, we must complete exactly one milestone from one project.
The problem provides two relational tables, one named Employees and another named Salaries, both keyed by employeeid. Each employee may or may not appear in both tables. The Employees table contains the employee’s name, while the Salaries table contains the employee’s salary.
This problem asks us to take multiple very small binary search trees (BSTs), each with at most three nodes, and attempt to merge them into a single valid BST. Each tree is represented by its root node in the array trees.
We are given up to 50 points on a 2D plane. Each point has integer coordinates, and some points may coincide. The task is to find the largest Euclidean distance between any pair of points.
We are given a single string containing English letters in arbitrary capitalization. The task is to rewrite the string so that characters at odd positions become uppercase and characters at even positions become lowercase. The positions are counted starting from 1, not from 0.
The problem defines a special array called powers. This array is built from the binary representation of n. Every positive integer can be uniquely represented as a sum of powers of two.
We process a sequence of tiles, each tile being either 2 or 4. A tile starts far to the right and slides left. When it touches an equal value, the two merge into a doubled value and the new tile keeps moving.
This problem asks whether it is possible to rearrange k contiguous equal-length substrings of a string s to form another string t, given that s and t are anagrams.
This problem asks us to equalize the water levels in a series of buckets while accounting for a spill loss. Each bucket initially contains some quantity of water given in an array buckets, and every time we pour water from one bucket to another, a percentage of that water…
We have an array of non-negative integers. In one move, we pick an index i with a[i] 0, decrease a[i] by 1, and increase some position j by 1, where j - i must be even and j = i. The parity restriction is the core of the problem.
The problem gives us a string called blocks, where each character represents the color of a block. A character 'B' means the block is black, and a character 'W' means the block is white. We are also given an integer k.
We are given a string of lowercase English letters and a sequence of queries. Each query specifies a substring, and for each substring, we are asked to rearrange its letters into a palindrome if possible.
The problem asks us to count the number of contiguous subarrays of a given array nums whose score is strictly less than a given threshold k. The score of a subarray is defined as the product of its sum and its length.
The problem asks us to analyze a chat log containing n messages and identify which sender has written the most words overall. Each message is represented as a string in the messages array, and the corresponding sender of that message is at the same index in the senders array.
This problem gives us a database table named Experiments. Each row represents a single experiment performed by a participant. Every experiment belongs to one platform and one experiment category.
We can think of the students as vertices of an undirected graph. Every friendship relation means two students must belong to the same team. Since each team has exactly three students, every connected component of this graph must fit entirely inside one team.
This problem gives us an array of integers representing balls. Balls with the same value are indistinguishable for grouping purposes, and every group must contain balls of only one value. The challenge comes from the balancing constraint.
The problem gives us two database tables, Candidates and Rounds, that together describe interview performance for job candidates. The Candidates table contains one row per candidate.
This problem asks us to compute the year-on-year, often abbreviated as YoY, growth rate of total spending for every product in the usertransactions table. Each row in the input table represents a single transaction.
We are working with a conceptual multiplication grid where the cell in row i and column j contains the value i × j. Instead of explicitly building this table, we imagine listing all n × m values and sorting them in non-decreasing order.
The problem asks us to maximize the difference between two numbers that can be created from a given integer num through a specific remapping operation.
This problem gives us a lowercase English string s and a list of shift operations. Each operation is represented as: The operation affects every character from index start to index end, inclusive.
The problem asks us to find the length of the longest subsequence of a given binary string s such that the resulting binary number is less than or equal to a given integer k. A subsequence is any selection of characters from s in their original order, possibly skipping some.
The problem gives us three integers: - zero, the exact number of 0s that must appear in the array - one, the exact number of 1s that must appear in the array - limit, the maximum allowed length of any consecutive block of identical values We must count how many binary arrays…
Codeforces 409B: Mysterious Language
The problem gives us a positive integer n and asks us to construct the smallest positive integer whose digits multiply together to exactly n. For example, if n = 105, we need to find some integer whose digits have product 105.
This problem gives us a directed graph with n nodes and exactly n - 1 edges. The important guarantee is that if we ignore the direction of every edge, the graph becomes a tree. That means the underlying structure is connected and acyclic. Each edge is currently directed one way.
In this problem, we are given an array of strings called sentences. Each string represents a sentence composed of lowercase English words separated by exactly one space. The problem asks us to determine the maximum number of words that appear in any single sentence.
The problem requires us to merge overlapping events in the same hall. Each row of the HallEvents table represents an event in a particular hall with a startday and an endday. Events are considered overlapping if they share at least one day in common.
This problem is the classic mathematical game known as Nim. We are given an array piles, where each element represents the number of stones in a pile. Two players, Alice and Bob, take turns removing stones.
We are given a single positive integer $n$, and we need to count how many integers from 1 up to $n$ have a very specific property: there exists a pair of digits $x$ and $y$ such that every digit in the number’s decimal representation is either $x$ or $y$.
The problem asks us to calculate the total appeal of all substrings of a given string s. The appeal of a string is defined as the number of distinct characters it contains.
This problem asks us to implement a function that returns an infinite-method object. The defining property of this object is that any method name can be accessed and called, and when invoked, it should return the exact name of the method.
We work with arrays of length n, where every element is an integer from 0 to 2^m - 1. A sequence is called "wool" if there exists some contiguous subarray whose xor is 0. We are asked to count how many sequences are not wool, meaning every contiguous subarray has non-zero xor.
The problem gives us a singly linked list with an even number of nodes. For every node at index i, there is a corresponding twin node at index n - 1 - i, where n is the total number of nodes in the list. We define the twin sum as: - node[i].val + node[n - 1 - i].
The problem asks us to count how many binary strings satisfy a very specific structural rule. We are given four integers: - minLength, the minimum allowed length of a valid string - maxLength, the maximum allowed length of a valid string - oneGroup, the required divisibility…
Qwerty wants to maximize profit by buying items on one planet and selling them on another. Each planet offers multiple types of items, with known buying and selling prices and stock limits. Qwerty's ship has a fixed capacity, so he cannot carry more than k items.
We have an undirected graph representing a computer network. The vertices are computers and the edges are cables. The edges are stored in a fixed order from 1 to m.
The problem gives us two arrays, nums and target, both of the same length. We are allowed to perform operations on nums until it becomes exactly equal to target.
We are asked to simulate a dynamic table derived from an initial row of integers. Each subsequent row is generated such that the entry in column p of row i equals the count of how many times the value in column p of the previous row has appeared so far in that row.
This problem asks us to implement a time-limited wrapper around an asynchronous function. We are given an input function fn that returns a Promise and a time limit t in milliseconds. The task is to return a new function that executes fn but enforces a maximum execution time.
We need to build a rectangular box with integer side lengths a, b, and c. Its volume must equal a given number V, and among all such integer triples we want the one with minimum surface area.
The problem asks us to find all pairs of indices (i, j) in an array hours such that i < j and the sum of hours[i] + hours[j] is an exact multiple of 24, which is considered a "complete day.
The problem asks us to transform a given array arr into a K-increasing array using the minimum number of operations. A K-increasing array is defined such that for every index i where i = k, the condition arr[i-k] <= arr[i] holds.
We are given a string s and an integer k. The user initially registers a username t which is simply the string s repeated k times consecutively. After that, a sequence of edits occurs, each removing the p-th occurrence of a specified character from the current string.
We are given a noisy black-and-white image represented as an n × n grid. A value of 1 means a black pixel, and 0 means a white pixel. The image originally contained only circles and squares drawn in black on a white background.
The task is to determine how many trailing zeros appear at the end of the factorial of a given integer n. A factorial, denoted n!, is the product of all integers from 1 up to n.
The problem presents a matrix grid of size m x n with distinct positive integers. The goal is to transform this matrix so that every element is replaced with another positive integer while maintaining the relative order in each row and column.
We start with a string of length l. In one move we may choose any contiguous substring that is a palindrome and whose length k is allowed, meaning a[k] != -1. After deleting it, the remaining characters concatenate together. The score gained from this move is a[k].
The city is represented by a grid. Every cell is either a building, a street tile with a traversal cost from 1 to 9, or a junction labeled by a lowercase letter. Movement rules are unusual.
This problem asks us to design a small task management system, similar to a lightweight todo application. We need to implement a TodoList class that supports adding tasks, marking tasks as completed, retrieving all pending tasks for a user, and filtering pending tasks by tag.
We are given a triangular pyramid structure made of cells arranged in rows. The first row has one cell, the second row has two, and so on, up to n rows. Each cell can hold a value.
We are asked to fill an $n times m$ grid with positive integers not exceeding $10^8$. The constraint is not about individual cells but about structure: for every row and every column, if we square all numbers in that line and sum them, the result must itself be a perfect square.
The problem describes a bank floor plan as a binary matrix, represented by an array of strings. Each row corresponds to one row in the bank, and each character in the string represents a cell.
The problem presents a street with n plots on each side, for a total of 2 n plots. The goal is to count the number of ways to place houses on these plots such that no two houses are adjacent on the same side of the street.
The problem asks us to compute the maximum factor score of an array of integers, where the factor score is defined as the product of the GCD (greatest common divisor) and LCM (least common multiple) of all elements in the array.
This problem gives us a binary tree that always contains exactly three nodes. There is one root node, one left child, and one right child. We must determine whether the value stored in the root node is equal to the sum of the values stored in its two children.
The problem defines a repeated string transformation process over lowercase English letters. Every character does not simply become one new character, instead it expands into multiple characters depending on the value stored in nums.
We are given all integers from 0 to n, and we must arrange them into a permutation p. Each position i contributes a value equal to i XOR p[i], and the goal is to maximize the total sum of these contributions over all positions.
We are asked to count the number of binary strings containing exactly n zeros and m ones that can be reduced, through a sequence of specific operations, to a single character g.
This problem asks us to determine, for each query, the maximum number of elements we can select from an array nums such that their sum does not exceed a given value.
The problem requires calculating the number of unique subjects each teacher teaches in a university, given a table that maps teachers to subjects and departments. Each row in the Teacher table represents a specific combination of a teacher, a subject, and a department.
We are given a triangular region of a square matrix called an n-degree staircase. In this staircase, each cell is accessible except for the area above the secondary diagonal, which runs from the top right to the bottom left.
We are given a string containing lowercase letters, spaces, and a limited set of punctuation marks: comma, dot, exclamation mark, and question mark.
This problem describes a simulation of robots moving on a one-dimensional line, each with a given starting position, health, and direction. Every robot moves at the same speed, and collisions occur when two robots meet at the same position.
The problem asks us to transform a given array of positive integers into a palindrome using the minimum number of operations, where each operation consists of taking two adjacent elements and replacing them with their sum.
The problem gives us an integer array nums with an even number of elements. We repeatedly simulate a game between Alice and Bob until the array becomes empty. In each round, the following sequence happens: 1. Alice removes the smallest remaining number. 2.
We are given a single positive integer and must print its representation in base 2. In other words, instead of expressing the number as powers of 10, we express it as powers of 2 using only digits 0 and 1.
We are given a line of length $n$, where each position can either already contain a bomb, already contain a number, or be undecided. A bomb is fixed as a .
The task is to simulate the exception-handling behavior of a simple programming language. The program consists of three types of statements: try, catch(type, message), and throw(type).
We are given an array of distinct integers. At every operation, we inspect the first element of the current array. If that first element is currently the smallest remaining value in the array, we remove it. Otherwise, we rotate it by moving it from the front to the back.
In this problem, we are given two strings of equal length, each representing a sequence of actions taken by two competing teams in a hypothetical game. Each character corresponds to a distinct type of action, and each action has a predetermined score.
The problem gives us a database table named Weather that stores temperature readings for different cities on specific days in the year 2022.
We are given a tree with n vertices, meaning a connected graph with no cycles, and a number k. The task is to count how many pairs of distinct vertices are separated by exactly k edges.
The problem is asking us to vertically concatenate two DataFrames into a single unified DataFrame. In simpler terms, given two tables df1 and df2 with identical columns and types, we need to stack the rows of df2 below the rows of df1.
The problem describes a scenario where a robber wants to steal from houses lined along a street, but with the constraint that adjacent houses cannot both be robbed.
The Friends table stores friendship relationships between users on a social platform. Each row contains two user IDs, user1 and user2, indicating that those two users are friends with each other.
We are given a sequence of digits written in base k. Every digit is between 0 and k - 1, and one special digit b is called lucky. For every substring of the digit sequence, we interpret that substring as a base-k number, allowing leading zeroes.
This problem asks us to find the maximum sum of a balanced subsequence from a given integer array nums. A subsequence is a selection of elements from the array in their original order, possibly skipping elements.
We have exactly five students standing in a line for a shower. While one student is showering, the remaining students wait in line and talk in adjacent pairs. The first and second students talk, the third and fourth students talk, and so on.
This problem asks us to implement a utility function called promisify. The purpose of this function is to convert a callback-based asynchronous API into a Promise-based API. The input is a function fn.
The problem asks us to count how many substrings of a given string s contain at least one character that appears at least k times within that substring. A substring is a contiguous portion of the string.
The problem asks us to split an array of integers nums into contiguous subarrays such that each subarray satisfies a validity condition: the greatest common divisor (GCD) of the first and last elements of the subarray must be strictly greater than 1.
The problem gives us a sorted array prizePositions, where each value represents the position of a prize on the X-axis. Multiple prizes may exist at the same position. We are also given an integer k. We may choose exactly two segments on the number line.
Codeforces 424C: Magic Formulas
This problem is unusual compared to standard Codeforces tasks because it is not asking for a mathematical algorithm or a data structure. We are given a document, consisting of a title and body text, and we must classify it into one of three categories.
The problem asks us to partition a given string s into k contiguous substrings and modify the characters minimally so that each substring becomes a semi-palindrome.
The problem asks us to determine the maximum number of good people in a group given a set of statements about each other. Each person can either be good (always tells the truth) or bad (may lie or tell the truth).
The problem gives us a string s consisting only of digits from '0' to '9'. We must count how many distinct substrings satisfy a special condition: Every digit that appears in the substring must appear the same number of times. A substring is a contiguous portion of the string.
Each cross is defined by two axis-aligned rectangles centered at the same cell (x0, y0). The first rectangle extends a cells vertically and b cells horizontally from the center, so its size is: $(2a+1)(2b+1)$ The second rectangle is defined similarly using c and d.
This problem asks us to count the number of distinct ways to move from startPos to endPos on an infinite number line using exactly k steps. At each step, we may move either one position to the left (-1) or one position to the right (+1).
We are given a very small computational system that starts from two registers: the first register begins at 1 and the second at 0. From this starting point we can repeatedly apply two operations. One operation increments the second register by 1.
The problem gives us a sorted integer array nums in non-decreasing order. Our task is to count how many numbers are positive and how many numbers are negative, then return the larger of the two counts.
We are given a connected undirected simple graph and we need to break its edge set into groups of three vertices, where each group forms a path of length two.
The problem asks us to take an integer array nums and sort its elements based on their indices in two separate ways: the elements at even indices should be sorted in non-decreasing order, and the elements at odd indices should be sorted in non-increasing order.
We are given a permutation of the numbers from 1 to n, and a sequence of m operations. Each operation targets two positions in the array.
The problem is asking us to determine the minimum total cost to paint n walls using two painters with different constraints. We are given two arrays: cost and time, both of size n.
The problem asks us to find the maximum number of points that can be contained in a square centered at the origin (0, 0) such that no two points inside the square share the same tag. The square's edges are parallel to the axes, and points on the edges are considered inside.
The problem asks us to construct the lexicographically smallest subsequence of length k from the string s, while ensuring that a specific character, letter, appears at least repetition times in the resulting subsequence.
The problem gives us a string word containing uppercase and lowercase English letters. We must count how many letters are considered "special". A character c is special if two conditions are true: 1. The lowercase version of the letter appears somewhere in the string. 2.
This problem is asking us to decode a secret message using a substitution cipher defined by a key string. The key string may include spaces and contains every lowercase letter of the English alphabet at least once.
The problem requires us to insert spaces into a given string s at specific positions described by the array spaces. Each element in spaces represents an index in the string before which a space should be inserted.
The problem gives us an integer array nums, and we may perform a special operation any number of times. In one operation, we choose an index i and any non-negative integer x, then replace: Our goal is to maximize the bitwise XOR of all elements in the array after performing as…
The problem asks us to compute the sum of powers of all subsequences of length k from an array nums. A subsequence is any subset of elements taken in order from the array without reordering.
Each participant receives a real-valued score chosen uniformly from an interval $[li, ri]$. Scores are independent. After all scores are generated, participants are sorted by score. Smaller score means better rank, so the participant with the largest score finishes last.
We are tasked with distributing T-shirts to a line of engineers where each engineer has a probability distribution over which T-shirt size fits them. We know how many engineers, n, there are, and the total number of T-shirt sizes, m.
The problem gives us a binary matrix where each cell represents a piece of land. A value of 1 means the land is fertile, while 0 means barren. We need to count every valid pyramidal plot and inverse pyramidal plot formed entirely from fertile cells.
Lesha has a problem description made of n distinct words, written in a fixed order. Each archive problem is another sequence of words, but archive descriptions may repeat words many times. We want to compare Lesha’s description against every archive description.
We are given a linear congruential generator, a classic pseudorandom sequence formula: $$ri = (a cdot r{i-1} + b) bmod m$$ The sequence starts from r0, and every next value is computed from the previous one.
We are asked to distribute a collection of n words among k Keepers in such a way that every Keeper receives a subset of words satisfying three conditions. First, no two Keepers can share a word. Second, every word must be assigned to some Keeper.
We are given two integer arrays, nums1 and nums2, along with a positive integer k. A pair of indices (i, j) is considered good if: In other words, nums1[i] must be divisible by nums2[j] k. The task is to count how many such index pairs exist.
We have a list of n strings, and every string has the same length m. An operation chooses two strings and a prefix length k, then swaps the first k characters between those two strings. We only care about the string that eventually appears in position 1.
This problem asks us to implement an Encrypter class that can both encrypt and decrypt strings according to custom character mappings. The keys array provides the characters that can be encrypted, and values provides the corresponding 2-character strings that each key maps to.
The problem asks us to minimize the maximum Manhattan distance between any two points on a 2D plane after removing exactly one point from the input list. The input points is an array of integer coordinates [[x1, y1], [x2, y2], ..., [xn, yn]].
We are asked to find the shortest distance between two points on a cone, where the cone has a circular base of radius r and height h, and the points may lie anywhere on the cone's lateral surface or the base.
The problem gives us a binary matrix grid, where each cell contains either 0 or 1. Our goal is to place exactly three non-overlapping axis-aligned rectangles so that every cell containing 1 is covered by at least one rectangle.
The problem asks us to determine, for each person in a 2D grid of heights, how many other people they can see according to a specific line-of-sight rule.
The problem gives us a robot located on a two dimensional grid. The robot starts at startPos = [startrow, startcol] and wants to reach homePos = [homerow, homecol]. The robot can move one cell at a time in four directions: up, down, left, and right.
The problem requires calculating a score for all prefixes of an array. Given an integer array nums, we first define a conversion array for any prefix of nums.
The problem requires finding the maximum sum of a pair of numbers in an array such that the sum of the digits of both numbers is equal. Specifically, you are given a 0-indexed array nums containing positive integers. You can choose two distinct indices i and j such that i !
The problem starts with a single integer n written on a board. Every day, we examine every number currently on the board. For each number x, we look for all integers i such that: - 1 <= i <= n - x % i == 1 Whenever such an integer i exists, we add it to the board.
In this problem, we are given a pile of integers represented as an array nums, where nums[0] is the current top element of the pile. We must perform exactly k moves, and in each move we are allowed to do one of two operations: 1. Remove the current top element from the pile. 2.
The problem asks us to find the longest common subsequence (LCS) among multiple sorted integer arrays. In simpler terms, we are given a list of arrays where each array is sorted in strictly increasing order, and we need to identify the sequence of numbers that appears in all…
We are given an array nums consisting of distinct integers. For every position i, we need to determine the maximum possible length of a contiguous subarray in which nums[i] is the largest element. More formally, for each index i, we want to find the longest subarray nums[l..
The problem asks us to simulate the execution of tasks on a single-threaded CPU. Each task has an enqueue time and a processing time.
You are given two arrays, nums1 and nums2, both of length n. You are allowed to perform operations only on nums1. In a single operation, you may swap any two indices in nums1, and the cost of that operation is the sum of the two indices involved in the swap.
The problem gives us an integer array nums and an integer k. For every contiguous subarray of length k, we must determine how many unique values appear inside that window. A subarray is a continuous portion of the array.
This problem gives us a 2D grid representing a shop layout. Every cell in the grid has one of three meanings: - 0 means the cell is blocked by a wall and cannot be traversed. - 1 means the cell is empty space and can be walked through.
We are given a matrix grid with m rows and n columns, along with an integer k. The task is to count how many rectangular submatrices satisfy two conditions simultaneously: 1. Every value inside the submatrix must be at most k. 2.
The problem gives us an integer array nums, and we need to find the maximum distance between the indices of any two prime numbers in the array. More specifically, we are interested in indices i and j such that both nums[i] and nums[j] are prime numbers.
The car started somewhere on the infinite 2D grid and eventually reached the ice-cream stall at (0, 0). We know the number of moves and the order of the GPS records, but each record only tells us a set of possible directions for that step.
The problem gives us two chessboard coordinates, such as "a1" or "h8", and asks whether the two squares have the same color on a standard 8 x 8 chessboard. A chessboard alternates between black and white squares.
The problem asks us to transform a string s into a target string t using a very specific operation: choosing a non-empty suffix of s (not the whole string) and moving it to the front.
We have a vertical pole with bars attached at heights 1..n. Every bar points in one of four directions. A child can move between two bars if both bars point in the same direction and their height difference is at most h.
We have n cubes. Every cube has a color and a size. We want to build a tower using cubes from exactly two distinct colors, and adjacent cubes in the tower must always have different colors. There are no restrictions on cube sizes or ordering.
The problem gives us an m x n binary matrix named grid, where every cell contains either 0 or 1. We are allowed to perform two kinds of operations: 1. Flip an entire row 2. Flip an entire column Flipping means changing every 0 into 1 and every 1 into 0.
The problem gives us a 2D integer matrix called grid and an integer k. We need to count how many submatrices satisfy two conditions: 1. The submatrix must contain the top-left cell of the original matrix, which is grid[0][0]. 2.
Each participant in the company meeting points to two other people and claims that the culprit is one of those two. From this we can think of the input as an array of length $n$, where each index $i$ stores an unordered pair $(xi, yi)$.
The problem asks us to find the shortest distance from a given starting index in a circular array of strings to a target string.
This problem asks us to take a 3x3 grid of integers representing stones in each cell and redistribute them so that every cell contains exactly one stone. Each move consists of moving a stone from one cell to a directly adjacent cell (sharing a side, not a diagonal).
The problem asks us to find the number of non-empty substrings in a given string s that satisfy two conditions simultaneously. First, the substring must have an equal number of vowels and consonants.
We are given a sequence of non-negative integers and must split it into two groups. One group remains with Petya and the other is given to Masha. For each group we compute the XOR of all numbers assigned to it, and we call these values x1 and x2.
Codeforces 403E: Two Rooted Trees
The problem gives us two strings, s and p, and a list of indices removable. The string p is guaranteed to be a subsequence of s, meaning all characters in p appear in s in the same order, though not necessarily consecutively.
In this problem, we are given the starting mass of a planet and a list of asteroid masses. The planet can collide with the asteroids in any order we choose.
The author has to take n exams. Every exam receives an integer grade between 2 and 5 inclusive. A grade of 2 means the exam is failed and must be retaken. The total sum of all grades must equal exactly k.
We are given an array of $n$ integers, each between 1 and 5000, and we are asked to transform this array into a permutation of the numbers from 1 to $n$. A permutation is a sequence where each integer from 1 to $n$ appears exactly once.
This problem asks us to find every starting index where the substring a appears inside the string s, subject to an additional proximity condition involving another substring b. More specifically, an index i is considered beautiful if two conditions hold: 1.
Vasya receives food every morning. The food from day i can only be eaten on day i or day i + 1. Every day Vasya himself must consume exactly v kilograms.
We are asked to generate "beautiful" IP addresses under a very specific definition. Each IP address consists of four decimal numbers between 0 and 255, written without leading zeroes, separated by periods.
The problem gives us a rectangular maze represented by a 2D grid. Each cell contains either: - '.', meaning the cell is empty and can be walked on - '+', meaning the cell is a wall and cannot be crossed We are also given the coordinates of an entrance cell.
This problem gives us a collection of equations of the form: Each variable is represented as a string, and each equation defines a multiplicative relationship between two variables. The task is to determine whether all equations can simultaneously be true.
The problem gives us a table named Purchases, where each row represents a purchase made by a user. Every record contains three fields: - userid, identifying the customer - timestamp, indicating when the purchase happened - amount, representing how much money was spent The pair…
The problem requires identifying users who have taken both a "Refinance" loan and a "Mortgage" loan. The input is a table Loans containing loanid, userid, and loantype. Each row represents one loan taken by a user, and loanid is unique.
We are given an $n times n$ grid where every cell is initially unclean. Some cells are marked as forbidden for casting a spell. When we cast a purification spell on a cell $(r, c)$, we clean every cell in row $r$ and every cell in column $c$.
Codeforces 251C: Number Transformation
We are given a small array of non-negative integers, at most three in length. Two players, BitLGM and BitAryo, take turns reducing these integers in one of two ways: either they pick a single number and subtract any positive amount, or they pick an amount and subtract it from…
We know the second bag contains y potatoes. The first bag originally contained some positive number x, but that value was lost. The only remaining information is that the total number of potatoes, x + y, was divisible by k and did not exceed n.
LeetCode 2541: Minimum Operations to Make Array Equal II (Medium)
This problem gives us a rooted tree with n nodes, rooted at node 0. Every node has a numeric value, and the sum of a subtree is defined as the sum of all node values inside that subtree, including the root of the subtree and all descendants.
This problem asks us to multiply two arrays that are already stored in run-length encoded form, without fully expanding them into normal arrays. A run-length encoded array stores consecutive repeated values compactly.
The problem gives us two arrays, nums1 and nums2, both of the same length n. We are allowed to repeatedly perform a right circular shift on nums1. A right shift moves every element one position to the right, and the last element wraps around to the beginning.
This is a Type A - “Find all X” problem. A valid proof must do two things: 1. Verify that every claimed solution actually satisfies the conditions. 2. Prove that no other solutions exist. The proposed solution does not complete either direction fully.
The problem gives us an array power, where each element represents the damage value of a spell. Every spell can be used at most once, and multiple spells may share the same damage value. The restriction is the important part of the problem.
We are given a string s of even length n. For every query, we are allowed to independently rearrange two specific substrings: - One substring lies completely inside the left half of the string. - The other substring lies completely inside the right half of the string.
This problem asks us to compute a dominance score for each football team across two halves of a match, based on passing outcomes. The input is represented using two database tables, Teams and Passes. The Teams table maps every player to exactly one team.
The problem presents two m x n binary matrices, grid1 and grid2, representing maps of land (1) and water (0). Each matrix may contain multiple islands, where an island is defined as a connected group of 1s that are adjacent vertically or horizontally.
We are asked to simulate the spread of a chain letter through a network of friends. Each person is represented as a node in a graph, and the connections between people-who sends letters to whom-are given as an adjacency matrix of "0" and "1" characters.
The problem asks us to count all subarrays of a given array nums such that if we remove the subarray, the remaining array becomes strictly increasing. A strictly increasing array is one where each element is less than the next element.
We have a staircase where the height of stair i is a[i], and the heights are already sorted in non-decreasing order. Boxes are dropped one after another. A box with width w covers exactly the first w stairs, meaning its horizontal span is above stairs 1..w.
The problem gives us an integer n representing the size of an n x n matrix. Initially, every cell in the matrix contains 0. We are also given a list of queries.
We are given an array containing only 1 and -1. For every query [l, r], we look at the subarray from index l to r.
The problem gives us a table named Tasks, where each row represents a task assigned to an employee. Every task has a start timestamp and an end timestamp. Multiple tasks may overlap in time for the same employee. We need to compute two values for every employee: 1.
We have a four-team football group where every pair of teams plays exactly once, so the full tournament contains six matches. Five results are already known, and the only remaining match is the one involving BERLAND. Each match contributes points in the usual way.
The problem asks us to transform a given integer array nums into either a non-decreasing or a non-increasing array with the minimum number of operations. Each operation allows increasing or decreasing an element by exactly 1.
This problem asks us to repeatedly remove words from an array when two adjacent words are anagrams of each other. We are given a 0-indexed string array words, where every string contains only lowercase English letters.
We are given a list of item prices and a collection of discount types. Each discount type describes how many items must be placed in a “paid group”.
This problem asks us to analyze a singly-linked list of integers and produce a new linked list containing the frequencies of each distinct element in the input list.
This problem asks us to compute the sum of the XOR values of every possible subset of a given array. A subset is formed by choosing any combination of elements from the array, including the empty subset and the full array itself.
The problem asks us to find two indices i and j in an integer array nums that satisfy two conditions simultaneously: the absolute difference between the indices must be at least indexDifference (abs(i - j) = indexDifference) and the absolute difference between the values at…
We are given a text editor that stores several lines of text. Each line has a known length, and the cursor is allowed to sit not only on characters but also in the gap positions before the first character and after the last character.
The problem asks us to find the largest color value along any path in a directed graph of n nodes, where each node has a color represented by a lowercase English letter. The graph may contain cycles, in which case the function should return -1.
We must construct n pentagram-shaped stars in the plane. Each star comes from a regular pentagon with side length 10, and the painted segments are exactly its five diagonals. The output is not just geometry. We also need a valid drawing order.
This problem gives us a sorted integer array nums in non-decreasing order. We may repeatedly remove pairs of elements under one important rule: for a chosen pair (i, j), the values must satisfy nums[i] < nums[j].
The problem asks us to repeatedly replace adjacent non-coprime numbers in an array with their Least Common Multiple (LCM) until no more adjacent non-coprime pairs exist. Two numbers are non-coprime if their greatest common divisor (GCD) is greater than 1.
We are given an m x n matrix called board, where each cell contains an integer value. We must place exactly three rooks on the board. A rook attacks every cell in the same row and the same column.
The problem presents a scenario where a wall is decorated with multiple concentric circles, some of which are painted red while others are blue in an alternating pattern. The outermost area beyond the largest circle is always blue.
We are given the arrival times of customers visiting a cafe during one day. Every customer is served in less than one minute, so the only time a queue can appear is when several customers arrive at exactly the same minute.
We are given an array of integers, and for every element we must find another array element whose bitwise AND with it is zero. Two numbers are compatible exactly when they do not share any bit set to 1.
The problem asks us to design a service that works on a square matrix, where every value in the matrix is unique. For any given value, we must efficiently compute two different types of neighbor sums.
We are given a sequence of non-negative integers representing a game board. The task is to repeatedly perform moves that decrease one number and increase another number two steps ahead.
The problem asks us to divide a given array nums into k contiguous subarrays in a very specific way. Each subarray has a cost defined as its first element. The objective is to minimize the total cost, which is the sum of the first elements of all k subarrays.
We are given an undirected weighted graph of planets connected by stargates. Jack starts on planet 1 at time 0 and wants to reach planet n as early as possible. Moving through a stargate takes a fixed positive amount of time. The unusual part is the waiting rule.
You are given an array nums and an integer k. For every subsequence of nums, we define its power as the number of subsequences inside it whose sum equals k. The task is to compute the total power across all subsequences of the original array.
In this problem, we are given a string s and a robot that manages another temporary string t. Initially, t is empty. The robot repeatedly performs one of two allowed operations: First, it may remove the first character from s and append it to the end of t.
In this problem, we are given a two dimensional matrix called grid. Each cell contains a positive integer. From any cell, we may move in four directions: up, down, left, or right. Diagonal movement is not allowed.
We are given up to 1000 points on the plane, and we must remove exactly k of them. After removing those points, the remaining set should have the smallest possible diameter. The diameter of a set is the maximum Euclidean distance between any two remaining points.
The problem gives us a total number of working days, numbered from 1 to days, along with a list of meeting intervals. Each interval [start, end] represents a meeting that occupies every day from start through end, inclusive.
This problem asks us to implement a generator that traverses a multi-dimensional array and yields integers in the same order as an inorder traversal. The input is not a normal one-dimensional list.
This problem asks us to count the number of 2 x 2 blocks in a grid based on how many black cells they contain. We are given the dimensions of a grid, m rows and n columns, and a list of coordinates representing black cells. Every cell not listed is white.
In this problem, we are given an n x n matrix where every value is guaranteed to be between 1 and n. A matrix is considered valid if every row contains all integers from 1 through n exactly once, and every column also contains all integers from 1 through n exactly once.
Valera already owns four horseshoes, and each horseshoe has a color represented by an integer. He wants all four horseshoes to have distinct colors before going to the party. If some colors repeat, he must buy new horseshoes to replace the duplicates.
We are given a sequence of statements about an unknown integer value $y$. Each statement restricts $y$ relative to some integer threshold $x$, and is either strict or non-strict. After each statement, we are also told whether that statement is true or false.
The problem gives us an array nums containing positive integers. In a single operation, we are allowed to remove either: - Exactly 2 equal elements, or - Exactly 3 equal elements. Our goal is to remove all elements from the array using the minimum possible number of operations.
The problem gives us a binary string s, consisting only of characters '0' and '1'. We are allowed to repeatedly perform a specific operation: - Find an index i such that: - s[i] == '1' - s[i + 1] == '0' - Move that '1' to the right until it either: - reaches the end of the…
The problem asks us to simulate the behavior of switching between open windows using the classic Alt + Tab operation. We are given an array called windows, which represents the current ordering of open windows.
We are given two permutations, one of length n and another of length m (n ≤ m). A permutation is a sequence containing all numbers from 1 to its length exactly once.
This problem gives us an array nums containing non-negative integers. We want to transform the entire array into zeros using a specific operation.
The problem gives us an array called questions, where each element contains two integers: - questions[i][0] represents the number of points earned if we solve question i - questions[i][1] represents how many subsequent questions must be skipped after solving question i We must…
We are given several scientists, each producing a sequence of computational tasks. Every task has a fixed resource requirement, and within each scientist’s list the tasks must be executed in the given order.
We are designing a two-part medal. The outer part is a ring with outer radius $r1$ and inner radius $r2$. The inner part is a solid disk of radius $r2$. Both parts have the same thickness, so their masses depend only on area and density.
We are counting connected black-cell shapes on an n × m grid with a very strong geometric restriction. Take all painted cells as vertices of a graph, where edges connect side-adjacent painted cells. The shape must satisfy three conditions. First, at least one cell is painted.
This problem asks us to rearrange an array around a given pivot value while preserving relative ordering inside certain groups. We are given an integer array nums and an integer pivot. The goal is to reorganize the array into three consecutive sections: 1.
We are asked to simulate movement across a 2D plane from a starting point $A$ to a destination $B$ while avoiding a periodic laser. The laser alternates between charging and firing, with durations $a$ and $b$ seconds, respectively.
We are given two strings of equal length. From each string we independently pick a substring, and both substrings must have the same length. After choosing them, we compare them character by character and count how many positions match.
This will be a very large, multi-thousand-word technical guide because of the required depth, full proofs, worked examples, Python and Go implementations, detailed walkthroughs, complexity reasoning, and exhaustive tests.
The problem asks us to count how many pairs of indices (i, j) satisfy two conditions: 1. i < j 2. The sum nums[i] + nums[j] lies within the inclusive range [lower, upper] We are given an integer array nums and two bounds, lower and upper.
The problem presents a binary string s representing a sequence of train cars. Each character '0' or '1' denotes whether a train car contains illegal goods ('1') or not ('0').
The problem asks us to find the maximum min-product of any contiguous subarray of a given integer array nums. The min-product of a subarray is defined as the minimum value in that subarray multiplied by the sum of all elements in that subarray.
We are given an m × n matrix values where each row represents a shop and each column represents an item in that shop. The important property is that every row is sorted in non-increasing order: When buying from a shop, we are not allowed to choose any arbitrary item.
The problem asks us to design a data structure that supports two operations efficiently: 1. Assign a number to a given index. 2. Find the smallest index currently assigned to a given number.
The problem asks us to determine whether a number remains unchanged after performing two digit reversals. A digit reversal means reading the digits of a number from right to left.
We are given a username consisting only of lowercase English letters. The task is to count how many different characters appear in that username. The decision rule is simple. If the number of distinct characters is even, we print "CHAT WITH HER!".
This problem gives us a singly linked list and asks us to process the nodes in groups whose sizes follow the natural number sequence: - Group 1 contains 1 node - Group 2 contains 2 nodes - Group 3 contains 3 nodes - Group 4 contains 4 nodes - And so on However, the list may…
We are given a tree with n nodes labeled from 1 to n. A tree is a connected acyclic graph, so every pair of nodes has a unique simple path between them. The task is to count the number of pairs of paths (a, b) and (c, d) that do not share any node.
We start with three piles of pixels, one red, one green, and one blue. A fight can only happen between two different colors. When that happens, one pixel survives and immediately changes into the third color.
The problem asks us to identify which employee worked the task with the longest duration given a sequence of tasks completed by employees. Each task is represented in logs[i] = [idi, leaveTimei], where idi is the employee id and leaveTimei is the time the task was completed.
We want to count strings of length n built from an alphabet of size m, under one strong restriction: every substring of length k must be a palindrome.
This problem gives us an array of strings, words, and allows us to swap any character with any other character across the entire collection of strings. The swaps are completely unrestricted.
In this problem, we are given several fruit piles placed on an infinite one dimensional number line. Each pile is represented by a position and the number of fruits available at that position. The input array fruits is already sorted by position, and every position is unique.
Two tanks fight each other. Each tank has hit points, a reload time, a damage interval, and a probability that its shot fails to penetrate armor. Whenever a tank fires, one of two things happens.
We are given a set of points in the plane, and we want to place two infinite straight lines that must always be perpendicular to each other. One of these lines is constrained in orientation: it must make a 45-degree angle with the positive x-axis.
This problem provides a Pandas DataFrame named students with four columns: | Column | Type | | --- | --- | | studentid | int | | name | object | | age | int | | grade | float | The task is straightforward: the grade column is incorrectly stored as floating-point values, and we…
The problem asks us to design a data structure that continuously tracks scenic locations ranked by two rules: 1. A higher score means a better location. 2. If two locations have the same score, the lexicographically smaller name is considered better.
The problem requires us to recursively traverse a nested object or array and replace all occurrences of undefined with null. In JavaScript, undefined and null behave differently when serializing objects to JSON. Specifically, JSON.
We are given an array of positive integers. For every number x, a recursive function f(x) is defined as: $f(0)=0,quad f(2x)=f(x),quad f(2x+1)=f(x)+1$ The task is to count how many index pairs (i, j) with i < j satisfy f(ai) = f(aj).
LeetCode 2539: Count the Number of Good Subsequences (Medium)
The problem asks us to manipulate an array of integers, nums, in a very specific way to minimize a metric called the "score.
The problem asks us to compute a ranking of students based on textual feedback they receive. We are given two lists of words: positivefeedback and negativefeedback.
The problem gives us an integer array nums and an integer k. We must count how many contiguous subarrays have a least common multiple, LCM, exactly equal to k. A subarray is any continuous segment of the array. For every possible subarray, we compute the LCM of all its elements.
We are given a rectangular grid of non-negative integers. For every valid center cell (x, y), we define a diamond-shaped region of radius k - 1. The function f(x, y) is the sum of all values inside that rhombus. The task is not to compute every value explicitly and print them.
=== 1996-G1 === Origin: GBR Let triangle have orthocenter , and let be a point on its circumcircle, distinct from , , . Let be the foot of the altitude , let and be parallelograms, and let meet in . Prove that is parallel to .
This problem provides a table named Servers that records status changes for multiple servers over time. Each row contains three values: - serverid, which identifies the server - statustime, which indicates when the status change occurred - sessionstatus, which is either…
We are asked to analyze a two-player game on a rectangular grid where each cell may contain a toy soldier from Shrek (green) or Donkey (red), or be empty. The grid has dimensions n × m, and on each row there are at most two soldiers.
We are given an undirected graph where every vertex has a very strong local condition: each node is connected to at least $k$ other distinct nodes. From this structure we are asked to extract a simple cycle whose length is not just positive, but at least $k+1$.
Greg performs exercises in a fixed repeating order. The first exercise trains the chest, the second trains the biceps, the third trains the back, then the pattern repeats again: chest, biceps, back, and so on.
We need to compute the sum of the number of divisors of every product i j k, where i ranges from 1 to a, j ranges from 1 to b, and k ranges from 1 to c. For each triple (i, j, k), we evaluate d(i j k), where d(x) means the number of positive divisors of x.
The problem is asking us to find the intersection of multiple arrays. Specifically, given a 2D array nums, where each nums[i] is a non-empty array of distinct positive integers, we want to identify which integers appear in every array in nums.
The problem gives us two strings, s and t, of equal length. We want to transform s into t character by character using cyclic alphabet shifts. For every character in s, we are allowed to repeatedly perform one of two operations: 1.
This problem asks us to repeatedly transform a numeric string until its length becomes less than or equal to a given integer k.
The problem asks us to compute the sum of total strengths across every possible contiguous subarray of the input array strength.
This problem gives us the head of a singly linked list where every node contains a positive integer. Our task is to modify the list by inserting a new node between every pair of adjacent nodes.
The problem presents a two-dimensional matrix grid of size m x n with integer values in the range 0 to 9. The task is to transform this matrix using the minimum number of operations, where each operation allows you to change the value of any cell to any non-negative integer.
The problem gives us a list of unique strings called ideas. We must repeatedly choose two different strings, swap their first characters, and determine whether the newly formed strings are both absent from the original list.
We are given a string composed of lowercase English letters, and our goal is to select a subsequence from it that is lexicographically the largest possible. A subsequence is formed by taking zero or more characters in order without reordering them.
The problem asks us to compute the permutation difference between two strings, s and t. Both strings contain unique characters, and t is a permutation of s.
Polycarpus is running a simple monitoring routine for two servers, which we can call server a and server b. Each "ping" command sends exactly ten packets to one of the servers.
The problem gives us an undirected graph with n nodes. Each node has a score, and the graph edges define which nodes are directly connected. We must find a valid sequence of exactly four distinct nodes such that every adjacent pair in the sequence has an edge between them.
This problem describes a dungeon represented as an n x m grid of rooms, where each room has a constraint on the earliest time you can enter it, given by the 2D array moveTime.
The problem gives us an integer array nums and an integer k. We must perform exactly k operations. During each operation, we choose one element from the array, add its value to our score, remove it from the array, and then insert a new element equal to the chosen value plus one.
The problem requires identifying users who verified their signup exactly on the second day after registering. We have two tables: emails and texts. The emails table contains the emailid, the userid, and the signupdate.
Petya can decide in which months he waters the flower. Each month contributes a fixed amount of growth, and skipping a month contributes nothing. The goal is to reach at least k centimeters of total growth while using as few months as possible.
The problem asks us to determine the number of ways to select a subset of students from a class such that every student is happy according to the given rules. Each student has a number nums[i] that represents a threshold.
The problem gives us a binary string s, where each character is either '0' or '1'. We start at index 0, and the problem guarantees that s[0] == '0', meaning the starting position is always valid.
In this problem, we are given a string s that contains lowercase English letters and digits. The task is to repeatedly remove digits according to a very specific rule. Whenever we encounter a digit, we must delete two characters: 1. The digit itself 2.
This problem asks us to determine the largest number of candies that every child can receive equally, given a collection of candy piles and a number of children k. We are given an integer array candies, where candies[i] represents the size of the i-th pile.
The problem gives an integer array cards, where each number represents the value written on a card. Two cards are considered matching if they contain the same value.
The problem gives us an n x n matrix called grid. Every value in the matrix should normally contain all integers from 1 to n² exactly once. However, there is one mistake in the matrix: - One number appears twice. - One number is completely missing.
The problem asks us to construct a target string target by repeatedly appending strings from a given array words, where each word has an associated cost in costs. The goal is to determine the minimum total cost to construct target exactly.
In this problem, we are asked to simulate a sequence of events in a feudal hierarchy represented as a tree. Each feudal owns a castle, and except for the king, each feudal reports to exactly one superior.
The problem gives us an integer array nums and two integers, m and k. We must examine every contiguous subarray whose length is exactly k. Among those subarrays, we only consider the ones that contain at least m distinct values. These are called almost unique subarrays.
The problem describes a game played by a beaver with n pebbles. On each turn, the beaver arranges all remaining pebbles into a rectangle with more than one row and some number of columns. He then selects a single row and discards the rest.
Before I generate it, there is one important issue to resolve: LeetCode 2701 is a SQL-only Database problem, so there is no valid Python or Go LeetCode-submittable solution signature for this problem.
We are asked to count the number of integers that can be formed by permuting the digits of a given number $n$, do not start with zero, and are divisible by a given modulus $m$. The input $n$ can be as large as $10^{18}$, which means up to 18 digits, and $m$ is at most 100.
We are given two arrays: - nums, where nums[i] represents the initial value associated with index i + 1 - changeIndices, where at second s, we are allowed to perform a special operation on index changeIndices[s] Every index in nums starts as unmarked.
The system describes a rectangular box where a laser beam enters through one small hole on the left wall and must exit through another hole on the right wall. Inside the box, there are horizontal mirror segments placed either on the floor or on the ceiling.
This problem involves analyzing two database tables, Listens and Friendship, to determine pairs of "similar friends." In plain terms, we are asked to find pairs of users who are friends and who listened to at least three different songs on the same day.
We are given a directed acyclic graph. Some vertices are sources, meaning no edge enters them. Some vertices are sinks, meaning no edge leaves them. The number of sources and sinks is guaranteed to be equal.
The problem gives us an undirected tree with n nodes rooted at node 0. Every node has an associated positive value. We may repeatedly perform an operation where we choose a node, add its current value to our score, and then permanently set that node's value to 0.
The problem asks us to compute a special value called the punishment number for a given integer n. For every integer i in the range [1, n], we square the number and examine the decimal representation of i i.
This problem asks us to simulate the process of entering a cooking time into a microwave while minimizing the total finger movement and button pressing cost. The microwave accepts at most four digits.
The problem asks us to design a data structure that can efficiently answer frequency queries on subarrays. We are given an integer array arr, and we need to support repeated queries of the form: - Given left, right, and value - Return how many times value appears in the…
The problem asks us to split a string into consecutive groups, where every group must contain exactly k characters. We process the string from left to right. The first k characters form the first group, the next k characters form the second group, and so on.
We are asked to transport a group of schoolchildren through a sequence of regions along a single road. Each region has a fixed outdoor temperature and a maximum tolerable bus temperature. Every child inside a bus above that tolerable temperature triggers a monetary penalty.
This problem asks us to work with JavaScript promises and asynchronous execution. We are given two promises, promise1 and promise2, and each promise is guaranteed to eventually resolve to a numeric value.
This problem asks us to find a path in a matrix whose product contains the maximum possible number of trailing zeros. A trailing zero in a number is created by a factor of 10, and every factor of 10 comes from one factor of 2 paired with one factor of 5.
The problem asks us to find the smallest number that satisfies several constraints relative to a given number num and a target integer t. The input num is a string representing a positive integer, and the number must be zero-free, meaning none of its digits are zero.
We are given three inputs: - source, the original string of length n - pattern, a string that is guaranteed to already be a subsequence of source - targetIndices, a sorted list of indices in source that are eligible for removal An operation consists of removing a character…
The problem asks us to determine the minimum number of sequential queries required to transform an array nums into a Zero Array, where all elements are zero. Each query specifies a subarray [li, ri] and a value vali.
The problem is asking for the minimum time required to reach the bottom-right room (n - 1, m - 1) in a dungeon represented as an n x m grid. Each room (i, j) has a moveTime[i][j], which specifies the earliest time at which the room can be entered.
Paw the Spider's web consists of n main threads radiating from the center, dividing the plane into n equal sectors. Each sector may have bridges connecting its bounding main threads. Every bridge has attachment points at the same distance from the center.
The order of tanks changes after every message. At any moment, the column is some cyclic rotation of the original sequence. Suppose a tank currently stands at position i in the column and wants to send the message to a tank at position j i.
We are given a lowercase string and must choose some characters, in order, to form a subsequence that is both a palindrome and lexicographically as large as possible. A subsequence does not need to stay contiguous.
Each football team owns two uniforms, a home uniform and a guest uniform. During a match, the host team normally wears its home uniform while the visiting team wears its guest uniform. There is one special situation.
The problem asks us to transform a given string word of length n into a k-periodic string using the minimum number of operations. A string is k-periodic if it can be formed by repeating a substring s of length k multiple times.
The problem asks us to find the maximum XOR value of a strong pair in a given array of integers nums. A pair (x, y) is considered strong if it satisfies the condition |x - y| <= min(x, y).
The problem asks us to find the largest integer x such that: - x <= n - The bitwise AND of every integer in the inclusive range [x, n] equals 0 In other words, we want the widest possible suffix ending at n whose cumulative bitwise AND becomes zero, while keeping x as large as…
The problem describes a string manipulation game between Alice and Bob. Initially, Alice starts with a string word = "a". Bob provides a list of operations, represented by the integer array operations.
We are given the final exam results of a college where every student receives exactly one of three grades: 3, 4, or 5. Every student with the same grade must receive the same scholarship amount.
We are given one long lowercase string s. For every query, we are also given a set of characters C. Among all substrings of s, we only care about those whose set of distinct characters is exactly C. From those substrings, we must count how many are maximal by inclusion.
The problem gives us a complete binary tree where nodes are labeled in the same way as a binary heap. For every node with value x: - Its left child is 2 x - Its right child is 2 x + 1 The tree contains all node values from 1 to 2^n - 1.
We are given a Boolean formula consisting of constants 0 and 1, unknown variables denoted by ?, and the logical operators AND (&), OR ( The task is to determine whether it is always possible to uniquely identify the species of each colony, no matter how the unknowns are…
We are given two integer arrays, nums1 and nums2, of the same length n, along with an integer diff. We must count how many index pairs (i, j) satisfy both: - i < j - nums1[i] - nums1[j] <= nums2[i] - nums2[j] + diff The goal is to return the total number of valid pairs.
The problem asks us to determine the minimum time needed for all hens to eat all grains when both hens and grains are located on a one-dimensional line.
The problem asks us to compute the maximum Hamming distance for each element in an array of integers. The Hamming distance between two integers is defined as the number of positions where their binary representations differ.
We are working with an undirected graph where each vertex represents a social network profile. Two profiles are considered "doubles" if every other profile sees them identically. For any third vertex k, either both i and j are connected to k, or neither is connected to k.
We are given a rectangular dressing room divided into n × m squares, where each square contains a single shoe. Every shoe belongs to exactly one pair, and each pair is represented by a unique integer appearing exactly twice in the grid.
We have a collection of sheep, each identified by a unique number from 1 to n. Every sheep has a range of meadow regions it likes, represented as an interval $[li, ri]$. In the morning, the first shepherd ties any pair of sheep whose preferred regions overlap.
The problem gives us two sorted 2D integer arrays, nums1 and nums2, where every element is a pair of integers in the form [id, value]. Each id uniquely identifies an entry inside its own array, and the corresponding value represents that id's associated number.
We build a directed weighted forest incrementally. Every vertex has at most one outgoing edge, so if we start from any vertex and repeatedly follow outgoing edges, we eventually reach a root. When vertex i is added, the input gives several pairs (v, x).
This problem asks us to repeatedly search for a value inside an integer array and double that value whenever it is found. We begin with the integer original. If that number exists anywhere in the array nums, we multiply it by two and repeat the process using the new value.
The problem asks whether a program written in the joke language HQ9+ will produce any output. The program is a string containing between 1 and 100 printable ASCII characters. Only four instructions matter: "H", "Q", "9", and "+".
The problem describes a taxi traveling along a one directional road from point 1 to point n. Along the way, passengers request rides. Each ride is represented as [start, end, tip], meaning the passenger wants to travel from start to end and will additionally pay a tip.
This problem asks us to calculate the percentage of a given character, letter, in a string s, and return it as an integer rounded down to the nearest whole percent.
The problem gives us three inputs: - banned, an array of integers that are not allowed to be selected - n, which defines the valid integer range [1, n] - maxSum, the maximum total sum allowed for all chosen integers We must choose as many distinct integers as possible while…
We are given a permutation, which we can think of as a row of numbered positions, where each position contains a distinct value from 1 to n. A swap operation exchanges two positions, and after a sequence of swaps we obtain another permutation.
We have a farm with a straight road and several hills numbered from 1 to n. Each hill is separated from the previous one by a known distance. There are m cats, each of which finishes wandering on a particular hill at a certain time, and waits there for a feeder.
Codeforces 262B: Roma and Changing Signs
The problem asks us to count n-digit integers that are "good" with respect to a given integer k. A "good" integer is one whose digits can be rearranged to form a k-palindromic integer.
We are given two stacks of cards, each card either face up or face down. The goal is to merge the decks into a single stack in a way that preserves the relative order of cards within each original deck, and then perform a series of operations to turn all cards face down using…
This problem is asking us to generate all the Excel-style cell references within a rectangular range given in string format. Each cell is identified by a column letter and a row number, for example "A1" or "K2".
The problem gives us a database table named Transactions. Each row represents a transaction made by a customer on a specific date.
We are given a rectangular grid where some cells are painted with . All painted cells initially form one connected component using 4-directional adjacency, meaning movement is allowed only through shared sides. We may delete painted cells one by one.
The problem asks us to rearrange a given integer array nums such that positive and negative numbers alternate, starting with a positive number. The array has an even length and contains an equal number of positive and negative integers.
We are given a binary string s, an integer k, and many range queries. For each query [l, r], we only consider the substring s[l..r]. Among all substrings completely contained inside this range, we must count how many satisfy the k-constraint.
We are given three positive integers representing the areas of three faces of a rectangular parallelepiped that meet at a single vertex.
This problem involves a network of servers where server 0 is the master and all other servers are data servers. Each data server initially sends a message to the master, and the master instantly responds upon receiving the message.
The problem presents a sequence of DNA letters and a series of mutations that sequentially expand the sequence. Each mutation activates a contiguous subsequence, duplicates it, and mangles the copy in a specific way: all letters at even positions come first, followed by all…
The problem gives us an integer array nums that contains either positive integers or the value -1. We process the array from left to right while maintaining two conceptual arrays: - seen, which stores previously encountered positive integers - ans, which stores the answers for…
The problem asks us to determine the size of a DataFrame named players. Specifically, it requires computing two values: the number of rows and the number of columns.
This problem asks us to create a Pandas DataFrame from a given two dimensional list named studentdata. Each element of studentdata is itself a list containing exactly two values: 1. A student ID 2. The student's age For example: represents four students.
Alexander receives cards from friends one by one. Friend i sends card i, so card numbers and friend numbers are the same thing. At any moment Alexander may decide to send cards to some friends. He never creates new cards, he only reuses cards he has already received.
The problem asks us to determine if we can remove exactly one character from a string word such that all remaining characters in the string have the same frequency. The input string consists of lowercase English letters and has a length between 2 and 100.
We are asked to simulate a three-dimensional billiard-like scenario where a ball travels inside a rectangular corridor with perfectly reflecting walls, floor, and ceiling.
The problem is asking us to simulate a "reduce" operation on an array of integers without using the built-in Array.reduce method. The input consists of three elements: an array nums, a function fn, and an initial value init.
The problem asks us to consider every possible non-empty subarray of the input array nums. For each subarray, we compute how many distinct values appear inside it. We then square that distinct count, and finally sum the squared values across all subarrays.
The problem is asking us to identify two numbers that appear twice in an otherwise consecutive list of integers ranging from 0 to n - 1.
We have a directed graph of cities and roads. City 1 is the capital. Every road is either already usable or broken. A broken road may be repaired, after which it behaves like a normal directed edge. The government wants every city to become reachable from the capital.
We are given a simplified HTML-like document. Every tag is either an opening tag like <a, a closing tag like </a, or a self-closing tag like <a/. The document is guaranteed to be properly nested, so every opening tag has exactly one matching closing tag.
The problem gives us an integer array nums, where some positions contain -1. These -1 values represent missing elements that must be replaced. The key restriction is that we are allowed to choose exactly two positive integers, x and y, one time globally for the entire array.
In this problem, we are given an array nums and two integers x and y, where x y. During each operation, we choose one index i and apply two different decrements: - nums[i] decreases by x - every other element decreases by y The goal is to determine the minimum number of…
We are given a single English word and need to modify it so that only the first character becomes uppercase. Every other character must stay exactly as it originally appeared. The input contains just one non-empty string.
We are given a sorted sequence $x1, x2, dots, xn$. We want to construct another sequence $y1, y2, dots, yn$ such that every adjacent difference stays inside a fixed interval: $$a le y{i+1} - yi le b$$ and every value remains inside $[1, q]$.
We are given an n × n board where every cell contains an integer. For each cell, we compare two quantities. The first quantity is the sum of all numbers in that cell’s row. The second quantity is the sum of all numbers in that cell’s column.
The problem asks us to find the sum of the n smallest k-mirror numbers, where a k-mirror number is a positive integer that reads the same forwards and backwards both in base-10 and in base-k.
The problem asks us to take a string s and an integer k and compute the maximum number of partitions we can create from s under a specific set of operations. First, we are allowed to change at most one character in the string to any other lowercase English letter.
The problem presents two integer arrays, nums1 and nums2, of equal length n. You are allowed to select a contiguous subarray from both arrays and swap them exactly once, or choose not to swap at all.
The problem gives us a binary string s and a list of queries. Each query contains two integers, first and second. For every query, we need to find a substring of s whose decimal value satisfies: where ⊕ represents the bitwise XOR operation.
We are asked to construct three permutations $a$, $b$, and $c$ of length $n$ such that for every index $i$ the sum of $a[i]$ and $b[i]$ modulo $n$ equals $c[i]$ modulo $n$.
This problem gives us a 2D grid where every cell is either empty or blocked by an obstacle. A value of 0 means we can move through the cell freely, while a value of 1 means the cell contains an obstacle that can be removed.
The problem asks us to count the number of ways we can split an array of positive integers, nums, into two arrays, arr1 and arr2, such that they satisfy a set of monotonic conditions.
That will be quite long and detailed given your required structure and depth. To ensure quality and avoid truncation, I will provide it in a carefully structured, complete editorial format.
Here is a complete, detailed technical solution guide for LeetCode 2350 - Shortest Impossible Sequence of Rolls, following your requested format precisely.
The problem defines the product sum of two arrays as the sum of the products of corresponding elements at the same indices. If we have arrays a and b, then the product sum is: We are given two arrays, nums1 and nums2, both of the same length n.
The problem asks us to count the number of operations required to reduce either of two non-negative integers num1 or num2 to zero. An operation is defined as subtracting the smaller number from the larger one (or subtracting either if they are equal).
We are asked to organize a sequence of plants along a one-dimensional greenhouse so that each species occupies a contiguous segment, numbered left to right from 1 to m. Every plant has a species label and a fixed position on the line.
This problem asks us to process an array of strings and split each string using a given separator character. After performing all splits, we must collect every resulting substring into a single output array while preserving the original order.
The problem gives us two strings, s1 and s2, each containing exactly four lowercase English letters. We are allowed to repeatedly perform a very specific swap operation on either string.
We are given a single starting integer $X$, and we repeatedly expand it into a sequence. One expansion step replaces every number in the sequence by all of its positive divisors, written in increasing order.
The problem asks us to find the length of the longest contiguous subarray in an array of integers nums such that the frequency of every element in that subarray does not exceed a given integer k. In other words, in the resulting subarray, no number appears more than k times.
Edit This problem asks us to retrieve the k-th character from a string represented by a special binary tree called a rope tree. Instead of storing one large string directly, the string is distributed across the tree structure.
This problem models currency conversions as graph traversal across two separate days. You begin with exactly 1.0 unit of initialCurrency, and you may perform any number of exchanges on day 1 using the first set of conversion rates, followed by any number of exchanges on day 2…
We are given two binary strings, a and b. Our task is to transform a into b using only two operations. The first operation appends the parity of the current string a to its end, where parity is 1 if the number of 1s in a is odd, and 0 if even.
This problem models an excavation process on a square n x n grid. Several rectangular artifacts are buried in the grid, and each artifact occupies one or more cells.
The problem gives us an array of lowercase strings called words and two integer indices, left and right. We need to count how many strings within the inclusive range [left, right] are considered vowel strings.
Here is a fully detailed technical solution guide for LeetCode 1924 - Erect the Fence II, following your formatting instructions: The problem asks us to compute the minimum enclosing circle for a set of points in 2D space, where each point represents a tree.
That is a large, highly structured guide. To make sure I target the exact problem and keep the response complete in a single message, can you confirm you want the full solution guide for LeetCode 2368 - Reachable Nodes With Restrictions?
This problem asks us to identify how many numbers in an unsorted array are still guaranteed to be found by a randomized binary-search-like process. The array nums contains unique integers.
This problem is asking you to plan the consumption of two energy drinks over n hours to maximize the total energy boost. Each drink has a known energy contribution per hour, given by two arrays, energyDrinkA and energyDrinkB. You can drink only one energy drink per hour.
The problem asks us to rearrange a distinct integer array so that no element is equal to the average of its neighbors. The input is a zero-indexed array nums of length at least 3, and all elements are guaranteed to be distinct.
We are given a fixed number of shelves and a collection of rewards split into two categories, cups and medals. Each category is further divided into three ranks, but for the placement logic those ranks do not matter beyond counting total items in each category.
This problem asks us to minimize the total amount of money spent when buying candies under a special discount rule. For every two candies that are purchased, we may take one additional candy for free.
The problem presents a binary string floor representing a row of tiles, where '0' corresponds to a black tile and '1' corresponds to a white tile.
The problem asks us to count how many integers lie between two given numeric strings num1 and num2 (inclusive) such that the sum of their digits is between minsum and maxsum.
The problem gives a 0-indexed array nums of non-negative integers and allows a specific operation: choose an index i (where 1 <= i < n) such that nums[i] 0, then decrease nums[i] by 1 and increase nums[i - 1] by 1.
We are given a set of cupboards, each with two doors: left and right. Each door can be either open or closed. The initial state of each door is given in the input. Karlsson wants all left doors to be in the same position and all right doors to be in the same position.
We are given an undirected, connected graph representing cities and roads. Every pair of cities is reachable, and each road has equal travel time. Among all pairs of cities, we are especially interested in shortest routes between city 1 and city n.
The problem gives us an initial string s and a sequence of update operations. Each update changes exactly one character in the string. After every update, we must determine the length of the longest contiguous substring that contains only one repeating character.
We are asked to determine the earliest square where the opposition can hold a demonstration given the interference of the city administration. There are n squares arranged by increasing distance from the city center, with square 1 being the most central.
The problem asks us to minimize the length of a given string s by repeatedly performing two types of deletion operations. In the first operation, we can pick a character at some index and delete the closest identical character to its left if one exists.
The problem asks us to process a sequence of removal queries on an array nums, with an optional initial operation where we can replace nums with any subsequence of itself to optimize query processing. Each query specifies a threshold value.
The problem asks us to compute the largest three distinct rhombus sums in a given m x n integer grid. A rhombus in this context is a square rotated 45 degrees, whose corners align with grid cells. The rhombus sum includes only the border cells of this shape, not the interior.
The problem is asking us to count the number of good paths in a tree. A tree is a connected acyclic graph with n nodes and n - 1 edges. Each node has an integer value assigned by the array vals.
We are asked to determine whether one polygon is completely contained inside another. Polygon A is strictly convex, meaning all internal angles are less than 180 degrees and no three consecutive points are collinear.
That is a very large, detailed reference document with multiple long sections, full walkthroughs, two language implementations, worked examples, test suites, and edge case analysis for LeetCode 2737.
The statement is intentionally misleading. We are not asked to solve the Hamiltonian path problem ourselves. Instead, we must reproduce the output of Petya’s supposedly correct program. That changes the task completely.
This problem asks us to process a sequence of update queries on an array. After each update, we must compute the maximum possible sum of a subsequence where no two selected elements are adjacent in the array. The key detail is that the subsequence does not need to be contiguous.
This problem asks us to determine whether a given string s is exactly the acronym formed from an array of words. An acronym is created by taking the first character from each word in the words array and concatenating those characters in the same order as the words appear.
We are given a lowercase string and a classification of the 26 English letters into two groups, good and bad. A substring is considered valid if it contains at most k bad characters. Among all such valid substrings, we must count how many different strings appear.
The problem gives us an integer array nums, along with two integers, k and numOperations. We are allowed to perform exactly numOperations operations. In each operation: - We must choose an index that has not been used before.
This problem describes a tournament of n teams, where every pair of teams has a clear winner. The input is given as a square matrix called grid, where grid[i][j] tells us whether team i is stronger than team j.
The problem requires transforming a given string s into a good string, where a good string is defined as one in which every character appears the same number of times.
The problem gives us an array of strings called words. For every pair of indices (i, j) where i < j, we must determine whether words[i] is both a prefix and a suffix of words[j]. A string is a prefix of another string if it appears at the beginning.
The problem gives us an integer n, representing a conceptual array nums of length n, indexed from 0 to n - 1. We are also given a list of inclusive ranges, where each range marks positions in the array as covered.
The problem presents a rectangular piece of wood of size m x n and a list of specific prices for certain subrectangles. Each entry in the prices array, [hi, wi, pricei], indicates that a piece of wood of height hi and width wi can be sold for pricei dollars.
This problem asks us to identify users who requested confirmation messages at least twice within a 24 hour time window. We are given two database tables: The Signups table contains one row per user and records when the user signed up. The userid column is unique.
We need to compute the n-th pentagonal number. Pentagonal numbers form a sequence generated by a direct mathematical formula: $$Pn = frac{3n^2 - n}{2}$$ The input contains a single integer n, and the output is the value of this formula for that position in the sequence.
The problem gives us two strings, s and target. We are allowed to take characters from s and rearrange them in any order to form copies of target. Each character in s can only be used once.
We are given a network of containers, each holding some amount of water, and a set of bidirectional pipes connecting them.
The problem gives us a binary matrix grid with m rows and n columns. Every cell contains either 0 or 1. We must construct another matrix called diff, where each position diff[i][j] depends on how many ones and zeros appear in row i and column j.
The problem is asking us to count the number of interesting subarrays in a given array nums. An interesting subarray is defined by a modular counting condition: within any subarray nums[l..r], count the number of elements nums[i] where nums[i] % modulo == k.
This problem asks us to count the number of paths in a 2D integer matrix from the top-left corner (0, 0) to the bottom-right corner (m - 1, n - 1) such that the sum of the values along the path is divisible by a given integer k.
The problem asks us to find all days that are "good" for robbing a bank based on the number of guards on duty over a period of days.
This problem requires generating friend recommendations for users on the Leetcodify platform based on their listening habits. We are given two tables: Listens and Friendship.
The problem asks us to express each of a set of positive integers as a sum of exactly six numbers that only contain the digits 0, 4, or 7. These “lucky” numbers include zero, so numbers like 0, 4, 40, 47, 400, or 7074 are all valid.
We are given two sequences of stones, each colored red, green, or blue, represented as strings s and t. Liss starts on the first stone of the first sequence and Vasya on the first stone of the second sequence.
The problem asks us to determine the number of ways to split a given 0-indexed array nums into two non-empty contiguous subarrays such that the sum of the first part is greater than or equal to the sum of the second part.
We need to count how many integer-sided right triangles exist such that all three sides are at most n. A right triangle with sides (a, b, c) satisfies the Pythagorean equation: $a^2+b^2=c^2$$a$$b$$c = sqrt{a^2 + b^2} approx 21.21$$a^2 + b^2 = c^2 approx 225.00 + 225.00 = 450.
This problem asks us to compute the maximum total distance a truck can travel given fuel stored in two tanks: - mainTank, the truck's primary fuel tank. - additionalTank, a reserve tank that can transfer fuel into the main tank under specific conditions.
We are given a single compressed string that represents a song that was modified by repeatedly inserting the marker string "WUB" before, after, and between the original words.
The problem asks us to calculate the number of ways to distribute n candies among exactly three children such that no child receives more than a specified limit of candies.
The park is an undirected multigraph. Glades are vertices, trails are edges. Self-loops are allowed, and multiple edges between the same pair of vertices are also allowed. Vasya wants to start at vertex 1, traverse every edge exactly once, and return to vertex 1.
The problem asks us to count the number of index pairs (i, j) such that i < j and the sum of elements at these indices in nums1 is greater than the sum of elements at the same indices in nums2.
We maintain a dynamic sequence of integers. Initially the sequence contains only one value, 0. Each operation changes the sequence in one of three ways. We may add some value x to the first a elements, append a new number to the end, or remove the last element.
The problem gives us a list of integer ranges, where each range represents all integers between start and end, inclusive. We must divide all ranges into exactly two groups. Either group may be empty. The important restriction is that overlapping ranges cannot be separated.
The problem is asking us to place the minimum number of food buckets on empty spaces in a linear arrangement of cells so that all hamsters are fed. Each cell in the string hamsters represents either a hamster ('H') or an empty spot ('.').
The problem asks us to determine, for each spell, how many potions it can pair with to achieve a product of at least success. The arrays spells and potions represent the strengths of spells and potions, respectively.
We are given an integer array nums and a non-negative integer k. We must find the maximum possible length of a subsequence such that the number of adjacent unequal pairs is at most k.
The problem asks us to count how many substrings of length exactly three contain only distinct characters. A substring is a continuous portion of the string, so for every position in the string, we can examine the next three consecutive characters and determine whether all…
The video consumes a units of data every second while being watched. The internet connection downloads only b units per second, and a b, so if the users start immediately, the buffer will eventually run out.
We are given a graph of n houses arranged in a straight line. Normally, every house i is connected to i + 1, so the graph forms a simple path: 1 - 2 - 3 - ... - n In addition to these standard edges, there is one extra street connecting house x and house y.
In this problem, we are given a string s that represents characters typed on a faulty keyboard. Most characters behave normally and are appended to the current text on the screen. However, whenever the character 'i' is typed, the keyboard does not insert the character itself.
The problem asks us to find the minimum possible difference between the highest and lowest scores when selecting exactly k scores from an array of student scores.
The problem gives us a string num that contains only numeric digits from '0' to '9'. We must determine whether the string is "balanced". A string is considered balanced when the sum of digits located at even indices is equal to the sum of digits located at odd indices.
The problem gives us a binary array nums, meaning every element is either 0 or 1. We are allowed to divide the array at any index i where 0 <= i <= n, and that division creates two parts: - The left part contains elements from index 0 to i - 1 - The right part contains…
This problem asks us to sort an array using a custom sorting rule. Instead of sorting elements directly by their own value, we are given a function fn that transforms each element into a numeric value, and that numeric value determines the order.
We are given an undirected graph with n nodes and a list of edges. The graph may contain multiple disconnected components. Our task is to divide all nodes into ordered groups numbered from 1 to m. The key constraint is based on graph edges.
In this problem, we are given an array called forts, where each position represents one of three possible states: - 1 means the fort belongs to us. - 0 means there is an enemy fort. - -1 means the position is empty.
The problem asks us to calculate the sum of the integer division results, floor(nums[i] / nums[j]), for every pair of elements (i, j) in a given array nums. Here, floor() represents the largest integer less than or equal to the division result.
Here is a comprehensive, detailed technical solution guide for LeetCode 2593 - Find Score of an Array After Marking All Elements, following your requested format precisely. The problem provides an array of positive integers, nums.
We are given an array of integers and a value h. Every pair of elements contributes a value depending on whether the two elements are placed into the same group or different groups.
The problem asks us to identify, from a list of divisors, the integer that has the maximum divisibility score with respect to a given array of numbers. The divisibility score of a divisor is defined as the count of elements in nums that are divisible by that divisor.
We are asked to count numbers in a given range [l, r] whose binary representations are periodic. A number is periodic if its binary string has a repeating pattern of length k that divides the total length n of the string, meaning every segment of length k repeats exactly…
The problem describes a scenario where you visit a sequence of n rooms, starting from room 0 on day 0. The order of subsequent visits is determined by a rule that depends on how many times you have visited the current room.
We are given a set of containers, each containing some fixed amount of mercury. Over time, two kinds of operations happen. The first operation changes the mercury amount in a single container.
This problem asks us to compare two integer arrays and identify the unique values that appear in one array but not the other. We are given two 0-indexed integer arrays, nums1 and nums2.
The problem requires identifying users who made at least two purchases within a 7-day window. We are given a Purchases table with purchaseid, userid, and purchasedate. Each row represents a single purchase made by a user.
Each client describes a fixed amount of work Valera must perform: a certain number of low quality photos and a certain number of high quality photos.
The contest platform numbers every round consecutively by start time. A round can either be a standalone Div2 round, or a pair of simultaneous rounds where Div2 gets identifier i and Div1 gets identifier i + 1. Sereja only participates in Div2 rounds.
We are given a weighted tree where every edge has a cost, and we can think of it as a structure connecting n labeled nodes. Between any two nodes x and y, there is exactly one simple path, and we define a function g(x, y) as the largest edge weight along that path.
Here is a complete, detailed technical solution guide for LeetCode 2244 - Minimum Rounds to Complete All Tasks following your formatting and content requirements. The problem provides a 0-indexed array tasks, where each element represents a task's difficulty level.
The problem gives us an infinite two dimensional grid and asks whether we can travel from the starting point (1, 1) to a target point (targetX, targetY) using a specific set of operations.
The problem is asking us to find a binary string of length n that does not exist in a given list nums of unique binary strings, where each string in nums also has length n.
The problem asks us to design a graph data structure that supports two operations efficiently: 1. Dynamically adding directed weighted edges. 2. Finding the shortest path cost between two nodes. We are given a directed weighted graph with n nodes labeled from 0 to n - 1.
The problem asks us to determine how many strings in the array patterns appear as substrings within a given string word. A substring is defined as a contiguous sequence of characters, meaning the characters must appear in order and without gaps inside word.
The problem gives us an n x n square matrix called grid, where each cell contains an integer value. Our task is to construct a new matrix called maxLocal, whose dimensions are (n - 2) x (n - 2).
We are given a single integer and must decide whether it satisfies a very specific prime property. A number is considered truncatable if every suffix formed by repeatedly removing the leftmost digit is still prime.
We are given a rectangular grid of characters, each cell containing a lowercase English letter. The task is to count how many axis-aligned subrectangles have two properties at the same time.
The problem gives us a 0-indexed integer array nums. We must count how many pairs of indices (i, j) satisfy 0 <= i < j < n and have the following property: - Take the first digit of nums[i]. - Take the last digit of nums[j].
Two players stand on integer points of a number line. On each turn, the current player chooses a new integer coordinate inside a fixed interval relative to their current position. If the first player is at position x, he may move to any integer point in [x + a, x + b].
The problem asks us to calculate the arrival time of a train given its scheduled arrivalTime and the amount of delayedTime. Both times are expressed in hours using the 24-hour clock format. The goal is to determine the new arrival time after accounting for the delay.
We are given a square matrix with non-negative integer entries. You can think of it as a weighted directed graph with n nodes where the entry a[i][j] tells you how strongly node i influences node j.
We need to compute $$sum{i=1}^{a}sum{j=1}^{b}sum{k=1}^{c} d(i cdot j cdot k)$$ where $d(x)$ is the number of positive divisors of $x$.
The problem gives us a string s that contains only two possible characters, 'a' and 'b'. We must determine whether every 'a' appears before every 'b'. Another way to think about the requirement is this: once we encounter a 'b', we should never see another 'a' later in the string.
Each poem is divided into quatrains, groups of four lines. Two lines rhyme if the suffix starting from the k-th vowel from the end is identical in both lines. For example, with k = 1, we compare suffixes starting at the last vowel.
We can model the students as an undirected graph. Each student is a vertex, and every pair of enemies creates an edge. We want to split the remaining students into two teams of equal size such that no edge stays inside one team.
We are asked to simulate a chase on a two-dimensional grid between Princess Vlada and her playful Shadow. The grid is infinite but sparse trees are present, which act as obstacles. Both the Princess and Shadow start at distinct integer coordinates.
The problem asks us to simulate a game in which n children, numbered from 0 to n - 1, stand in a straight line. Child 0 starts with a ball, and every second the child holding the ball passes it to the next child in the current direction.
The problem requires us to detect the first occurrence of a binary pattern within a theoretically infinite stream of bits. The stream is accessed sequentially using the next() method, which returns one bit at a time.
The problem asks us to count the number of increasing quadruplets (i, j, k, l) in a 0-indexed array nums of size n, where nums is a permutation of the integers from 1 to n.
The problem gives us an integer p and defines an array containing every number from 1 to 2^p - 1. Each number is represented in binary using exactly p bits. We are allowed to repeatedly perform a special operation.
This problem provides a table named Orders, where each row represents a single purchase event. Every order contains an orderid, a productid, the purchased quantity, and the purchasedate. The goal is to identify all products that satisfy two conditions simultaneously: 1.
We are given exactly four integers in a sequence, each between 1 and 1000. The goal is to determine whether this sequence forms an arithmetic progression or a geometric progression. If it does, we must compute the next element of the progression.
This problem asks us to repeatedly process a string that contains lowercase English letters and star characters (). Every star represents a removal operation. When we encounter a star, we must remove two things: 1. The star itself. 2.
The problem gives us a mathematical expression string s containing only: - Single digit numbers 0-9 - Addition operators + - Multiplication operators The expression is guaranteed to be valid.
This problem is intentionally unusual for Codeforces. There is no mathematical formula, graph structure, or dynamic programming hidden in the statement. Instead, we are asked to classify a text document into one of three categories using a training dataset.
The problem asks us to count the number of special permutations of a given array of distinct positive integers nums.
The problem asks us to select the longest subsequence of words from a given array such that adjacent words in the subsequence satisfy two conditions: first, their corresponding groups values are different, and second, the words are of the same length and differ by exactly one…
The problem asks us to schedule an event where Devu, a singer, performs n songs of varying lengths, and Churu, a comedian, tells jokes of fixed 5-minute duration.
This problem asks us to compute a 3-day rolling average of daily step counts for each user. The input is a table named Steps, where each row contains: - userid, identifying a user. - stepscount, the number of steps taken on a particular day.
We need to construct a permutation of numbers from 1 to n with two conditions. The first condition is p[p[i]] = i for every position i. Applying the permutation twice must return us to the original index. This means every element points back to its partner.
Here’s the complete, detailed technical solution guide for LeetCode 3011 - Find if Array Can Be Sorted, following your requested formatting and style: The problem provides a 0-indexed array of positive integers nums.
We are asked to maintain an array of integers while supporting two types of operations: updating a single element to a new value, and computing a weighted sum over a subarray where the weights follow a small repeating zigzag pattern determined by a factor z.
The problem gives us three relational database tables: - students, which stores each student's ID, name, and major - courses, which stores all courses along with the major they belong to - enrollments, which stores which students took which courses and the grade they received…
We are asked to count the number of substrings of a string s that can be transformed into an anagram of a given string p. The string s can contain question marks ?, which can be replaced by any lowercase letter.
The Teams table contains the names of all teams participating in a league. Each row represents exactly one team, and the teamname column is guaranteed to contain unique values.
This problem asks us to implement a throttling mechanism for a function. We are given a function fn and a delay interval t in milliseconds. We must return a new function, called a throttled function, that controls how often fn is allowed to execute.
The problem gives us an m x n matrix where each cell contains an integer value. We may start from any cell, and from the current cell we are allowed to move only within the same row or the same column.
We are given the root of a binary tree and an integer k. Our goal is to find the size of the k-th largest perfect binary subtree contained anywhere within the tree. A perfect binary tree has two defining properties: 1. Every internal node has exactly two children. 2.
The problem asks us to find the length of the longest subarray within a given array nums such that the bitwise AND of all elements in that subarray is maximized.
We are given two arrays of the same length: - words[i] is a string. - groups[i] is either 0 or 1. We want to select a subsequence of words. A subsequence preserves the original order of elements, but we may skip any number of elements.
We are given a sequence of integers, and the task is to compute a "beauty" metric for any subsequence of it. The beauty is defined as the minimum number of operations needed to remove all elements, where each operation can remove a subsequence of equally spaced equal numbers.
We start with some small matrix b. One mirroring operation doubles its height. The top half stays unchanged, and the bottom half becomes the rows of the top half written in reverse order.
We are asked to maximize the total number of flamingos that can be observed from a row of binoculars on the x-axis. Each binocular sits at position (i,0) and can be aimed in any direction.
We are given one large text string s, then many query strings x. For every query, we must count how many substrings of s are rotations of x. If x = "abcd", then all of these strings are considered equivalent: abcd, bcda, cdab, dabc.
We are given two circles on the plane. Each circle represents an enemy ring surrounding a city. A radar can be placed at any point, and it detects everything within distance r from its position. The radar must be able to detect at least one point from each circle.
We have a collection of schoolchildren organized into groups, and each group wants to travel together in a taxi. Each group has between one and four children, and each taxi can carry at most four children.
LeetCode 2536: Increment Submatrices by One (Medium)
The game gives us two types of coins: - x coins worth 75 - y coins worth 10 Two players, Alice and Bob, take turns. Alice always moves first. On every turn, the current player must select coins whose total value is exactly 115.
The problem asks us to determine whether a given password string satisfies a set of security requirements. We are given a single string, password, and we must return true if every condition is satisfied, otherwise return false.
This problem asks us to assign jobs to workers in a way that minimizes the total number of days needed to finish all jobs. We are given two integer arrays of equal length: - jobs[i] represents the amount of work required for the i-th job.
This problem asks us to design a data structure that dynamically maintains a collection of integer intervals and efficiently reports how many distinct integers are covered by at least one interval. Initially, the interval set is empty.
The problem is asking us to determine the maximum number of minutes n computers can run simultaneously using a given set of batteries. Each battery has a fixed amount of energy in minutes, and you can initially assign at most one battery per computer.
The problem gives us an array of positive integers and asks us to choose the largest possible subset that can be rearranged into a very specific symmetric structure. The required structure looks like this: where is a power of two.
The problem presents a sequence of n cities numbered from 0 to n - 1 with an initial chain of unidirectional roads such that city i is connected to city i + 1 for all valid i. You are given a list of queries where each query adds a new road from city ui to city vi.
The problem asks us to count all substrings of a given string word that satisfy two conditions simultaneously: first, the substring must contain all five vowels 'a', 'e', 'i', 'o', and 'u' at least once; second, the substring must contain exactly k consonants.
The problem is asking us to simulate a painting process on a 2D matrix. We are given a 1D array arr of integers and an m x n matrix mat, both containing all integers from 1 to m n exactly once.
We are asked to identify T-primes in a list of positive integers. A T-prime is defined as a number that has exactly three distinct positive divisors. Thinking about divisors, the only way a number can have exactly three is if it is the square of a prime number.
The problem asks us to maximize the number of distinct elements in an integer array nums by performing a limited set of operations. Each element can be modified at most once by adding an integer in the range [-k, k].
This problem presents a 2D square matrix grid of size n x n where each cell contains a non-negative integer. Initially, all cells are white. An operation consists of selecting a cell (i, j) and coloring black all cells in column j from the top (row 0) down to row i.
The problem gives us an m x n binary matrix where every cell contains either 0 or 1. Our goal is to remove all 1s using the minimum number of operations. An operation can only be performed on a cell (i, j) that currently contains a 1.
This problem provides a pandas DataFrame named students with three columns: | Column | Type | | --- | --- | | studentid | int | | name | object | | age | int | Some rows contain missing values in the name column.
The problem asks us to transform a circular integer array so that every subarray of length k has the same sum. A circular array means the end wraps around to the beginning, so subarrays can cross the boundary of the array.
We are given an array of length n, initially filled with zeroes. Each query sets a single element to a value between 0 and 3. Zero represents an unset element, while 1, 2, or 3 are actual values.
The problem requires calculating the total distance traveled by each user based on ride data stored in a relational database. We are given two tables: Users and Rides. The Users table contains userid and name, where userid is unique.
The problem gives us a 0-indexed integer array called nums. We need to find an index such that the sum of all elements to the left of that index is equal to the sum of all elements to the right of that index. More formally, for an index i: - Left sum = nums[0] + nums[1] + ...
The problem asks us to find the shortest subarray within a given array nums such that the bitwise OR of all elements in that subarray is at least k. A subarray is any contiguous sequence of elements in nums.
This problem asks us to compute the minimum total cost required to move from index 0 to index n - 1 in an array. The movement rules are unusual because whether a jump is valid depends not only on the values at the start and destination indices, but also on every element in…
The problem gives us a string representing a time in 12-hour format using the pattern "HH:MM". Some characters may already be fixed digits, while others are replaced with "?". Our task is to replace every "?
The problem asks us to find the minimum possible score of a path between city 1 and city n in a graph defined by n cities and roads. Each road connects two cities bidirectionally and has an associated distance.
We start with exactly one card (x, y) where 1 ≤ x < y ≤ m. Three operations are available. The gray horse increases both numbers by one: (a, b) → (a + 1, b + 1). The white horse works only when both numbers are even: (a, b) → (a / 2, b / 2).
The problem asks for a function that returns the number of arguments passed to it. Essentially, we need to measure the length of the input in a dynamic, variadic sense. The input is presented as a JSON array in the examples, representing all the arguments passed to the function.
This problem asks us to create a wrapper around an existing function fn such that the wrapped version can only execute the original function one time. In other words, we are given a function fn, and we must return a new function.
We are given two arrays, nums1 and nums2, both of length n. Each index j represents a point: and has an associated value: For every query [xi, yi], we must find an index j such that: and Among all indices satisfying both constraints, we want the maximum possible value: If no…
We have a complete directed graph on n cities. For every car type, we know the travel time between every ordered pair of cities. These travel times are not guaranteed to be symmetric, so going from u to v may cost something different than going from v to u.
The problem asks us to compute, for every index i in the array nums, the total distance between i and every other index j where nums[j] == nums[i]. More formally, for each position i, we need to calculate: for all indices j such that: - nums[j] == nums[i] - j !
The problem asks us to create a generator for a circular array, arr, starting from a given index startIndex. A generator is a construct that yields a value each time it is called. The first call to the generator should return the element at startIndex.
This problem asks us to identify all users who qualify for a discount based on their purchase history stored in the Purchases table.
This problem asks us to analyze parking transaction records and compute aggregated statistics for each car. Every row in the ParkingTransactions table represents one parking session, containing the parking lot ID, the car ID, the entry and exit timestamps, and the fee paid for…
The problem describes a painting laid out on a number line. Each segment of the painting is represented as a half-closed interval [start, end) and is painted with a unique color value.
The problem asks us to count how many integers in the inclusive range [a, b] contain only unique digits. A number has unique digits if no digit appears more than once within that number. For example, the number 123 has unique digits because 1, 2, and 3 each appear exactly once.
The problem gives us two arrays, nums1 and nums2, and asks us to compute the XOR of every possible pair formed between the two arrays. For every element in nums1, we pair it with every element in nums2 exactly once.
We start with a non-negative integer. In one operation, we may choose any digit that currently appears in the number and subtract that digit from the number itself.
The problem asks us to implement a mechanism to run a generator that yields promises, with the additional ability to cancel the execution at any time.
This problem asks us to compare two integer arrays and count how many elements from one array appear in the other. More specifically, we need to compute two values: - answer1 is the number of indices i in nums1 such that nums1[i] appears at least once in nums2.
The problem is asking us to determine whether there exists a single original string that could have been encoded into two different given strings, s1 and s2. Each encoded string may contain letters and digits.
We are given an array a of length n. From each position i, we are allowed to choose a value bi such that 1 ≤ bi ≤ ai. Once we choose all values, we look at two quantities: the maximum element of the chosen array b, and the least common multiple of all elements in b.
This problem asks us to determine whether a matrix remains identical to its original form after applying a specific cyclic shifting operation exactly k times. We are given an m x n integer matrix mat, where m is the number of rows and n is the number of columns.
The problem asks us to count how many contiguous subarrays of nums match a given relationship pattern. Instead of comparing exact values, the pattern describes how adjacent numbers should relate to each other.
We are given two positive integers and need to compute their least common multiple. The least common multiple, usually abbreviated as LCM, is the smallest positive integer that both numbers divide evenly.
The problem asks us to calculate the average of all integers in an array nums that satisfy two conditions simultaneously: they must be even and divisible by 3.
The array nums is 1-indexed, meaning the first element corresponds to index 1, the second element corresponds to index 2, and so on. We want to select a subset of indices such that for every pair of selected indices i and j, the product i j is a perfect square.
The problem presents a string expression in the form "<num1+<num2", where both <num1 and <num2 are positive integers represented as strings.
The problem is asking us to find employees in a company who meet two specific conditions. First, their salary must be strictly less than $30,000.
The problem asks us to detect the first occurrence of a given binary pattern within an infinite stream of bits. We are given an object stream that allows us to read one bit at a time using the next() function.
We are maintaining a growing collection of intervals, each interval identified by its insertion order. Along with building this collection, we also define a directed reachability relation between intervals.
We are given a square matrix of size n×n, where n is guaranteed to be odd. Each cell of the matrix contains a non-negative integer. The task is to sum the "good" elements of this matrix.
We have an n × m chessboard-like grid, and we want to place as many soldiers as possible. Two soldiers conflict if the squared Euclidean distance between their cells is exactly 5. The only integer pairs whose squared distance equals 5 are (1, 2) and (2, 1) up to sign.
We are given a convex polygon with integer coordinates. Inside this polygon, including its boundary, there are finitely many lattice points. We choose two distinct lattice points uniformly at random, then construct a square whose diagonal is the segment between those two points.
The problem gives us a shuffled sentence, where every word has a number appended to its end. That number represents the word's original position in the sentence, using 1-indexed ordering.
The problem asks us to enhance all JavaScript arrays so that they have a convenient method, last(), which returns the last element of the array. If the array is empty, it should return -1.
This problem asks us to implement function composition. We are given an array of functions and must return a new function that combines all of them into a single callable function.
We are asked to determine whether a robot, which can only make fence corners at a fixed angle a, can construct a regular polygon. A regular polygon is defined as a closed shape with all sides and all angles equal.
We are asked to count permutations of length n where exactly k positions are "good." A position is good if the value at that position differs from the index by exactly 1.
The problem gives a hierarchical file system represented as a list of absolute paths, where each path is an array of folder names from the root to a leaf folder.
We have a collection of cards, each with a front color and a back color. Initially, all cards lie with the front side up. The goal is to make at least half of the cards show the same color on the upper side.
The problem asks us to determine the optimal hour to close a shop to minimize a penalty based on customer arrivals. The input is a string customers where each character represents an hour: 'Y' means customers arrive, and 'N' means no customers arrive.
The problem gives a circular array nums of length n. At each second, every element in the array can be replaced simultaneously by either itself, its previous neighbor, or its next neighbor.
The problem asks us to determine the minimum number of coins required to purchase all fruits in a market, given a special offer. You are provided with a 1-indexed array prices, where prices[i] denotes the number of coins needed to buy the ith fruit.
We have a circular running track of length l. Each runner starts at a fixed position on the circle, and independently chooses one of two directions with equal probability. Everyone runs at speed 1.
We are given a sequence of towers standing in a straight line, where the height of the tower at position i is h[i]. The goal is to make the sequence non-decreasing from left to right using a set of allowed operations.
We are asked to implement a Caesar cipher on an input string of uppercase Latin letters. Conceptually, this means each letter is shifted forward in the alphabet by a fixed number of positions, denoted by k. If the shift goes past 'Z', it wraps around to 'A'.
We are given a tree of n nodes, rooted conceptually at node 1, which starts painted red. All other nodes are blue. We have to handle two types of queries: first, paint a blue node red; second, for a given node, report the distance to the nearest red node.
The input describes a collection of countries, where each country is followed by a list of superstition names observed in that country. Each superstition is written as a line starting with an asterisk, and names may contain multiple words separated by irregular spacing.
The problem is asking us to construct an array nums3 of length n from two given arrays nums1 and nums2 of the same length. For each index i, we can choose either nums1[i] or nums2[i] as the value for nums3[i].
The problem gives us two separate undirected trees. The first tree contains n nodes and the second tree contains m nodes. A tree is an acyclic connected graph, so every pair of nodes has exactly one simple path between them.
The problem gives us two arrays of positive integers, nums and numsDivide. We are allowed to delete any number of elements from nums, and our goal is to make the smallest remaining element in nums divide every number in numsDivide.
This problem asks us to transform a wide table into a normalized row-based format. The database table Products contains one row per product, and each store has its own dedicated column. The value inside a store column represents the product's price in that store.
The problem provides a singly linked list where the nodes are sorted in non-decreasing order by their absolute values, rather than their actual values. The task is to rearrange this list so that the nodes are sorted in non-decreasing order according to their actual values.
Every student points to exactly one best friend, and every student is pointed to by exactly one other student. That means the friendship relation forms a permutation p. On day 1, notebook i stays with student i. On every following day, notebooks move according to p.
The problem is asking us to count the number of maximal contiguous blocks of equal numbers in a very large array nums. A block is maximal if it contains all consecutive occurrences of the same number, and numbers are guaranteed to appear in consecutive segments, i.e.
The problem gives us an integer array nums and several range queries. For each query [li, ri], we must examine every possible subarray fully contained inside nums[li..ri] and return the maximum XOR score among them.
The problem asks us to count the number of special characters in a given string word. A character is defined as special if it appears in both lowercase and uppercase forms within the same string.
Here is a comprehensive technical solution guide for LeetCode 2045 following your requested format: The problem asks us to find the second minimum time to travel from vertex 1 to vertex n in a weighted, undirected graph, where the weight of every edge is the same (time).
In this problem, we are given a binary matrix called grid. Every cell contains either: - 0, meaning the cell is empty - 1, meaning the cell is blocked or occupied We are also given a rectangular stamp with dimensions: - stampHeight - stampWidth The goal is to determine whether…
We are given two binary arrays, nums1 and nums2. Each element is either 0 or 1. We must replace every value with a positive integer according to its parity: - Every 0 must become an even positive integer. - Every 1 must become an odd positive integer.
This problem asks us to determine the best candidate for every project based on required skills and a scoring system. We are given two database tables. The Candidates table stores information about each candidate's skills and proficiency levels.
The equation in the statement is $$leftlfloor frac{x}{y} rightrfloor + leftlfloor frac{y}{x} rightrfloor = z$$ and we want all positive integers $z$ for which this equation has no solution in positive integers $x,y$.
The problem is asking whether it is possible to disconnect a path from the top-left corner (0, 0) to the bottom-right corner (m - 1, n - 1) in a binary matrix by flipping at most one cell from 1 to 0 or 0 to 1. A path only allows moves down or right into cells containing 1.
The problem gives us an array of strings where every string has the same length. For each string, we construct a difference array by subtracting the alphabet position of consecutive characters.
The problem asks us to determine whether a given integer n has exactly three positive divisors. In other words, we want to check if there are precisely three distinct integers d such that n % d == 0 and d 0.
The problem asks us to process two kinds of operations on an integer array: 1. Query how many peaks exist inside a subarray. 2. Update a single element in the array. A peak is defined as an element that is strictly greater than both its immediate neighbors.
We are asked to compute the number of ways to distribute a set of good and not-so-good events across a sequence of days divided into three consecutive segments: a white segment, a black segment, and a second white segment.
The problem asks us to find the minimum constant speed (in kilometers per hour) required to travel a sequence of train rides and reach the destination within a given floating-point hour.
The problem asks us to process a set of suffix matching queries efficiently. We are given two arrays: - wordsContainer, which contains candidate strings - wordsQuery, which contains query strings For every query string, we must find the index of the string in wordsContainer…
The problem gives us a 0-indexed integer array nums and asks whether it is possible to make the array strictly increasing after removing exactly one element. A strictly increasing array means that every element must be greater than the previous one.
Understood. I will provide the complete, detailed reference guide for LeetCode 2702 - Minimum Operations to Make Numbers Non-positive, following your formatting rules exactly. The problem provides a 0-indexed integer array nums and two integers x and y.
The problem gives us an m x n matrix called points. We must choose exactly one cell from every row. The value of the chosen cell is added to our score. However, there is a movement penalty between consecutive rows.
The problem is asking us to determine how many rods, labeled from 0 to 9, have all three colors of rings placed on them.
We are asked to navigate a forest represented as a grid. Each cell can be empty, contain a tree, or contain one or more mikemon breeders. Our goal is to move from our starting position to a designated exit while minimizing the number of battles we are forced to engage in.
We are asked to compute the expected number of shelves that have no untasted honey pots after a series of actions. Each shelf starts with some number of honey pots. Winnie moves a small number of pots from one shelf to another, tasting them in the process.
The problem asks us to transform the capitalization of every word in a given title string according to a specific rule based on word length. We are given a string called title, which contains one or more words separated by single spaces.
Here is the complete, detailed technical solution guide for LeetCode 2592 - Maximize Greatness of an Array following your formatting requirements.
We are asked to compute the minimum cost to paper all walls in an apartment where each room is a rectangular prism. For each room, the length, width, and height are given. The perimeter of a room determines how many strips of wallpaper are needed.
The problem is asking us to construct an integer x that satisfies two conditions. First, x must have the same number of set bits (1's in the binary representation) as a given integer num2. Second, the XOR of x with another integer num1 must be minimized.
The problem gives us the root of a binary search tree, abbreviated as BST, along with a list of query values. For every query, we must determine two numbers: - The largest value in the BST that is less than or equal to the query.
The problem is asking us to compute the total number of distinct sequences of dice rolls of length n that satisfy two constraints: the greatest common divisor (GCD) of any two consecutive rolls must be 1, and any repeated value in the sequence must be separated by at least two…
We are given an array of integers, and one operation consists of choosing a contiguous segment and increasing every element inside that segment by exactly one. We may repeat this operation any number of times.
We are asked to construct a set of points on a 2D integer grid with coordinates between 0 and n along the x-axis and 0 and m along the y-axis, excluding the origin (0,0). The set must satisfy the property that the Euclidean distance between any two points is not an integer.
The problem asks us to implement a memoized version of a given function fn. A memoized function is one that caches the results of previous calls and returns the cached result if the same inputs are passed again.
We have a mushroom-growing contest with two phases separated by a break. Each participant has two speeds, and the problem is that we do not know the order they will use them. During the first phase of length t1, mushrooms grow at one speed.
We are given a single integer n, and the statement guarantees that n itself is a Fibonacci number. The task is to represent n as the sum of three Fibonacci numbers.
The problem gives us a string s containing only lowercase English letters. We need to return the first letter whose second occurrence appears earliest in the string. This detail is extremely important.
The problem provides a relational table named Relations, where each row indicates a directed following relationship: a user identified by followerid follows another user identified by userid.
We are given a grid where some cells are blocked and the rest are usable land. On each usable cell we may place one building, either a blue tower worth 100 population or a red tower worth 200 population.
The problem is asking us to simulate a simple bank account deduction after making a purchase. You start with an initial balance of 100 dollars. The purchase amount is given as an integer, purchaseAmount, representing the cost of an item in dollars.
We are asked to find the number of students in a class who are the best in at least one subject. Each student has grades for multiple subjects, with each grade being a single-digit number between 1 and 9.
The problem asks us to determine the minimum positive integer $n$ such that the factorial of $n$, divided by the sum of the factorials of $n - ai$ for a given sequence of integers $a1, a2, dots, ak$, results in a positive integer.
We are given a grid of integers. From this grid we may cut away some rows from the top and bottom, and some columns from the left and right. After cropping, the remaining part must still be a rectangle with at least two rows and two columns.
The problem gives us an undirected graph with n nodes and a list of edges. Each edge connects two different nodes, and the graph may contain multiple disconnected components.
The problem asks us to find a minimum index at which a given integer array nums can be split into two non-empty contiguous subarrays, such that both subarrays share the same dominant element as the original array.
We have an array representing the message and another smaller array representing the encryption key. The encryption process slides the key across the message from left to right.
The problem asks us to transform a given integer n into a beautiful integer by adding the smallest non-negative integer x. An integer is defined as beautiful if the sum of its digits is less than or equal to a given target.
The problem requires designing a movie rental system for multiple shops, where each shop carries at most one copy of each movie.
The problem asks us to determine how many lattice points lie inside at least one of a set of circles on a 2D grid. Each circle is represented by a triplet [xi, yi, ri], where (xi, yi) is the circle's center and ri is its radius.
Each bus is active at exactly one moment in time. A bus starting at stop s and ending at stop f can carry any passenger whose trip interval [l, r] is fully contained inside [s, f]. A person arrives at stop l at time b, so they can only use buses whose time t satisfies t = b.
We are given two permutations containing the numbers from 1 to n. The first permutation is the current arrangement, and the second permutation is the target arrangement we want to reach. The allowed operation is unusual.
This problem describes a queue of n people, each wanting to buy a certain number of tickets. The input is an array tickets where tickets[i] represents how many tickets the i-th person wants.
The problem asks us to transform an array of integers nums by replacing each element with the sum of its digits. After this transformation, we need to determine the minimum element in the resulting array.
We have an array of non-negative integers. In one move, we choose an index i with a[i] 0, decrease a[i] by one, and increase some position i + 2t by one. The destination index must have the same parity as i, because the distance moved is always even.
We need to find a positive integer n such that inside the interval (n, 2n], exactly m numbers have exactly k ones in their binary representation.
The problem provides a 0-indexed array of positive integers nums and a positive integer k. The task is to count all distinct pairs (num1, num2) such that both numbers exist in nums and the sum of the number of set bits in num1 OR num2 and num1 AND num2 is at least k.
The problem asks us to implement a function that filters an array based on a custom condition defined by another function fn.
The problem describes an undirected weighted graph with n vertices, represented by an array of edges. Each edge connects two vertices and has a weight.
The problem asks us to remove beans from bags in such a way that all non-empty bags have the same number of beans while minimizing the total number of beans removed. Each element in the input array beans represents a bag with a positive number of beans.
This problem gives us two arrays, basket1 and basket2, where each element represents the cost of a fruit in a basket. Both baskets contain exactly n fruits.
The problem asks us to implement a custom string encryption algorithm. We are given a string s and an integer k. For each character in s, we need to replace it with the character that is k positions ahead in the string, in a cyclic manner.
The problem asks us to transform a string s that may contain the '' character into a lexicographically smallest string by repeatedly applying a deletion operation. Specifically, for each '' in the string, we must remove it along with the smallest non-'' character to its left.
The problem asks us to find the minimum possible sum of a "mountain triplet" within a given array of integers. A mountain triplet is defined as three indices (i, j, k) such that i < j < k, nums[i] < nums[j], and nums[k] < nums[j].
We have an array of integers and a strange operation that keeps the array length unchanged. In one operation, we look at the current k-th element, append a copy of it to the end, then remove the first element.
We are given a multiset of positive segment lengths. Any permutation of these lengths defines a walk along the number line starting from 0, where we cumulatively add each chosen segment.
This problem asks us to compute the running balance for every bank account after each transaction. We are given a table named Transactions where each row represents a single transaction performed by an account on a particular day.
We are given an n × m grid where some cells are painted with and the others are empty .. The picture is supposed to come from painting exactly two rectangular frames. A frame is not a filled rectangle. Only the border cells of the rectangle are painted.
This problem asks us to distribute a set of integers into numbered slots in a way that maximizes a scoring function based on bitwise AND operations.
We are given a world of n countries connected by n-1 directed roads. Ignoring the direction of these roads, the countries form a tree. Each brother wants to establish rule in some country and can control every country reachable via directed roads.
The problem gives us an integer array nums containing exactly three types of elements: 1. n - 2 special numbers 2. One element equal to the sum of all special numbers 3. One outlier The task is to return the largest possible value that could serve as the outlier.
This problem asks us to extend JavaScript's built in Date object by adding a new method called nextDay(). Once implemented, any valid Date instance should be able to call this method and receive a string representing the next calendar day in the format YYYY-MM-DD.
We have n possible university orders. Every order has two values. The value a[i] measures how many grey hairs the chairperson gets if she obeys that order. The value b[i] measures how unhappy the directors become if she refuses that order.
Let's go through a full, detailed technical solution guide for LeetCode 1959 following your formatting and style rules. This problem asks us to minimize the total wasted space when resizing a dynamic array multiple times.
The problem asks us to examine an array of strings, words, and identify the first string that is a palindrome. A palindrome is a string that reads the same forward and backward.
We start with a collection of distinct integers. On each move, the current player picks any two numbers from the set and inserts their absolute difference, but only if that difference is not already present.
The problem gives us a grayscale image represented as an m x n matrix called image. Every value in the matrix is an integer between 0 and 255, representing the intensity of a pixel. We must examine every possible 3 x 3 subgrid inside the image.
Each scientist gives us a sequence of tasks. Inside one scientist's sequence, the order is fixed and cannot be changed. Between different scientists, we may interleave tasks however we want.
The problem asks us to count how many subsequences of the given array have an odd sum. A subsequence is formed by choosing any subset of elements while preserving their original order. Unlike subarrays, subsequences do not need to be contiguous.
In this problem, we are given a positive integer n, and we must find the smallest integer greater than or equal to n that is considered "fair". A number is fair when the count of even digits is exactly equal to the count of odd digits.
We are given an $n times n$ binary matrix with exactly $n-1$ ones. We may swap any two rows or any two columns. The goal is to rearrange the matrix so that every one ends up strictly below the main diagonal. In other words, if a one is located at $(r,c)$, we need $r c$.
The column of tanks changes after every message. At any moment, the current first tank must send a message to the current last tank through a sequence of intermediate tanks. Suppose a tank currently stands at position j in the column and its receiving radius is a[x].
We are asked to place two types of restaurants on a tree-shaped city map in such a way that no two adjacent junctions host different types, each junction hosts at most one restaurant, and each network has at least one restaurant.
We are given a connected graph with $n$ nodes and $n$ edges, meaning it is a single connected component with exactly one cycle.
This problem gives us two integer arrays: - a, which always has exactly 4 elements. - b, which has length at least 4 and can be as large as 100,000. We must select exactly four indices from b: The score obtained from such a selection is: Our goal is to maximize this score.
The problem provides a database table Logins that records user login events, with columns userid and timestamp. Each combination of (userid, timestamp) is unique, ensuring that every row corresponds to a distinct login.
Polycarpus needs to organize daily profit reports into folders, where each folder contains consecutive days. The key restriction is that a folder cannot contain three or more days with negative profit, because the boss cannot tolerate more than two loss days per folder.
We have a grid-aligned rectangle of size n × m. Every valid point has integer coordinates between (0, 0) and (n, m). We must choose another axis-aligned rectangle inside it. The rectangle is described by four integers (x1, y1, x2, y2).
We are given three uppercase strings. The first two strings are the names written on the door, and the third string is the pile of letters found the next morning after somebody mixed everything together.
The problem gives us a set of unique points on a 2D plane. We need to determine the largest possible axis-aligned rectangle that can be formed using exactly four of those points as its corners.
The problem asks us to transform a given integer array nums into a non-decreasing array by performing a sequence of operations. In each operation, we are allowed to replace any single element with any two elements that sum to it.
The problem asks us to count how many integers within the inclusive range [low, high] satisfy two independent conditions. The first condition is based on the digits of the number. A number is considered beautiful only if it contains an equal number of even digits and odd digits.
The problem gives us an m x n matrix called grid, along with an integer k. Both dimensions of the matrix are guaranteed to be even numbers. The task is to rotate every layer of the matrix counter-clockwise exactly k times.
The problem asks us to count how many permutations of the numbers 1 through n satisfy a special condition called self-divisible. We start with the array: We must rearrange these numbers into every possible permutation, then determine whether the permutation is valid.
The problem presents an interactive scenario where there is a hidden number n in the range [1, 2^30 - 1]. We are provided with an API commonSetBits(num) which returns the count of bits that are set in both n and num when performing a bitwise AND.
The problem describes a one dimensional road containing cars positioned from left to right. Every car has one of three possible states: - 'L', meaning the car moves left - 'R', meaning the car moves right - 'S', meaning the car stays stationary All moving cars travel at the…
The problem asks us to find the kth smallest sum among all possible non-empty contiguous subarrays of a given integer array nums. Each subarray is formed by taking a contiguous sequence of elements from nums, and its sum is the sum of the elements in that subarray.
The problem provides two arrays, keysArr and valuesArr, which always have the same length. Each position in the arrays represents a potential key-value pair.
Joe is starting on the top floor of a multi-story building represented as a grid of cells. Each floor is a row of m cells, and each cell can either be empty, contain breakable bricks, or be an unbreakable concrete wall.
We are given an n × n grid where each cell is either: - A positive integer, representing a valid cell with that value. - -1, representing a blocked cell. Movement is allowed only between non-blocked cells that share an edge, meaning up, down, left, or right.
Vasya scribbled an Internet address in his notebook, but he was in a hurry and omitted all punctuation characters like :, /, and ..
The problem is asking us to process queries on a tree structure. Each query gives a path between two nodes starti and endi and a third node nodei.
The problem asks us to compute the number of same-end substrings within specified subranges of a given string s. A substring is same-end if its first and last character are identical. We are given multiple queries in the form [li, ri], each representing a substring s[li..ri].
This problem is unusual for Codeforces because it is not a traditional algorithmic task. We are given a single document and must classify it into one of three categories using a provided training dataset. The input contains an identifier, a title, and the full document text.
Vasya has a line of items, each with a specific weight, and he wants a robot to pick all of them using its two arms. The left arm can take the leftmost item and the right arm can take the rightmost item.
The problem is asking for the sum of two integers num1 and num2. In simpler terms, you are given two numbers, and you need to calculate their total. The input integers can be negative, zero, or positive, and the output should be a single integer representing the sum.
The problem asks us to compute a hashed string from an input string s by dividing it into equal-length substrings and then mapping each substring to a single character using a simple hashing function.
This problem asks us to compute how much rainwater can be trapped between vertical bars after rainfall. The bars are represented in a database table named Heights, where each row contains an id and a height.
The problem gives us a range of rented floors in a building, from bottom to top, inclusive. Within this range, some floors are marked as special floors and cannot be counted as regular office floors.
The problem gives us a binary string s, meaning the string contains only the characters '0' and '1', along with an integer k. A substring is considered valid if it satisfies the k-constraint.
The problem models a simplified memory allocation scenario. We have a computer’s RAM represented as a sequence of cells. Some of these cells are already occupied, and the remaining empty consecutive sequences form memory clusters. Each cluster is described by its size in cells.
This problem gives us two arrays, nums1 and nums2, both of the same length n, along with a list of queries. Each query modifies one of the arrays or asks for information about them.
We are asked to compute the minimum and maximum bus fare that a group of passengers could pay under specific rules. There are two groups of passengers: grown-ups and children. Every grown-up pays one ruble for themselves, and they can each bring at most one child for free.
The problem asks us to determine the minimum number of adjacent swaps required to rearrange a string of black and white balls such that all white balls (0s) are grouped on the left and all black balls (1s) are grouped on the right.
We are given exactly $n^2$ integers and must place them into an $n times n$ grid so that every row, every column, and both diagonals all have the same sum. The multiset of numbers cannot change, each value must appear exactly as many times as it appears in the input.
We are given an undirected graph where each vertex has a color, represented by an integer. The goal is to determine which color has the most diverse set of neighboring colors.
The problem asks us to divide an integer array nums of size n (where n is guaranteed to be a multiple of 3) into smaller arrays of exactly size 3, such that within each array, the difference between the largest and smallest element does not exceed a given integer k.
We are asked to count the number of distinct circular contours that can be formed on a plane where two black-painted rings are placed. Each ring is defined by two concentric circles, an inner radius and an outer radius.
An IPv6 address in its full form is a fixed structure made of eight blocks. Each block represents 16 bits and is written as exactly four hexadecimal characters, including leading zeros when necessary.
The problem asks us to compute the distinct difference array for a given integer array nums. For each index i in the array, we calculate the difference between the number of distinct elements in the prefix nums[0, ...
The problem asks us to determine if a given string s, consisting solely of digits '0' through '9', can be split into consecutive value-equal substrings such that exactly one substring has length 2 and all remaining substrings have length 3.
The problem asks us to find the integer that appears most frequently immediately after a given key value in the array. More specifically, we are given an integer array nums and an integer key, which is guaranteed to exist in the array.
The problem gives us an array of strings called words. We must count how many pairs of indices (i, j) satisfy two conditions: - i < j - words[i] and words[j] are similar Two strings are considered similar if they contain exactly the same set of distinct characters, regardless…
We have a linear arrangement of h cells, numbered from 1 to h. Some of these cells contain treasures, each with a positive dollar value.
The problem asks us to compute a specialized sum, called the x-sum, over all subarrays of length k in a given integer array nums.
The problem asks us to analyze a maze represented as an undirected graph of n rooms connected by corridors. Each corridor allows travel in both directions, and the input corridors lists all such connections.
The problem defines a special type of array called a "good" array. A good array must be a permutation of: This means the array must contain: - Every integer from 1 to n - 1 exactly once - The integer n exactly twice - No other numbers - Total length equal to n + 1 The input is…
The problem asks us to implement a data structure representing an infinite set of positive integers, initially containing all integers starting from 1.
Gena has a set of problems, and a group of friends who can each solve some subset of them. If Gena hires a friend, that friend will solve all problems they are capable of solving, covering multiple problems at once.
The problem gives us a list of enemies, where each enemy has an associated energy value. We also start with some initial amount of energy called currentEnergy. We begin with zero points, and every enemy starts as unmarked.
We are building a growing collection of intervals, and after each addition we may be asked whether one interval can “reach” another through a chain of moves.
The problem asks us to maximize the total energy gained from a sequence of magicians arranged in a line, where each magician provides a certain energy value, which can be negative or positive.
This problem asks us to find a path from the top-left corner (0,0) to the bottom-right corner (n-1,n-1) that maximizes its safeness factor. The grid contains thieves, represented by cells with value 1, and empty cells, represented by 0.
We are given several prime numbers. Each prime represents the first row of a square matrix of digits. If the prime has length n, then the matrix is n × n. The matrix must satisfy two conditions. First, every row interpreted as a decimal number must itself be prime.
The problem asks us to take two integer arrays of equal length, nums1 and nums2, and rearrange the elements of nums2 to minimize the XOR sum. The XOR sum is computed as (nums1[0] XOR nums2[0]) + (nums1[1] XOR nums2[1]) + ... + (nums1[n - 1] XOR nums2[n - 1]).
This problem gives us a pointer to an arbitrary node inside a doubly linked list. Unlike many linked list problems, we are not guaranteed to receive the head of the list. Instead, we may receive any node somewhere in the middle or even the tail.
The problem asks us to design a data structure that manages a 3D binary matrix of size n x n x n. Every cell initially contains 0, and we must support three operations efficiently: 1. setCell(x, y, z) sets a specific cell to 1. 2.
We are given a set of friends, where each friend independently succeeds in producing a contest problem with a known probability. Andrey will invite some subset of these friends. Once invited, each friend either contributes a problem or does not, independently of the others.
Each problem in the contest has two stages. You may solve the Small version first, gaining some fixed score. After that, you may continue and attempt the Large version, gaining additional score if it succeeds.
We are given a sequence of integers and asked to find the longest subsequence that forms an almost arithmetical progression.
Codeforces 407B: Long Path
We are asked to count substrings of a binary string that contain exactly k ones. A substring is any contiguous sequence of characters within the string, and different occurrences at different positions count separately.
We are given a tree with $n$ cities connected by $n-1$ roads. Each road has a positive length. The tree structure guarantees a unique path between every pair of cities.
Before I write the full guide, I want to confirm one detail: for LeetCode 3094 - Guess the Number Using Bitwise Questions II, there are no public input/output examples in the problem statement because it is an interactive problem.
This problem models a tournament as a directed acyclic graph (DAG). Each node represents a team, and each directed edge u - v means team u is stronger than team v. A team is considered the champion if no other team is stronger than it.
We are asked to find a point in three-dimensional space, with non-negative coordinates, such that the sum of the coordinates does not exceed a given value $S$.
The problem asks us to find the largest possible length k such that there exist two adjacent subarrays of length k in a given array nums, where both subarrays are strictly increasing.
The problem gives us an array called colors of length n. Initially, every element is 0, which represents an uncolored position. We then process a sequence of queries.
The problem asks whether it is possible to sort an integer array nums into non-decreasing order using a very specific type of swap. You can swap any two elements nums[i] and nums[j] if and only if their greatest common divisor (GCD) is greater than 1.
The problem asks whether a given integer n is strictly palindromic. A number is strictly palindromic if, for every base b between 2 and n - 2 (inclusive), its representation in base b reads the same forwards and backwards.
We are given a huge implicit grid, up to $10^9 times 10^9$, but only a small subset of cells are usable. Each usable region is given as a horizontal segment: a row number and a contiguous interval of columns.
The problem is asking us to determine whether a given string s can be formed by concatenating the first k elements of an array words, where k is some positive integer less than or equal to the length of words.
The problem is about scheduling meetings into a fixed number of rooms, where each meeting has a unique start time and a defined duration. We have n rooms numbered from 0 to n - 1, and we are given a list of meetings represented as [start, end) intervals.
Petr has a book with n pages. Starting from Monday, he reads a fixed number of pages each day of the week. The input gives those seven daily reading capacities in order from Monday to Sunday. We need to determine on which day Petr finishes the book.
The problem asks us to count all non-empty subsets of a given array nums such that the product of the elements in each subset is square-free. A square-free integer is an integer not divisible by the square of any prime greater than 1.
We are given a linear memory array of size n and a set of m instructions, each of which sets a contiguous block of memory to the value 13. The instructions are indexed in the input order. Some instructions may overlap, fully or partially, with others.
We are given a fully connected directed graph with weighted edges, represented as an adjacency matrix. Each vertex has an edge to every other vertex, including a zero-weight self-loop. Greg wants to play a game where he removes vertices one by one according to a given sequence.
This problem gives us the root of a special binary tree. At first glance, it looks like a normal binary tree, but there is an unusual property involving the leaf nodes.
The game starts with n pebbles. In one move, we split the current number of pebbles into a equal rows of size b, where a 1 and a b = current. After forming the rows, we keep exactly one row and throw away the others.
The problem requires calculating the maximum number of servers that can be upgraded at each data center independently, given the number of servers, upgrade costs, potential revenue from selling servers, and available money.
The problem is asking us to take a given m x n matrix, potentially containing -1 values, and produce a modified matrix in which each -1 is replaced by the maximum value in its respective column.
We have a path graph with n platforms and n - 1 bridges between consecutive platforms. Bridge i connects platform i and i + 1, and can be crossed exactly a[i] times before disappearing permanently.
The problem gives us a 0-indexed 2D array called variables, where each element contains four integers: - ai - bi - ci - mi For each index i, we must evaluate the following mathematical expression: If the result equals the given integer target, then index i is considered a good…
We have two sets of bicycle gears. The front gears are attached to the pedals, and the rear gears are attached to the back wheel. If the chain connects front gear a[i] to rear gear b[j], the resulting gear ratio is b[j] / a[i].
We are given several integers written in an arbitrary base between 2 and 36. Digits above 9 are represented with uppercase letters, so in base 16 the digit sequence continues as A, B, C, D, E, F, and in larger bases it may continue up to Z.
The problem gives us a linked list and asks us to place its values into an m x n matrix in clockwise spiral order. We begin filling from the top-left corner of the matrix, which is position (0, 0). The traversal direction follows the standard spiral pattern: 1.
We are given an array nums where every element is either 1, 2, or 3. In one operation, we may remove any element from the array. Our goal is to make the remaining array non-decreasing while performing the minimum possible number of removals.
The array contains exactly 3 n elements. We must remove exactly n elements, leaving 2 n elements behind. The remaining elements keep their original relative order because we are removing a subsequence, not rearranging the array.
We have a rectangular grid with n rows and m columns. Vasya starts at position (xc, yc). Then he processes k movement vectors one by one. For a vector (dx, dy), he repeatedly moves: He keeps moving in that direction until the next move would leave the grid.
We are given a social graph of people. Some pairs are friends, and some pairs dislike each other. Friendship and dislike are both undirected relations. The Beaver wants to invite a set of people satisfying three conditions at the same time.
The problem requires calculating the percentage of time each age group spends on two types of snap activities: sending and opening. We are given two tables: Activities and Age.
The problem asks us to calculate the number of days Alice and Bob spend together in Rome based on their respective arrival and departure dates. Each date is represented as a string in the format "MM-DD".
The problem provides an m x n integer matrix grid and asks for the maximum sum of an hourglass shape within the matrix. An hourglass is defined as a 3x3 structure with the top and bottom rows fully included, and only the center element from the middle row.
The problem describes how a secret spreads over time. On day 1, exactly one person knows the secret. Every person who learns the secret behaves according to two rules: 1. They must wait delay days before they can begin sharing the secret. 2.
Codeforces 405B: Domino Effect
The grid is formed by n + 2 horizontal bars and m + 2 vertical bars. These bars divide the plane into unit squares. The bars are numbered starting from 1. Some horizontal bars listed in hBars may be removed, and some vertical bars listed in vBars may also be removed.
We are organizing a programming competition where the goal is to select at least $n cdot m$ finalists. Participants can qualify in three ways: first, by winning one of the main elimination rounds, which has $c$ problems and produces $n$ winners; second, by winning one of the…
We are given a set of people labeled from 1 to n, and each person may specify exactly one other person who must come immediately after them in a line. If this value is zero, the person does not specify who follows them, meaning their successor is not fixed.
The problem asks us to compute a special average for every index in the array. For each position i, we want to look at a subarray centered at i with radius k. That means we include all elements from index i - k through i + k, inclusive.
We are given an array nums and an integer k. We must choose a subsequence of exactly 2 k elements while preserving the original order of the array. After selecting the subsequence, we split it into two equal halves: - The first k selected elements form the left group.
The problem gives us an undirected weighted graph with n nodes and m edges. Each edge connects two nodes and has a positive weight. We need to determine which edges belong to at least one shortest path from node 0 to node n - 1.
We are given an undirected social network where each user is identified by a string name and friendships are given as pairs of names.
The problem presents us with a binary array called derived of length n. This array is constructed from another binary array original of the same length using the bitwise XOR operation on adjacent elements.
The Sessions table stores information about user activity sessions on a platform. Each row represents a single session and contains: | Column | Meaning | | --- | --- | | userid | The user who performed the session | | sessionstart | When the session began | | sessionend | When…
This problem asks us to implement a utility function called debounce. The function receives two inputs: - A function fn - A delay value t in milliseconds The goal is to return a new function, called the debounced version of fn.
The problem presents a rooted tree where each node is uniquely identified by an integer from 0 to n-1. This integer also represents the genetic value of the node.
We are asked to determine whether a given sequence of numbers appears as consecutive elements in some row of a hypothetical GCD table. The table has n rows and m columns, where each element at row i and column j is the greatest common divisor of i and j.
This problem asks us to answer multiple independent queries on a binary tree. For each query, we temporarily remove an entire subtree rooted at a specific node, then compute the height of the remaining tree.
We are given a set of rooms connected by doors, where each door has two mood values: one for each direction. Petya can move from room to room, and every move adds to his mood according to the direction-specific value.
The problem gives us an integer array nums, and we must choose two numbers x and y from the array such that they form a strong pair.
We are given a single string that represents a sum of small integers. The expression is written using digits and plus signs, where every number is guaranteed to be either 1, 2, or 3.
This problem asks us to determine if a given object or array is empty. In JavaScript terms, an object is empty if it has no key-value pairs, while an array is empty if it has no elements.
We are given a binary string s, containing only '0' and '1'. We must count how many substrings satisfy a special condition called "dominant ones".
The problem asks us to choose the largest possible set of people to invite to a party, subject to two conditions. First, if a person is invited, all their friends must also be invited. Second, no two people in the invited set may dislike each other.
The problem gives us a binary string s, which means the string contains only the characters '0' and '1'. We need to determine whether the longest contiguous sequence of 1s is strictly longer than the longest contiguous sequence of 0s.
We have a bag containing white and black mice. The princess moves first, then the dragon, then the princess again, and so on. Whoever draws a white mouse immediately wins. If someone draws a black mouse, the game continues. There is one extra rule during the dragon’s turn.
We are given a sequence of characters consisting of digits 0-9 and the symbols < and . This sequence is interpreted as a simple tape program. The interpreter has two pointers: the current character pointer (CP) and the direction pointer (DP).
The input array differences describes how consecutive values in an unknown array change from one position to the next.
We are asked to color an $n times m$ grid such that every contiguous region of the same color forms a square. Each square can be of any size, as long as it is a perfect square in shape and does not overlap another square of the same color.
We are asked to count how many sequences of fixed length we can build from integers between 1 and n, with two constraints. First, the sequence is non-decreasing. Second, every element must divide the next one in the sequence.
The problem describes a circular typewriter containing all lowercase English letters from 'a' to 'z'. A pointer moves around this circular arrangement, and initially the pointer starts at 'a'. To type a character, the pointer must currently point at that character.
This problem gives us a database table named Salary that stores employee salary records. Each employee may appear multiple times because older salary records are still kept in the table.
We are given an array of length $2^n$, where $n$ is at most 20, and a sequence of queries. Each query specifies a number $qi$, which tells us to repeatedly split the array into blocks of size $2^{qi}$, reverse each block, and then join the blocks back together.
We are looking at every contiguous segment inside the interval $[a, b]$. For a chosen length $l$, every segment of exactly $l$ consecutive integers must contain at least $k$ prime numbers. The task is to find the smallest such $l$. If no segment length works, we print $-1$.
The problem asks us to find, for each string in a given array, the shortest substring that does not appear in any other string in the array. If multiple shortest substrings exist, we must choose the lexicographically smallest one.
The problem gives a tree with n nodes labeled 0 to n-1, rooted at node 0. The tree is represented using a parent array, where parent[i] indicates the parent of node i. Node 0 has no parent, so parent[0] == -1.
We simulate a hash table that resolves collisions using linear probing with step size m. When an object with hash value t is inserted, we first try cell t. If it is occupied, we try (t + m) mod h, then (t + 2m) mod h, and continue until we find an empty position.
The problem gives us an undirected graph with n vertices labeled from 0 to n - 1. The graph is represented using an edge list, where each edge connects two vertices in both directions. Our task is to find the length of the shortest cycle in the graph.
We are given two integer arrays, nums1 and nums2. The array nums2 was created from nums1 using two operations: 1. Remove exactly two elements from nums1. 2. Add the same integer x to every remaining element.
The problem asks us to count all possible sub-multisets of an array nums such that the sum of elements in each sub-multiset is within a given inclusive range [l, r]. A sub-multiset is like a subset, but it accounts for repeated elements in the original array.
The problem asks us to design a data structure that tracks uploaded videos and continuously reports the length of the longest uploaded prefix. A prefix of uploaded videos means that every video from 1 through i has already been uploaded. The goal is to return the maximum such i.
The problem presents a sequential game consisting of n levels, where each level inflicts a specific amount of damage to the player. You are given a list damage where damage[i] represents the damage from level i.
The problem asks us to find the number of pairs of nodes in an undirected graph that cannot reach each other via any path. You are given an integer n representing the total number of nodes labeled from 0 to n - 1 and a list of edges representing connections between nodes.
The problem gives us an array of strings called words and a list of range queries called queries. For each query [li, ri], we must determine how many strings in words between indices li and ri, inclusive, both start and end with a vowel.
This problem asks us to analyze employee salary data within departments and extract all employees earning the second-highest salary in each department. The input is a table employees with three columns: empid, salary, and dept.
The problem gives us a database table named Tweets with two columns: | Column | Description | | --- | --- | | tweetid | Unique identifier for each tweet | | content | The text content of the tweet | We need to identify all tweets that are considered invalid.
The problem gives us an array of distinct integers and defines two possible operations: 1. If the first element is currently the smallest value in the array, we remove it. 2. Otherwise, we move the first element to the end of the array.
The problem asks us to find the longest self-contained substring in a given string s. A substring is self-contained if it does not share any characters with the rest of the string.
The problem gives us an undirected tree with n nodes. Every node has a price associated with it, and we are also given several trips between pairs of nodes. Since the graph is a tree, there is exactly one simple path between any two nodes.
Codeforces 407C: Curious Array
The problem asks us to count unique palindromic subsequences of length 3 in a given string s. A palindromic string reads the same forwards and backwards, and in this case, we only care about strings of exactly three characters.
This problem asks us to rename the column names of a Pandas DataFrame according to a fixed mapping. We are given a DataFrame named students with four columns: - id - first - last - age The goal is to return a DataFrame where these columns have been renamed to more descriptive…
That is a long, multi-section technical reference document. I can provide the full guide, but it will be quite extensive.
We are given a rational number written as a normal fraction p / q, and another number written as a finite continued fraction.
There are n kids sitting in a fixed cyclic order. During the buying process, the chosen package sizes must form a strictly increasing sequence globally across all turns. After kid n, the next turn goes back to kid 1.
Codeforces 261B: Maxim and Restaurant
You are given an integer array nums and must choose three indices (i, j, k) such that: - i < j < k - The triplet value is (nums[i] - nums[j]) nums[k] Among all valid ordered triplets, we want the maximum possible value.
The problem gives us a binary string s, containing only '0' and '1' characters. We are allowed to rearrange the bits in any order we want, but we must use exactly the same bits that appear in the original string.
The problem asks us to transform a given integer array nums into a new array result following specific movement rules. Each element in nums determines how many steps to move in a circular manner, either to the right (if positive) or left (if negative).
We are asked to compute the average distance a tourist would walk if they visited all destinations on a straight road in every possible order, starting from kilometer zero. Each destination is at a distinct, positive integer distance from the starting point.
The problem is asking us to analyze a table of user permissions where each user's permissions are encoded as an integer. Each bit in this integer represents a distinct access level or feature.
The problem asks us to find all pairs of prime numbers (x, y) such that both numbers are between 1 and n inclusive, their sum equals n, and x <= y.
We are given a sequence of words that originally described a type in a fictional language. The language has only two valid constructions. The simplest type is int. The second type is pair<type,type, where each side is itself another valid type.
This problem asks us to compute a minimum score requirement (cutoff score) for each school such that the number of students who meet or exceed that score does not exceed the school’s capacity, while also making that number as large as possible.
The problem asks us to find the longest contiguous subarray where all elements are equal, after we are allowed to delete at most k elements from the original array.
The problem asks us to take a string that represents a sentence in a fictional language, and a list of forbidden pairs of letters. Each forbidden pair consists of two distinct letters that cannot appear next to each other in the string in any order.
The problem gives us a country road network that forms a tree. A tree is a connected graph with no cycles, which means there is exactly one path between any two cities. The cities are numbered from 0 to n - 1, and city 0 is always the capital.
We are given two arrays, nums and target, of equal length. In a single operation, we choose two different indices and add 2 to one element while subtracting 2 from another element.
The problem asks us to design a data structure that supports a stream of operations on a dynamic collection of integers. Numbers are added over time, and the oldest inserted number can also be removed.
The problem asks us to find a node in a directed graph with a very specific property: it must be reachable from two given starting nodes (node1 and node2) such that the maximum distance from either starting node to this node is minimized.
This problem asks us to identify employees whose monthly working time is less than the required number of hours. Each employee has a minimum number of hours they must work during October 2022 to avoid salary deductions.
We are choosing a path from a western village at the origin to a river bank located at a vertical line $x = a$, then crossing the river in a straight line to another point on the eastern bank $x = b$, and finally following a pre-determined path back into the eastern village.
The problem gives us a 0-indexed array of positive integers and asks us to repeatedly perform a specific operation to reduce the array's length as much as possible.
We are working with a rooted infinite tree where every node always has exactly $k$ outgoing edges to children. Each of those $k$ edges has a fixed weight: the first is 1, the second is 2, and so on up to $k$.
The problem gives us an array named digits, where every element is a single decimal digit from 0 to 9. We must use exactly three elements from this array to build valid three digit integers.
We are given an undirected, simple, connected graph with a fixed number of vertices and edges. On top of that, a subset of vertices is marked.
This problem asks us to determine the minimum total damage Bob will receive while fighting a group of enemies. Each enemy has two attributes: - damage[i], the amount of damage they inflict on Bob every second while alive. - health[i], the amount of health they start with.
We are asked to simulate the operation of a short message service center (SMSC) that receives tasks, each consisting of a timestamp and a number of messages to send. Each second, if there are messages waiting in the queue, the SMSC sends exactly one message.
The problem asks us to find the longest subsequence of a string s that can be repeated k times while still being a subsequence of s. A subsequence is derived by deleting zero or more characters from a string without changing the order of the remaining characters.
The problem places us in a grid-like parking lot of size 4·n by 4·m meters, divided into squares of 4 by 4 meters, each containing a car with a known "rarity" value.
We are given a string containing uppercase and lowercase English letters. We want to transform it into a "fancy" string where every uppercase letter appears before every lowercase letter.
We are given a single formatted string that represents a set of lowercase English letters. The set is written in a very specific textual form: it starts with an opening brace, ends with a closing brace, and inside the braces letters are listed separated by comma and space.
We are given an undirected graph where every edge already belongs to one of two categories. Roads marked "S" are narrow roads that the Elf clears, and roads marked "M" are wide roads that Santa clears. We must choose a subset of roads satisfying two conditions at the same time.
Here’s a complete, detailed solution guide for LeetCode 2443 - Sum of Number and Its Reverse, fully following your formatting rules.
This problem models a social network with friendship restrictions. We are given n people labeled from 0 to n - 1. Initially, nobody is connected to anyone else. Over time, friendship requests arrive one by one, and each request must be processed immediately.
This problem asks us to make all elements in an integer array equal by repeatedly reducing the largest elements to the next largest element in the array.
We are given a grid of pixels with n rows and m columns. Every cell is either black () or white (.). We want to repaint the minimum number of cells so that the final picture satisfies two rules. First, every column must become monochromatic.
The problem gives us an even-length integer array nums. We repeatedly perform the following operation until the array becomes empty: 1. Remove the smallest element. 2. Remove the largest element. 3. Compute their average. 4. Store that average in another array called averages.
This problem gives us an array nums and an integer k. We repeatedly perform an operation where we remove the last element of the array and add it to our collection.
We are given two strings, s and pattern. The task is to find the smallest starting index in s such that the substring of length len(pattern) is "almost equal" to pattern.
The problem asks us to compute the total digit difference across every pair of numbers in the array. All numbers have the same number of digits. For any two numbers, their digit difference is defined as the number of positions where the digits are different.
This problem requires simulating the distribution of elements from a 1-indexed array nums into two separate arrays arr1 and arr2 under specific rules.
The problem asks us to process a DataFrame representing customers and remove rows that have duplicate email addresses. Specifically, if multiple rows share the same email value, only the first occurrence should be kept, and all subsequent duplicates should be discarded.
The problem gives us the root of a binary tree and an integer k. We must determine how many nodes in the tree are considered "great enough". A node is great enough if two conditions are satisfied: 1. Its subtree contains at least k nodes. 2.
The problem gives us an integer array nums, and our goal is to make all remaining elements distinct. The only allowed operation is removing exactly 3 elements from the beginning of the array. If fewer than 3 elements remain, we remove everything that is left.
This problem asks us to implement a Trie data structure, also known as a prefix tree, which is designed to store strings efficiently in a way that allows fast lookups and prefix queries. We are required to implement the following operations: 1.
We are given a long strip of vertical planks, each with a fixed height. For any contiguous segment of planks of fixed length $k$, Vasya paints a rectangle whose height is determined by the shortest plank inside that segment.
Mashmokh wants to pick a sequence of n distinct integers so that his boss, Bimokh, gains exactly k points in a game.
The problem involves a set of people, each with an appointment on a unique day in the next n days. You do not know who is scheduled on which day, but you can query the organization in forms that list up to m names.
We are given a set of distinct integers. We may delete at most k of them, where k ≤ 4. After deleting, we want all remaining numbers to produce different remainders modulo some positive integer m. Two numbers collide modulo m exactly when their difference is divisible by m.
The problem asks us to compute the maximum possible alternating sum of any subsequence of the given array. An alternating sum is calculated after the chosen subsequence is reindexed starting from index 0.
The problem asks us to find the kth distinct string in an array of strings. A string is considered distinct if it appears exactly once in the entire array. The important detail is that the order matters. We are not sorting the strings or rearranging them in any way.
We are given a set of distinct points on a 2D plane. The task is to count how many unordered triples of points form a configuration where one point lies exactly at the midpoint of the segment formed by the other two.
The problem is asking us to take an input integer finalSum and split it into the maximum number of unique positive even integers such that their sum equals finalSum.
We have an array of integers, and we are allowed to pick two different elements and simultaneously increase one by 1 and decrease the other by 1. We can perform this operation as many times as we like.
We are given a connected undirected graph representing cities and roads in the Roman Empire. Each merchant wants to transport goods between two cities.
This problem involves simulating passengers arriving at a bus station and boarding buses as they arrive. We are given two tables: Buses and Passengers. Each bus has a unique busid and an arrivaltime, and each passenger has a unique passengerid and an arrivaltime.
Let's dive into a comprehensive solution guide for LeetCode 3154 - Find Number of Ways to Reach the K-th Stair. This problem involves Alice navigating a staircase starting at stair 1, aiming to reach stair k.
The problem gives us an integer array nums and an integer k. We must divide the array into one or more subsequences such that every number belongs to exactly one subsequence, and for every subsequence, the difference between its maximum value and minimum value is at most k.
This problem asks us to classify every store member into a category based on their shopping behavior. The classification depends on how often a member makes a purchase after visiting the store. We are given three database tables: - Members contains the list of all members.
The problem asks us to determine how many steps it takes to make a given array nums non-decreasing by repeatedly removing elements that break the non-decreasing property. Specifically, for each step, any element nums[i] where nums[i - 1] nums[i] is removed.
The problem gives us a circular array called colors, where each value represents a tile color. A value of 0 means red, and a value of 1 means blue. We are also given an integer k. We must count how many groups of exactly k contiguous tiles form an alternating sequence.
We are given a string of digits, which represents the concatenation of the scores Vasya achieved in three rounds of Robot Bicorn Attack. Each round produces a non-negative integer not exceeding 1,000,000, and numbers cannot have leading zeros unless the number itself is zero.
The problem asks us to count occurrences of query strings inside Fibonacci strings. Fibonacci strings are defined recursively: the first string is "a", the second is "b", and each subsequent string is the concatenation of the previous string followed by the one before that.
The problem asks us to simulate a scenario in which k workers transport n boxes from a right-side warehouse to a left-side warehouse across a bridge.
We are working inside a rectangular football field where the left side contains a goal segment on the vertical line $x = 0$, and the right side contains a horizontal wall at height $y = yw$.
The problem asks us to transform a given string word into a k-special string using the minimum number of deletions. A string is considered k-special if, for every pair of characters in the string, the difference in their frequencies does not exceed k.
The problem asks us to compact a JSON-like object or array by removing all falsy values. Falsy values are those that evaluate to false in a Boolean context, such as null, 0, false, "", undefined (though JSON does not have undefined), and NaN.
The problem asks us to determine how many valid placements of Alice and Bob exist on a 2D grid of points such that Alice can build a rectangular fence with her position as the upper left corner and Bob’s position as the lower right corner.
The problem gives us a binary tree where every node contains a unique value from 1 to n. We are also given two node values: - startValue, the node where the path begins - destValue, the node where the path must end We must return the shortest sequence of directions needed to…
The problem is asking us to modify a given DataFrame named employees by adding a new column called bonus. Each value in the bonus column should be exactly double the corresponding value in the salary column.
We are given a line of cups, each cup carrying a unique label from 1 to n. The initial left-to-right order of these labels is unknown. What we do know is the exact sequence of m operations performed on this line.
The problem requires us to select cells from a 2D matrix grid such that no two selected cells are in the same row, and all selected values are unique. Our goal is to maximize the sum of these selected values.
The problem asks us to determine, for a list of people arriving at certain times, how many flowers are in full bloom at the time of their arrival. Each flower has a start and end time representing the period it is in full bloom, inclusive of both endpoints.
Codeforces 405C: Unusual Product
This problem provides a list of numbers represented as strings, all of equal length. The task is to answer multiple queries, where each query asks for the index of the k-th smallest number after trimming every number in the list to its last trimi digits.
The problem asks us to find the maximum gcd-sum of a subarray of a given integer array nums with the constraint that the subarray has at least k elements.
This problem is very different from a standard algorithmic task. We are not given arrays, graphs, or numeric constraints. Instead, we receive a text document and must predict which of three categories it belongs to. The input contains three parts.
The problem requires computing the value of the last element in an array after a series of sequential cumulative sum operations over a fixed number of seconds. You start with an array a of length n where all elements are initialized to 1.
The problem is asking us to implement a polyfill for JavaScript's built-in Function.prototype.bind method. Specifically, we need to create a bindPolyfill method that can be called on any function.
The problem asks us to design a mutable tree-based data structure that supports three operations: lock, unlock, and upgrade. Each node in the tree may either be unlocked or locked by exactly one user. The operations must follow strict rules about when a node may change state.
This problem is asking us to determine whether there exists a path from the top-left corner (0, 0) to the bottom-right corner (m-1, n-1) of a 2D grid containing only '(' and ')' characters, such that the sequence of characters along the path forms a valid parentheses string.
We are given a set of points positioned along a one-dimensional line. Petya wants to count how many triplets of points can be chosen such that the distance between the leftmost and rightmost points in the triplet does not exceed a given value d.
The problem is asking us to maximize the number of consecutive integers we can select from an array after we are allowed to increase any element by at most 1.
The problem gives us an n x n matrix called lcp, where: - lcp[i][j] represents the length of the longest common prefix between: - the suffix starting at index i - the suffix starting at index j If the unknown string is word, then: - suffix i is word[i:] - suffix j is word[j:]…
The problem asks us to determine how many employees in a company have worked at least a minimum number of hours, specified by target. We are given a 0-indexed array hours, where hours[i] represents the total hours worked by employee i.
We are given a team of n football players, each with a unique number from 1 to n. The coach wants to organize practice games so that every pair of players has faced each other on opposing teams at least once.
That is a detailed, long-form solution guide request. Before I generate it, I need one missing detail: do you want the solution centered on the optimal DP + graph/component approach (grouping numbers by modulo k and solving independent chains with house-robber style DP), or…
The problem gives us an array of positive integers and allows a special merge operation between adjacent elements.
The problem gives us a 2D integer matrix nums. Each row contains several integers, and we repeatedly perform a special removal process until every element has been removed. During each operation, we do two things: 1. From every row, remove the largest remaining element. 2.
The problem asks us to reconstruct a rooted tree given constraints on the sizes of the subtrees for each node. You are given an array c of length n, where c[i] represents the total number of nodes in the subtree rooted at node i.
We are given a bipartite graph. One side contains Berland cities, the other side contains Beerland cities, and every flight is an undirected edge between the two countries. Each edge must be assigned to one of t private companies.
This problem asks us to compute the total cost of all possible three-topping pizza combinations using a list of available toppings from a database table. Each topping has a name and a cost, and toppings are unique.
This problem asks us to count the number of passengers in a list who are strictly older than 60. Each passenger's information is compressed into a fixed-length string of 15 characters.
This problem asks us to implement a custom version of setInterval, but with a key difference. Instead of using a fixed delay between executions, the delay grows linearly according to the formula: where count starts at 0 and increases after every execution.
Let the inradius of $\triangle ABC$ be $r$, and let its area be $\Delta$.
The problem asks to prove the exact trigonometric identity
The motion takes place on two adjacent faces of the cube.
The problem asks for the locus of the centroid of a triangle whose vertices are the midpoints of three segments joining fixed points above a plane to arbitrary points on the plane.
The geometry becomes transparent after reducing the three-dimensional configuration to a two-dimensional meridian section through the axis of the cone.
The problem asks for two loci inside a cube.
We are asked to analyze functions $f:\mathbb{Q} \to \mathbb{Q}$ satisfying the following property: for every $x,y \in \mathbb{Q}$, at least one of the two identities
The board has $2024$ rows and $2023$ columns.
We must prove
For each $n>N$, the sequence satisfies
For each real number $\alpha$, define
The problem is asking us to track cumulative game activity for each player. Specifically, we are given an Activity table where each row represents a single login session for a player on a specific device and date, along with the number of games played during that session.
This problem asks us to simulate the operation of a Centennial Wheel and determine the minimum number of rotations requi
The problem asks us to query a database table Patients and return the records of patients who have Type I Diabetes. Each
The problem asks us to identify products that were sold only during the first quarter of 2019, meaning between 2019-01-01 and 2019-03-31, inclusive. We are provided with two tables: Product and Sales.
The problem requires implementing a BSTIterator class that allows forward and backward traversal over the in-order sequence of a Binary Search Tree (BST). In-order traversal of a BST visits nodes in ascending order.
The problem asks whether two strings, word1 and word2, are "close" according to two allowed operations. Operation 1 allows swapping any two existing characters in the string, which means the relative order of characters is flexible.
The problem is asking us to compute the distance value between two arrays arr1 and arr2 given a threshold d. Specificall
Each event has a range of possible days when it could have happened. For the -th event, any integer day between and is acceptable. We must assign exactly one day to every event, and no two events may share the same day.
The problem gives two arrays, houses and heaters, where each value represents a position on a one dimensional horizontal line.
This problem extends the mechanics introduced in earlier Maze problems, but adds two important complications. First, we are no longer looking for a simple reachable or unreachable answer. Instead, we must find the shortest path by travel distance.
The problem asks us to reverse only the vowels in a string while leaving all non-vowel characters in their original positions.
Here is the complete, detailed technical solution guide for LeetCode 1044 - Longest Duplicate Substring, following your formatting rules exactly. The problem asks us to find the longest duplicated substring in a given string s.
Given a non-negative integer n, we must count how many times the digit 1 appears in every number from 0 through n, inclusive. The important detail is that we are not counting how many numbers contain the digit 1.
This problem asks us to determine the maximum number of vowels that appear in any substring of a given length k within a
The problem asks us to compute the diameter of a binary tree. The diameter is defined as the length of the longest path between any two nodes, measured in number of edges. Importantly, this path does not need to pass through the root of the tree.
This problem asks us to transform a given sentence into a fictional language called Goat Latin, following a specific set of string manipulation rules. The input is a string called sentence, where words are separated by a single space.
The problem is asking to find the maximum dot product of two non-empty subsequences of arrays nums1 and nums2 such that
This problem gives us a database table named Point that contains integer coordinates on the X-axis. Each row represents one point, and the column x is unique because it is the primary key. The task is to compute the smallest absolute distance between any two points in the table.
The problem describes a binary watch that uses LEDs to represent time. The watch has two sections. The top section contains 4 LEDs for the hour, and the bottom section contains 6 LEDs for the minutes. Each LED represents a binary digit.
The problem asks us to determine whether subarrays of a given array can be rearranged to form an arithmetic sequence. An
The problem asks us to design an authentication system that manages tokens with expiration times. Each token is valid for a fixed timeToLive seconds starting from the moment it is generated or renewed.
The problem is asking us to find the smallest integer k = 2 such that a given number n can be represented in base k with all digits equal to 1. In other words, we want n to be a sum of consecutive powers of k, like 1 + k + k^2 + ... + k^(m-1) for some integer m = 2.
In this problem, we are given the root node of a binary tree. Every node belongs to a specific level in the tree. The root is at level 1, its direct children are at level 2, their children are at level 3, and so on.
We are asked to simulate a tiny GUI layout system. There are three kinds of widgets. A plain Widget has a fixed width and height. HBox and VBox are container widgets that can store other widgets. An HBox places children horizontally, while a VBox places them vertically.
We are given the current month as a string and an integer k representing how many months later a new game release will happen. The task is to determine the month after advancing exactly k months forward in the calendar.
The problem gives your current position on a 2D Cartesian grid as (x, y) and a list of other points. Each point is represented as [ai, bi].
The game is played on a one-dimensional stripe of squares, each either black or white. Vasya paints the initial configuration, and Petya can then perform moves to achieve an alternating pattern, where no two adjacent squares share the same color.
This problem asks us to implement a compressed string iterator. We are given a string where each character is immediately followed by a number representing how many times that character appears consecutively in the uncompressed version. For example, "a3b2" represents "aaabb".
In this problem, we are given two arrays, quality and wage, where each index represents a worker. The value quality[i] describes how much work or contribution the i-th worker provides, while wage[i] describes the minimum amount that worker is willing to accept.
The problem gives an m x n matrix and asks us to return all elements in spiral order. Spiral order means we start from the top-left corner and move in a clockwise spiral pattern: 1. Traverse the top row from left to right 2. Traverse the right column from top to bottom 3.
The problem gives us n identical dice, where each die has faces numbered from 1 to k. We roll all n dice and want to count how many different sequences of rolls produce a total sum equal to target. The important detail is that order matters.
The problem asks us to compute the XOR coordinate value for each element in a given 2D matrix and then find the kth largest among them.
The problem gives us a string s consisting only of lowercase English letters. We must find the index of the first character that appears exactly once in the entire string. If every character appears more than once, we return -1.
The problem asks us to find the largest palindrome number that can be written as the product of two n-digit integers. A palindrome is a number that reads the same forward and backward. For example, 9009 is a palindrome because reversing its digits still gives 9009.
The problem asks us to sort a singly linked list in ascending order and return the head of the sorted list. The input is the head node of a linked list. Each node contains an integer value and a pointer to the next node.
The problem gives us an n x n square matrix that represents an image. Each element in the matrix corresponds to a pixel or value in the image. Our task is to rotate the entire image by 90 degrees clockwise. The important requirement is that the rotation must happen in-place.
We have a single tram traveling along a line with n stops, starting empty at the first stop and ending empty at the last stop. At each stop, a certain number of passengers exit before any new passengers enter.
This problem gives us an undirected weighted graph with n nodes. Each edge connects two nodes and has a distance value associated with it.
We are given a string representing an email address, but all the . symbols have been spelled out as dot and all the @ symbols as at. Our goal is to reconstruct the original, valid email address in its shortest form.
We are given a sequence of marks, but the subject for each mark was lost. There are only two subjects, and we know exactly how many marks belong to each one. If subject 1 must receive a marks, then subject 2 automatically receives b = n - a marks.
You included two different problems in one message, and the second problem (LeetCode 1408) appears to replace the first
The problem asks us to determine whether a binary tree is symmetric around its center. A tree is symmetric if the left subtree is a mirror reflection of the right subtree.
The problem gives us a binary matrix of size m x n, where every cell contains either 0 or 1. We must count how many squa
This problem asks us to determine the minimum initial health a knight needs in order to safely travel through a dungeon and rescue a princess.
In this problem, we are given two strings, name and typed. The string name represents the intended sequence of characters your friend wanted to type. The string typed represents the actual characters that appeared on the screen.
We are given a string made only of '0' and '1'. Each character represents the team of a football player standing in a line. A dangerous situation happens if at least seven consecutive players belong to the same team.
We are given a string consisting of characters a, b, and c, and we are allowed to repeatedly perform two types of operations: copy the left character of any adjacent pair onto the right, or copy the right character onto the left.
The problem asks us to build a spellchecker with three levels of matching priority. For every query word, we must search the wordlist and return the best matching word according to a strict precedence order. The first and highest priority rule is an exact match.
The problem asks us to design a custom stack data structure that behaves like a normal stack, while also supporting an additional operation called getMin(). This operation must return the minimum value currently stored in the stack, and it must do so in constant time, O(1).
The problem gives us an integer array nums of length n + 1, where every value is guaranteed to be in the range [1, n]. Since there are n + 1 numbers but only n possible distinct values, at least one number must appear more than once. The task is to return that duplicate value.
We are asked to count the number of distinct binary search trees (BSTs) that have exactly n nodes labeled from 1 to n, with the additional constraint that the height of each tree is at least h.
We are given a string of lowercase letters that contains no spaces. The task is to split this string into exactly k consecutive pieces. Every piece must have length between a and b, inclusive. The order of characters cannot change. We are only deciding where to cut the string.
We are asked to count how many numbers from 1 to n consist only of the digits 0 and 1 in their decimal representation. In other words, Hexadecimal's memory only stores numbers that, when written in base 10, contain no digits other than 0 or 1.
The problem asks us to determine whether one string can "break" another string through some permutation. Given two strin
We are asked to navigate a graph of cities connected by bidirectional roads, with a twist: certain sequences of three consecutive cities are forbidden due to superstition. Formally, there are n nodes and m edges, all unweighted, representing cities and roads.
This problem asks us to minimize the number of unique integers in an array after removing exactly k elements. In other words, we are given a list arr and a number k, and we need to strategically remove k elements so that the count of distinct integers left in the array is as…
The problem gives us a table named Transactions where each row represents a financial transaction. Every transaction has a unique transactionid, a timestamp stored in the day column, and an integer amount.
The problem asks us to generate an n x n matrix and fill it with numbers from 1 to n^2 in spiral order. A spiral order traversal means we begin at the top-left corner and move in a clockwise pattern: 1. Move left to right across the top row 2.
This problem asks us to merge two already sorted singly linked lists into one new sorted linked list. The important detail is that we are not creating entirely new nodes for the merged result. Instead, we reuse the existing nodes by reconnecting their next pointers.
The problem asks us to generate all integers of length n such that the absolute difference between every two consecutive digits is exactly k.
The task is to print a given sequence of integers using a limited number of variables and two operations: assigning a variable to an integer, and printing a variable. Each assignment carries a cost equal to the number of set bits in the assigned number, while printing is free.
The problem gives us an absolute Unix-style file path and asks us to convert it into its canonical, simplified form. An absolute path always starts from the root directory, represented by /.
We have n mounds placed on a straight line at positions 1, 2, ..., n. The frog wants to visit every mound exactly once, so we must output a permutation of these positions.
The problem asks us to transform a string s containing only characters 'X' and 'O' so that all characters become 'O'. A move consists of selecting three consecutive characters and converting them to 'O'. If a character is already 'O', it remains unchanged.
The problem asks us to count the number of contiguous subarrays in a binary array nums whose sum equals a given integer goal. A binary array only contains 0s and 1s.
In this problem, we are given the coordinates of several darts thrown onto a 2D plane. Each dart is represented as a poi
There are m + n + 1 distinct cards in total. Shrek initially knows his own m cards, Donkey knows his own n cards, and one card is hidden on the table. Nobody knows the hidden card directly. Players alternate turns, with Shrek moving first.
Each skyscraper is an axis-aligned rectangle sitting on the ground. A building with parameters (h, l, r) occupies every point with l ≤ x ≤ r and 0 ≤ y ≤ h.
The problem asks us to identify the majority element in a given array of integers nums. The majority element is defined as the number that appears more than half of the times in the array, i.e., more than ⌊n / 2⌋ times where n is the length of the array.
This problem asks us to find the largest rectangular area that can be formed inside a histogram. A histogram is represented as an array of integers called heights, where each integer describes the height of a bar, and every bar has a width of exactly 1.
The problem gives us a string s and asks us to divide it into substrings such that every substring is a palindrome. A palindrome is a string that reads the same forward and backward. We must return all possible valid ways to partition the string.
We are given two sequences of balls, labeled by lowercase letters: the original sequence A and the target sequence B. We want to transform A into B using four types of operations: inserting a ball, deleting a ball, replacing a ball, or swapping two adjacent balls.
Each item has two properties. If Bob pays for that item, the cashier spends t[i] seconds processing it and Bob also spends c[i] money. During those t[i] seconds, Bob can steal other items, one item per second.
The problem asks us to find the length of the shortest reference string that can encode a list of words. A reference string is formed by concatenating some of the words with a '' character at the end of each word.
Each process repeatedly executes two atomic instructions:
The problem requires identifying all people in a group whose list of favorite companies is not a subset of any other per
The problem gives us an array called colors, where each position contains one of three possible values: 1, 2, or 3. Each value represents a color assigned to that index. We are also given a list of queries.
We are asked to determine how many ways the organizers of a winter school can buy exactly n liters of cola using a limited supply of bottles in three sizes: 0.5-liter, 1-liter, and 2-liter.
This problem asks us to compute the smaller angle formed between the hour hand and the minute hand on a standard 12-hour
We have three points on a plane: the cinema, the house, and the shop.
The problem asks us to determine the minimum number of parentheses insertions required to make a given string of parentheses valid.
The problem gives us a two dimensional grid representing a forest. Every cell contains one of three possible values: - 0 means the cell is blocked and cannot be entered. - 1 means the cell is empty and can be walked through.
We are given an undirected multigraph. Floors are vertices, staircases are edges. Harry starts at floor 1 and wants to visit every floor at least once. The graph is dynamic. Between Harry's walks, Ron and Hermione may modify staircases.
We are given a string made only of ( and ). Among all contiguous substrings, we need to find the maximum length of a substring that forms a valid bracket sequence. We also need to count how many substrings achieve that maximum length.
This problem asks us to generate every possible generalized abbreviation of a given word. A generalized abbreviation is created by replacing some characters in the string with their count, while preserving the order of the remaining characters.
The problem gives us an array of strings where every string has exactly the same length. If we place all strings vertically, character by character, the strings form a rectangular grid.
The problem gives an integer array nums where exactly half of the elements are even numbers and the other half are odd numbers. The task is to rearrange the array so that every even index contains an even number and every odd index contains an odd number.
The problem asks us to find the length of the longest turbulent subarray inside a given integer array arr. A subarray is considered turbulent if the relationship between every adjacent pair of numbers alternates between greater than () and less than (<).
The problem asks us to convert a given integer num into its representation in base 7, returning the result as a string.
The problem asks us to design a mutable range sum data structure. We are given an integer array nums, and we must efficiently support two operations: 1. Updating the value at a specific index. 2. Querying the sum of elements within a range [left, right].
The problem describes a scenario in which you have a sequence of balls numbered consecutively from lowLimit to highLimit, inclusive. Each ball must be placed into a box, where the box number is determined by the sum of the digits of the ball’s number.
The problem asks us to count how many words in a given list are subsequences of a string s. A subsequence is formed by deleting zero or more characters from the string without changing the order of the remaining characters.
We know the current championship standings before the final race. Every racer already has some number of points, and the last race distributes additional points to the top m finishers.
The problem gives us an array of lowercase strings called words. We must return all characters that appear in every string in the array, including duplicate occurrences. The important detail is that duplicates matter.
We are asked to determine if a given positive integer can be represented as a triangular number. Triangular numbers are formed by arranging dots into an equilateral triangle, so the _n_-th triangular number is the sum of the first _n_ positive integers.
This problem asks us to construct a mathematical expression using only the number x repeatedly, combined with the operators +, -, , and /, such that the final expression evaluates exactly to target.
The problem asks us to reconstruct a binary matrix with exactly two rows and n columns given three constraints: the sum
We are asked to evaluate a string expression in a small function-based language. Every string literal is enclosed in quotes, and there are only four types of functions: concat, reverse, and substr in two forms. Each function operates only on strings or integers as indices.
The problem gives us a list of tasks, where each task is represented as [actual, minimum]. For every task: - minimum is the amount of energy we must currently have before we are allowed to start the task.
This problem asks us to find the titles of movies that satisfy three separate conditions at the same time. First, the content must be marked as kid-friendly. In the Content table, this is represented by the column Kidscontent = 'Y'. Second, the content must actually be a movie.
We are asked to solve a quadratic equation of the form , where , , and are integers in the range . The goal is to find all real roots of the equation, count them, and print them in ascending order with high precision.
The problem is asking us to take an existing m x n matrix and "reshape" it into a new matrix with r rows and c columns. Reshaping means rearranging the elements in the matrix in row-major order (left-to-right, top-to-bottom) without changing the order of the elements themselves.
The problem requires us to take a string s containing multiple words separated by single spaces and return a list of strings representing the words written vertically.
The problem asks us to randomly pick an integer point from a set of non-overlapping rectangles in 2D space. Each rectangle is defined by its bottom-left (ai, bi) and top-right (xi, yi) corners.
The problem gives us a sorted integer array nums in strictly increasing order and asks us to convert it into a height-balanced binary search tree (BST). A binary search tree is a binary tree where: - Every node in the left subtree contains values smaller than the current node.
This problem asks us to determine whether a given array of integers is monotonic. An array is considered monotonic if it is entirely non-decreasing (monotone increasing) or entirely non-increasing (monotone decreasing).
The problem asks us to find the smallest element that appears in every row of a given m x n matrix mat, where each row is sorted in strictly increasing order. In other words, we need an element that is common to all rows and is the minimum among all such elements.
We have a rectangular board where some cells are already blocked by and the remaining cells . must be covered by polyominoes. The allowed pieces are extremely flexible: any connected shape consisting of 2, 3, 4, or 5 cells may be used, with arbitrary rotations and reflections.
We are asked to construct a special kind of binary sequence, called a recurrent binary sequence. Each term in this sequence is either 0 or 1, and for a given integer k, there exist coefficients c₁, c₂, …, cₖ, also 0 or 1, such that every term from the k-th onward is a…
The problem asks us to design a data structure that supports efficient range sum queries on a fixed array. We are given an integer array nums, and after initialization, the array never changes.
This problem asks us to find the length of the longest contiguous subarray that can contain only 1s after flipping at most k zeroes into ones.
The problem asks us to determine whether the elements of a given array can be rearranged so that they form an arithmetic
The problem asks us to supply water to all houses in a village in the most cost-effective way. Each house has two options: either build a well in the house with a fixed cost, or connect the house to another house via pipes, where each pipe has its own construction cost.
We are given up to 100 triangles on the plane. Each triangle represents a black spot on a white square. Triangles may overlap, intersect, or even completely cover one another. The task is to compute the perimeter of the union of all black regions.
The problem gives us a database table named MyNumbers that contains a single column, num. The table may contain duplicate values because there is no primary key restriction. Our task is to find the largest number that appears exactly once in the table.
The problem gives us a positive integer num and allows us to swap digits only if the two digits have the same parity. In
The problem asks us to compute the integer square root of a non negative integer x. More specifically, we need to return the largest integer k such that: This means we are not looking for the exact decimal square root. Instead, we must round down to the nearest integer.
The problem asks us to determine how far Bob, a photographer, must move along a straight racetrack to take pictures of every sportsman. Each sportsman runs back and forth along a fixed segment of the racetrack, defined by two positions ai and bi.
We need to count stripe sequences built from four colors: black, red, white, and yellow. A sequence is valid if it satisfies several local restrictions. Equal adjacent colors are forbidden. White cannot touch yellow. Red cannot touch black.
The problem asks us to compute the minimum number of edit operations needed to transform one string into another. The allowed operations are insertion, deletion, and replacement of a single character. Each operation has a cost of exactly one.
We are given a single string that represents multiple email addresses concatenated together with no separators. Each email address has the form A@B, where A and B are non-empty strings consisting of lowercase Latin letters.
The problem is asking for the minimum time required to collect all apples in a tree, where each edge traversal takes 1 s
The problem asks us to determine whether a given array of integers arr represents a valid sequence from the root to a leaf in a binary tree.
The problem gives us an array of integer pairs, where each pair represents an interval-like relationship of the form [left, right]. Every pair satisfies the condition left < right. We want to build the longest possible chain of pairs.
The problem asks us to find the smallest positive integer divisor for a given array nums such that when each element of
We are given an undirected weighted graph with exactly n vertices and n edges. Since a connected graph with n vertices and n edges contains exactly one cycle, the graph is a unicyclic graph, a tree with one extra edge.
We are asked to model the random movements of two friends inside a museum represented as an undirected connected graph with n rooms and m corridors. Each room has a probability of staying in place for a minute, and otherwise the person moves uniformly to a neighboring room.
We are given a line of people, each identified as a boy or a girl, and each with a numeric dancing skill. The line evolves over time as couples consisting of one boy and one girl who are adjacent and have the smallest difference in skill leave to dance.
The problem requires us to remove all elements from an integer array arr in the minimum number of moves, where a move consists of removing a contiguous palindromic subarray. A palindromic subarray reads the same forwards and backwards.
This problem gives us the head of a singly linked list whose values are not sorted. Our task is to remove every node whose value appears more than once anywhere in the list. The important detail is that we are not removing duplicate occurrences while keeping one copy.
The problem asks us to find the next "lucky" ticket number that is strictly greater than a given ticket. Each ticket is a string of digits with even length, denoted 2n.
The problem asks us to determine whether a given searchWord occurs as a prefix of any word within a sentence. A sentence is defined as a string of lowercase English letters separated by single spaces, and a prefix is any contiguous leading substring of a word.
The problem gives us a list of meeting intervals where each interval is represented as [start, end]. Each interval describes the time range during which a meeting occupies a conference room.
This problem asks us to find the minimum cost to buy a set of items given individual prices and optional special offers.
The problem asks us to construct the longest possible string using only the characters 'a', 'b', and 'c', while satisfyi
The problem asks for the maximum sum of a subarray in a circular array. In simpler terms, we are given a list of integers nums where the end of the list wraps around to the start.
The problem asks for the total number of distinct substrings of a given string s. A substring is any contiguous sequence of characters from the string, including single-character substrings and the string itself.
This problem gives us a tree with n nodes numbered from 0 to n-1, rooted at node 0. Each node has an associated value from the array nums, and the tree structure is given as a list of edges.
We need to construct the smallest possible lucky number whose digits add up to a given value n. A lucky number may contain only digits 4 and 7. For example, 447 is valid because every digit is either 4 or 7, while 45 is invalid because digit 5 appears.
The problem asks us to find the length of the longest contiguous substring in a given string s such that the substring contains at most k distinct characters. A substring is a continuous portion of the string.
We need to construct the smallest number that is at least n and satisfies two conditions simultaneously. Every digit must be either 4 or 7, and the total count of 4s must equal the total count of 7s. The input is a decimal string that can be extremely long, up to 10^5 digits.
Here is the complete, detailed technical solution guide for LeetCode 1061 following your exact formatting requirements.
We are given two lowercase strings of the same length. We may repeatedly transform characters using directed conversion rules. A rule like a -> b with cost 5 means we can change one occurrence of a into b by paying 5.
We are asked to simulate a farm in which animals arrive one per day over n days. Each animal has a fixed daily food requirement starting from the day it arrives. The farm starts with X tons of food.
The problem asks us to find the length of the longest palindrome that can be constructed using the characters of a given string s. A palindrome is a string that reads the same forwards and backwards.
The problem asks whether Vasya, who typed a string of lowercase letters, effectively managed to say "hello". The goal is not to check if the typed string is exactly "hello", but whether we can remove some letters (possibly zero) to produce the sequence h, e, l, l, o in order.
The student is already riding a bus that moves along the x-axis from left to right. The bus stops at fixed positions (xi, 0) in increasing order. At any stop except the first one, the student may get off and run directly to the university located at (xu, yu).
We start with a strictly increasing array of problem complexities. The imbalance of the set is defined as the largest difference between two neighboring elements after sorting.
The problem asks us to determine how many connected components exist in an undirected graph. A connected component is a group of nodes where every node can reach every other node through some path.
The problem is asking us to determine the maximum number of water bottles a person can drink given two integers: numBott
The problem asks us to find the smallest contiguous substring inside string s that contains every character from string t, including duplicate occurrences. A substring must consist of consecutive characters. We are not allowed to reorder characters or skip positions.
The problem asks us to compute the total number of unique characters across every possible substring of a given string. A character is considered unique inside a substring if it appears exactly once within that substring.
The problem asks us to find the longest contiguous subarray whose sum is exactly equal to a given integer k. A subarray is a continuous portion of the array. This means we cannot reorder elements or skip positions.
We have a tree where each vertex initially contains some number of beavers. The robot starts at a fixed vertex s. Every time it traverses an edge from u to v, it immediately eats exactly one beaver at v. If v already has zero beavers left, the move is impossible.
This problem is asking us to identify all customers from a Customers table who have never placed an order according to the Orders table. In other words, we are looking for entries in Customers that do not have a corresponding customerId in the Orders table.
This is a comprehensive, multi section technical guide that will be quite long if completed properly with all requested sections, detailed explanations, two implementations, walkthroughs, worked examples, complexity analysis, exhaustive test cases, and edge case discussions.
The problem gives us two integer arrays, nums1 and nums2, and asks us to return their intersection. The intersection consists of all values that appear in both arrays. However, the result must contain only unique elements, even if a number appears many times in either array.
The problem asks us to determine whether a given integer n can be represented as a power of three. In mathematical terms, we need to check whether there exists an integer x such that: A power of three sequence looks like this: The input is a single integer n, and the output…
The problem presents an array of integers values, where each element represents the attractiveness or value of a sightseeing spot. The task is to find a pair of spots (i, j) with i < j such that the score values[i] + values[j] + i - j is maximized.
We have employees and directed supervisor offers between them. An offer (a, b, c) means employee a is willing to supervise employee b for cost c. Qualifications are strictly decreasing along every offer, so q[a] > q[b].
The problem is asking us to find two disjoint palindromic subsequences from a given string s such that the product of their lengths is maximized. A subsequence is derived by deleting zero or more characters from the original string while keeping the relative order intact.
The problem gives a list of online courses, where each course is represented as [duration, lastDay]. The duration tells us how many consecutive days are required to complete the course. The lastDay tells us the latest possible day by which the course must be finished.
The problem is asking us to generate the coordinates of cells in a 2D grid that we would visit if we started at a given cell (rStart, cStart) and walked in a clockwise spiral pattern.
This problem asks us to transform a string of parentheses into a special kind of balanced parentheses string using the m
The problem asks us to rearrange the characters of a given string s such that no two adjacent characters are the same. The input is a string of lowercase English letters with a length between 1 and 500.
The problem is asking us to determine whether it is possible to use exactly four numbers, each between 1 and 9 inclusive, to form a mathematical expression that evaluates to exactly 24. The numbers are given in an array called cards of length 4.
The problem asks us to find every element in an integer array that appears more than ⌊ n/3 ⌋ times, where n is the length of the array. The floor notation means we round down to the nearest integer.
In this problem, we are given a list of travel days during a single year and the costs of three different train passes. Each pass covers a consecutive range of days: - A 1-day pass covers exactly one day. - A 7-day pass covers seven consecutive days.
This problem describes a two player game played on an array of stones. Each stone has a positive integer value, and the
This problem asks us to build a new array called target by following a sequence of insertion operations. We are given tw
The problem gives us two arrays, sensor1 and sensor2, representing readings collected simultaneously by two sensors. Under normal circumstances, both sensors should produce the same sequence of values. However, one sensor may be defective.
The problem models a circular dial, represented by the string ring, where each character is engraved at a position around the circle. Another string, key, represents the sequence of characters we must spell.
You included two different problems in one message, and the second prompt supersedes the first. I will provide the detai
This problem asks us to find the lowest common ancestor (LCA) of two given nodes, p and q, in a binary tree, under the c
The problem asks us to find the length of the longest awesome substring in a given string s consisting of digits. A subs
The boss starts with max health and regenerates reg health every second. We own several scrolls. Each scroll can only be activated when the boss health is at most some percentage of the original health.
The problem asks us to compute the maximum width among all levels of a binary tree. The important detail is that the width is not simply the number of non-null nodes at a level.
The problem asks us to calculate the total time it takes for a piece of urgent information to propagate through a compan
This problem provides a database table named Calls, where each row represents a phone call between two people. The table contains three columns: | Column | Meaning | | --- | --- | | fromid | The caller | | toid | The receiver | | duration | Duration of the call | The important…
We are given a string of digits representing a car number. The number is considered beautiful if at least k positions contain the same digit. We may change any digit into another digit, and changing digit a into digit b costs The task has two objectives.
The problem gives us a square matrix mat of size n x n. A square matrix means the number of rows and columns are the sam
The problem asks us to determine the minimum number of times a string a must be repeated such that another string b becomes a substring of the repeated version of a. A substring is a consecutive sequence of characters within a string.
The problem gives an integer array nums and asks for the length of the shortest continuous subarray such that, if only that subarray is sorted in non-decreasing order, the entire array becomes sorted.
We are asked to find the shortest distance between two points lying on the perimeter of a square of side length n. The square is aligned with the axes, so its corners are at (0,0), (n,0), (0,n), and (n,n). The two points are guaranteed to lie on the edges, not in the interior.
We are given the maximum length of a single SMS message and one complete text consisting of sentences separated by spaces. A sentence always ends with one of ., ?, or !. Words contain only letters.
This problem asks us to find all words that are considered "uncommon" between two sentences. Each sentence is made up of lowercase words separated by single spaces. A word is considered uncommon if it satisfies two conditions: 1. It appears exactly once in one sentence. 2.
The problem asks us to process a string s consisting of lowercase and uppercase English letters, removing pairs of adjac
The problem asks us to modify an integer array so that it becomes the next lexicographically greater permutation of its current arrangement. A permutation is simply an ordering of the elements. Lexicographical order works the same way dictionary order works for words.
This problem is an interactive problem. We do not receive the actual binary array directly. Instead, we can only gather information through the ArrayReader API. The hidden array nums contains only 0 and 1.
This problem gives us an n x n binary matrix called board, where every cell contains either 0 or 1. In one operation, we are allowed to swap any two rows or swap any two columns.
This problem models a very large chessboard-like grid where certain cells contain active lamps. A lamp illuminates four directions simultaneously: - Its entire row - Its entire column - Its main diagonal, identified by row - col - Its anti-diagonal, identified by row + col For…
We are given an $n times m$ table filled with consecutive integers starting from 1. The filling is done row by row, left to right.
This problem describes a two-player turn-based game. Players take turns choosing a number from a shared pool of integers ranging from 1 to maxChoosableInteger. Once a number is chosen, it cannot be used again for the rest of the game.
The problem gives us a screen with a fixed number of rows and cols, along with a sentence represented as an array of words. We need to determine how many complete times the sentence can be written on the screen while following strict formatting rules.
This problem asks us to convert a non-negative integer n into its representation using base -2, instead of the usual positive bases such as base 2, base 10, or base 16. In standard binary representation, numbers are written as powers of 2. For example: which becomes "1101".
This problem asks whether one integer array, arr, can be transformed into another array, target, by repeatedly reversing
The problem gives us a binary array nums consisting of 0s and 1s and an integer k. The task is to transform the array so that all elements are 1s using the minimum number of operations called k-bit flips.
The problem gives us a string s where each character represents a student's attendance record for one day. There are only three possible characters: - 'A' means the student was absent. - 'L' means the student was late. - 'P' means the student was present.
This problem asks us to find the maximum number of non-overlapping subarrays in a given integer array nums such that the
The problem is asking us to find the shorthand RGB color that is most similar to a given full-length RGB color string.
The cylinder has n slots arranged in a circle, and exactly k of them contain bullets. Before the game starts, the cylinder is rotated uniformly at random, so every cyclic shift is equally likely. After the rotation, Sasha shoots first.
This problem asks us to validate whether a given string represents a correctly structured code snippet according to a custom XML-like syntax. At first glance, it resembles parsing HTML or XML tags, but the validation rules are stricter and more specialized.
The problem asks us to count how many contiguous subarrays of length at least three form an arithmetic sequence. An arithmetic sequence is one where the difference between every pair of adjacent elements is identical.
The problem gives us an array of strings, wordsDict, and two target words, word1 and word2. We must find the minimum distance between any occurrence of these two words in the array.
This problem gives us an array named cost, where each element represents the cost of stepping onto a particular stair. If cost[i] equals 10, that means stepping on stair i requires paying 10. From any stair, we are allowed to move either one step or two steps upward.
The problem asks us to transform a given integer array nums into a strictly increasing array using the minimum number of operations. An operation consists of incrementing any element of the array by 1.
The problem asks us to count the number of good meals that can be formed from a given list of food items. A good meal is defined as a pair of two different items whose combined deliciousness is a power of two.
This problem asks us to find the names of all salespeople who have never handled an order for the company named "RED". We are given three database tables: - SalesPerson contains information about each salesperson, including their unique salesid and name.
This problem asks us to generate a complete report showing how many times every student attended every subject examinati
The problem gives us an m x n matrix where every cell initially contains 0. We are also given a list called indices, whe
We are asked to calculate how many hexagonal cells a single archer can fully cover with a circular attack of radius k, where the archer is positioned at the center of a cell. Each hexagon has side length 1.
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Your requested guide is substantial enough that a high quality version will be very long, likely exceeding a single mess
The problem gives us three integer arrays, arr1, arr2, and arr3. Each array is already sorted in strictly increasing order. Strictly increasing means there are no duplicate values inside the same array, and every next element is larger than the previous one.
The problem asks us to implement a URL shortening service, similar to TinyURL. Given a long URL, our system should generate a short, unique URL that maps back to the original URL. When the short URL is accessed, the system should return the original long URL.
Here’s a complete, detailed technical solution guide for LeetCode 1478 following your requested format and requirements.
The problem asks us to remove comments from a C++ program represented as an array of strings. Each string represents a line of the program, split by newline characters.
The problem gives us a binary matrix mat of size m x n, where every cell contains either 0 or 1. We need to find the length of the longest consecutive sequence of 1s that appears in any of four directions: 1. Horizontal, from left to right 2. Vertical, from top to bottom 3.
We are asked to prepare a set of additional undirected edges, called wormholes, on top of an unknown tunnel system.
This problem asks us to find the largest axis-aligned plus sign made entirely of 1s in an n x n binary grid. Initially, every cell in the grid contains 1. However, some positions are marked as mines, meaning those cells contain 0.
In this problem, we are given two strings: - word, the original full word - abbr, a supposed abbreviation of that word We must determine whether abbr is a valid abbreviation of word. An abbreviation works by replacing one or more non-empty substrings with their lengths.
The problem asks us to minimize the length of a run-length encoded string after deleting at most k characters from the o
The problem asks us to take an array of integers nums that is already sorted in non-decreasing order and return a new array containing the squares of each number, also sorted in non-decreasing order.
This problem gives us a directed acyclic graph representing course dependencies. Each course is a node, and a prerequisi
The problem asks us to identify all words in a given list that can be formed by concatenating at least two other words from the same list. In other words, each concatenated word must be fully composed of smaller words that exist in the input array.
This problem asks us to design a custom data structure called RandomizedSet that supports three operations: 1. Insert a value into the set. 2. Remove a value from the set. 3. Return a random element from the set.
The problem asks us to determine the minimum number of frogs needed to produce a given sequence of croaks represented by
The problem gives us an array of phrases, where each phrase is a string made of lowercase English letters and spaces. Every phrase is well-formed, meaning there are no leading or trailing spaces and no consecutive spaces.
The problem defines a recursively constructed binary string sequence: - S1 = "0" - Si = Si-1 + "1" + reverse(invert(Si-1
The problem describes a variation of the classic Josephus problem. We have n friends sitting in a circle, numbered 1 through n clockwise. Starting from the first friend, we count k friends clockwise (inclusive of the starting friend).
This problem gives us a set of cities positioned on a number line. The array locations stores the coordinate of each cit
The task looks trivial at first glance: read two integers and print their sum. The catch is hidden inside the constraints. Each number can contain up to 500 decimal digits, far larger than the range of standard 32-bit or 64-bit integers in many languages.
We are given a single line containing several lowercase words separated by spaces. Every valid word in Petya's language belongs to exactly one grammatical category and exactly one gender, determined entirely by its suffix.
Here’s a fully detailed technical solution guide following your requested format for LeetCode 471 - Encode String with Shortest Length.
The problem gives us a single word consisting of uppercase and lowercase Latin letters. This word may have been typed with the Caps Lock key unintentionally engaged. We are asked to correct such accidental capitalization.
In this problem, we are given an array called costs, where each element represents the travel cost for one person to two different cities.
The N-Queens problem asks us to place n queens on an n x n chessboard such that no two queens can attack each other. In chess, a queen can attack horizontally, vertically, and diagonally.
The problem asks us to implement a simplified web crawler. We are given a starting URL and access to an HtmlParser interface that can retrieve all URLs linked from a given webpage.
We are given a rectangular board filled with playing cards. The board size is at most $17 times 17$, but the total number of placed cards never exceeds 52 because the deck has only 52 regular cards plus two jokers. A valid 3×3 square must satisfy one of two properties: 1.
The problem asks us to construct the string target using the fewest possible subsequences of the string source. A subsequence preserves relative order, but characters do not need to be contiguous.
The problem is asking us to generate all possible word squares from a given list of unique words. A word square is a square arrangement of words such that the word at row i is identical to the word at column i for every i.
This problem asks us to identify second-degree followers from a social network represented by a single table Follow. The table contains two columns: followee and follower. Each row indicates that a user (follower) follows another user (followee).
We have a rooted tree with root at city 1. Every city initially contains exactly one military division. Division i starts in city i and has priority a[i], where a smaller value means higher priority. Each edge has a capacity.
The problem asks us to determine whether two strings are exactly one edit apart. An edit can be one of three operations: 1. Insert a single character 2. Delete a single character 3. Replace one character with a different character The key detail is the word exactly.
This problem gives us an undirected graph representing a network of servers. Each server is identified by an integer from 0 to n - 1, and each pair [a, b] in connections represents a bidirectional edge between server a and server b.
We are given a decimal string and an operation count k. Each operation looks for the leftmost occurrence of the substring "47". Suppose the substring "47" starts at position x using 1-based indexing. If x is odd, we replace both digits with '4', so "47" becomes "44".
We have n positions arranged in a circle. The flea starts on one position. After the first minute it moves forward by 1 step, after the second minute by 2 steps, after the third minute by 3 steps, and so on.
We are asked to construct a square matrix of size _n_ × _n_, where _n_ is an even number. The matrix must satisfy four properties. The main diagonal contains only zeroes.
We are given a string that represents a simplified HTML table language. The language contains only three kinds of tags: <table>, <tr>, and <td>, together with their matching closing tags.
In this problem, we are given an n x n board where each cell contains one of four possible values: - 'E', the ending position located at the top-left corner - 'S', the starting position located at the bottom-right corner - A digit character '1' through '9', representing points…
We are given a list of n lessons, each with a start and end time, scheduled in a single room. Two lessons overlap if one starts before another ends. The goal is to find which single lesson can be removed so that the remaining lessons have no overlaps.
We have an $n times m$ grid, and every cell must be painted with one of $k$ colors. The restriction is about every vertical cut between columns.
The problem asks us to design a logging system that controls how frequently identical messages can be printed. Each message is associated with a timestamp, and the same message is only allowed to be printed once every 10 seconds.
We are given a collection of envelopes, each with a width and height. A postcard already has fixed dimensions, and we want to build the longest possible nesting chain of envelopes such that:
The problem asks us to find the longest special substring that appears at least three times in the given string s. A sub
The problem gives us an n x n binary matrix called grid. Each cell contains either 1 or 0. A value of 1 represents land, and a value of 0 represents water. Land cells that are connected vertically or horizontally form an island.
This problem asks us to compute the average selling price for every product based on two tables: Prices and UnitsSold. T
The problem gives us a tree with n nodes numbered from 0 to n - 1. The tree is represented by an array of edges, where each edge connects two nodes ai and bi. Each node also has a label, represented as a string labels, where labels[i] is the label of node i.
We are given an absolute Unix-style path. The path is split into components by /, and every component represents either a normal directory or file name, ".", or "..". A normal name means “go into this directory or file”. The component ".
The problem asks us to rearrange the rows of a Genders table in a specific repeating order while maintaining internal so
The problem asks us to design a data structure that supports two operations efficiently: 1. Adding words into the structure. 2. Searching for words, where the search pattern may contain the wildcard character '.'. The wildcard '.' represents any single lowercase English letter.
The problem gives us the root of a Binary Search Tree, abbreviated as BST, and a floating point target value. Our task is to return the integer value stored in the BST that is numerically closest to the target.
This problem is asking us to simulate a single move in a simple two-player game. The input is a string currentState consisting only of '+' and '-' characters.
This problem asks us to take a string s consisting of lowercase English letters and the character '?' and replace every '?' with a lowercase letter so that no two consecutive characters in the resulting string are the same.
This problem asks us to simulate how text is formatted in a text editor when using full justification. We are given an array of words and a target line width called maxWidth.
The problem gives us a wall made of multiple rows of bricks. Each row is represented as an array of integers, where each integer describes the width of a brick. Every brick has height 1, and all rows together form a rectangle with the same total width.
We are given several intervals of integers. For every interval $[l, r]$, we must count how many numbers contain two lucky digits, either 4 or 7, whose positions differ by at most k. Positions are counted inside the decimal representation of the number.
This problem asks us to compute the total number of apples and oranges across all boxes, taking into account that some boxes may contain a chest. The Boxes table gives us the count of apples and oranges in each box, and optionally the chestid if a chest is present in that box.
We are asked to analyze a two-player game with a single pile of n stones. The players alternate turns, starting with Serozha.
This problem is asking us to determine how many apples we can fit into a basket with a maximum carrying capacity of 5000 units of weight. We are given an array weight where each element represents the weight of a single apple.
The problem gives us a string s that contains English letters and spaces. Inside this string, words are separated by spaces, and there may also be extra spaces at the beginning or end of the string. Our task is to return the length of the last word in the string.
We have dominoes placed on a number line. Each domino stands at coordinate x[i] and has height h[i]. If a domino falls to the right, it reaches every point from x[i] + 1 up to x[i] + h[i] - 1.
We are given a collection of ticket pieces. Each piece is a number, representing the fragment of a ticket that was originally divisible by three. The problem asks us to reconstruct the maximum number of lucky tickets we can from these pieces.
This problem asks us to find the earliest overlapping time slot of a given duration between two people, given their individual availability schedules.
This problem asks us to multiply two matrices, but with an important detail: both matrices are sparse. A sparse matrix is a matrix where most entries are zero. The goal is to take advantage of this property so that we avoid unnecessary computation involving zero values.
This problem asks us to determine whether a given chessboard square is white or black based on its coordinate. The input is a string of length two, where the first character is a lowercase letter from 'a' to 'h', representing the column, and the second character is a digit…
The problem asks us to maximize the advantage of one array over another. Given two arrays nums1 and nums2 of equal length, the advantage is defined as the number of positions i where nums1[i] nums2[i]. Our task is to rearrange nums1 to maximize this advantage.
We have a grid of size where each cell is 1 centimeter square. A flea starts at some cell and can jump exactly centimeters either vertically or horizontally, staying inside the board.
The problem asks us to count how many attendance records of length n satisfy two award eligibility rules. Each attendance record is a string made up of three possible characters: - 'P' means the student was present. - 'A' means the student was absent.
We are given two strings, a and b, of equal length up to one million characters. The task is to find indices i and j in a such that if we perform a specific transformation, we get b.
That is a long, structured technical guide that will exceed a practical single-message limit if done comprehensively with all requested sections, detailed walkthroughs, Python and Go solutions, worked examples, full test suite, and edge case analysis.
The problem is asking us to process a string containing lowercase letters and parentheses, and remove the minimum number of parentheses so that the remaining string is valid.
That is a long, structured reference document. To make sure I target the exact problem correctly, please confirm the Lee
The problem asks us to take a string s and an integer array indices of the same length, and produce a new string where e
The problem is essentially about simulating the process of repeatedly smashing stones together until at most one stone remains. Each stone has a positive integer weight.
We are asked to count the number of ways to line up Caesar’s soldiers, consisting of a given number of footmen and horsemen, so that no more than a fixed number of the same type stand consecutively.
The problem gives us an array of non-negative integers and a starting index. From any position i, we are allowed to jump
The problem asks us to implement a queue using only stack operations. A queue follows the FIFO, First In First Out, principle. The first element inserted into the queue must be the first element removed.
The problem asks us to compute the sum of the minimum value of every possible contiguous subarray of a given array. For an array arr, every contiguous slice of the array is considered a subarray. For each subarray, we determine its minimum element.
The problem asks us to count the number of valid boomerangs among a set of distinct 2D points. A boomerang is defined as a tuple (i, j, k) where the distance between point i and point j is equal to the distance between point i and point k.
We have a strip of cells, and each cell contains an integer. We may cut the strip only between adjacent cells, which splits the array into a left part and a right part. The task is to count how many cut positions produce two non-empty parts with equal sums.
The problem is asking for the minimum number of operations required to generate exactly n characters 'A' on a notepad starting with a single 'A'.
We are given a connected undirected graph representing the road network of Bertown, where junctions are nodes and roads are edges.
This problem asks us to find the most recent order for every product that has been ordered at least once. We are given t
We are asked to construct a graphical progress bar as an array of squares, where each square has a saturation value. The bar has a total of n squares, and the maximum saturation is k.
This problem asks us to generate monthly statistics for the year 2020 using information from three database tables: Driv
Aryo wants to choose a title for a new album from a list of candidate names. Some names have already been used in previous years. His decision rule has two layers. If a candidate name has never been used before, that is the best possible choice.
The knights sit on a circle, equally spaced. Each position is marked either 1 for a knight in a good mood or 0 for a knight in a bad mood.
The problem gives us an integer array nums and a window size k. A sliding window of length k starts at the beginning of the array and moves one position to the right at a time. For every position of this window, we must determine the maximum value inside that window.
The problem provides an m x n binary matrix, grid, where each element is either 0 or 1. You are allowed to perform a move, which consists of selecting any row or column and flipping all its values - turning every 0 into a 1 and every 1 into a 0.
The cableway sends one cablecar every minute, and the colors repeat in a fixed cycle: red - green - blue - red - ... Each cablecar can carry at most two students, and every ride takes exactly 30 minutes to reach the top. We are given three groups of students.
The problem asks us to find the number of ordered pairs of indices (i, j) in a list of digit strings nums such that concatenating nums[i] and nums[j] produces exactly the string target.
This problem gives us an undirected weighted graph with n nodes and a list of edges. Each edge connects two nodes and has an associated distance value. The graph may contain multiple edges between the same pair of nodes, and it may also be disconnected.
The problem asks us to count how many valid partition points exist in an array after optionally changing at most one element to a given value k.
The problem asks us to simulate a robot moving on an n x n grid, starting from a specified position startPos = [startrow, startcol].
This problem asks us to find all root-to-leaf paths in a binary tree where the sum of the node values equals a given targetSum. A binary tree is provided through the root node, and every node contains an integer value. We are also given an integer targetSum.
We have an grid. Some cells contain pies, and several pies may share the same cell.
The problem asks us to make a string containing only characters 'a' and 'b' balanced by deleting the minimum number of characters. A string is considered balanced if there are no occurrences where a 'b' comes before an 'a' in the string.
The problem describes a town containing n people labeled from 1 to n. Among these people, there may exist a special person called the town judge. The judge must satisfy two strict conditions: 1. The judge trusts nobody. 2. Every other person trusts the judge.
This problem gives us the root of a binary tree and a list of node values that must be deleted from the tree. Every node value in the tree is unique, which is important because it means we can identify nodes directly by value without ambiguity.
The problem is a two-player turn-based game played on a row of stone piles. Each pile has a positive number of stones, and players take turns taking stones from the start of the remaining piles.
The problem gives us a database table named Problems. Each row represents a LeetCode problem and contains three columns:
We have a square room with side length n, so the total area is n × n. We also have k square carpets, each with side length n1. Every carpet always stays axis-aligned because rotation is forbidden, but since the carpets are squares, rotation would not actually change anything.
In this problem, we are given a list of log strings. Each log contains an identifier followed by one or more words separated by spaces. The first token is always the identifier, and everything after it represents the content of the log.
The problem asks us to determine whether a given integer n is an ugly number. An ugly number is defined as a positive integer whose prime factors are limited to only 2, 3, and 5.
We are given a digital time in HH:MM format and need to determine how far the analog clock hands must rotate from the initial position 12:00. The clock starts with both hands pointing at 12. We may rotate each hand independently, and only in the clockwise direction.
This problem asks us to multiply two complex numbers represented as strings and return the result in the same string format.
We are given a three-dimensional grid representing the inside of a plate. The grid has k layers, each layer has n rows and m columns. Every cell is either empty . or blocked . Water starts entering from one specific cell on the top layer.
This problem gives us a special type of linked list called an immutable linked list. Unlike a normal linked list problem
This problem asks us to reorder a singly linked list so that all nodes located at odd indices appear first, followed by all nodes located at even indices. The important detail is that the grouping is based on the node's position in the list, not the node's value.
This problem asks us to count the number of triples (i, j, k) from an integer array nums such that the bitwise AND of the three numbers at these indices equals zero. In other words, we want all combinations where nums[i] & nums[j] & nums[k] == 0.
This problem asks us to reverse a singly linked list in groups of size k. Instead of reversing the entire list at once, we only reverse consecutive chunks containing exactly k nodes.
The problem defines an infinite cyclic string built from the lowercase English alphabet: This means the sequence continues forever, and after 'z' comes 'a' again. Any substring that follows this cyclic ordering exists somewhere in the infinite wraparound string.
The problem gives us an array of strings called wordsDict, along with two distinct target words, word1 and word2. Both target words are guaranteed to exist somewhere in the array.
The problem gives us a string s whose length is guaranteed to be even. We split the string into two equal parts: - The first half is called a - The second half is called b We must determine whether both halves contain the same number of vowels.
This problem asks us to write a SQL function that returns the nth highest distinct salary from the Employee table. The table contains two columns: Column Meaning --- --- id Unique employee identifier salary Employee salary The key detail is that the salary must be distinct.
The problem asks us to modify a given palindromic string in such a way that it is no longer a palindrome while ensuring
Each person points to exactly one other person. If someone learns the news, they call the person they point to, who then calls the next person, and so on. We may add extra directed edges of the form x -> y, meaning person x must also call y.
The problem gives us a string representing a nested ternary expression and asks us to evaluate it. A ternary expression follows the familiar format: The condition is always either 'T' or 'F'. If the condition is 'T', the expression evaluates to the value before the ':'.
In this problem, we are given the root node of a binary tree, and we need to find the length of the longest ZigZag path
Here’s a complete, detailed technical solution guide for LeetCode 1537 - Get the Maximum Score following your requested
The problem gives us a permutation of integers from 1 to n, and we insert the numbers into an initially empty Binary Search Tree (BST) in the exact order they appear in the array. A BST has the following property: - Values smaller than the current node go to the left subtree.
The problem gives us n cities labeled from 1 to n. Two cities are directly connected if they share a common divisor that
The problem gives us an undirected tree with n nodes labeled from 0 to n - 1. A tree is a connected graph with exactly n - 1 edges and no cycles. Each edge connects two nodes bidirectionally.
This problem asks us to generate a specific path for Bob from the top-left corner (0, 0) to a destination (row, column)
The problem asks us to take a string s containing words separated by spaces and return a new string where the words appear in reverse order. A word is defined as any sequence of non-space characters, and words can be separated by multiple spaces.
That is a very large, multi-section request covering a full detailed guide for LeetCode 1292 with comprehensive explanat
The problem gives us a database table named Scores that contains two columns: id and score. Each row represents the score achieved in a game.
The problem asks us to determine the minimum number of deletion steps required to make two strings identical. We are given two input strings, word1 and word2, and in each step, we can delete exactly one character from either string.
This problem asks us to determine whether two binary trees can be made identical by performing a special operation called a flip. A flip operation can be applied at any node, and it simply swaps that node’s left and right child subtrees.
This problem asks us to identify the seller or sellers who generated the highest total sales revenue. We are given two database tables: Product and Sales. The Product table contains information about products, including their IDs, names, and unit prices.
The problem asks us to navigate a m x n integer matrix grid from the top-left corner (0, 0) to the bottom-right corner (
We are asked to calculate the area of the floor in a house that is illuminated by two light sources placed symmetrically outside a horizontal strip representing the house.
The problem asks us to determine how many trailing zeroes appear at the end of n!, where n! represents the factorial of n. A factorial is defined as: For example: - 5!
The problem is asking us to identify a redundant edge in a graph that started as a tree. A tree is a connected graph with n nodes and exactly n-1 edges, meaning it has no cycles. The input graph has n nodes and n edges, so by definition, it contains exactly one cycle.
The problem asks us to determine the maximum depth of a binary tree. A binary tree consists of nodes where each node can have up to two children, a left child and a right child.
This problem asks us to determine whether a given positive integer num is a perfect number. A perfect number is defined as a number whose sum of positive divisors, excluding the number itself, equals the number.
The problem gives us the head of a singly linked list and asks whether the sequence of values stored in the list forms a palindrome. A palindrome is a sequence that reads the same forward and backward.
The problem gives us a sorted array of lowercase English letters and a target character. Our task is to find the smallest character in the array that is strictly greater than the target character in lexicographical order.
The problem asks us to balance a string composed of exactly four types of characters: 'Q', 'W', 'E', and 'R'. A string is balanced when each character occurs exactly n / 4 times, where n is the length of the string.
This problem asks us to determine the minimum number of swaps needed to make two strings, s1 and s2, equal. Both strings are of the same length and consist only of the characters "x" and "y".
The problem asks us to reorder an array of integers based on the frequency of each value. Specifically, elements with lo
The problem is asking us to query a database table called Cinema and return a filtered set of movies based on two conditions: the movie ID must be odd, and its description must not be "boring". The result must then be sorted in descending order by the movie's rating.
This problem asks us to determine the length of the shortest transformation sequence between two words, beginWord and endWord, under a strict transformation rule.
The problem describes a two-player game played on an array of stone piles. Each pile contains a positive number of stones, and the piles are arranged in a row. Alice and Bob alternate turns, with Alice always moving first.
The problem asks us to remove a given interval toBeRemoved from a list of non-overlapping, sorted intervals intervals. E
This problem describes a two player game played on an array of integers. The numbers are written on a chalkboard, and players take turns erasing exactly one number. Alice moves first, then Bob, and both players play optimally. The key rule is unusual.
The problem asks us to count how many separate word-like groups exist inside a string. In this problem, a "segment" is defined as a continuous sequence of characters that are not spaces.
We are given up to 200 distinct lowercase words. For every word, we must choose a short abbreviation whose length is between 1 and 4 characters, and whose characters appear in the original word in the same order.
The garden is represented as an n × m grid. Each cell contains either 0 or 1. A 1 means there is a tree in that square, while 0 means the square is empty.
We are asked to count numbers between 1 and that have exactly two distinct prime factors. For instance, 6 is almost prime because it can be factored as , and both 2 and 3 are prime. Numbers like 4 or 8 are not almost prime because they are powers of a single prime.
We are asked to count all pairs of clients whose assigned numbers are exact opposites. Each client has a number between -10 and 10, and a pair is valid if one client has number $x$ and the other has number $-x$.
This problem asks us to generate all integers that can be written in the form: where: - i = 0 - j = 0 - the resulting value is less than or equal to bound The inputs are three integers: - x, the base of the first exponential term - y, the base of the second exponential term -…
The problem asks us to merge n piles of stones into a single pile under a very specific merging rule. Each pile is represented as an integer in the array stones, where stones[i] is the number of stones in the i-th pile.
This problem asks us to find the length of the longest consecutive sequence path in a binary tree. Unlike the simpler version of the problem where the sequence must move strictly downward from parent to child, this version allows the path to move in a child-parent-child…
Each crater is described by a center coordinate c and a radius r. Since the robot moves along a line, every crater can be represented on that line by its interval: $[c-r, c+r]$ Professor Okulov’s rule says that any two selected craters must either be completely disjoint or…
This problem asks us to convert a fraction, represented by an integer numerator and an integer denominator, into its decimal string representation. If the decimal expansion contains a repeating fractional sequence, we must enclose the repeating part in parentheses.
This problem asks us to remove the minimum number of elements from an array so that the remaining elements form a valid
The problem asks whether it is possible to transform a starting point (sx, sy) to a target point (tx, ty) using a defined set of operations. Specifically, from a point (x, y), you can either move to (x, x + y) or (x + y, y).
The problem asks us to implement a SnapshotArray, a data structure that behaves like an array but also allows taking snapshots of its state at any moment and retrieving the value of any element at a specific snapshot.
We are given an undirected weighted graph. Every edge connects two vertices and has a positive cost. The task is to start at vertex 1, reach vertex n, and print one shortest path. If no route exists, we print -1.
The problem asks us to evaluate a Lisp-like expression represented as a string. The expression can contain integers, variables, and three special operations: let, add, and mult. An expression evaluates to a single integer value.
Each paper describes an undirected edge between two cities. We are told that all papers came from exactly two travel journals.
This problem asks us to determine how many times a given string word can be repeated consecutively while still appearing
We need to evaluate a sum over an interval [l, r]. For every integer x in that range, we compute next(x), where next(x) means the smallest lucky number greater than or equal to x. A lucky number is a positive integer whose decimal digits are only 4 and 7.
This problem asks us to evaluate a mathematical expression represented as a string. The expression may contain: - Non-negative integers - Addition (+) - Subtraction (-) - Multiplication () - Division (/) - Parentheses (( and )) The goal is to compute the final integer result…
The problem gives us two soups, A and B, each starting with exactly n milliliters. At every turn, one of four serving operations is chosen uniformly at random. Each operation removes different amounts from A and B simultaneously.
We start from the sorted permutation 1 2 3 ... n. Someone chooses exactly one contiguous segment and reverses it. We are given the final permutation and must determine whether it could have been produced by exactly one such reversal. The task is not to sort the array.
We have a theatre club with n boys and m girls. A performance group must contain exactly t people, with two extra restrictions: at least 4 of them must be boys, and at least 1 must be a girl. The task is to count how many different valid groups can be formed.
We have three transformation spells that convert one material into another. The first spell turns a grams of sand into b grams of lead. The second spell turns c grams of lead into d grams of gold. The third spell turns e grams of gold into f grams of sand.
The problem requires us to process a string word that contains both lowercase English letters and digits. We are asked to extract all the sequences of digits as integers, ignoring any non-digit characters.
This problem gives us the root node of a Binary Search Tree, commonly abbreviated as a BST, along with an integer value val. Our goal is to locate the node whose value is exactly equal to val and return that node.
The problem requires computing a matrix block sum. Given a matrix mat of size m x n and an integer k, the task is to pro
This problem asks us to traverse a binary tree and identify every node that is considered "lonely". A node is lonely if
The problem asks us to find the greatest common divisor (GCD) of two strings. In this context, a string t divides another string s if s can be formed by concatenating t multiple times. The goal is to find the largest string x that divides both str1 and str2.
The problem asks us to calculate the perimeter of an island represented in a 2D grid. Each cell in the grid is either land (1) or water (0). The island consists of one or more connected land cells, where connectivity is strictly horizontal or vertical.
The problem asks us to count how many contiguous, non-empty subarrays in an integer array nums have a sum equal to a target value k. A subarray is different from a subsequence.
The problem asks us to count how many contiguous subarrays contain exactly k distinct integers. We are given: - An integer array nums - An integer k A subarray is any continuous portion of the array.
Please provide the missing problem number or title you want the guide for. Your prompt contains the formatting/template
This problem gives us two arrays, nums1 and nums2, where both arrays are already sorted in non-decreasing order. We are also given two integers, m and n, which tell us how many valid elements exist in each array. The important detail is that nums1 has extra space at the end.
We are given a connected graph representing rooms in Arnie’s apple house. Each room is a node, corridors between rooms are edges, and the graph has no self-loops or multiple edges between the same pair of rooms.
The problem gives two inputs: a pattern string and a space-separated string of words. We need to determine whether the sequence of words follows the same structure as the sequence of characters in the pattern.
This problem is the two dimensional version of the classic "Trapping Rain Water" problem. Instead of a one dimensional array of heights, we are given an m x n grid where each cell represents the elevation of a block in a terrain.
The problem asks us to find the longest contiguous subarray that appears in both input arrays. A subarray is different from a subsequence because the elements must remain adjacent.
Let's dive deep into a detailed technical solution guide for LeetCode 691, following your formatting rules exactly. The problem asks us to construct a target string using letters cut from an unlimited supply of given stickers, where each sticker is a lowercase word.
The problem gives us the root of a binary tree and asks us to identify all duplicate subtrees inside it. A subtree is defined as any node together with all of its descendants.
The problem asks us to design a data structure that continuously receives integers from a stream and returns the average of the most recent values within a fixed-size sliding window. A sliding window means that we only care about the latest size elements.
The problem asks us to find the integral coordinate on a 2D plane where the network quality is maximized. We are given s
The problem asks us to calculate the average daily percentage of posts that were removed after being reported as spam. We are given two tables: Actions and Removals.
The problem gives two strings, secret and guess, representing two numbers of equal length. We need to compare them and return a hint in the format "xAyB". A "bull" is a digit that matches in both value and position.
The problem is asking us to determine, for each row in a table called Triangle, whether the three given line segments can form a valid triangle. Each row contains three integers x, y, and z representing the lengths of the segments.
We are given two integer arrays. We need to build the longest sequence that satisfies two conditions at the same time.
The Skyline Problem asks us to compute the visible outer contour formed by a collection of rectangular buildings when viewed from far away. Each building is represented by three integers: a left x-coordinate, a right x-coordinate, and a height.
This problem asks us to find the shortest path to collect all keys in a 2D grid maze. The grid contains walls, open spaces, keys, locks, and a starting point. You can move in the four cardinal directions but cannot move diagonally, through walls, or outside the grid.
This problem gives us a database table named Products. Each row represents a product and contains three columns: | Column | Meaning | | --- | --- | | productid | Unique identifier for the product | | lowfats | 'Y' if the product is low fat, otherwise 'N' | | recyclable | 'Y'…
The problem asks us to convert a positive integer into its Roman numeral representation. Roman numerals use combinations of specific symbols to represent values, and the rules for combining those symbols are strict.
The problem requires finding the water cell in a given n x n grid that is farthest from any land cell, using Manhattan distance. The grid consists only of 0s and 1s, where 0 represents water and 1 represents land.
The problem presents a lock with four wheels, each wheel labeled from '0' to '9', which can rotate forwards or backwards. The lock starts at '0000', and we are given a list of "deadends," which are lock states that will cause the lock to freeze if reached.
We are given several lines of text. Every line may contain letters, digits, and spaces inside the line, but never at the beginning or end. The task is to print all lines inside a rectangular frame made of * characters.
The problem asks us to design a mutable sequence data structure that supports four operations efficiently: 1. Append a v
We are asked to help Ciel move from the bottom-left corner of a castle grid, coordinate (1,1), to the top-right corner (n,n), while avoiding being caught by a system of sensors. The castle grid allows only two types of moves: right (R) or upward (U).
The problem gives us an array of positive integers, and the array elements are combined using division operations in order from left to right. For example, if the input is: the default expression becomes: which evaluates as: because division is left associative.
We are given a text file written in INI format. Every meaningful line is either a section declaration such as [network] or a key-value assignment such as port=8080. Spaces around keys, values, and section brackets are irrelevant and must be removed in the final output.
This problem can be interpreted as a graph traversal challenge. Each room is a node in a graph labeled from 0 to n-1, and the keys inside a room represent directed edges to other nodes that can be unlocked.
The problem asks us to find the length of the longest substring that appears at least twice in a given string s. A substring is a contiguous sequence of characters inside a string. The repeated substrings may overlap with each other, which is an important detail.
The problem gives us two inputs: an array of strings called words, and a target string s. Every occurrence of every word from words inside s must become bold by surrounding that substring with <b and </b tags.
The bank stores diamonds in a row of cells. After every minute, the security system checks the sums of every adjacent pair: $$a1 + a2, a2 + a3, dots, a{n-1} + an$$ If any of these sums changes compared to the previous check, the alarm triggers.
The problem asks us to design a special data structure that stores string keys together with their occurrence counts.
The problem gives us a sentence s and an integer k. A sentence is guaranteed to contain words separated by exactly one space, with no extra spaces at the beginning or end. Our task is to return a new sentence that contains only the first k words from the original sentence.
The problem asks us to calculate the maximum absolute sum of any contiguous subarray in a given integer array nums. A subarray is a sequence of consecutive elements, and its absolute sum is defined as the absolute value of the sum of all elements in that subarray.
The problem gives us two integer arrays, nums1 and nums2. We must count the number of valid triplets that satisfy one of
We are given two lowercase strings. The first string is a word in one language, and the second string is supposed to be its translation into another language where every word is written backwards.
The problem gives us a string s consisting only of the characters 'I' and 'D'. This string describes the relationship between adjacent numbers in a hidden permutation.
We have n fish in a lake. Every day, exactly one unordered pair of currently alive fish is chosen uniformly at random. When fish i meets fish j, fish i eats fish j with probability a[i][j], and fish j eats fish i with probability a[j][i] = 1 - a[i][j].
This problem models communication in a social network where each user knows one or more languages. Two users can communicate directly if they share at least one common language.
The problem asks us to design a reversible encoding system for a list of strings. We need two functions: - encode, which converts a list of strings into a single string - decode, which reconstructs the original list from that encoded string The important requirement is that…
The problem describes a competitive game played on an array of distinct integers. At every round, only the first two ele
We are given a binary string representing Vasya's observations over several days. A character '1' means a signal was received on that day, while '0' means no signal appeared.
This problem is asking us to take a starting IPv4 address and a number n of consecutive IP addresses and convert them into the minimal set of CIDR blocks that exactly covers that range. An IPv4 address is a 32-bit number written in the familiar dotted decimal format (e.g., "192.
We are asked to tile an rectangular floor with three types of 2x2 square tiles: black, white, and mixed tiles that have a black and a white section in a diagonal pattern.
We are given a string that represents the sequence of station flags seen while traveling from city A to city B. Peter woke up twice during the trip and wrote down two substrings he saw, in chronological order.
The problem asks us to determine in how many distinct ways an integer n can be expressed as a sum of consecutive positive integers.
The problem asks us to count the number of elements in an array arr for which the value plus one also exists in the array. In other words, for every element x in the array, we check whether x + 1 is also present. If it is, we include x in our count.
The problem asks us to find the k-th smallest number in lexicographical order among all integers from 1 to n. Lexicographical order means dictionary order, not numerical order.
We have four integers describing the number of fingers on two alien hands. The Venusian girl has al fingers on her left hand and ar on her right hand. The Marsian boy has bl and br fingers. They want to hold exactly one pair of hands.
In this problem, we are given two strings, jewels and stones. The string jewels represents all stone types that are considered jewels. Each character is a unique jewel type. For example, if jewels = "aA", then both lowercase 'a' and uppercase 'A' are jewel types.
The problem asks us to generate the nth term of the "count-and-say" sequence. This sequence is built recursively, where each term is created by describing the previous term.
We can model the students and shoelaces as an undirected graph. Each student is a vertex, and every shoelace between two students is an edge. In one round, every student whose degree is exactly 1 gets removed at the same time.
The problem gives us two integers, a and b, representing how many 'a' characters and 'b' characters we must place into a string.
The problem gives us an array of distinct positive integers called nums and a target integer called target. We must determine how many different ordered sequences of numbers from nums sum exactly to target. The key detail is that order matters.
The problem gives us a list of intervals, where each interval is represented as [li, ri], corresponding to the half open
We have a set of points on a 2D plane representing cows, each with integer coordinates. Vasya, the shepherd, must walk a closed path around all the cows in such a way that every cow lies strictly inside the path. The goal is to minimize the number of moves needed.
This problem asks us to compute the overall acceptance rate of friend requests in a social network system. Two database tables are provided. The first table, FriendRequest, stores all friend requests that users sent to each other.
We are given a row of n trees, each with a certain height, and our task is to adjust some of their heights so that the row forms a “beautiful” sequence.
We are given a board of size n by m, with a spider on every cell. Each spider can move to any adjacent cell or stay in place, as long as it stays inside the board. All spiders move simultaneously, and multiple spiders can occupy the same cell after moving.
The problem asks us to count all substrings of a given string s that contain at least one occurrence of each character '
This problem asks us to count how many times a given digit d appears in every integer within a range [low, high]. For instance, if d = 1 and the range is [1, 13], we count all occurrences of the digit 1 in the numbers 1 through 13.
Each camel stands at a unique coordinate on a number line. A camel at position x spits exactly toward position x + d. If another camel stands there, it gets hit. We need to determine whether there exists a pair of camels such that each one hits the other.
The problem asks us to determine whether an array contains two equal values whose indices are close to each other.
This problem asks us to return every cell in a matrix, ordered by its Manhattan distance from a given center cell.
This problem asks us to find the minimum number of columns to delete from a set of equal-length strings so that each individual string becomes lexicographically sorted.
This problem describes a two-player perfect-information game played on a small grid. One player controls the mouse and the other controls the cat. Both players move optimally and alternate turns, with the mouse always moving first.
The problem presents a simulation scenario with a row of exactly 8 prison cells, where each cell is either occupied (1) or vacant (0). Each day, the state of a cell changes based on the states of its immediate neighbors.
The problem provides a database table named Products, where each row represents a price update for a product on a specific date.
The problem asks us to compute the total tilt of an entire binary tree. Every node in the tree has its own tilt, and the final answer is the sum of all individual node tilts.
The problem asks us to remove the minimum number of invalid parentheses from a string so that the remaining string becomes valid. The input string may contain lowercase English letters in addition to parentheses.
The problem gives an array nums where each value represents the maximum distance you can jump forward from that position. You start at index 0, and your goal is to reach the final index using the minimum number of jumps possible.
The problem asks us to determine whether a given string can be formed by repeating one of its substrings multiple times. In other words, we want to know if there exists some substring pattern such that concatenating that pattern repeatedly recreates the entire string exactly.
This problem asks us to simulate repeated shift operations on a 2D matrix. We are given an m x n grid and an integer k,
We have a row of chocolate bars, and each bar takes a certain amount of time to eat. Alice starts from the left end and keeps moving right. Bob starts from the right end and keeps moving left.
We are asked to count the number of ways an arithmetic expression can be made unambiguous with parentheses so that, if all parentheses are removed, the expression is exactly the string given in the input.
We are given a string representing a character's name in a game, and we can change at most k of its letters to maximize a score called euphony. The euphony is computed as the sum of bonuses for every consecutive pair of letters in the string.
We have two people represented as points on the plane, Victor and Peter. The room contains exactly two objects, a solid wall segment and a double-sided mirror segment. Victor wants to know whether he can see Peter either directly or through a single reflection in the mirror.
This problem asks us to determine how many students are enrolled in each department, including departments that currently have no students. We are given two tables: Student and Department.
The problem asks us to count the number of corner rectangles in a binary m x n matrix grid. A corner rectangle is defined as a set of four 1s in the grid that form the corners of an axis-aligned rectangle. Importantly, only the corners need to be 1; the interior cells can be 0.
We are given a table representing the economy of Berland over n days and m months. Each cell in the table contains either 1 or -1, indicating a profit or a loss for that day of that month.
This problem asks us to compute the maximum nesting depth of parentheses in a valid parentheses string. A valid parentheses string, often abbreviated as VPS, is a string where every opening parenthesis '(' has a matching closing parenthesis ')', and the parentheses are…
We start with an array where position i initially contains value i. Each position also has a fixed jump distance d[i]. A swap is allowed between positions i and j only if |i - j| = d[i].
We are given a convex polygon defined by a list of points in clockwise order. For each query point in the plane, we need to count how many triangles formed by the polygon’s vertices contain that point strictly inside.
The problem gives us an integer array nums, a maximum allowed absolute value limit, and a target sum called goal.
Here is a complete, detailed technical solution guide for LeetCode 1660 - Correct a Binary Tree, formatted exactly as re
This problem asks us to maximize the total number of vacation days over k weeks while traveling between n cities under specific flight constraints. We start in city 0 on the Monday morning of week 0.
Edit This problem asks us to count the number of valid ways to divide a rectangular pizza into exactly k pieces such tha
The problem gives us a database table named Customers, where each row contains a unique customerid and a corresponding customername. Our task is to identify all missing customer IDs between 1 and the maximum customerid currently present in the table.
This problem asks us to examine a list of integers, arr, and determine whether there exists a sequence of three consecut
Each amulet is a 2 x 2 square filled with numbers from 1 to 6. We can think of it as four cells:
The problem asks us to determine whether certain substrings of a given string s can be rearranged and partially modified to form a palindrome. A palindrome is a string that reads the same forward and backward.
The problem asks us to reverse a string. Given a sequence of characters, the output should be the same sequence but in the opposite order, so that the first character becomes the last, the second becomes the second-to-last, and so on.
The problem gives a string containing only six possible characters: (, ), {, }, [ and ]. Each opening bracket must eventually be matched with the correct closing bracket, and the order of matching matters.
The test contains several questions that must be answered in order. Each question has multiple answer choices, and exactly one of them is correct. Vaganych does not know any correct answers in advance, but after making a mistake he remembers which options were wrong.
This problem asks us to compute the average number of sessions per user over a specific 30-day period ending on 2019-07-27. Each row in the Activity table represents an activity a user performed in a session on a particular date.
This problem gives us a sorted, 1-indexed array of integers called numbers and a target integer target. Our task is to find exactly two numbers in the array whose sum equals target, then return their 1-based indices.
The problem asks us to sort an integer array in ascending order without using any built in sorting functions. The result must contain the same elements as the input, but arranged from smallest to largest. The input is an array nums containing integers.
Each laptop has four values: processor speed, RAM size, HDD size, and price. A laptop is considered outdated if there exists another laptop that is strictly better in all three technical characteristics at the same time.
The problem asks us to calculate the total duration that Ashe is poisoned by Teemo's attacks. Each attack at a given second t causes Ashe to be poisoned for exactly duration seconds, and the poisoning time is inclusive, meaning that the poison effect lasts from time t to t +…
The problem describes a two-player turn-based game between Alice and Bob. The game starts with a single integer n written on a chalkboard. Alice always takes the first turn.
The problem asks us to identify customers who meet a very specific purchasing pattern. We are given two tables: Customer
The problem gives four points in a 2D coordinate system and asks whether those four points form a valid square. The points are not provided in any guaranteed order, which means we cannot assume adjacent points or diagonally opposite points are already grouped correctly.
The problem gives us the head of a singly linked list and asks us to reorder the nodes in a very specific alternating pattern. A normal linked list looks like this: We must transform it into: The important detail is that we are not allowed to change node values.
The problem gives us an integer array nums, and we must perform exactly one operation on the array. The operation allows us to choose a single index i and replace the value nums[i] with its square, nums[i] nums[i].
This problem asks us to determine if a given target integer is a majority element in a sorted array nums. A majority element is defined as an element that appears more than half of the array's length. In other words, if nums.
This problem asks us to determine whether a given binary tree is height-balanced. A binary tree is considered height-balanced if, for every node in the tree, the height difference between its left and right subtrees is at most 1.
We are given a sequence of chat server events. A user can join the chat, leave the chat, or send a message. Every time someone sends a message, the server delivers that message to every user currently inside the chat, including the sender.
The problem asks us to design a data structure that manages a fixed pool of phone numbers. Initially, every number from 0 to maxNumbers - 1 is available. The directory must support three operations efficiently. The get() operation should assign and return an available number.
In this problem, we are given a list of strings where each string represents a visit count paired with a domain name. Each input entry has the form: For example: means the domain discuss.leetcode.com was visited 9001 times.
We are asked to compute the expected lifespan of a dynamic particle moving on a 2D grid with static obstacles. The grid has n rows and m columns. Static particles occupy certain cells, never sharing a row, column, or diagonally adjacent cell.
This problem asks us to analyze relationships between customers, their contacts, and invoices. We are given three tables: Customers, Contacts, and Invoices. The Customers table identifies each customer by customerid along with their name and email.
The problem is asking us to simulate a modified game of Snakes and Ladders on an n x n board. The board is labeled from 1 to n² in a boustrophedon pattern, which means the numbering starts from the bottom-left, alternates direction every row, and ends at the top-right.
This problem asks us to place introverts and extroverts inside an m x n grid in a way that maximizes the total happiness score. Every grid cell can either remain empty, contain one introvert, or contain one extrovert.
The problem gives us a string consisting only of digits, and asks us to split it into a sequence of integers that behaves like a Fibonacci sequence. A Fibonacci-like sequence follows three important rules: 1.
We are given a tournament graph, which is a special type of directed graph where for every pair of distinct vertices, there is exactly one directed edge connecting them.
The problem asks us to return the k most frequently occurring elements from an integer array. We are given an array nums, which may contain duplicates, and an integer k. Our task is to identify which values appear most often and return exactly k of them.
The problem asks us to transform a table of student names and their continents into a pivoted report, where each continent becomes a column and the student names appear alphabetically under their respective continents.
The task is to normalize the case of a single word so that either all letters are lowercase or all are uppercase.
The problem describes a robot moving on an infinite two dimensional grid. The robot begins at coordinate (0, 0) and initially faces north, which corresponds to the positive Y direction.
The problem describes a turn-based game between Alice and Bob with a row of stones, each having a numeric value. On each turn, a player removes any stone of their choice.
This problem asks us to interpret a Goal Parser command string. The string command consists of the characters "G", "()",
We are given the positions of two chess pieces on a standard 8 × 8 board, one rook and one knight. Their starting positions are guaranteed to be safe, meaning the rook does not attack the knight and the knight does not attack the rook.
Here is a complete, detailed technical solution guide for LeetCode 1582 - Special Positions in a Binary Matrix, followin
The UserVisits table stores records of when users visited a retailer. Each row contains a userid and a visitdate. A user may appear multiple times because they can visit on many different days.
This problem asks us to find the k values in a Binary Search Tree, or BST, whose values are numerically closest to a given floating point target. A BST has a very important property: - Every value in the left subtree is smaller than the current node.
The problem asks us to determine whether a given string s can be split into exactly three non-empty substrings, each of which is a palindrome. A palindrome is a string that reads the same forward and backward. For example, the string "aba" is a palindrome, whereas "abc" is not.
The problem provides a Customer table with three columns: id, name, and refereeid. Each row represents a customer, their unique identifier (id), their name, and optionally the id of the customer who referred them.
The problem gives an integer array nums with n elements. In a single move, you are allowed to increment exactly n - 1 elements by 1. Your goal is to determine the minimum number of such moves required to make every element in the array equal.
We are given a set of circular targets, all lying on the Ox axis, each defined by its center coordinate and radius. Valera shoots multiple times, and each shot has an (x, y) coordinate.
The problem gives us an undirected connected graph where each node represents a city, and each city has a three letter name. We are also given a target sequence of names called targetPath. Our goal is to construct a valid path through the graph such that: 1.
The language in this problem has only two real base types, void and errtype. Every other type is defined through typedef, and every query asks us to evaluate a type expression with typeof. A type expression is built from a base name plus some number of and & operators.
This problem gives us a two dimensional grid where every cell contains a directional sign. The sign tells us which neigh
The problem is asking us to design a custom stack-like data structure that supports two operations: pushing values onto the stack and popping the most frequent element.
We are given a tree of n cities, meaning each city is connected in such a way that there is exactly one simple path between any two cities. Roads are undirected, and each road has a length of 1.
We are given n marbles on a one-dimensional axis, each with a position x[i] and a pin cost c[i]. You can stick a pin in some marbles, paying the associated cost, and unpinned marbles will roll left until they hit the nearest pinned marble.
We have a rectangular cornfield represented as a grid of size n × m, where each cell contains a certain number of tons of corn. The father wants to divide this field among three sons in such a way that each son receives exactly a predetermined amount of corn: A, B, or C tons.
The problem gives us a sorted array nums and an integer maximumBit. For every query, we must choose a number k such that: - 0 <= k < 2^maximumBit - The value of: is as large as possible.
The problem asks us to construct the lexicographically smallest subsequence of a string that contains every distinct character exactly once. A subsequence is formed by deleting zero or more characters without changing the relative order of the remaining characters.
We start with one register, eax, containing some unknown value x. Every other register contains 0. The goal is to produce n x in any register using the minimum possible number of lea instructions. The instruction set is surprisingly limited, but also surprisingly powerful.
This problem asks us to determine the smallest contiguous subarray we can remove from a given array of positive integers
We have three coins labeled A, B, and C. Every pair of coins has already been compared once using a balance scale, and each comparison tells us which coin is heavier.
We are given two vertices of a unit cube. Each vertex is described by three coordinates, and every coordinate is either 0 or 1. Since each coordinate can only take two values, these coordinates represent the eight corners of the cube. Two flies stand on two different vertices.
Here’s a complete, detailed technical solution guide for LeetCode 2034 - Stock Price Fluctuation following your formatting and content requirements. The problem presents a stream of stock price records where each record consists of a timestamp and a price.
We are given two positive integers, and then many range queries. For each query [low, high], we must find the largest integer inside that interval that divides both numbers. The first observation is that we are not really working with a and b independently.
The problem gives us two strings, s1 and s2, and asks us to make them equal by deleting characters from either string. Every deleted character contributes its ASCII value to the total cost. Our goal is to minimize this total deletion cost.
The board has N rows and M columns. A robot starts at cell (i, j) and repeatedly performs one random move. From an interior cell it has four equally likely choices: stay in place, move left, move right, or move down.
The problem asks for the minimum number of perfect square numbers whose sum equals a given integer n. A perfect square is a number that can be written as x x for some integer x. Examples include 1, 4, 9, 16, and 25.
The problem gives us an encoded array where each element represents the XOR of two consecutive elements from an unknown original array. Specifically: We are also given the first element of the original array, first = arr[0].
We are given a sequence of nails in the plane, each with integer coordinates, and a collection of straight rods. The nails define the vertices of a closed polyline that only moves along horizontal or vertical segments.
This problem asks us to maximize profit from stock trading over a sequence of days. We are given an array prices, where prices[i] represents the stock price on the ith day.
This problem asks us to count how many numbers within a given inclusive range are "super-palindromes". A number is considered a super-palindrome if both of the following conditions are true: 1. The number itself is a palindrome. 2. The number is the square of another palindrome.
This problem gives us an array of lowercase English words and asks us to find the longest word that can be constructed one character at a time using other words from the same dictionary. More specifically, suppose we have a candidate word like "world".
The traversal is by reference, not value.
This problem asks us to repeatedly simulate a process involving stones with different weights. At every turn, we must select the two heaviest stones, smash them together, and update the collection based on the result.
The problem asks us to split a binary search tree (BST) into two separate subtrees based on a target value. Specifically, the first subtree should contain all nodes with values less than or equal to the target, and the second subtree should contain all nodes with values…
The problem asks us to analyze a system that runs one task per day over a fixed period, from 2019-01-01 to 2019-12-31. Each day, the task either succeeds or fails.
The problem gives us the root of a binary tree where every node contains a single digit from 0 to 9. Every path starting from the root and ending at a leaf node forms a number by concatenating the digits along that path.
We are given the top view of a robotic vacuum cleaner as a convex polygon. The room corner is the usual 90 degree corner formed by two perpendicular walls. We may rotate the vacuum cleaner by any angle and then push it as far as possible into the corner.
The problem defines a mysterious function func(arr, l, r) that computes the bitwise AND of all elements in the subarray
The problem is asking us to partition an integer array nums into exactly k subsets of equal size, such that no subset co
We are asked to maintain a dynamic set of positive integers under three operations: adding a number, deleting a number, and computing the sum of medians of every consecutive group of five elements after sorting the set.
The problem presents a list of m arrays, each individually sorted in ascending order. You are asked to pick one element from two different arrays and compute the absolute difference between these two elements, which is defined as the distance.
The problem asks us to find the number of non-empty subsequences of a given array nums such that for each subsequence, t
We are asked to count the number of arrays of length n containing integers from 1 to n such that the array is either entirely non-decreasing or entirely non-increasing.
This problem asks us to maximize the number of children who can be satisfied with the available cookies. Each child has a greed factor, and each cookie has a size. A child becomes content only if they receive a cookie whose size is greater than or equal to their greed factor.
Valera has a list of subjects he knows how to solve. Every subject takes a fixed amount of working time. Students come with requests: each request has a subject, an exam deadline, and a reward. If Valera finishes the solution strictly before the exam starts, he gets paid.
Here’s a full, detailed technical solution guide for LeetCode 585 following your exact formatting rules: The problem presents an Insurance table containing policyholder information, including a unique policy ID (pid), investment values for 2015 (tiv2015) and 2016 (tiv2016)…
The problem gives us three arrays of equal length: - startTime[i] represents when the i-th job starts - endTime[i] represents when the i-th job finishes - profit[i] represents the money earned if we complete that job Our goal is to select a subset of jobs that do not overlap…
The problem asks us to count the number of valid ways to divide an array into three contiguous, non-empty parts: - left - mid - right The split must satisfy two conditions: 1. sum(left) <= sum(mid) 2.
Here is a complete, detailed technical solution guide for LeetCode 1276 - Number of Burgers with No Waste of Ingredients
That is a long-form technical guide with multiple required sections and detailed worked examples. I can provide the complete reference document, but it will be quite lengthy.
This problem gives us an array called heights, where each element represents the height of a building. The buildings are arranged in a straight line from left to right, and the ocean is located to the right side of the last building.
The problem asks us to transform a binary string into a monotone increasing string using the minimum number of character flips. A binary string is considered monotone increasing if all 0s appear before all 1s. In other words, once a 1 appears, no 0 can appear after it.
This problem asks us to find the in-order successor of a given node p inside a Binary Search Tree, abbreviated as BST. An in-order traversal of a BST visits nodes in sorted order: 1. Traverse the left subtree 2. Visit the current node 3.
This problem is asking us to transform an initial binary string s (all zeros) into a target binary string target using a
The problem asks us to find the k-th smallest fraction that can be formed by dividing one element of a sorted array arr by another element later in the array. The array arr is strictly increasing, starts with 1, and all elements after the first are prime numbers.
We are given a set of enemy objects, each at a fixed 2D coordinate, and a fixed strike point. The objective is to determine the minimum impact radius of a nuclear warhead such that the probability of failing to deactivate at least K of these objects is at most ε per mils.
The problem asks us to rearrange the characters of a string so that characters with higher frequency appear before characters with lower frequency. The output string must group identical characters together, and those groups must appear in descending order of occurrence count.
The problem gives us a database table named Queries. Each row represents the outcome of running a particular query against a database.
The problem asks us to compute: The complication is that the exponent b is extremely large. Instead of being given as a normal integer, it is provided as an array of digits.
This problem asks us to generate the shortest possible string that guarantees a safe will unlock at some point while typing it. The safe password has exactly n digits, and each digit can be any value from 0 to k - 1. The safe does not validate the entire entered sequence at once.
This problem asks us to assign students to seats such that every student occupies exactly one seat and no two students share the same seat. Each student can move left or right on a number line, and every movement by one position costs exactly one move.
That is a long-form technical guide request with multiple required sections and complete code. Before I write it, I want to confirm one formatting detail because your template is very specific: Do you want the solution to focus on the standard Digit DP + Fibonacci-style DP on…
The problem asks us to determine whether two binary expression trees are equivalent, where equivalence is defined as the trees representing the same arithmetic expression, modulo the order of addition since addition is commutative.
The problem gives an integer array nums and allows only one type of operation, decreasing any element by 1. We may perform this operation as many times as needed on any elements. Our goal is to transform the array into a zigzag array with the minimum number of moves.
This problem asks us to prune a binary tree by removing insufficient nodes. A node is insufficient if every root-to-leaf path passing through it has a sum strictly less than the given limit. The input is the root of a binary tree and an integer limit.
The problem gives us k sorted integer lists, and we must find the smallest inclusive range [a, b] such that the range contains at least one element from every list. Each list is already sorted in non-decreasing order, which is a very important property.
The problem gives us an array called piles, where each element represents the number of coins in a pile. The length of the array is always divisible by 3, meaning there are exactly 3n piles for some integer n. In every round, three piles are selected.
The Employee table represents a company hierarchy. Every row corresponds to one employee and contains four pieces of information: Column Meaning --- --- id Unique employee identifier name Employee name department Department the employee belongs to managerId The id of this…
The problem asks us to traverse a binary tree and return the node values level by level, but in bottom-up order. Normally, a level order traversal collects values from the root down to the leaves.
We are asked to determine whether a given number is almost lucky. A number is almost lucky if it is divisible by at least one lucky number. Lucky numbers are positive integers composed entirely of the digits 4 and 7, like 4, 7, 44, 47, 74, and so on.
This problem asks us to connect n cities together using a set of possible bidirectional connections. Each connection has a cost, and our goal is to connect every city while minimizing the total cost.
We are given a sequence of log entries in the exact order they were written. Originally every entry had both a date and a time, but the dates were lost, so only the 12-hour clock timestamps remain.
The problem asks us to find the length of the longest substring that lies between two identical characters in a given string s. The substring does not include the two identical characters themselves. If no character occurs at least twice, we must return -1.
This problem is a two-player combinatorial game problem where Alice and Bob alternately remove stones from a single pile. On each turn, a player may remove any positive number of stones that is a perfect square (1, 4, 9, 16, etc.
This problem asks us to determine how many non-negative integers x exist such that the factorial of x ends with exactly k trailing zeroes. The function f(x) counts trailing zeroes in x!. For example, f(3) = 0 because 3! = 6 has no trailing zeroes, and f(11) = 2 because 11!
This problem asks us to identify whether there is a "celebrity" among n people at a party. A celebrity has a very specific property: - Every other person knows the celebrity. - The celebrity knows nobody else. We are not given the entire relationship graph directly.
This problem asks us to determine the first day each player logged into a game based on a table called Activity. Each row of this table contains a playerid, the deviceid used, the eventdate on which the player logged in, and the number of gamesplayed during that session.
The problem provides a table named Products, where each row represents the price of a specific product in a specific store.
The problem presents a string time formatted as hh:mm, where h and m are digits representing hours and minutes, respectively. Some digits may be hidden and are represented by a question mark ?. The goal is to replace the ?
The problem gives us two arrays, nums1 and nums2, both of the same length. At every index i, we are allowed to either keep the values as they are, or swap nums1[i] with nums2[i]. Our goal is to make both arrays strictly increasing while performing the minimum number of swaps.
We are given a list of integers where every number is either 1, 2, or 3. The goal is to transform the sequence so that all numbers are the same, and we want to do this using the smallest number of replacements possible.
This problem asks us to perform an inorder traversal on a binary tree and return the values of the visited nodes in the correct order. A binary tree is a hierarchical data structure where each node can have at most two children, a left child and a right child.
The problem gives us an m x n binary matrix, where every element is either 0 or 1. We are allowed to choose any set of columns and flip them. Flipping a column means changing every 0 in that column to 1, and every 1 to 0.
This problem describes an infinite binary tree where nodes are normally arranged level by level, but with a twist in how labels are assigned. In a standard binary tree, nodes in each level are labeled from left to right.
The problem gives a list of money transfers between people. Each transaction is represented as [from, to, amount], meaning one person paid a certain amount to another person.
The problem asks us to count how many non-empty contiguous subarrays have a sum that is divisible by a given integer k. A subarray is any continuous segment of the array. For example, in the array [1,2,3], the subarrays include [1], [2], [3], [1,2], [2,3], and [1,2,3].
This problem asks us to select the largest possible subset of binary strings while staying within two resource constraints. Each string consumes a certain number of 0 characters and a certain number of 1 characters.
The problem asks us to design a data structure that behaves like an infinite row of stacks, where every stack has the same fixed capacity. Instead of working with a single stack, we maintain many stacks indexed from left to right starting at 0.
We have a circle of n players, numbered 1 through n clockwise. The ball starts with player x. There are m throws, and each throw has a distance r_i and a remembered direction c_i. The direction could be clockwise (0), counterclockwise (1), or unknown (?).
This problem gives us a calculator that starts with an integer startValue and allows only two operations: 1. Multiply the current number by 2 2. Subtract 1 Our goal is to transform startValue into target using the minimum number of operations.
The problem gives us an integer array nums and a target integer target. We must select exactly three elements from the array, using distinct indices, and compute their sum. Among all possible three-number sums, we want the one whose value is closest to the target.
We are given two dates in the format DD.MM.YY: one representing the day of the Codeforces World Finals and the other representing Bob's date of birth. Bob can rearrange the components of his birth date-the day, month, and year-but only as complete numbers, not individual digits.
The problem gives us a binary matrix M with dimensions height x width. Every cell in the matrix must contain either 0 or 1. There is one important restriction: Every square submatrix of size sideLength x sideLength may contain at most maxOnes cells equal to 1.
This problem asks us to determine whether three given points in a 2D Cartesian plane form a valid boomerang. A boomerang is defined as a set of three points that are all distinct and not in a straight line. In other words, the points must not coincide and must not be collinear.
We are asked to color a collection of intervals on the number line such that no three intervals with the same color form a "triple overlap pattern.
The problem asks us to simulate a painter working on an 8×8 chessboard. Every square starts white, and the painter can perform only two operations: paint an entire row black or paint an entire column black. Painting the same square multiple times has no effect beyond the first.
The problem gives us an integer array nums and a maximum jump distance k. We start at index 0, and from any position i,
The problem asks us to find the number of distinct non-empty substrings of a given string text that can be expressed as
We have several jars of honey, and each jar starts with some amount of honey. Winnie repeatedly chooses the jar that currently contains the most honey. When Winnie picks a jar, two things may happen.
We are given the source code of a tiny recursive function f(n) written in a heavily restricted subset of C++. The function only contains sequential if (...) return ...; statements and unconditional return ...; statements.
The problem asks us to implement a multi-threaded web crawler that starts from a given URL startUrl and visits all URLs reachable from it, but only those that share the same hostname.
This problem describes a square room with perfectly reflective walls. The room has side length p, and there are three receptors placed at three corners of the square: - Receptor 0 is at the southeast corner - Receptor 1 is at the northeast corner - Receptor 2 is at the…
The problem asks us to design a class called SubrectangleQueries that operates on a two dimensional integer matrix, refe
We are given a list of integers representing problem scores in a contest. A contestant may solve any non-empty subset of these problems, and the final score becomes the product of all chosen values.
We start with an array of positive integers. We must change exactly one element to a different value, still between 1 and 10^9. After that single replacement, we sort the array in non-decreasing order.
This problem gives us an n x n grid where each cell represents the height of a building in a city. The city can be viewed from four directions: north, south, east, and west. From these viewpoints, the skyline is determined by the tallest building visible in each row or column.
The problem asks us to generate every possible string that can be formed by independently changing the case of each alphabetic character in the input string. Digits cannot be modified, so they always remain the same in every generated result.
The problem asks us to determine whether an array contains three numbers that form a strictly increasing subsequence.
We are given several 10 × 10 Battleship boards. Each cell is either empty or occupied by part of a ship. The task is to verify whether every occupied cell belongs to a valid fleet configuration.
We have three convex polygons on the plane. From each polygon we must choose one point, and the average of these three chosen points must equal the position of the idol.
The problem is asking us to determine the number of posts that were reported yesterday grouped by the reason for the report.
This problem is an interactive shortest path problem on a hidden weighted grid. We control a robot that starts somewhere in an unknown grid, and we must determine the minimum total movement cost required to reach a hidden target cell.
This problem asks us to control a car moving on an infinite one dimensional number line. The car starts at position 0 with speed +1.
The problem requires calculating sums of specific subsets of an integer array nums based on queries. Each query [xi, yi] specifies a starting index xi and a step yi. The sum for this query includes all elements nums[j] such that j starts at xi and increases in steps of yi (i.e.
The problem is asking us to count the number of valid subarrays in a given integer array nums. A subarray is valid if its leftmost element is not larger than any other element in the subarray.
The problem is asking us to combine two relational database tables, Person and Address, in such a way that we report each person's first name, last name, city, and state.
This problem asks us to count how many subarrays have sums that fall within a given inclusive range [lower, upper]. A range sum S(i, j) represents the sum of all elements from index i through index j in the array.
The problem asks us to compute the maximum alternating subarray sum for a given integer array. A subarray is any contiguous sequence of elements from the array.
The problem gives us a table named Friendship, where every row represents a friendship relationship between two users. E
In this problem, we are given two arrays, nums1 and nums2, both of length n. For every index i, we must choose exactly o
The problem asks us to calculate the sum of divisors for numbers in a given integer array nums, but only for those numbe
Each student group attends exactly two lessons. For a group, the classroom used in the first lesson must not exceed the classroom used in the second lesson.
The problem asks us to determine whether a given string s can be formed by substituting each character in a pattern with a non-empty string, under a bijective mapping.
We are trying to reconstruct a hidden binary string of length n. Every guess Vasya made is another binary string of the same length, together with a number saying how many positions matched the real code exactly.
The problem asks us to take a string containing only digits and determine every possible way to insert exactly three dots so that the resulting string becomes a valid IPv4 address. A valid IP address has four numeric segments separated by dots.
This problem asks us to determine which day of the week corresponds to a given calendar date. The input consists of three integers: - day, representing the day within the month - month, representing the month number from 1 to 12 - year, representing the year We must return the…
This problem asks us to search for a target value inside a sorted array, but with an important limitation: we do not know the size of the array, and we cannot access the array directly. Instead of normal array indexing, we interact with the array through an ArrayReader interface.
The problem asks us to find the intersection of two integer arrays, but with an important detail: duplicates matter. If a number appears multiple times in both arrays, it must appear in the result as many times as it appears in both.
This problem asks us to generate every possible unique combination of numbers from the given candidates array such that the sum of the chosen numbers equals target. There are several important details in the problem statement: - Every number in candidates is distinct.
We have a circular route with n post offices, each with a gas station. Each station i has a[i] liters of gasoline available, and the distance from station i to i+1 is b[i] kilometers, wrapping around at the end.
We have houses numbered from 1 to n. Every house must receive a color. For each color, if we add together all house indices painted with that color, the result must be a prime number.
The problem gives us a positive integer num that contains only the digits 6 and 9. We are allowed to change at most one
The problem gives us a two dimensional grid called board, where each cell contains either 'X' or 'O'. The goal is to modify the board in place by capturing every region of 'O' cells that is completely surrounded by 'X'.
The problem gives us an integer array nums whose length is exactly 2 n. Among all the numbers in the array, there are n + 1 distinct values. One special value appears exactly n times, while every other value appears only once.
The problem asks us to design a data structure that determines whether a word's abbreviation is unique within a given dictionary.
We are given a string of digits representing a phone number. The task is to split this string into pieces where every piece has length either 2 or 3. The groups must appear in the original order, and they are joined with - characters in the output.
The problem gives us an integer array nums and two fixed subarray lengths, firstLen and secondLen. We must select exactly two contiguous subarrays such that: - One subarray has length firstLen - The other subarray has length secondLen - The two subarrays do not overlap - The…
This problem provides a compressed representation of a dataset instead of listing every number individually. The table Numbers contains two columns: - num, the actual number - frequency, how many times that number appears If we expanded the table into a full sorted array, each…
The problem asks us to maximize the total score obtained by performing n operations on an array nums of size 2 n.
This problem asks us to maximize profit from stock trading under a special restriction called a cooldown period. We are given an integer array prices, where prices[i] represents the stock price on day i. On any day, we may choose to buy one share, sell one share, or do nothing.
We are given an n × m grid, and two distinct interior cells. The task is to construct the longest possible simple path between them. A simple path means we may visit each cell at most once. Consecutive cells in the path must share a side.
The problem is asking us to reconstruct a binary tree given two traversal orders: preorder and inorder. In a preorder traversal, nodes are visited in the order: root, left subtree, right subtree.
The problem requires counting the numbers within a given inclusive range [left, right] that have a prime number of set bits in their binary representation. The set bits of a number refer to the number of 1s in its binary representation.
We are given a set of islands connected by bidirectional roads. The islands form regions: each region is a connected component, and islands in different regions have no path between them.
The problem gives us an integer array arr and asks whether it can be divided into exactly three non-empty contiguous parts such that all three parts have the same sum. The partitions must preserve the original order of the array. We are not allowed to rearrange elements.
Each stamp describes a direct transfer between two cities. If the envelope contains n stamps, then the letter passed through exactly n + 1 cities in sequence. The route never revisits a city, so the entire journey forms a simple path.
The sequence starts with two fixed values:
This problem asks us to maximize the number of successful invitations between boys and girls under a one-to-one matching constraint. We are given an m x n binary matrix called grid.
This problem gives us the head of a singly linked list and two integers, m and n. We must traverse the linked list while
Each tooth belongs to exactly one row. When Valerie eats one crucian using a row, every tooth in that row loses one unit of viability. A row becomes unusable as soon as at least one tooth inside it would drop below zero.
The problem gives us a directed graph of cities connected by roads. There are n cities numbered from 0 to n - 1, and exactly n - 1 roads. Since the graph contains n - 1 edges and there is exactly one path between any pair of cities, the graph forms a tree.
The problem requires parsing and evaluating a mathematical expression that includes integers, variables, addition, subtraction, multiplication, and parentheses, while also applying a substitution map for certain variables.
The problem asks us to compute the vertical order traversal of a binary tree. In simpler terms, we are asked to “look at the tree from the side” and collect the nodes that align in the same vertical column.
The problem asks us to find the length of the longest subsequence in an array such that the differences between consecutive elements alternate between positive and negative.
This problem gives us the root of a Binary Search Tree, abbreviated as BST, and asks us to rearrange the tree into a very specific form.
The problem asks us to find the area of the largest rectangle containing only 1s in a binary matrix. The matrix consists of characters '0' and '1', where '1' represents a valid cell that may belong to a rectangle and '0' represents a blocked cell that cannot be included.
We are asked to distribute milk from a set of bottles into cups such that each bottle contributes to at most two cups, and all cups end up with the same total volume. Specifically, we have n bottles, each containing w units of milk, and m friends who each receive one cup.
The problem asks us to find the minimum Hamming distance between two arrays, source and target, after performing any number of swaps on source at positions allowed by allowedSwaps. The Hamming distance is defined as the number of indices i for which source[i] != target[i].
We are given a sequence of integers representing the yearly income of a company starting from 2001. The first number is the income in 2001, the second in 2002, and so on. These values may be negative if the company incurred a loss that year.
The problem asks us to find the minimum number of Fibonacci numbers whose sum equals a given integer k. Fibonacci number
The problem gives us a sorted integer array nums in non-decreasing order. Because the array is already sorted, any duplicate values will always appear next to each other.
We have a scenario where a cannon at the origin shoots a number of balls with the same initial speed, each at a given angle. The goal is to determine where each ball lands after either hitting a vertical wall or reaching the ground.
We are given a year in the Gregorian calendar and a day number inside that year. The task is to determine the actual calendar date corresponding to that day number.
The problem asks us to find the shortest contiguous subarray of an array nums that has the same degree as the original array. The degree of an array is defined as the maximum frequency of any element in that array.
This problem simulates a simplified stock exchange order book. Each incoming order is either a buy order or a sell order, and each order contains a price and an amount. Orders are processed strictly in the order they appear in the input.
There are n boxes arranged in a straight line. Exactly one of them may contain the cereal flakes. The roommate leaves statements of two possible forms. If the hint says "To the left of i", then the flakes must be somewhere strictly before box i. Box i itself is also impossible.
The problem asks us to group all integers from 1 to n based on the sum of their digits. Every number belongs to exactly
The problem requires rearranging spaces in a given string text so that all words are separated by the maximum possible e
The problem asks us to modify a binary tree by adding a new row of nodes with a given value val at a specified depth depth. The input consists of the root of a binary tree, the integer value val to insert, and the target depth depth.
We have three cups arranged in a line, and a ball starts under one of them. The performer performs exactly three swaps. Each swap exchanges the positions of two cups. After all swaps finish, we must determine which cup contains the ball.
The problem requires constructing a binary expression tree from a postfix arithmetic expression and implementing a metho
The problem asks us to compute the minimum sum of a falling path with non-zero shifts in a square n x n matrix. A fallin
This problem asks us to paint a row of houses while satisfying two constraints simultaneously: 1. Every house must end u
The problem asks us to design a data structure that continuously receives numbers from a stream and can efficiently compute a special average called the MKAverage. For every query, we only care about the most recent m elements in the stream. From those m elements: 1.
Zeyad wants to commit exactly n crimes in a sequence so that he avoids any punishment. Each crime type is represented by a capital letter, and for some crimes there are conditions describing multiplicities: committing that crime a number of times divisible by its multiplicity…
This problem asks us to simulate the movement of balls dropped into a 2-D box represented by a grid. Each cell in the grid has a diagonal board that directs a ball either to the right (1) or to the left (-1).
I can do that, but the full guide at the level of detail you requested, including complete prose explanations, worked tr
This problem asks us to sort arr1, but not in ordinary ascending or descending order. Instead, the sorting order is partially dictated by another array, arr2. The key requirement is that every number appearing in arr2 must appear in arr1 in exactly the same relative order.
This problem presents a maze represented as a 2D grid of size m x n where each cell is either an empty space (0) or a wall (1). A ball starts at a given position and can roll up, down, left, or right.
The Products table stores product prices across three different stores. Each row represents one product, identified by productid, and each store column contains the product's price in that specific store.
This problem asks us to determine whether a car can successfully complete a series of passenger trips without ever exceeding its seating capacity.
Each viewing mode groups the inventory into pages of size ai. If the inventory contains k items, then the game shows $bi = leftlceil frac{k}{ai} rightrceil$ pages in that mode. Vasya does not know the actual value of k, only that 2 ≤ k ≤ x.
The problem asks us to build an index mapping from nums1 to nums2, where nums2 is guaranteed to be an anagram of nums1. Since an anagram means the same elements appear in both arrays, but possibly in a different order, every value in nums1 must appear somewhere in nums2.
We are given a set of N random integers, where each integer i can take any value in a given inclusive range [L_i, R_i], all values equally likely. The task is to compute the probability that at least K% of these N integers start with the digit 1.
This problem asks us to determine the maximum profit that can be made from a single stock transaction. A transaction consists of buying one stock on one day and selling it on a later day. The key restriction is that the selling day must come after the buying day.
We are given three original strings. A student's answer is considered correct if it can be formed by concatenating these three strings in any order, after ignoring two kinds of differences. The first difference is letter casing.
Each row of the table contains integers, and from every row we must take a positive-length prefix. If we choose c[i] cells from row i, then the selected cells in that row are exactly the first c[i] entries.
This problem asks us to efficiently compute the dot product between two sparse vectors. A normal vector is simply an array of numbers. The dot product of two vectors is computed by multiplying corresponding elements and summing the results.
We need to build an undirected graph on n cities such that no triangle exists. A triangle means three distinct cities where every pair is directly connected.
The problem is asking us to partition a string s into as many contiguous parts as possible such that no letter appears in more than one part. In other words, each character in the string should be confined to a single segment.
We have a group of soldiers, each with a rank between 1 and k. During one training session, soldiers are grouped by equal rank. From every group whose rank is still below k, exactly one soldier is promoted by one level.
The problem asks us to find the leftmost value in the last row of a binary tree. A binary tree is a hierarchical data structure where each node has at most two children: a left child and a right child.
We are asked to maximize profit from a set of potential races in a linearly connected kingdom. Each race occupies a contiguous set of roads and provides a payment if all the roads it uses are repaired.
This problem asks us to convert an Excel-style column title into its corresponding numerical index. In Excel spreadsheets, columns are labeled alphabetically: The labeling system works similarly to a positional number system, except instead of digits 0-9, it uses letters A-Z…
The problem asks us to find the maximum sum of elements from an integer array nums such that the sum is divisible by thr
We start with an undirected multigraph. Multiple edges are allowed, and loops are also allowed. We may add new edges, and the goal is to transform the graph into a very specific structure.
This problem asks us to find the maximum length of a contiguous subarray in a binary array where the number of 0s and 1s are equal. A contiguous subarray means the elements must appear consecutively in the original array. We are not allowed to rearrange elements or skip indices.
The problem gives us an array called coins, where each element represents a coin denomination, and an integer called amount, which represents the target sum we want to construct. Our goal is to determine the minimum number of coins needed to make exactly amount.
The problem describes a repeating weekly saving pattern. Hercy deposits money into the LeetCode bank every day, and the amount increases in a structured way. On the very first Monday, he deposits 1 dollar. Each following day in the same week increases by 1.
We have exactly five people. Some pairs of people know each other, and the input lists all such acquaintance relations. The task is to determine whether there exists either: 1. Three people where every pair knows each other. 2. Three people where no pair knows each other.
The problem gives an integer array nums and asks whether any number appears more than once. If at least one value is repeated, we return true. If every value appears exactly once, we return false. The input is a list of integers.
The problem asks us to generate the sequence of ugly numbers and return the nth value in that sequence. An ugly number is defined as a positive integer whose only prime factors are 2, 3, and 5.
The problem asks us to compute the number of set bits, also called population count or popcount, for every integer from 0 through n. A set bit is a bit with value 1 in the binary representation of a number.
We are given two positive integers, a and b. First we compute their normal sum c = a + b. Then we apply the same transformation to all three numbers: remove every digit 0 from their decimal representation. The question is whether the transformed equation still holds.
This problem asks us to reconstruct a target string by repeatedly applying a smaller string called stamp. We begin with a string s that has the same length as target, but every character is initially '?'.
The problem is asking to calculate the percentage of immediate food deliveries from a delivery table. An immediate delivery is defined as one where the customerprefdeliverydate matches the orderdate.
We are given two arrays of positive integers, arr1 and arr2. The task is to determine the maximum length of a common dig
We are given a flag represented as an _n_×_m_ grid where each cell is painted with one of 26 colours labeled a to z. The goal is to repaint as few squares as possible so that two conditions hold. First, each row can use at most two different colours.
This problem provides a database table named Sales that stores the number of fruits sold on each day. Each row represents a single fruit sale summary for a specific date.
The problem asks us to design a data structure that simulates an n x n Tic-Tac-Toe game between two players. We need to implement a class that supports two operations: - Initializing a game board of size n - Processing moves one at a time and immediately determining whether…
The problem asks us to convert a decimal number given as a string into its Hexspeak representation. Hexspeak is derived
The problem gives us a list of unique points on a 2D X-Y plane. Each point is represented as a pair of integers [x, y]. Our task is to choose any three distinct points and compute the area of the triangle formed by those points.
The problem asks us to generate a sales report that summarizes the total quantity of items sold for each category on eac
The problem asks us to generate every possible subset of a given integer array. A subset is any selection of elements from the array, including the empty subset and the subset containing all elements.
The problem asks us to examine the binary representation of a positive integer n and determine the largest distance between two adjacent 1 bits. A binary number is made up of 0s and 1s.
The problem asks us to count how many non-empty rectangular submatrices inside a 2D matrix have a sum equal to a given target. A submatrix is any contiguous rectangular region within the matrix.
This problem asks us to determine the maximum number of ice cream bars a boy can buy with a limited number of coins. The input consists of an array costs, where costs[i] represents the price of the i-th ice cream bar, and an integer coins representing the total coins available.
We are asked to mix hot and cold water to achieve a target bath temperature as close as possible to a given value, while filling the bath as quickly as possible.
The problem asks us to compute how many distinct values can appear as the GCD of some non-empty subsequence of the given array. A subsequence is formed by deleting zero or more elements while preserving order.
This problem asks us to find the smallest contiguous substring of s1 such that s2 appears inside that substring as a subsequence. A subsequence does not require characters to be adjacent, but they must appear in the same relative order.
This problem models processes in an operating system as a tree structure. Every process has exactly one parent, except for the root process, which has no parent and is identified by ppid[i] = 0.
The keyboard is arranged in a fixed grid layout containing the 26 uppercase English letters. Each letter has a coordinat
The problem asks us to construct a special integer sequence using numbers from 1 to n. The resulting sequence has length 2 n - 1 because: - The number 1 appears exactly once. - Every number from 2 to n appears exactly twice.
The problem asks us to determine the minimum number of positive deci-binary numbers required to sum up to a given decima
We are given a sequence of integers, and we are asked to extract the shortest subsequence that is _not ordered_. A sequence is considered ordered if it is either entirely non-decreasing or entirely non-increasing.
This problem asks us to construct heights for n buildings arranged in a straight line while satisfying several constraints. Each building has a non-negative integer height. Building 1 must always have height 0.
I can do that, but the guide will be very long and may get truncated in a single message because of the required depth a
The problem gives us a string s that contains lowercase English letters and numerical digits. Our task is to find the second largest distinct digit that appears anywhere in the string. The important detail is that we care about distinct digits, not frequency.
The problem asks for a preorder traversal of an n-ary tree. In a preorder traversal, the order of visiting nodes is: first the root node, then recursively all the children from left to right.
In this problem, we are given an integer n, and we must determine whether it is an Armstrong number. An Armstrong number is defined as follows: - Let k be the number of digits in n. - Take every digit in the number. - Raise each digit to the power k.
This problem asks us to determine whether a given word can be legally placed into a crossword board while respecting crossword placement rules.
The input is a numeric string num that is guaranteed to already be a palindrome. The task is to rearrange its digits to create another palindrome that is strictly larger than the original number, while also being the smallest such palindrome possible.
The problem asks us to count how many substrings of a given string contain only one distinct character. A substring is a contiguous section of the string. For example, in the string "aaaba", the substring "aaa" is valid because every character is 'a'.
The problem gives us a database table named Events. Each row represents how many times a certain type of event occurred for a particular business.
The problem asks us to determine whether a sequence of integers represents a valid UTF-8 encoded byte stream. Each integer in the input array represents one byte, meaning only its lowest 8 bits matter.
The problem asks us to determine whether a given integer target exists inside an m x n matrix. Unlike a completely unsorted matrix, this matrix has two important ordering guarantees: 1. Each row is sorted in ascending order from left to right. 2.
We are tasked with counting the number of ways to cover an n × m grid using standard dominoes of size 1 × 2, where each domino can be placed either vertically or horizontally.
This problem asks us to count how many different ways a ball can leave the boundaries of a grid within a limited number of moves.
This problem asks us to determine the popularity of product features based on user survey responses. We are given a list of features where each element is a single-word feature name, and a list of responses where each element is a string of space-separated words that users…
The problem asks us to count how many different ways we can assign either a '+' or '-' sign to every number in the array nums such that the resulting arithmetic expression evaluates to target.
This problem asks us to remove the nth node counted from the end of a singly linked list and return the modified list. A singly linked list is a sequence of nodes where each node contains a value and a pointer to the next node.
The problem gives us a string containing digits and the '' character. This string encodes lowercase English letters usin
The snitch moves along a fixed polyline in 3D space. It starts at the first vertex and travels segment by segment at constant speed vs. Harry starts at another point and can move in any direction at constant speed vp, where vp = vs.
This problem asks us to delete a node from a singly linked list, but with an unusual restriction: we are not given access to the head of the list. Instead, we are only given a reference to the node that should be deleted.
The problem asks us to find an index in the array such that the sum of all elements strictly to the left of that index is equal to the sum of all elements strictly to the right of that index. More formally, for an index i: - Left sum = nums[0] + nums[1] + ...
We are asked to determine whether a positive integer $x$ is extraordinarily nice. By the problem's definition, a number is extraordinarily nice if it has exactly the same number of even divisors as odd divisors. The input is a single integer $x$ between 1 and 1000.
We are given a simple polygon on the plane, representing the incubator area. DravDe starts at point , which is guaranteed to lie strictly outside the polygon.
This problem asks us to rearrange a list of non-negative integers so that, when concatenated together, they form the largest possible number. At first glance, this may look like a straightforward sorting problem where we simply sort the numbers in descending order.
The problem asks us to build a very large binary number by concatenating the binary representations of every integer fro
The problem asks whether there exists a pair of different indices (i, j) in the array such that two conditions are satisfied at the same time. The first condition is about index distance: This means the two elements must be relatively close together in the array.
The problem presents a simulation scenario involving k servers, each uniquely identified from 0 to k-1. Every server can
The problem gives us an array called matchsticks, where each element represents the length of a matchstick. The goal is to determine whether all of the matchsticks can be arranged to form exactly one square.
Each row of the river can be treated independently. Along a chosen row, we want to place supports in some columns so that the first and last columns always contain supports, and the number of cells skipped between two neighboring supports is at most d.
The problem gives us the root of a binary search tree, but exactly two nodes in the tree have had their values swapped accidentally. Our task is to restore the tree so that it once again satisfies the binary search tree property, without modifying the tree structure itself.
The problem asks us to select three items from a store such that their prices are strictly increasing and their indices
The problem gives us two traversal orders of the same binary tree: - preorder, which visits nodes in the order: root → left subtree → right subtree - postorder, which visits nodes in the order: left subtree → right subtree → root Our task is to reconstruct and return the…
The problem asks us to find the length of the smallest positive integer that satisfies two conditions: 1. The integer contains only the digit 1 2. The integer is divisible by k Such numbers are commonly called repunits.
The n-queens puzzle asks us to place n queens on an n x n chessboard so that no two queens can attack each other. In chess, a queen can move horizontally, vertically, and diagonally.
The problem requires computing a rolling 3-month average of ride distance and ride duration from ride data in a ride-hailing company database. We are given three tables: Drivers, Rides, and AcceptedRides.
We are asked to simulate planting crops on a rectangular field, but with a catch: some cells are wasteland. The field is represented as an n by m grid, with rows numbered from 1 to n and columns from 1 to m.
We are asked to model the spread of hyperdrive news across a galaxy of planets, where ships move along straight lines at uniform speed.
The problem gives us an integer array nums and asks us to compute the sum of all elements that appear exactly once in the array. An element is considered unique only if its frequency is exactly one.
The problem gives us a list of events, where each event is represented as a pair [startDay, endDay]. An event is available to attend on any single day within that inclusive range.
The problem gives us an integer array nums, and we want to maximize the total number of points earned by repeatedly deleting elements. When we delete a number x, we gain x points. However, deleting x also forces the removal of every occurrence of x - 1 and x + 1.
This problem asks us to generate a monthly financial summary for each country using information from two database tables: Transactions and Chargebacks. The Transactions table stores incoming transactions.
We are asked to calculate the probability that if Petya and Vasya each pick an integer randomly from their respective intervals, the interval between the two chosen numbers contains exactly k lucky numbers. Lucky numbers are those containing only the digits 4 and 7.
The problem asks us to design a data structure that supports efficient searches based on both a prefix and a suffix. We are given an array of words, where each word has an implicit index based on its position in the array. The WordFilter class must support two operations: 1.
Vasya walks into a bar and sees several customers. For each customer, he only knows one piece of information: either the person’s age or the drink they ordered. He wants to determine how many people must still be checked to guarantee that nobody under 18 is drinking alcohol.
This problem asks us to analyze user interaction data stored in the SurveyLog table and determine which question has the highest answer rate. The table records actions performed by users during survey sessions. Every row represents a single interaction with a question.
This problem asks us to create a deep copy of an undirected connected graph. We are given a reference to one node in the graph, and we must return a completely independent copy of the entire graph structure.
We are given an array of positive integers and many interval queries. For each query [l, r], we look only at the subarray between those indices and compute a value called its power. If a number x appears k times inside the subarray, then x contributes k² x to the answer.
This problem asks us to design a reversible transformation between two different tree structures: - An N-ary tree, where each node can have any number of children - A binary tree, where each node has at most two children The important requirement is not how the encoding looks…
This problem asks us to sort an array that contains only three distinct values: 0, 1, and 2. These values represent colors: - 0 represents red - 1 represents white - 2 represents blue The goal is to rearrange the array so that all 0s come first, followed by all 1s, followed by…
The problem asks us to determine whether two strings s and t, each representing a rational number in decimal notation, correspond to the same numerical value. These numbers can be expressed as integers, finite decimals, or decimals with repeating parts denoted by parentheses.
The problem asks us to calculate the total number of valid sequences generated by rolling a six-sided die n times, with the added constraint that each face i cannot appear more than rollMax[i] consecutive times.
The problem asks us to analyze two database tables, Visits and Transactions, and compute how many visitors performed a given number of transactions per visit. Each row in the Visits table represents a unique visit by a user on a specific date.
We are given a subway system of n stations connected by exactly n passages, each passage connecting two distinct stations. The system forms a connected graph where each station can reach every other station.
This problem gives us a special type of array called a mountain array. A mountain array strictly increases until it reaches a single peak element, then strictly decreases afterward. In other words, there exists some index i such that: - arr[0] < arr[1] < ...
The problem gives us a string called licensePlate and an array of candidate words called words. We must find the shortest word that satisfies all the letter requirements contained in licensePlate. The key detail is that only alphabetic characters matter.
In this problem, we are given an integer array nums and a specific index k. We must find a contiguous subarray that contains index k, and among all such subarrays, maximize the following score: The minimum element inside the chosen subarray determines the limiting value of the…
We start with a monitor whose dimensions are a × b. Both values are integers. We want to shrink this monitor so that the new dimensions keep the exact aspect ratio x : y.
We start with an n × m chessboard. The top-left cell is black, so the coloring alternates exactly like a normal chessboard. Only the initially black cells participate in the repainting process.
The problem asks us to determine whether we can select disjoint subarrays from a given array nums such that each subarray matches exactly one of the subarrays in the groups array, in order.
The problem gives us an array heights, where each value represents the height of a building. You start at building 0 and
The problem asks us to determine, for each element in an array, how many elements in the same array are strictly smaller than it. In other words, for an element nums[i], we count all elements nums[j] such that nums[j] < nums[i] and j != i.
The problem asks us to divide a chocolate bar into k + 1 consecutive pieces such that we maximize the minimum total sweetness among those pieces. Each element in the input array sweetness represents the sweetness of a single chunk.
We have up to 50 subjects. Each subject has three properties. The interval $[ai, bi]$ describes how many homework exercises this subject may assign. We are free to choose any value inside that interval. The value $ci$ is the subject complexity.
This problem asks us to reconstruct an array that was compressed using a simple run-length encoding format. The input array nums always contains an even number of elements.
This problem asks us to analyze participation counts for different activities and return only the activities whose parti
The problem gives us a non-empty array of digits that together represent a large integer. Each element in the array is a single decimal digit, and the digits are stored in left-to-right order from the most significant digit to the least significant digit.
The problem requires merging two input strings word1 and word2 in an alternating fashion, starting with the first character of word1. This means we take one character from word1, then one character from word2, and repeat this process until one or both strings are exhausted.
This problem asks us to find the smallest element in a sorted array that has been rotated some number of times. A sorted array in ascending order might originally look like this: After rotation, it could become: or remain unchanged: The important observation is that the array…
This problem asks us to count the number of arrays of length n where each element is between 1 and m inclusive, and the
The problem asks us to design a circular double-ended queue, also called a deque. A deque is a data structure that allows insertion and deletion from both the front and the rear.
The problem asks us to find the rectangular submatrix whose sum is as large as possible while still being less than or equal to a given integer k. A rectangle in a matrix is any contiguous block of cells formed by choosing a range of rows and a range of columns.
This problem asks whether we can transform one digit string, s, into another digit string, t, using a special operation. In one operation, we choose any non-empty contiguous substring of s and sort that substring in ascending order.
The problem gives two strings, s and p, both containing only lowercase English letters. We need to find every starting index in s where a substring is an anagram of p.
The problem gives us a string s and an integer k. We must rearrange the characters in the string so that identical characters are separated by at least k positions.
This problem asks us to coordinate multiple concurrent threads so that they form water molecules correctly. A water molecule contains exactly two hydrogen atoms and one oxygen atom, so the synchronization logic must ensure that threads proceed only in groups of three…
This problem asks us to find all shortest transformation sequences from beginWord to endWord, where each transformation changes exactly one character and every intermediate word must exist in wordList. We can think of the problem as navigating through a graph.
The problem gives us an even integer n and defines an initial permutation: This means the array initially looks like: We repeatedly apply a transformation rule to build a new array arr: - If the index i is even: - If the index i is odd: After constructing arr, we replace perm…
The problem asks us to compute the probability that a chess knight remains on an n x n board after making exactly k moves. A knight starts at position (row, column).
This problem gives us two Quad-Trees, where each tree represents a binary matrix containing only 0s and 1s. Our goal is to compute the logical bitwise OR of the two matrices and return the result as another Quad-Tree. A Quad-Tree is a recursive spatial data structure.
The problem gives us a two dimensional grid representing a server center. Each cell contains either: - 1, meaning there
We have n + 1 cities. The first n cities lie on the x-axis at positions (x1, 0), (x2, 0), ..., (xn, 0). One additional city is somewhere off the axis at (x_{n+1}, y_{n+1}).
The problem describes a frog trying to cross a river by jumping across stones placed at specific positions. The input array stones contains the positions of all stones in sorted ascending order. The frog starts on the first stone, which is always at position 0.
The problem asks us to count all possible non-empty sequences that can be formed from a set of letter tiles, where each tile has a single uppercase letter. The input is a string tiles representing the letters available on the tiles.
This problem asks us to coordinate three independent threads so that they execute in a strict order, regardless of how the operating system schedules them. We are given a class with three methods, first(), second(), and third().
The problem gives us the numbers from 1 to n, and asks us to find the kth permutation when all possible permutations are ordered lexicographically.
We need to compute the remainder when $a^b$ is divided by $c$. The input gives three integers, one per line. The first value is the base, the second is the exponent, and the third is the modulus. The direct interpretation is straightforward.
The problem gives us the head of a singly linked list and asks us to swap every pair of adjacent nodes. The important detail is that we are not allowed to modify the values stored inside the nodes. We must physically rearrange the node connections by changing pointers.
We have a burglar who can carry exactly n matchboxes. In the warehouse, there are m containers. Each container i has a_i matchboxes, and every matchbox in that container contains b_i matches.
The problem asks us to cover a sporting event that lasts time seconds using a set of video clips. Each clip is defined by its start and end times, [starti, endi], and clips can overlap or extend beyond each other.
We are asked to tile a rectangular floor of size n × m using three types of parquet planks. The first type is a 1×2 horizontal plank, the second is a 2×1 vertical plank, and the third is a 2×2 square plank.
We are playing the classic Bulls and Cows game with four-digit numbers whose digits are all distinct. Leading zeroes are allowed, so 0123 is valid, but repeated digits such as 0012 or 1223 are not. Each previous guess comes with two values.
The problem requires us to manipulate a string s in a structured way. Specifically, we need to reverse the first k characters for every consecutive block of 2k characters in the string. If a block has fewer than k characters, we reverse all of them.
The Spending table records purchases made by users on an e-commerce platform. Every row represents one user's spending on a specific date using either the mobile platform or the desktop platform.
This problem asks us to determine which children could potentially have the greatest number of candies if we distribute a fixed number of extra candies to any single child.
The problem gives us a rectangular grid called heights, where each cell represents the elevation of a piece of land. Water can flow from one cell to another if the neighboring cell has a height less than or equal to the current cell.
The problem gives us a list of intervals that are already sorted by starting value and guaranteed to be non-overlapping. Each interval represents a continuous range, written as [start, end].
We are asked to construct a path for an ant on a tree. The tree has n vertices, with vertex 1 as the root, and n-1 edges connecting them so the graph is connected.
Here is a complete, detailed technical solution guide for LeetCode 1382 - Balance a Binary Search Tree, following all yo
The problem asks us to design a dictionary-like data structure that supports two operations. First, we load a collection of distinct words into the structure.
The problem asks us to remove vowels from a given string s. Specifically, the vowels are defined as the lowercase characters 'a', 'e', 'i', 'o', and 'u'. The input string consists only of lowercase English letters, and its length ranges from 1 to 1000.
The problem asks us to perform a level order traversal on a binary tree. A binary tree is a hierarchical structure where each node can have at most two children, commonly referred to as the left child and the right child.
This problem asks us to select a subsequence of strings from the input array arr and concatenate them together such that the resulting string contains only unique characters. Among all valid concatenations, we must return the maximum possible length.
The problem asks us to determine whether a given list of strings forms a valid word square. A word square is a special arrangement of words such that the kth row and the kth column contain the same sequence of letters for every valid index k.
The problem requires reversing the characters of each individual word in a string while maintaining the original order of the words and the spacing between them. In other words, the sentence structure remains the same, but each word is mirrored in place.
We have a rectangular grid of trees. A fire starts simultaneously from several cells, and every minute it spreads to neighboring cells that share a side. The task is to find any cell whose burning time is as large as possible, meaning it catches fire later than every other tree.
We are given a binary grid. A valid object is not a filled square, it is only the border of a square drawn with 1s. Every cell not belonging to the border must be 0.
In this problem, we are given a collection of cards where each card has two numbers, one on the front and one on the back. Initially, every card is placed with its front side facing upward.
We have a fixed sequence of trucks. For every truck we know its value, the number of people inside it, and two fear constraints.
The problem gives an array of unique integers called preorder. This array is supposed to represent the preorder traversal of a binary search tree, and we must determine whether such a BST could actually exist. In a preorder traversal, nodes are visited in this order: 1.
The problem asks us to design two operations for a Binary Search Tree, abbreviated as BST: - serialize(root) converts the BST into a string representation. - deserialize(data) reconstructs the exact same BST from that string.
The problem gives us an integer array nums and asks us to find the index of any peak element. A peak element is defined as an element that is strictly greater than its immediate neighbors.
We are given a lowercase string and must find the substring that appears the largest number of times inside it. Occurrences may overlap. Among all substrings with the same maximum frequency, we choose the longest one.
This problem asks us to query a database table named Users and return only the rows corresponding to users whose e-mail
This problem gives us a two dimensional matrix with two very important ordering properties: 1. Each row is sorted from left to right in non-decreasing order. 2. The first element of every row is greater than the last element of the previous row.
The problem gives us a fence with n posts and k available colors. Every fence post must be painted using exactly one of those colors. The important restriction is that we are not allowed to have three or more consecutive posts painted with the same color.
The problem asks us to find the maximum possible score of a path from the top-left corner (0, 0) to the bottom-right corner (m-1, n-1) of a 2D grid. Each cell in the grid contains an integer value, and the score of a path is defined as the minimum value along that path.
The problem asks us to reverse a string that is represented as an array of characters. Instead of returning a new reversed string, we must modify the original array directly.
The problem gives us two integer arrays, nums1 and nums2. We are allowed to draw lines between matching values in the two arrays, but with an important restriction: the lines cannot cross each other. A line can only connect nums1[i] with nums2[j] when the values are equal.
This problem gives us a two dimensional integer array called accounts. Each row represents a customer, and each column represents one of that customer's bank accounts. The value accounts[i][j] represents how much money the i-th customer has in the j-th bank account.
This problem asks us to determine how many structurally different Binary Search Trees, or BSTs, can be formed using the integers from 1 to n. A Binary Search Tree has an important property: - Every value in the left subtree is smaller than the root.
This problem asks us to simulate how water droplets behave when poured onto a one-dimensional terrain. The terrain is represented by the array heights, where each element describes the height of a column at that position.
We are given a multiset of numbers and asked whether it is possible to arrange them in a circle so that every pair of adjacent numbers differs by exactly one. Conceptually, this means each number is a vertex on a cycle, and the absolute difference between neighbors must be 1.
This problem gives us an array of unique strings called dict, where every string has the same length. We must determine whether there exists at least one pair of strings that differ by exactly one character at the same position.
This problem asks us to maximize a score obtained by performing a sequence of exactly m operations on an array nums of length n using multipliers from another array multipliers of length m.
Please provide the full problem statement or LeetCode number you want the detailed solution guide for. You mentioned Lee
In this problem, we are given a list of directed paths between cities. Each entry in paths has the form [cityA, cityB],
We are given a string consisting of 0, 1, and ?. Each character represents a card in a row. During the game, players alternately remove one card until only two cards remain.
We are asked to maximize the amount of money Vasya can have at the end of _n_ days if he starts with a given sum of bourles and can buy and later sell Martian dollars.
We are asked to process a sequence of queries, each defined by two integers, x and y. For each query, we need to count how many positive divisors of x do not divide any of the y numbers immediately preceding x in the sequence, that is, the numbers x-y, x-y+1, ..., x-1.
The problem asks us to find the length of the longest contiguous subarray whose product of elements is strictly positive. We are given an integer array nums, which may contain positive numbers, negative numbers, and zeros.
The problem gives an integer array nums, and for every index i, we must compute the product of every element in the array except nums[i].
We are given a string and we consider every palindromic substring inside it. Each occurrence matters separately, even if two substrings have the same text. For example, in "aaa" there are three different occurrences of "a".
We need to construct n distinct positive integers with two simultaneous properties. First, every pair of numbers must share a common divisor larger than 1. In other words, for every pair (ai, aj), their gcd cannot equal 1. Second, the gcd of the entire set must equal 1.
The problem asks us to determine the winner in each group of players based on their accumulated points across multiple matches. We are given two tables: Players and Matches.
The problem describes a car traveling from position 0 to a destination located target miles away. The car starts with startFuel liters of fuel, and every mile driven consumes exactly one liter of fuel.
The problem gives a sequence of points on a 2D coordinate plane. Each point is represented as [x, y], where x and y are
We have a ticket seller, Charlie, who sells race tickets costing 10 euros each. Customers arrive in some random order: some have only 10 euro banknotes, some have only 20 euro banknotes. Charlie initially has k 10 euro banknotes.
We have a square box of size 2^n × 2^n. Inside this box, we repeatedly place special triangular cookies. A cookie of size k occupies the upper triangular part of a k × k square, including the main diagonal.
The problem asks us to simulate one iteration of Conway's Game of Life on a two-dimensional grid. Each cell in the grid is either alive, represented by 1, or dead, represented by 0. Every cell changes state simultaneously according to the number of live neighbors surrounding it.
Each historical event is represented by a time interval [a, b], where a is the starting year and b is the ending year. One event is considered contained inside another if it starts later and ends earlier.
This problem asks us to analyze a database table named Submissions and determine how many unique comments belong to each post.
The problem asks us to compute the complement of a positive integer by flipping every bit in its binary representation. A binary complement means changing every 1 bit into 0, and every 0 bit into 1. For example, the integer 5 is represented in binary as 101.
The problem asks us to sort an integer array arr based on the number of 1 bits in the binary representation of each elem
In this problem, we are given two database tables, Employee and Bonus, and we need to produce a result table containing employees who meet one of two conditions: 1. Their bonus is less than 1000. 2. They do not have a bonus record at all.
The problem asks us to reverse the order of words in a character array s in-place. Each word is a contiguous sequence of non-space characters, and words are separated by exactly one space.
We are asked to analyze a two-player game on an rectangular board. The players take turns moving a single chess piece starting in the top-left corner. On each turn, a player can move the piece one cell right, one cell down, or diagonally cells down-right.
We are given the results of an entire Formula One season. Each race lists drivers from first place to last place. The championship winner depends on one of two ranking systems.
We are given a rectangular grid representing a country’s flag. Each cell in the grid has a color, encoded as a digit from 0 to 9. The dimensions of the flag are rows by columns. The goal is to determine whether the flag is “striped” according to the new ISO standard.
The problem gives us the head of a singly linked list whose values are already sorted in ascending order. We must convert this linked list into a height balanced Binary Search Tree, usually abbreviated as BST.
The problem asks us to determine whether a given string s can be broken into a sequence of valid words from a given dictionary wordDict. Essentially, we need to check if we can insert spaces into s such that every substring separated by these spaces exists in the dictionary.
The problem asks us to design a miniature spreadsheet system similar to Microsoft Excel. The spreadsheet contains cells arranged in rows and columns, where rows are numbered starting from 1 and columns are labeled using uppercase letters such as A, B, C, and so on.
This problem gives us the root of a Binary Search Tree (BST) and an integer k. We must determine whether there are two distinct nodes in the tree whose values add up to k.
This problem asks us to find the in-order successor of a given node inside a Binary Search Tree, abbreviated as BST. Unlike the classic version of the problem, we are not given access to the root of the tree.
We are given a collection of toy spiders. Each spider is a tree, represented by beads (nodes) connected with strings (edges). A spider with $k$ beads has $k-1$ strings connecting its beads so that all beads are connected and there are no cycles.
The problem gives us two arrays: - rowSum, where rowSum[i] represents the total sum required for row i - colSum, where c
The problem gives an integer array nums and asks us to select exactly three numbers whose product is as large as possible. We must return that maximum product value.
This problem asks us to maximize the profit from stock trading under a strict constraint: we can perform at most k transactions, where each transaction consists of buying and then later selling one share of the stock.
The problem asks us to rearrange an integer array so that it follows a specific alternating pattern called a wiggle sequence. The required relationship is: - nums[0] <= nums[1] - nums[1] = nums[2] - nums[2] <= nums[3] - nums[3] = nums[4] - and so on.
The problem is asking us to compute the total number of seats reserved for each flight given a list of flight bookings. Each booking specifies a range of consecutive flights [firsti, lasti] and the number of seats seatsi reserved for each flight in that range.
This problem provides a database table named Employee, where every row represents a single employee and the team they belong to.
This problem asks us to implement a concurrent synchronization mechanism for the classic Dining Philosophers problem. We have five philosophers sitting around a circular table, and between every pair of neighboring philosophers lies a single fork.
The problem asks us to round an array of decimal prices to integers such that the sum of the rounded numbers equals a given target, while minimizing the total rounding error. Each price can be rounded either down (Floor) or up (Ceil).
The problem describes a tournament where teams compete until only one winner remains. At every round, teams are paired t
We are given a small group of people and a list of pairs who cannot work together. We need to choose the largest possible subset such that every pair inside the chosen group is compatible. This is naturally a graph problem. Think of each volunteer as a vertex.
The problem asks whether it is possible to transform a given integer array into a non-decreasing array by modifying at most one element. A non-decreasing array means that every element is less than or equal to the element that comes after it.
We are given a rectangular grid where each cell is either empty or marked with . Every cell represents a point located at the center of that cell. We need to count how many right triangles can be formed such that:
We have several rectangles on the plane. Each rectangle represents a photo hanging on a wall. The rectangles may overlap, may share edges, may coincide completely, and may also be rotated arbitrarily.
We are given a sequence of integers and need to compute their total sum. The input starts with an integer n, which tells us how many numbers follow. Each of the next n lines contains one element of the array. The task is simply to add all of them together and print the result.
The problem provides a tree rooted at node 0, represented in two parallel arrays: parent and value. The parent[i] array
We start with a row of n panels, all turned OFF. The target password is another configuration where exactly k specific positions must be ON and every other position must remain OFF. One operation chooses a segment of consecutive panels whose length belongs to the array a.
This problem is asking whether a string containing the characters '(', ')', and '' can be considered a valid parenthesis string. In essence, '(' must be matched by a ')' in the correct order, and '' can act as a flexible placeholder that behaves as '(', ')', or an empty string.
The problem asks us to determine the maximum number of candies we can collect from a set of boxes with varying accessibi
This problem asks us to repeatedly apply character shifts to a string of lowercase English letters. We are given two inputs: - A string s, consisting only of lowercase English letters.
We are given a simple polygon listed in clockwise order. The first two vertices form a horizontal edge AB, and every other vertex lies strictly on the same side of that edge. Along the segment AB, every integer-coordinate point is a possible location for a watchtower.
The problem gives us a string representation of a nested integer structure and asks us to reconstruct the corresponding NestedInteger object hierarchy. A NestedInteger can represent one of two things: 1. A single integer value 2.
The problem describes a street as a number line, and each building occupies a half-closed interval [start, end) with a fixed height.
This problem is asking us to calculate the probability that, when tossing a set of coins, exactly a specific number of them come up heads. Each coin has its own individual probability of landing heads, which is given in the input array prob.
We are given a sequence of integers, and the task is to maximize its sum by performing two operations: first, we may choose any prefix of the sequence and multiply every element in it by -1; second, we may choose any suffix of the sequence and multiply every element in it by -1.
We are asked to find the closest prime numbers surrounding a given integer n. Concretely, we need two numbers, a and b, such that a ≤ n ≤ b and both a and b are prime. Among all prime intervals containing n, the one with the smallest length b - a is preferred.
This problem asks us to calculate the total quantity sold for each product based on records in the Sales table. The Sales table contains transaction level information.
The problem asks us to find the closest leaf node to a given target node k in a binary tree. The input is the root of a binary tree where each node has a unique integer value. A leaf node is defined as any node without children.
The problem asks us to split a given integer n into two positive integers a and b such that: - a + b = n - Neither a nor b contains the digit 0 anywhere in their decimal representation Such integers are called No-Zero integers.
The problem gives us a positive integer n and asks whether its digits can be rearranged to form a power of two. The rearrangement may keep the digits in their original order or place them in any other order, but the resulting number cannot contain a leading zero.
The problem asks us to compute minimal unique abbreviations for an array of distinct strings. An abbreviation replaces the middle characters of a word with a count, keeping the first and last characters.
The problem is asking us to filter and sort a list of restaurants based on multiple criteria. Each restaurant is represe
The problem gives us a single integer, area, which represents the area of a rectangle. Our task is to find two integers: - L, the length - W, the width such that: 1. L W == area 2. L = W 3.
We need to find the smallest number greater than or equal to a given integer such that: 1. Every digit is either 4 or 7. 2. The count of 4s equals the count of 7s. These numbers are called super lucky numbers. For example, 47 is valid because it contains one 4 and one 7.
The folders are displayed in a grid with exactly m columns per row. Folder 1 is in the top-left corner, folder 2 is next to it, and so on. After every m folders we move to the next row. A rectangular frame selection toggles every folder inside the rectangle.
The problem asks us to find the position on a straight street that is illuminated by the largest number of street lamps, which is referred to as its brightness.
This problem asks us to find the minimum value in a sorted array that has been rotated, while also allowing duplicate values. A rotated sorted array is created by taking an ascending sorted array and shifting some suffix of the array to the front.
The problem asks us to design an efficient cache that follows the Least Recently Used, or LRU, eviction policy. A cache stores a limited number of key-value pairs.
We can think of the party as an undirected friendship graph. Every person is a vertex, and an edge means two people are friends.
The problem gives two dates in the format YYYY-MM-DD and asks us to compute the absolute number of days between them. Each input string represents a valid calendar date. The year, month, and day are separated by hyphens.
The problem models a recursive elimination game played on a row of colored balls. The board is represented as a string where each character corresponds to a colored ball. The available colors are 'R', 'Y', 'B', 'G', and 'W'.
We are given a 2D grid of size _n_ by _m_, each cell either containing a star * or empty .. The task is to locate constellations shaped like a cross.
This problem gives us two database tables, Users and Rides. The Users table contains information about each user. Every user has a unique id and a corresponding name. The Rides table contains ride records.
The problem asks us to determine whether two nodes in a binary tree are cousins. Two nodes are considered cousins if they satisfy two conditions simultaneously: 1. They are located at the same depth in the tree. 2. They have different parents.
This is a long, structured technical guide with multiple substantial sections, worked examples, code, tables, and test c
The problem gives us two arrays of strings, word1 and word2. Each array represents a single larger string formed by conc
The graph has exactly n vertices and n edges. A connected graph with n vertices and n edges contains exactly one simple cycle. Every other edge belongs to a tree attached to that cycle. Initially every edge is turned off.
The problem asks us to design a cache that supports two operations: - get(key) returns the value associated with a key if it exists, otherwise -1 - put(key, value) inserts or updates a key-value pair Unlike a normal cache, eviction is not based only on recency.
This problem asks us to count how many distinct non-empty palindromic subsequences exist in a string. A subsequence is formed by deleting characters while preserving the relative order of the remaining characters. Unlike substrings, subsequences do not need to be contiguous.
Bob experiences one event per day. A day is either a prize day or a customer day.
This problem gives us two inputs: - arr, a one-dimensional array of distinct integers - pieces, a collection of smaller
The board is a fixed hexagon with 19 cells. Some cells contain chocolates, some are empty. Two players alternate moves, and a move consists of choosing a contiguous segment of chocolates that lies on a straight line parallel to one of the three hexagon directions.
This problem asks us to design an iterator that can traverse a deeply nested list structure as if it were a flat list of integers. The input is not a normal array.
The problem asks us to find the length of the longest sequence of consecutive integers in an unsorted array. A consecutive sequence means numbers that appear one after another numerically, regardless of their position in the array.
We have a collection of items, and every item belongs to exactly one of three equipment classes: weapon, armor, or orb. Each item has three base stats, attack, defense, and resistance, plus a capacity telling us how many residents it can hold. Residents also come in three types.
The problem gives us an initially empty m x n grid where every cell starts as water. We are then given a sequence of operations in positions, where each operation turns a specific cell from water into land.
The problem asks us to find the shortest superstring that contains all the given strings in the array words as substrings.
The problem asks us to find the largest connected component in a graph derived from an array of unique positive integers nums.
This problem asks us to maximize the length of a contiguous block of identical answers in a true/false exam answer key. The input string answerKey contains only two characters, 'T' and 'F', representing the answers for each question.
The problem asks us to count the number of ways to fill an array of size ni with positive integers such that the product of all elements equals ki. Each query in the input array queries is independent, meaning we compute the answer for each (ni, ki) pair separately.
This problem asks us to transform a Binary Search Tree, abbreviated as BST, into a "Greater Tree". In the transformed tree, every node's value should become: - its original value - plus the sum of all values greater than it in the original BST The structure of the tree does…
The problem asks us to design an iterator that traverses a two dimensional array as though it were a single flat sequence. Instead of returning nested lists, the iterator should expose elements one at a time through two methods, next() and hasNext().
This problem asks us to identify every (actorid, directorid) pair where an actor and a director have collaborated at least three times. The input is a database table named ActorDirector. Each row represents one collaboration event between an actor and a director.
That is a long, structured reference document and will exceed a single response comfortably. I can provide the complete
This problem gives us a database table named Point2D, where every row represents a unique point on a 2D Cartesian plane. Each point contains two integer coordinates, x and y. The pair (x, y) is guaranteed to be unique because it is the primary key.
The problem asks us to count how many numbers within the inclusive range [low, high] are strobogrammatic. A strobogrammatic number is a number that appears unchanged when rotated 180 degrees.
The problem asks us to perform a level order traversal of a binary tree, but with a twist. Instead of always traversing each level from left to right, we alternate the traversal direction at every level.
The problem gives us two strings, str1 and str2, which are guaranteed to have the same length. We want to determine whether it is possible to transform str1 into str2 using a sequence of character conversion operations.
Each leaf is described by two strings: the tree species and the leaf color. Alyona only keeps a leaf if she does not already have another leaf with the exact same pair of values. The task is simply to count how many distinct (species, color) combinations appear in the input.
This problem is asking us to assign a set of jobs, each with a specific time requirement, to k workers such that every job is assigned to exactly one worker, and we minimize the maximum total working time among all workers.
The problem asks us to determine if a sorted integer array nums can be divided into one or more disjoint increasing subsequences, where each subsequence has a length of at least k.
The Death Star is a sphere of radius R. Its center starts at point A and moves forever in a straight line with constant velocity vector v. Every mine consists of two kinds of geometry. The first part is a sphere centered at O with radius r. The second part is a set of spikes.
This problem asks us to simulate a sequence of string shift operations on a given string s. Each operation in the shift
The problem gives us three positive integers, a, b, and c. We are allowed to flip individual bits in either a or b. A flip means changing a bit from 0 to 1 or from 1 to 0. Our goal is to perform the minimum number of bit flips so that: The OR operation works bit by bit.
We are given two decimal integers, a and c. The computer in this problem does not use binary xor. Instead, it uses a ternary operation called tor. To apply tor, both numbers are written in base 3.
The problem asks us to minimize the deviation in an array of positive integers. The deviation is defined as the differen
The problem asks us to count all tuples (a, b, c, d) from a given array of distinct positive integers such that the product of the first two elements equals the product of the second two elements, a b = c d, and all elements are distinct.
We are given a sequence of n natural numbers, where n is at least 3 and at most 100, and each number is at most 100. Among these numbers, all except one share the same parity - either even or odd - and exactly one number differs.
This problem simulates a school cafeteria where students line up to take sandwiches from a stack. Each student has a preference for either a circular sandwich (0) or a square sandwich (1).
The problem gives the root node of a binary tree and asks us to invert the tree. Inverting a binary tree means swapping the left and right child of every node in the tree.
In this problem, we are given a circular pizza divided into 3n slices. Each slice has a size represented by the array sl
The problem describes a supermarket that sells products identified by unique integer IDs. Each product has a corresponding price. Customers purchase products in certain amounts, generating a bill that is the sum of the prices multiplied by the amounts purchased.
The problem gives us the results of some dice rolls and asks us to reconstruct the missing ones. We have a total of n + m rolls of a standard 6-sided die.
We are given a limited stock of T-shirts in five sizes: S, M, L, XL, and XXL. Each participant in the contest has a preferred size. Participants arrive in a fixed order and try to pick the T-shirt closest to their preferred size.
The problem provides a collection of employees where each employee contains three pieces of information: - A unique employee ID - An integer importance value - A list of IDs representing their direct subordinates We are also given the ID of one employee, and we must calculate…
The problem asks us to count how many pairs of indices (i, j) satisfy two conditions: 1. i < j 2. nums[i] 2 nums[j] These pairs are called reverse pairs.
The problem requires determining how many items can be stored in a warehouse with a limited square footage of 500,000. T
We have an array of cow weights. For every query (a, b), we repeatedly take positions a, a + b, a + 2b, ... until the index exceeds n, and we must output the sum of all visited values.
This problem asks whether a collection of smaller axis-aligned rectangles perfectly forms one larger rectangle, with no gaps and no overlaps.
In this problem, we are given an undirected graph where two players, Mouse and Cat, move alternately across the graph according to specific rules. The graph is represented as an adjacency list, where graph[a] contains all nodes directly connected to node a.
The problem asks us to split an array into two contiguous parts, left and right, such that every value in left is less than or equal to every value in right. Among all valid partitions, we must return the smallest possible size of left.
The problem presents a stream of points in a 2D plane and asks us to design a data structure that supports two operations efficiently: adding points and counting squares.
We have a directed acyclic graph on vertices 1...n. For every pair i < j, there is exactly one directed edge from i to j. Each pipe has three parameters. It must carry between l and h units of flow inclusive, even if we do not want to use that pipe.
The problem gives us an array that was originally sorted in ascending order, but may have been rotated at some pivot point. A rotation means that some prefix of the sorted array was moved to the end while preserving the relative order of elements.
We have a rectangular block made of unit cubes with dimensions x × y × z. A single cut is always made along the grid lines and must be parallel to one of the faces of the box.
Five people stand in a queue in a fixed order: Sheldon, Leonard, Penny, Rajesh, Howard. Whenever the person at the front buys a cola, that person immediately creates a copy of themselves, and both copies go to the back of the queue. The queue keeps growing forever.
We are given exactly four line segments on the 2D plane. Every segment is axis-aligned, or may even degenerate into a single point.
We are given a permutation A of numbers from 1 to n. For every value i, we know a number b[i]. This number describes how many elements appear before i in the permutation and are at least i + k. The condition is attached to the value itself, not to the position.
The problem gives us the head of a singly linked list and an integer k. We must divide the linked list into exactly k consecutive parts while preserving the original order of nodes. The important requirement is that the parts should be as evenly sized as possible.
The problem describes a mutation process between genetic sequences. Each gene is represented as a string of exactly 8 characters, and every character must be one of four possible nucleotides: 'A', 'C', 'G', or 'T'.
The problem describes a country, Berland, with n cities arranged in a perfect ring. Originally, each city had two-way roads connecting it to its two neighbors, so it was trivial to travel from any city to any other.
This problem asks us to traverse a matrix in a specific diagonal pattern and return all elements in the order they are visited. We are given an m x n matrix mat, where m represents the number of rows and n represents the number of columns.
The problem asks us to find a missing value from an array that originally formed an arithmetic progression (AP). An arithmetic progression is a sequence where the difference between consecutive terms is constant, i.e., arr[i + 1] - arr[i] is the same for every consecutive pair.
This problem asks us to maximize the total profit earned by assigning jobs to workers, under a specific rule: each worker can only perform jobs whose difficulty is less than or equal to their ability.
We are given a one-dimensional collider with n particles, each with a starting position x_i and a velocity v_i. Positive velocity means a particle moves right, negative velocity means it moves left.
The problem provides a special binary tree where each node either has zero or two children, and every internal node has a value equal to the smaller value among its two children.
The problem is asking us to maximize the sum of an integer array after performing exactly k negations. A negation operation consists of picking an index i and replacing nums[i] with -nums[i]. This operation can be applied to the same element multiple times.
We have a complete binary tree of height h. Every vertex may store some number of electrons, and queries gradually add more electrons to vertices. A decay operation chooses one leaf uniformly at random and deletes every edge on the path from the root to that leaf.
We are given a collection of sticks, where each stick has an integer length. A rectangular frame needs four sticks arranged as two equal pairs. If the frame is a square, then all four sticks must have the same length.
The problem requires computing the result of a mathematical expression containing fractions with addition and subtraction operators. The input is a single string, expression, which consists of positive or negative fractions in the format numerator/denominator.
This problem asks us to design a data structure that allows insertion into a complete binary tree while maintaining its completeness. A complete binary tree is one where all levels are fully filled, except possibly the last, which is filled from left to right.
We are asked to find the maximum number of "full lucky numbers" that can appear after adjusting segments of numbers. A lucky number is any positive integer whose digits consist only of 4 and 7. Each segment is a range [li, ri] on the number line.
The problem presents an undirected graph with n nodes and three types of edges: Type 1 for Alice, Type 2 for Bob, and Ty
The problem gives us a binary matrix called grid, where: - 0 represents water - 1 represents land We can move only in four directions, up, down, left, and right.
This problem gives us the root node of a Binary Search Tree, abbreviated as BST, and asks us to find the minimum absolute difference between the values of any two distinct nodes in the tree.
The problem gives us a sorted integer array arr, along with two integers, k and x. We need to return exactly k elements from the array that are closest to x.
We are given a house with a certain number of rooms connected by doors, and each door has a unique key. There are several residents in the house, each initially in some room with some keys. We also know the positions and key holdings of every resident at a later time.
The problem asks us to group strings based on a special equivalence property. Specifically, two strings are special-equivalent if you can swap characters at even indices among themselves and characters at odd indices among themselves any number of times to make the two strings…
In this problem, we are given two arrays: - boxes, where each value represents the height of a box - warehouse, where ea
This problem asks us to determine which words in a given list can be typed using letters from only one row of an American keyboard. The input is an array of strings, words, where each string represents a word consisting only of English alphabet characters.
The problem presents n gardens, labeled from 1 to n, and a list of bidirectional paths connecting pairs of gardens. Each garden must be planted with one of four types of flowers (1 through 4).
The problem asks us to find the number of triplets (i, j, k) in an array arr such that the XOR of elements from index i
The problem requires us to reorder a string s by repeatedly picking characters in a specific increasing and decreasing p
The problem is asking us to determine the number of provinces in a network of cities. Each city can be connected directly to other cities, and indirectly through chains of connections.
The problem gives us several colors of balls, where balls[i] represents how many balls exist for color i. The total numb
This problem asks us to create a completely independent copy of a linked list where each node contains two pointers: - next, which points to the next node in the list - random, which can point to any node in the list or null The key requirement is that the copied list must be…
The problem asks us to find all the integers in the range [1, n] that are missing from an input array nums of length n.
This problem gives us the root of a binary tree, a specific target node inside that tree, and an integer k. Our task is to return all node values whose distance from the target node is exactly k. The important detail is the definition of distance.
The problem gives us an alphanumeric string s that contains only lowercase English letters and digits. Our task is to re
The problem gives us a collection of distinct points on a 2D plane. Each point is represented as [x, y], where x is the horizontal coordinate and y is the vertical coordinate.
This problem asks us to identify suspicious accounts in the LogInfo table. Each row represents a login session for a user account, including the account ID, the IP address used during the session, and the login and logout timestamps.
The problem gives us an integer array arr and asks whether every distinct number appears a unique number of times. In other words, we first count how many times each value occurs in the array.
The problem gives us a numeric string num and an integer target. We must insert binary operators, specifically '+', '-', and '', between the digits of the string so that the resulting mathematical expression evaluates exactly to target.
The problem asks us to populate the next pointer for every node in a binary tree so that it points to the node immediately to its right on the same level. If no such node exists, the next pointer should remain NULL.
The problem asks us to find the shortest path from a starting location to any food cell in a 2D grid. The grid consists of four types of cells: '' indicating your starting position, '' representing food, 'O' as free space you can move through, and 'X' as obstacles you cannot…
The problem asks whether it is possible to construct a given target array from an initial array arr of the same length,
The problem gives us an array called changed, which was supposedly created from another array called original. The transformation process works like this: 1. Take every number in original. 2. Append its doubled value, meaning 2 x. 3. Shuffle all the numbers together.
The problem gives us a row of houses, where each house must be painted using exactly one of three colors: red, blue, or green. The input is provided as a two-dimensional array named costs, where costs[i][j] represents the cost of painting the i-th house with the j-th color.
We have 3n students ranked by personal performance. Higher-ranked students become captains earlier. When a captain forms a team, they choose two currently unassigned students according to their personal preference list.
Each student points to exactly one other student, the person they call whenever they hear news. This creates a directed functional graph, every node has out-degree exactly one.
The problem gives us two non-empty singly linked lists where each node stores a single digit of a non-negative integer. Unlike the classic "Add Two Numbers" problem, the digits are stored in forward order, meaning the most significant digit appears first.
This problem asks us to determine whether two sentences are considered similar based on a set of similarity relationships between words. Each sentence is represented as an array of strings, where each string is a single word.
We are asked to place a space rescue station in three-dimensional space such that the maximum distance from it to any of the given planets is minimized. Each planet is represented by its coordinates $(x, y, z)$.
This problem provides two database tables, Project and Employee. The Project table represents which employees are assigned to which projects. Each row contains a projectid and an employeeid.
We are given a list of integers of length _N_, which may contain negative numbers, zeros, or large positive numbers. The task is to transform this list into a non-decreasing sequence, where each element is at least as large as the previous one.
Edit This problem gives us a weighted, undirected graph where each node represents a city and each edge represents a bid
This problem models a matching process between workers and bikes on a two dimensional grid. Each worker and each bike has a unique coordinate, and every worker must eventually receive exactly one bike.
The problem gives us an integer array nums and a target value val. Our task is to remove every occurrence of val from the array, but we must do it in-place. This means we are not supposed to create another array and return it. Instead, we modify the original array directly.
We are asked to move a chess king from one square to another on a standard 8×8 board in the fewest number of moves. The king can move to any adjacent square in eight possible directions: vertically, horizontally, or diagonally.
Petr is standing somewhere in a line containing n people. Positions are numbered from 1 at the front to n at the back. He knows two things about his position. At least a people are standing in front of him, and at most b people are standing behind him.
We are asked to count all distinct real roots of quadratic equations of the form , where ranges from 1 to and ranges from 1 to . Each pair defines one quadratic. The output is the total number of distinct real roots across all these quadratics.
The problem asks whether the characters of a given string can be rearranged to form a palindrome. A palindrome is a string that reads the same forward and backward. Examples include "racecar", "abba", and "a".
The problem provides an array of pairs, where each pair [xi, yi] indicates that xi is either an ancestor of yi or yi is an ancestor of xi in a rooted tree. The key challenge is to determine how many different rooted trees satisfy all given pairs.
The problem asks us to distribute m balls into n baskets located at given positions along a line in such a way that the minimum distance between any two balls is maximized. The magnetic force between two balls is defined as the absolute difference of their positions.
The problem asks us to generate every possible permutation of the given array nums, while ensuring that duplicate permutations are not included in the final result. A permutation is an arrangement of all elements in a particular order.
We have a square board of size 2n × 2n, divided into unit cells. One cell is marked. We want to know whether it is possible to draw a cutting path along grid lines so that the board is split into two congruent parts after rotation, while the cutting path never touches the…
This problem asks us to compute the number of distinct active users for each day within a fixed 30 day window. The window ends on 2019-07-27, inclusive, which means we only consider activity dates from 2019-06-28 through 2019-07-27.
This problem asks us to count how many distinct subsequences of a string s are equal to another string t. A subsequence is formed by deleting zero or more characters from a string without changing the relative order of the remaining characters.
The problem asks us to group together all strings that are anagrams of each other. Two strings are considered anagrams if they contain exactly the same characters with the same frequencies, but possibly in a different order.
This problem asks us to find the k-th smallest distance among every possible pair of numbers in the array. For any pair (a, b), the distance is defined as: We are given an array nums, and we must consider every pair (nums[i], nums[j]) where i < j.
The problem gives a list of time points in 24-hour clock format, where every time is represented as a string in the form "HH:MM". Your task is to determine the smallest difference in minutes between any two time points in the list.
The problem asks us to generate all simplified fractions between 0 and 1 (exclusive) where the denominator does not exce
The problem asks us to compute the number of ways to distribute n distinct candies into k bags such that every bag has a
We have two independent lists, one containing names and one containing surnames. Every name must be paired with exactly one surname, and every surname must be used exactly once. After choosing the matching, we print all pairs in one comma-separated line.
We are given an interval of consecutive integers starting at a and having length l. For every number x, define F(x) as the count of lucky digits inside its decimal representation. Only digits 4 and 7 are considered lucky.
We are counting sequences of heights that describe a polyline. The x-coordinates are fixed as 1, 2, ..., n, so the whole shape is determined only by the sequence y1, y2, ..., yn.
This problem gives us access to a binary matrix where every row is sorted in non-decreasing order. That means every row
The problem asks us to find the diameter of a tree, which is defined as the number of edges in the longest path between any two nodes. We are given an undirected tree represented as a list of edges. Each edge connects two nodes, labeled from 0 to n - 1.
The problem asks us to design a data structure that processes a stream of characters one at a time. After each newly added character, we must determine whether any suffix of the stream matches one of the words from a predefined dictionary.
We have a line of students, and for every adjacent pair we know only the relative order of their marks. If the relation character between positions i and i + 1 is: - L, then student i must receive strictly more toffees than student i + 1 - R, then student i + 1 must receive…
We are given a string of lowercase letters and an integer k. The task is to generate all possible substrings of the string, sort them lexicographically, and return the k-th substring in that order.
The problem is asking us to find the next greater element for each element in nums1 within another array nums2. Formally, for each element in nums1, we need to locate its position in nums2 and then find the first element to its right in nums2 that is greater than itself.
This problem asks us to calculate the percentage of users who attended each contest. We are given two tables: Users and
The problem asks us to find the k-th missing number in a sorted, strictly increasing array of unique integers. Given nums and an integer k, we need to identify the number that is missing from the sequence formed by consecutive integers starting from nums[0].
The problem is asking us to completely cover a rectangle of size n x m with the fewest number of squares that have integer side lengths. The rectangle can be tiled with squares of any size, as long as each square fits entirely within the rectangle and there is no overlap.
We are asked to construct a fully connected graph of _N_ tourist attractions, where each road has a distinct positive integer cost not exceeding 1000.
This problem asks us to identify all customers who have purchased every product listed in the Product table. In other words, a customer is eligible for the output if, for each productkey in the Product table, there exists a corresponding row in the Customer table where that…
The problem provides a singly linked list and an integer k. The task is to swap the values of the kth node from the start with the kth node from the end of the list. The list is 1-indexed, meaning that the first node is position 1, not 0.
We are given a circular table divided into n sectors, each containing a quiz question. Some questions have already been asked, marked with a 0, and others are still available, marked with a 1.
The Stocks table records stock trading activity. Each row represents either a Buy or Sell operation for a specific stock
This problem asks us to compute the boundary traversal of a binary tree in a very specific order. The boundary is formed by combining four parts: 1. The root node 2. The left boundary, excluding leaves 3. All leaf nodes from left to right 4.
The problem gives us two axis-aligned rectangles, meaning their sides are parallel to the X-axis and Y-axis. Each rectangle is represented as: Where: - (x1, y1) is the bottom-left corner - (x2, y2) is the top-right corner The task is to determine whether these two rectangles…
The problem asks us to count how many permutations of the numbers 1 through n are valid derangements. A derangement is a permutation where no element remains in its original position. For example, when n = 3, the original array is [1, 2, 3].
This problem asks us to build a product suggestion system similar to what appears in e commerce search bars. We are give
The problem asks us to design a calendar system that supports booking time intervals while enforcing one important rule: no point in time may be covered by three events simultaneously. Each event is represented as a half open interval [startTime, endTime).
The problem asks us to compute the n-th number in the Tribonacci sequence. The Tribonacci sequence is very similar to the Fibonacci sequence, except that instead of summing the previous two numbers, each value is formed by summing the previous three numbers.
The problem requires finding the kth smallest sum obtainable by selecting exactly one element from each row of a matrix mat with sorted rows. Each row is sorted in non-decreasing order, and we must explore combinations of elements across rows to form sums.
We are asked to count integers $x$ in a given range $[a, b]$ that satisfy a certain remainder-based property. Petya has four distinct integers $p1, p2, p3, p4$.
The problem gives us a database table named UserActivity that stores activity periods for different users. Each row cont
The problem asks us to process a singly-linked list and, for each node, determine the value of the next node that has a strictly larger value.
The problem gives us a special type of doubly linked list where every node contains four fields: The next and prev pointers behave exactly like a normal doubly linked list. The extra child pointer introduces another linked list that branches downward from the current node.
The problem is asking us to design a log storage system that supports two primary operations: storing logs with unique IDs and timestamps, and retrieving logs based on a timestamp range and granularity.
This problem models a computer network as an undirected graph. Each computer is a node, and each ethernet cable is an ed
We are given a single lowercase string and need to find the longest string that satisfies three conditions at the same time. The chosen string must be a prefix of the original string, a suffix of the original string, and also appear somewhere strictly inside the string.
The problem asks us to design a data structure that tracks how many events, called "hits", occurred during the last 5 minutes. Every hit comes with a timestamp measured in seconds, and timestamps are guaranteed to arrive in chronological order.
We are given the final standings of a table bowling tournament. Every participant has a unique name and an integer score. The task is not just to sort the players, but also to assign ranking labels in the style used in real tournaments.
This problem asks us to modify a binary tree by removing every subtree that does not contain at least one node with value 1. A subtree consists of a node and all of its descendants. If an entire subtree contains only 0 values, then that subtree should be deleted from the tree.
We have an n × m grid. Some entire rows are empty, some entire columns are empty, and every empty cell belongs to at least one of those empty rows or empty columns. All remaining cells contain oil.
We have an 8 × 8 board. Maria starts in the bottom-left corner, Anna stays permanently in the top-right corner, and several statues occupy other cells. The game proceeds in rounds. Maria moves first, then every statue moves one row downward simultaneously.
The problem asks us to count how many palindromic substrings exist inside a given string s. A substring is any contiguous segment of the string. This means we cannot rearrange characters or skip positions.
The problem gives us a string s and a list of index pairs called pairs. Each pair [a, b] means we are allowed to swap the characters at positions a and b. The important detail is that swaps can be performed any number of times. This changes the nature of the problem completely.
This problem asks us to sort n items while satisfying two different kinds of constraints at the same time. The first constraint comes from dependencies between items. If item a appears in beforeItems[b], then item a must appear before item b in the final ordering.
This problem asks us to merge user accounts based on shared email addresses. Each account is represented as a list of strings. The first string is the user's name, and every remaining string is an email address associated with that account.
The problem is asking us to count all pairs of dominoes in a list that are equivalent, where two dominoes [a, b] and [c, d] are considered equivalent if one is a rotation of the other, meaning either (a == c and b == d) or (a == d and b == c).
That is a long, comprehensive reference document with multiple sections and full code in two languages. To keep quality
This problem gives us the root of a Binary Search Tree, abbreviated as BST, along with an integer k. We need to return the kth smallest value in the tree. A Binary Search Tree has a very important property: - Every value in the left subtree is smaller than the current node.
This problem asks us to compute a researcher's h-index from a sorted list of citation counts. The input array citations is sorted in non-decreasing order, meaning the citation counts appear from smallest to largest.
The Employees table stores information about employees inside a company. Each row represents one employee and contains four columns: | Column | Meaning | | --- | --- | | employeeid | Unique identifier for the employee | | name | Employee name | | reportsto | The manager this…
We are asked to calculate the probability that Dot wins a simple dice game against Yakko and Wakko. Each character rolls a standard six-sided die. Dot wins if her roll is at least as large as the maximum of Yakko’s and Wakko’s rolls.
In this problem, we are given a binary array called seats. Each position in the array represents a seat in a row: - 1 means the seat is occupied. - 0 means the seat is empty.
We are given three positive integers representing the prices of three presents. There are three sisters, ranked by age: eldest, middle, and youngest.
The problem asks us to rearrange an array so that all even numbers appear before all odd numbers. The relative ordering among even numbers does not matter, and the relative ordering among odd numbers also does not matter.
The problem asks us to find the lexicographically largest permutation that is still smaller than the given array, using exactly one swap operation.
The picture in the statement describes a recursive triangular arrangement of paths and blocked regions. The black segments form a planar graph, and the gray triangles represent forbidden forest areas.
The problem gives us a sorted array of unique integers and asks us to summarize consecutive values into compact range strings. A range represents a continuous sequence of integers.
The problem requires creating a masked version of a personal information string s that can either be an email address or a phone number. The goal is to obscure sensitive information while keeping enough data to identify the user minimally.
The problem gives a non-negative integer c and asks whether it can be represented as the sum of the squares of two integers.
The problem describes a one-dimensional garden that stretches from position 0 to position n. At every integer position i, there is a tap that may water some interval around it.
The problem is asking for the minimum number of steps required to reach the last index of an integer array, starting fro
We have a set of horizontal blind stripes of varying lengths, and our goal is to construct a rectangular blind for a window using these stripes. Each stripe can be cut into smaller pieces, but pieces cannot be shorter than a given minimum length, l.
The problem asks us to select exactly three non-overlapping subarrays from the input array nums, where each subarray has length k. Among all valid choices, we must maximize the total sum of all elements contained in those three subarrays.
The input is a tiny arithmetic expression written as a three-character string. The first and third characters are digits from 0 to 9, and the middle character is either + or -. The task is to evaluate the expression and print the resulting integer.
We are given an integer array and must classify it into one of three categories.
We are asked to construct a convex polygon with vertices that satisfies three conditions. First, all vertices must lie on lattice points, meaning each coordinate is an integer. Second, all sides must have distinct lengths.
This problem asks us to determine which key on a keypad was pressed for the longest duration during a test sequence. We
We have an n × n grid of non-negative integers. Starting from the top-left corner, we may move only right or down until we reach the bottom-right corner. Along a chosen path, we multiply every visited value together.
We are given six words and must arrange them into a very specific crossword shape. The shape looks like a rectangular infinity symbol. There are three horizontal words and three vertical words, and they intersect at fixed positions.
In this game, we are given an array stoneValue where each element represents the value of a stone. The stones are arranged in a row, and Alice repeatedly splits the current row into two non-empty contiguous parts.
We are given the final text printed by INTERCAL's strange "Turing Tape" output procedure. Each printed character was produced from one integer of an unknown array. The encoding process depends on the previous printed character, so every step is linked to the one before it.
We are given the trump suit for a game of Durak and two cards. The task is to decide whether the first card can beat the second card under the game rules. Each card has a rank and a suit. The ranks are ordered as: A card can beat another card in exactly two situations.
--- [LeetCode Problem 520](https://leetcode.com/problems/detect-capital/) Difficulty: 🟢 Easy Topics: String
We need to validate whether a string follows the exact syntax of a Jabber ID.
This problem asks us to generate the shortest possible abbreviation for a given target word such that the abbreviation cannot also represent any word in the dictionary. An abbreviation replaces one or more non-adjacent substrings with their lengths.
The problem describes a turn-based voting process between two parties in the Dota2 senate, the Radiant party represented by 'R' and the Dire party represented by 'D'. We are given a string senate where each character represents a senator and their party affiliation.
The problem asks us to generate all valid triplets of students representing a country from three schools: SchoolA, Schoo
We need to decide whether a given string can be interpreted as a valid domain name under a simplified set of rules. The string may only contain lowercase English letters, digits, and dots. Dots separate the string into segments.
The problem asks us to find a subsequence of a given integer array nums such that the sum of that subsequence is as close as possible to a given integer goal.
The problem presents a binary array nums, where each element is either 0 or 1. We are asked to compute the sequence of numbers formed by interpreting the subarray nums[0..i] as a binary number, denoted as xi.
The problem gives us an array called boxes, where each integer represents the color of a box. We may repeatedly remove groups of adjacent boxes that share the same color. If we remove a group containing k boxes, we earn k k points.
This problem asks us to determine whether one binary tree appears as an exact subtree inside another binary tree.
We are asked to analyze a robot's path on an infinite 2D grid. The robot can move up, down, left, or right, and its moves are recorded as a string of the characters U, D, L, R.
This problem asks us to compute the nth Fibonacci number. The Fibonacci sequence is defined recursively. The first two numbers are fixed: Every value after that is calculated as the sum of the previous two values: Given an integer n, we must return the value of F(n).
It looks like you want the full detailed solution guide, but the specific LeetCode problem number and statement are missing from your latest request template. You included the formatting instructions and structure, but not which problem to solve beyond prior context.
The problem asks us to convert a number from one base to another, with the twist that the target base may be either a standard positional numeral system (2 through 25) or the Roman numeral system.
The problem asks us to design a mutable data structure that simulates the classic Snake game. The game is played on a rectangular grid with dimensions height x width. The snake starts at the top-left corner (0, 0) and initially has length 1.
This problem takes place on an infinite two dimensional grid. You begin at coordinate (0, 0) and want to reach a destination called target. At the same time, several ghosts also move on the grid from their own starting positions.
The problem provides a database table named DailySales, where each row represents a sale event involving a product manuf
The problem describes a scenario with a series of apple trees that grow apples for n consecutive days. Each day, the tree produces a number of apples given by apples[i], and these apples have a limited lifespan, given by days[i].
The problem asks us to transform a binary tree into its "upside-down" version. In more precise terms, we are given a binary tree where every right node either has a left sibling or is absent, and no right node has children.
The problem gives a string of digits where each digit is between 2 and 9. Each digit corresponds to a set of letters on a traditional phone keypad: - 2 - "abc" - 3 - "def" - 4 - "ghi" - 5 - "jkl" - 6 - "mno" - 7 - "pqrs" - 8 - "tuv" - 9 - "wxyz" We must generate every possible…
This problem asks us to design a small tracking system for an underground railway network. The system must support three
The problem asks us to find the substring of a given string s that is lexicographically largest among all possible substrings. A substring is any contiguous portion of the string.
We are asked to simulate a turn-based game between two players, Simon and Antisimon, who each have a fixed integer, a and b respectively. There is a heap of n stones.
This problem asks us to process a list of folder paths and remove all sub-folders, returning only the top-level folders. A folder "/a/b" is considered a sub-folder of "/a" because it is nested inside it.
We are given a filesystem path as a string. In this operating system, multiple consecutive '/' characters are treated exactly the same as a single '/'. That means paths like ///home//user///docs and /home/user/docs refer to the same location.
The problem asks us to determine whether a given string of digits can be partitioned into a valid additive sequence. An additive sequence is a sequence of numbers where: - There are at least three numbers.
This problem asks us to find the shortest path from a robot's unknown starting position to a hidden target inside a grid. The major challenge is that the grid itself is not directly accessible.
The problem asks us to generate every possible full binary tree that contains exactly n nodes. A full binary tree is a special kind of binary tree where every node has either exactly two children or no children at all. In other words, a node can never have only one child.
The problem gives us an array named score, where each element represents the score earned by an athlete in a competition. Every athlete has a unique score, which means there are no ties to worry about when determining rankings.
The problem asks us to identify active users from a database containing two tables: Accounts and Logins. The Accounts ta
The problem provides a string traversal representing a preorder depth-first traversal of a binary tree, where each node is represented by its value preceded by D dashes, with D being the depth of the node in the tree. The root node has depth 0 and is not preceded by any dash.
This problem asks us to simulate a data stream where each element arrives with a unique ID in arbitrary order. Each element consists of an integer idKey (between 1 and n) and a string value.
We start with a vector A = (x1, y1) and want to transform it into another vector B = (x2, y2). Two operations are allowed. We may rotate the current vector by 90 degrees clockwise, and we may add vector C = (x3, y3) any number of times. The operations may be mixed in any order.
The problem gives us an integer array nums, and we are allowed to perform at most three moves. In each move, we may sele
In this problem, we are given an n x n grid where each cell contains a unique integer representing its elevation. Rain begins falling, and the water level rises over time.
Vasya currently holds rank a in the army and wants to eventually reach rank b. Moving from rank i to rank i + 1 requires a fixed number of years, stored in the array d.
The problem asks us to construct a permutation of the integers from 1 to n such that the array satisfies a special condition.
This problem asks us to determine whether a given integer n is an exact power of four. In other words, we need to check whether there exists some integer x such that: Examples of powers of four are: The input consists of a single integer n, and the expected output is a boolean…
We are given several sets of integers, each with an associated cost. We may choose any collection of these sets, including the empty collection. Let the number of chosen sets be $k$, and let the union of all chosen sets contain $u$ distinct integers.
This problem asks us to build the smallest possible team that collectively covers every required skill. We are given two inputs: - reqskills, a list of unique required skills - people, where people[i] contains the skills possessed by person i A team is considered sufficient if…
Your requested guide is very long and detailed, especially with all required sections, full prose explanations, worked traces, Python and Go implementations, complexity proofs, and extensive test coverage.
The problem describes a two-player game played on an integer array. Players alternate turns, and during each turn a player may only take a number from one of the two ends of the array. The chosen value is added to that player's score, and the element is removed from the array.
This problem asks us to classify nodes in a binary tree based on their role in the tree structure. The input is represen
The problem gives us a two dimensional grid where each cell contains an integer representing a color. We are also given a starting position, (row, col), and a new color value. The cell at grid[row][col] belongs to some connected component.
The problem gives a single integer w, the weight of a watermelon. We need to decide whether it can be split into two positive parts such that both parts are even numbers.
The problem asks us to repeatedly remove all leaf nodes from a binary tree and record the values removed during each round. A leaf node is a node with no left or right child.
We are asked to simulate a queue of people where each person has two properties: an importance value a[i] and a patience limit c[i].
This problem asks us to find the minimum number of steps required to move from the top-left corner of a grid to the bott
The problem gives us a sorted integer array nums and a target value target. Our goal is to find the first position where the target appears and the last position where the target appears.
The problem asks us to find the largest integer less than or equal to a given number n, such that its digits are monotone increasing. A number has monotone increasing digits when every digit is less than or equal to the digit that comes after it.
In this problem, we are given a circular track divided into n sectors, numbered from 1 to n. A marathon runner moves around this track in increasing numerical order, and after sector n, the runner wraps back to sector 1.
The problem gives us a database table named Views. Each row represents a single viewing event where a user viewed an article on a specific date.
The problem is asking us to perform addition between a number represented as an array of digits, num, and an integer k. The array-form of a number represents each digit in left-to-right order, so the first element corresponds to the most significant digit.
The problem gives us an array called encoded, which was generated from an unknown permutation array perm. A permutation of the first n positive integers means the array contains every integer from 1 to n exactly once. For example: - [1,2,3] is a valid permutation of 1..
The problem asks us to calculate the sum of widths for all non-empty subsequences of a given integer array nums. A subsequence is any sequence derived by removing zero or more elements from the original array while maintaining the order.
The problem presents an m x n matrix called grid where each element is a character representing either a wall 'W', an enemy 'E', or an empty cell '0'. The task is to determine the maximum number of enemies that can be eliminated by placing a single bomb in an empty cell.
We are given a single string containing printable ASCII characters. Some characters may be lowercase English letters, some may already be uppercase letters, and others may be symbols or digits.
The problem gives us a string s that represents several positive integers concatenated together without spaces. Original
We need to build a permutation of numbers from 1 to n. Position j contains volume p(j). A positive integer i is called a divisor of the disposition if there exists some position j such that both j and p(j) are divisible by i.
The problem asks us to detect cycles in a 2D grid of characters where all cells in the cycle must contain the same character. A cycle is defined as a path that starts and ends at the same cell and has a length of four or more.
This problem asks us to compute a cumulative salary summary for each employee based on their monthly salaries over the year 2020. The input is a table Employee where each row contains an employee id, the month (1 through 12), and the salary for that month.
This problem asks us to analyze two database tables, Subscriptions and Streams, and determine how many accounts purchased a subscription during the year 2021 but never streamed any content during 2021. The Subscriptions table stores subscription intervals for each account.
The problem is asking us to identify the minimum set of vertices in a directed acyclic graph (DAG) such that starting from any of these vertices, all other nodes in the graph are reachable.
The problem asks us to find the shortest path from the top-left corner (0, 0) to the bottom-right corner (n-1, n-1) of a given n x n binary matrix grid. A cell with a value of 0 is passable, while a cell with a value of 1 is blocked.
We move on an infinite grid of unit squares. From any square we may move one step up, down, left, or right. Some squares are marked as bad, and entering such a square costs one.
The problem asks us to repeatedly remove groups of exactly k adjacent identical characters from a string until no more such groups exist. A removal operation works as follows: - Find k consecutive characters that are all the same. - Remove those characters from the string.
The problem asks us to generate every unique way to express a number n as a product of integers greater than 1 and less than n. A factor combination is a list of integers whose product equals n. The order inside a combination does not matter.
The problem gives a string s and asks us to create the shortest possible palindrome by adding characters only at the beginning of the string. A palindrome is a string that reads the same forward and backward.
This problem gives us a database table named Tweets with two columns: | Column | Description | | --- | --- | | tweetid |
The problem asks us to determine the number of days in a specific month of a given year. The inputs are two integers: year, which ranges from 1583 to 2100, and month, which ranges from 1 (January) to 12 (December).
This problem asks us to find the lowest common ancestor, usually abbreviated as LCA, of two nodes in a binary tree. Unli
The problem gives us a string that represents the preorder serialization of a binary tree. Each value is separated by commas. A normal integer represents a real tree node, while the character '' represents a null pointer.
This problem asks us to evaluate an arithmetic expression written in Reverse Polish Notation, also called postfix notation. Instead of placing operators between operands as in standard infix notation, Reverse Polish Notation places operators after their operands.
The problem asks us to determine the length of the longest palindromic subsequence within a given string s. A palindromic subsequence is a sequence of characters that reads the same forwards and backwards and can be obtained by deleting zero or more characters from the…
The problem describes a set of cars driving toward the same destination, represented by the integer target. Each car starts at a unique position and moves at a constant speed. Cars cannot overtake each other.
The problem gives us a sorted array called stations, where each value represents the position of an existing gas station on a one dimensional x-axis. We are also given an integer k, representing how many additional gas stations we are allowed to add.
We have a parking segment represented by the interval [0, L]. Cars arrive one at a time, always driving from left to right, and each driver wants to park at the earliest possible position.
We have a university with several departments. Each department contributes some number of basketball players. Herr Wafa belongs to department h, and he is already guaranteed a place on the final team. The team must contain exactly n players including Wafa himself.
This problem asks us to implement a read function using a restricted API called read4. The file itself is hidden from us, and the only way to access its contents is by repeatedly calling read4.
The problem asks us to find the length of the longest subsequence in a strictly increasing array that forms a Fibonacci-like sequence. A Fibonacci-like sequence follows the rule: for every valid index in the sequence, and the sequence must contain at least three numbers.
We are asked to find the longest contiguous substring of a string s that avoids certain "boring" substrings. In other words, given a string s and a small list of forbidden patterns b1, b2, ...
We are given a rectangular map of size n × m, where each cell has a non-negative height. Peter wants to build cities of size a × b. To place a city, he must level the ground inside its rectangle by reducing all cells to the minimum height within that rectangle.
The problem gives us an integer array nums and asks us to determine the minimum positive starting value such that when w
This problem asks us to compute a complete league table from two database tables, Teams and Matches. The Teams table contains the identity of each team in the league. Each row represents one team and includes a unique teamid and the corresponding teamname.
The problem gives us a matrix called costs, where costs[i][j] represents the cost of painting house i with color j. We must paint every house such that no two adjacent houses use the same color, and we want the minimum total painting cost.
We are given a decimal integer as a string. The number can be extremely large, up to 100 digits long, so it may not fit into normal integer types in many programming languages. The task is to determine the smallest Java integer type that can store this value.
The problem is asking us to compute cumulative scores for each gender across different days in a competition. The input
We are asked to route two people, Bob and Alex, across a town represented as an undirected graph with n crossroads and m roads. Bob starts at node 1 and wants to reach node n, while Alex starts at node n and wants to reach node 1.
The problem asks us to find the length of the longest harmonious subsequence in an integer array. A harmonious array is defined as one where the difference between the maximum and minimum values is exactly 1.
We are given two integers, n and k. We look at all prime numbers from 2 up to n. Among those primes, we want to count how many can be written in the form:
This problem asks us to calculate, for every month in the year 2020, the percentage of drivers who actually worked durin
In this problem, a biker starts at altitude 0 and travels through a sequence of roads. The input array gain describes how the biker's altitude changes between consecutive points on the trip. If gain[i] is positive, the biker climbs upward between point i and point i + 1.
The problem gives us the root of a binary tree and asks us to count how many nodes are considered "good". A node is called good if, along the path from the root to that node, there is no node with a value greater than the current node's value.
The problem asks us to find the length of the longest strictly increasing contiguous segment in an array. The key detail is that the subsequence must be continuous, which means the elements must appear next to each other in the original array.
This problem gives us the root of a binary tree and an integer distance. We need to count how many pairs of leaf nodes s
We are asked to process a user-entered sequence of page numbers for printing. The input is a single string of positive integers separated by commas, such as 1,2,3,1,1,2,6,6,2. Some numbers may repeat, possibly non-consecutively.
The problem gives us a matrix called picture, where each cell contains either 'B' for a black pixel or 'W' for a white pixel. We are also given an integer target. We need to count how many black pixels qualify as "lonely pixels" under two strict conditions.
The problem gives an integer array nums, and asks us to choose two different indices i and j. For those two elements, we compute the expression: Our goal is to return the maximum possible value of this expression.
Here’s a full technical solution guide for LeetCode 1285 following your requested structure and formatting: The problem
The problem gives us a sorted integer array in non-decreasing order. We are guaranteed that exactly one element appears
The problem gives us a string s consisting only of uppercase English letters and an integer k. We are allowed to perform at most k replacement operations. In one operation, we can change any character in the string into any other uppercase English character.
We are looking at finite strings over an alphabet of size k. For every such string, define a special value: Take all substrings of the string. Among them, some substrings can also appear as a subsequence in a non-contiguous way.
We need to build a lowercase string of length n that satisfies a special condition on repeated letters. For every character, we look at all positions where it appears.
This problem gives us the root of a binary tree where every node contains either 0 or 1. Each path from the root node to a leaf node represents a binary number, where the root contributes the most significant bit and each child adds another bit to the right.
The problem asks us to interpret a string expression representing a set of words generated according to a specific grammar and return all distinct words sorted in lexicographical order. The expression can contain lowercase letters, curly braces {}, and commas ,.
We are given the scoreboard data for every participant in a Codeforces room. Each contestant has a handle, a number of successful hacks, a number of unsuccessful hacks, and the points earned from problems A through E.
The problem gives us an integer array nums and asks us to return every distinct subsequence that is non-decreasing and has length at least two. A subsequence is formed by deleting zero or more elements from the array without changing the relative order of the remaining elements.
We are given a party scenario where Petya invited n friends, each of whom gave exactly one gift to another friend. The input lists, for each friend in order, the friend they gave a gift to.
This problem asks us to analyze banking transaction data and determine which users currently have a balance greater than 10000. We are given two database tables: The Users table stores account information.
The problem describes a plank of length n on which ants are walking either to the left or to the right at a constant spe
The problem gives us an array ages, where each value represents the age of a person on a social media platform. We must determine how many friend requests are sent between people according to a specific set of rules.
The problem gives us a keyboard with four operations: - Press A, which inserts one character 'A' - Press Ctrl-A, which selects everything currently on the screen - Press Ctrl-C, which copies the selected text into a clipboard buffer - Press Ctrl-V, which pastes the clipboard…
The problem gives us a binary string s, meaning the string contains only the characters '0' and '1'. We must split this string into two non-empty parts, a left substring and a right substring.
Each player owns a multiset of artifacts. Some artifacts are basic and can be purchased directly. Some are composite and can be crafted from several basic artifacts in fixed quantities. Whenever a player buys a basic artifact, it is added to their inventory.
We are given an array of distinct integers representing the power of soldiers standing in a line. We need to count how many index triples (i, j, k) satisfy two conditions at the same time: - The positions are ordered as i < j < k - The values are strictly decreasing as a[i]…
We are given the indices of tests that already exist in the system. Every index is a positive integer, and all indices are distinct. The task is to find the smallest positive integer that does not appear in the list.
The problem asks us to transform a given integer array nums into a continuous array with the minimum number of operations.
The problem gives us an integer array nums and an integer k. For every element in the array, we must choose exactly one of two operations: - add k - subtract k After modifying every element, we calculate the score of the array, which is defined as: Our goal is to minimize this…
In this problem, we are given a string s consisting only of the characters 'I' and 'D'. The string has length n, and we must construct a permutation of the integers from 0 to n, inclusive. A permutation means every number in the range appears exactly once.
We are given a village represented as points along a one-dimensional line, each point being a house coordinate. The task is to place exactly three cellular base stations along this line so that every house lies within the coverage of at least one station.
The problem asks us to simulate the structure of a playoff tournament bracket. We are given n teams, numbered from 1 to n, where smaller numbers represent stronger teams.
This problem asks us to insert a new value into an existing Binary Search Tree, abbreviated as BST, while preserving the BST property.
We are given a rectangular plaza, the Theatre Square, with dimensions _n_ meters in length and _m_ meters in width. The city wants to pave the entire area with square flagstones of side _a_.
This problem requires analyzing a table of orders to calculate monthly statistics. Specifically, for each unique month present in the Orders table, we need to determine two metrics: the number of unique orders and the number of unique customers whose orders have an invoice…
The problem gives us an array nums, where each element represents a balloon containing a number. When we burst a balloon at index i, we gain coins equal to: The important detail is that the neighbors of a balloon change dynamically as balloons are removed.
You are given several bags of balls, where nums[i] represents how many balls are inside the i-th bag. You are allowed to perform at most maxOperations split operations.
The problem asks us to determine how many triplets from a given integer array nums can form a valid triangle. In geometric terms, a triangle is valid if the sum of any two sides is greater than the third side.
In this problem, we are given a binary string s that contains only the characters '0' and '1'. The string is guaranteed to begin with '1', which means there are no leading zeros. The task is to determine whether the string contains at most one contiguous segment of ones.
This problem asks us to count how many different valid paths exist for a robot moving through a grid while avoiding obstacles. The robot starts in the top-left corner of the grid at position (0, 0) and wants to reach the bottom-right corner at position (m - 1, n - 1).
We are given a positive integer and need to decide whether it is a nearly lucky number. A number is considered nearly lucky if the count of its lucky digits-digits equal to 4 or 7-is itself a lucky number. A lucky number contains only 4s and 7s in its decimal representation.
This problem asks us to maximize the number of “happy” customer groups by choosing the best possible ordering of the groups. The donut shop produces donuts in batches of exactly batchSize. A fresh batch starts only when the previous batch has been completely consumed.
The problem gives us a list of directory descriptions, where each string represents a directory and the files inside it. Each file entry includes both the filename and its content. For example: means: - The directory path is root/a - Inside this directory: - 1.
The problem asks us to generate an array called nums using a specific set of rules, then return the maximum value that a
The problem gives us a string s and two scoring rules: - Removing the substring "ab" earns x points. - Removing the substring "ba" earns y points. We may perform these removals as many times as possible, in any order we choose.
The problem gives us a string representing a mathematical expression containing non-negative integers, spaces, and the four operators +, -, , and /. Our task is to evaluate the expression and return the resulting integer value.
The problem asks us to segment a given string s into all possible sentences where each word in the sentence exists in a given dictionary wordDict.
In this problem, we are given a binary tree where every node contains some number of coins. The total number of coins across the entire tree is exactly equal to the number of nodes in the tree.
The problem asks us to design a data structure that supports two operations efficiently over a stream of integers. The first operation, add(number), inserts a number into the data structure. Numbers may appear multiple times, so duplicates must be handled correctly.
The problem requires reformatting a phone number string in a specific structured way. The input is a string number conta
The problem defines a modified sequence of positive integers where every number containing the digit 9 is removed.
This problem asks us to reconstruct the ordering of characters in an unknown alien language. We are given a list of words that are already sorted according to the alien language's lexicographical order.
This problem requires calculating the day one retention metric for a game based on user activity data. The input is an Activity table, where each row represents a login session of a player on a particular day, including the device used and the number of games played.
This problem asks us to simulate a sequence of brick removals in a 2D grid, while determining how many additional bricks become unstable and fall after each removal.
The problem asks us to simulate building a sorted array incrementally. We process the instructions array from left to ri
We are asked to divide an island into connected territories for multiple parties. The island is represented by two rectangles placed side by side: one of size a by b and another of size c by d.
The problem gives us a sentence where words are separated by single spaces, the very first character of the sentence is
The problem gives us an array nums containing non-negative integers. We need to determine whether there exists an intege
The problem asks us to construct a binary expression tree from a valid infix arithmetic expression string. An infix expr
This problem asks us to count how many different ways a pointer can end up back at index 0 after taking exactly steps mo
The problem presents an m x n matrix M initialized with all zeros. You are also given an array of operations ops, where each operation ops[i] = [ai, bi] instructs you to increment by one all elements in the submatrix defined by the top-left corner (0,0) and the bottom-right…
The problem asks us to count how many contiguous subarrays of a given array have a product strictly smaller than a target value k. A subarray is a continuous portion of the array.
We are asked to investigate whether the greedy algorithm for making change can fail with a given set of coin denominations.
The problem gives an array of strings and asks us to find the longest prefix shared by every string in the array. A prefix is a sequence of characters that appears at the beginning of a string.
The problem gives an integer array arr and asks whether there exist two different indices i and j such that: In simpler
This problem asks us to design a very small parking lot management system. The parking lot contains exactly three types
We are given a short Brainfuck program consisting of characters like +, -, , <, and so on. Each command corresponds to a fixed 4-bit binary string. After replacing every character with its binary code, we concatenate all those 4-bit chunks into one long binary number.
The problem gives us two arrays, tops and bottoms, representing a sequence of dominoes. Each domino has two values, one on the top half and one on the bottom half. For the ith domino, the top value is tops[i] and the bottom value is bottoms[i].
The problem asks us to find two different subarrays inside the given array such that: 1. Each subarray has a sum exactly
The problem gives us an n x n matrix where both rows and columns are sorted in ascending order. This means two ordering guarantees exist simultaneously: - Every row is sorted from left to right. - Every column is sorted from top to bottom.
The problem asks us to find the contiguous subarray within an integer array nums that produces the largest possible product. Unlike the classic maximum subarray sum problem, multiplication introduces additional complexity because negative numbers can completely change the result.
The problem gives us two lists of closed intervals. Each interval is represented as [start, end], meaning every value from start through end, inclusive, belongs to that interval.
We are asked to analyze a two-player game on an n × n chessboard. Petya and Gena take turns placing knights such that no knight can threaten another. A knight threatens positions in its standard L-shaped moves.
The problem gives us a two dimensional binary matrix where each cell contains either '0' or '1'. Our task is to find the largest square submatrix that contains only '1' values, then return the area of that square.
The problem asks us to design a stack data structure, MaxStack, which behaves like a normal stack with additional operations for efficiently accessing and removing the maximum element.
The problem gives an array called citations, where each element represents how many citations a research paper has received. Each index corresponds to one paper, and the value at that index is the number of times that paper has been cited.
We have a small rectangular grid where each cell is either empty, contains a pig, or contains a wolf. A wolf may eat one pig that is directly adjacent to it in one of the four cardinal directions. Once a pig is eaten, it disappears and cannot be eaten again.
This problem asks us to design a leaderboard system that supports three operations efficiently. Each player has a unique playerId and an associated score. The leaderboard begins empty, and players can be added dynamically as operations are performed.
This problem asks us to determine which words from a given list can be constructed using the characters available in another string, chars. A word is considered good if every character it needs exists in chars in sufficient quantity.
The problem gives us an array of distinct positive integers and asks us to find the largest subset where every pair of numbers satisfies a divisibility relationship. For any two elements a and b in the subset, either a % b == 0 or b % a == 0 must hold.
We are given a decimal number as a string. The number contains an integer part, then a dot, then a fractional part. The task is to simulate the kingdom's strange rounding rules. The rules are intentionally incomplete.
The problem asks us to identify all employees in a company who directly or indirectly report to the head of the company,
This problem asks us to find numbers in a table that appear at least three times consecutively in order of their id. The table Logs consists of two columns: id and num, where id is an auto-incrementing primary key and num is a string representing a number.
We are given a grid representing an island with impassable sea cells and traversable land cells. Some of the land cells contain unique local sights labeled with uppercase letters.
The problem asks us to determine whether two binary trees are exactly the same. We are given the roots of two binary trees, p and q. A binary tree consists of nodes where each node contains a value and references to a left child and a right child.
This problem gives us a special nested data structure called NestedInteger. Each element in the input can either be: 1. A single integer 2.
The problem asks us to determine the minimum number of swaps needed to transform one string into another, where both strings are anagrams of each other. A single operation consists of swapping any two characters in s1.
We are asked to simulate a simplified Piet interpreter on a small rectangular grid. Each cell is a pixel with a color between 0 and 9, where 0 is black and other digits are colored blocks.
This problem asks us to construct the numerically largest possible integer such that the total painting cost of its digi
The problem gives us a list of integers representing asteroids moving through space in a straight line. Each integer contains two pieces of information: - The absolute value represents the asteroid's size. - The sign represents the direction: - Positive numbers move to the right.
The problem asks us to format a non-negative integer so that every group of three digits is separated by a dot (.), starting from the right side of the number. For example, the number 1234567 should become "1.234.
This problem gives us a database table named Tree, where each row represents a node in a tree structure. Every node has a unique identifier id and a pid column that stores the identifier of its parent node.
The problem describes a subway system as an undirected graph where each vertex is a station and each edge represents a direct connection between two stations. Every edge is labeled with a color representing the subway line it belongs to.
The problem asks us to calculate the total time each employee spends in the office per day based on a table of their in and out times.
The problem asks us to find all pairs of numbers in an array that have the smallest absolute difference. In other words, given a list of distinct integers, we want to identify every pair [a, b] such that the difference b - a is minimized across all possible pairs in the array…
The problem asks us to determine the optimal rotation of an array nums so that the resulting array achieves the highest possible score, where a score is defined as the number of elements that are less than or equal to their index after rotation.
Tom uses his laptop during several disjoint time intervals. While he is actively using it, the laptop stays in normal mode and consumes P1 watts per minute. When he stops interacting with the laptop, the machine does not immediately switch to lower-power states.
The problem asks us to process a paragraph of text and determine which word appears most frequently, while ignoring a given list of banned words. The final answer must be returned in lowercase.
Each player initially owns cards of exactly one color, their own color. Player i starts with a[i] cards, all of color i. During a swap, two players exchange one card each. A player may only give away cards of their own color, and may never receive a color they already possess.
Here is a comprehensive, detailed technical solution guide for LeetCode 1152 - Analyze User Website Visit Pattern following your requested format exactly. The problem asks us to analyze user website visit sequences to determine the most common pattern of length three.
We are given an undirected graph where cities are vertices and roads are edges. Each connected component of the graph is called a province. Vasya may additionally build tunnels between cities, but tunnels have two restrictions.
We have n seats and n professors. Professor 1 is the most senior, professor n is the least senior. Some pairs of seats impose ordering constraints. If (a, b) is given, then the professor sitting in seat a must be more senior than the professor sitting in seat b.
The problem describes a network of people who meet at different times. Whenever a person who already knows a secret participates in a meeting, they immediately share the secret with the other participant. The key detail is that sharing is instantaneous within the same timestamp.
This problem asks us to find all repeated DNA subsequences of a fixed length, specifically length 10, inside a given DNA string. The input is a string s, where each character represents a DNA nucleotide.
The problem asks us to determine whether a given set of points in a 2D Cartesian plane all lie on a single straight line. The input is a list of integer coordinates, where each coordinate [x, y] represents a point.
We are given a rectangular chocolate bar with integer width _W_ and height _H_. Bob breaks the chocolate multiple times along vertical or horizontal lines that go from one edge to the opposite edge of a piece.
We are given an array of worm lengths. Each position represents a different worm form, and we need to find three distinct indices such that the length at one index equals the sum of the lengths at the other two indices.
We are asked to check whether a password entered on a 3×3 keypad is symmetric with respect to its central button. The keypad is represented as a 3×3 grid of characters, where "X" indicates a pressed button and "." indicates an unpressed one.
The problem gives us two water jugs with capacities x and y. We can perform only three types of operations: - Fill a jug completely - Empty a jug completely - Pour water from one jug into the other until either the source jug becomes empty or the destination jug becomes full…
The problem asks us to count how many permutations of numbers from 1 to n satisfy a specific condition: all prime numbers must appear at prime-numbered indices (1-indexed).
The problem gives us the root of a binary tree and asks us to find the smallest subtree that contains all of the deepest nodes in the tree. The depth of a node is defined as the number of edges between that node and the root. The root itself has depth 0.
The problem asks us to count how many distinct index triplets (i, j, k) satisfy two conditions: 1. The indices are ordered such that 0 <= i < j < k < n 2.
The problem gives us a database table named Customers with three columns: | Column | Meaning | | --- | --- | | customerid | Unique identifier for a customer | | year | The year associated with the revenue | | revenue | Revenue value for that customer in that year | The…
The problem asks us to design a class CombinationIterator that generates all combinations of a given length from a strin
In this problem, we are given a final colored matrix called targetGrid. We need to determine whether a strange printer could have produced this grid under two unusual restrictions.
Here is a comprehensive, detailed solution guide for LeetCode 1337 following your exact instructions. The problem provid
We have a rectangular canvas of size W × H. Initially the whole canvas is white. Then several black line segments are drawn on it. After that, a sequence of flood-fill operations is applied. A flood-fill chooses a point and a color.
The problem gives us an integer array arr and an integer target. We are allowed to choose a value x, then modify the arr
The problem gives us an array of integers called nums, where each integer represents a possible side length. Our goal is to select exactly three lengths that can form a valid triangle with a non-zero area, and among all valid triangles, return the maximum possible perimeter.
The problem gives us two arrays, target and arr. The target array contains distinct integers, which is extremely important. The arr array may contain duplicates. We are allowed to perform insert operations on arr.
We start with a lowercase string and may delete at most k characters from it. The remaining characters must stay in their original order, since deleting characters creates a subsequence.
This problem asks us to determine how many students are doing homework at a specific point in time, given arrays represe
Each DNA string evolves by repeatedly applying rules of the form a -> bc. One character is replaced by exactly two characters, and this operation can be repeated any number of times. Starting from some ancestor string, evolution only increases the length.
The problem gives us an undirected and connected graph with n nodes labeled from 0 to n - 1. The graph is represented as an adjacency list, where graph[i] contains all nodes directly connected to node i.
We are given an integer N. Consider all triples (A, B, C) where every value lies in the range [1, N].
This problem asks us to reconstruct a binary tree from a specially formatted string representation. The input string contains integer values and parentheses. Every integer represents a tree node, and parentheses represent child subtrees.
This problem asks us to select as many non-overlapping substrings as possible from a given lowercase string s, while sat
The game state is completely described by two numbers: how many heads and how many tails the dragon currently has. From a state (h, t) Ivan may choose one of two move types.
This problem asks us to compute the total area covered by a set of axis-aligned rectangles on a 2D plane. Each rectangle is defined by its bottom-left and top-right coordinates [xi1, yi1, xi2, yi2].
This problem describes a frog navigating a 3-lane road of length n. The frog starts at point 0 in lane 2 and wants to reach point n. Each point along the road may have at most one obstacle in one of the three lanes, represented by the obstacles array.
We are given two integers, _a_ and _b_, written in some unknown base _p_. Vasya wants to compute the sum _a + b_ in all valid bases and determine which base gives the sum with the largest number of digits.
The problem gives us n identical unit cubes that must be placed inside a cubic room. The goal is to minimize how many boxes directly touch the floor. The placement rule is the important part of the problem.
We are given three line segments on a 2D plane. We must decide whether these three segments can be interpreted as the shape of the capital letter A.
We are given exactly four triangles. Each triangle represents the shape of one spaceship. A landing platform is just a set of points in the plane, called columns. A ship can land if we can choose three columns that form a triangle congruent to the ship.
The problem asks us to take an integer num and find the maximum number we can create by swapping at most two digits. In other words, we can pick two positions in the number, swap their digits, and we want the resulting number to be as large as possible.
The problem defines a special binary grammar sequence that grows row by row. The first row contains only a single value: Every later row is generated from the previous one using these rules: - Replace every 0 with 01 - Replace every 1 with 10 This means the rows evolve like…
In this problem, we are given a two dimensional grid of characters called board and a target string called word. The task is to determine whether the word can be formed by traversing the grid under a strict set of movement rules.
The problem gives us an integer array nums and an integer k. We need to determine whether it is possible to divide all n
We start with four integers on the board. We also know the exact sequence of three operations that must be applied, where each operation is either addition or multiplication.
The problem presents an interactive scenario where ships are placed at integer coordinates on a Cartesian plane. You do
We have several photo albums. Album i contains a[i] photos. We must build a cyclic gallery containing exactly n photos. Instead of choosing concrete photo IDs, we only need to output the album number for each position. The gallery is circular, so every position has two neighbors.
The problem asks us to transform a given binary matrix mat into a zero matrix, where all elements are 0. Each operation
The problem asks whether an array of positive integers can be divided into two subsets such that both subsets have exactly the same sum. Suppose the total sum of all numbers in the array is S.
The problem defines a function f(s) for a non-empty string s. This function returns the frequency of the lexicographically smallest character in the string.
The problem asks us to count how many substrings of length k contain only unique characters. A substring is a contiguous portion of the string. For every possible substring of length k, we must determine whether all characters inside it are distinct.
This problem asks us to split a string into the largest possible number of substrings such that every substring is uniqu
Each bowl is a frustum, a cone with its tip cut off. A bowl is described by its height h, bottom radius r, and top radius R. The bowls are stacked in the given order, always sharing the same vertical axis.
You are given a 2D grid where each cell represents one of three possible states: - 0 means the cell is empty. - 1 means the cell contains a fresh orange. - 2 means the cell contains a rotten orange.
The problem gives us an array of distinct integers called nums, and asks us to generate every possible permutation of those numbers. A permutation is an arrangement of elements in a specific order.
This problem asks us to return the preorder traversal of a binary tree. A binary tree consists of nodes where each node may have a left child and a right child. The input root represents the root node of that tree.
Each athlete runs through a contiguous interval of sections. While an athlete is inside a section, they spend exactly t[i] time on that section, so the winner of a section is simply the active athlete with the smallest t.
The problem asks us to coordinate four separate threads so they collectively print the correct Fizz Buzz sequence in order from 1 to n. Unlike the classic single threaded Fizz Buzz problem, this version introduces concurrency.
This problem is an interactive backtracking problem where we must control a robot without directly seeing the room layout. Unlike traditional grid traversal problems, we are not given access to the actual room matrix during execution.
In this problem, we are given n friends, where n is always even. Every friend ranks all other friends in order of preference. The earlier someone appears in a person's preference list, the more that person is preferred. We are also given a final pairing arrangement.
Here is a complete, detailed technical guide for LeetCode 1410 following your formatting and style requirements. The pro
We start with a point on the plane at coordinates (x, y). Players alternate turns, and on each turn they may do one of two things. They may add one of the given movement vectors to the current position.
The problem gives us two parallel arrays, keyName and keyTime. Each index represents a single key-card usage event. For example: Together, they describe when a specific employee used their key-card during a single day.
We are given several short DNA fragments over the alphabet {A, C, G, T}. A longer DNA string is considered valid if every position of the string belongs to at least one occurrence of one of the given fragments. The fragments may overlap arbitrarily.
The problem is asking us to generate the first numRows of Pascal's triangle, which is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it.
This problem asks us to reverse only a specific portion of a singly linked list, rather than reversing the entire list. We are given the head of a linked list and two integer positions, left and right, where left <= right.
The problem gives us a list of rectangles, where each rectangle is represented as [li, wi]. Here, li is the rectangle's length and wi is its width. From each rectangle, we want to determine the largest square that can be cut from it.
The problem asks us to count all permutations of a given integer array nums such that the array is squareful, meaning that the sum of every pair of adjacent elements is a perfect square. In other words, for a permutation [a1, a2, a3, ...
This problem asks us to examine every possible subtree inside a binary tree and determine whether that subtree forms a valid Binary Search Tree, abbreviated as BST. Among all valid BST subtrees, we must return the maximum possible sum of node values.
This problem asks us to find the lowest common ancestor, abbreviated as LCA, of all the deepest leaf nodes in a binary tree. A leaf node is any node with no children. The depth of the root is 0, and every level downward increases the depth by 1.
We start with the array [1, 2, 3, ..., n]. A recursive transformation rearranges it by repeatedly taking all elements at odd positions, transforming that subarray, then taking all elements at even positions and transforming that subarray.
The problem is asking us to simulate a dieter’s performance over a sequence of days based on their calorie consumption. The input is an array calories where calories[i] represents the number of calories consumed on day i. We are also given three integers: k, lower, and upper.
We are given a string representing a simple arithmetic expression of the form a+b=c, where a, b, and c are integers.
The problem asks us to maximize the number of employee transfer requests that can be fulfilled under the constraint that
The problem gives us a 2 x n grid where each cell contains some number of points. Two robots move across this grid one after another. Both robots start at the top-left corner (0, 0) and must reach the bottom-right corner (1, n - 1). The movement rules are very restrictive.
The problem gives us an integer array nums and a collection of range requests. Each request [start, end] asks for the su
The problem gives two arrays of positive integers, nums1 and nums2, both of equal length n. The goal is to calculate the absolute sum difference between these arrays, which is the sum of the absolute differences at each index: |nums1[i] - nums2[i]|.
The problem asks us to determine whether a given binary tree is a complete binary tree. A complete binary tree has a specific structural property: every level, except possibly the last, is completely filled, and in the last level, all nodes appear as far left as possible.
This problem provides a 2D grid representing a map of land and water. Cells with 0 represent land, and cells with 1 repr
This problem asks us to locate the first occurrence of one string inside another string. The string we are searching for is called needle, and the larger string we search inside is called haystack.
We have an array arranged in a circle. Every operation works on a segment between two indices, but the segment may wrap around the end of the array.
The problem asks us to generate all integers from 1 to n, but not in normal numerical order. Instead, the numbers must appear in lexicographical order, also called dictionary order. Lexicographical order compares numbers as strings rather than as numeric values.
The problem asks us to compute the subtree sum for every node in a binary tree, then determine which subtree sum appears most frequently. A subtree rooted at a node includes that node and all of its descendants.
This is a SQL database problem where we need to calculate the total sales amount per product for each year, even when a
This problem asks us to count the number of valid paths in a grid under very strict movement rules. We are given a matrix where each cell has a special meaning: - 1 is the starting position - 2 is the ending position - 0 is an empty square we may walk on - -1 is an obstacle…
The problem gives us a sequence of product versions numbered from 1 to n. At some point, one version becomes bad, and every version after it is also bad.
The problem requires designing a system to manage seat reservations for n seats numbered from 1 to n. You need to implement a SeatManager class with two main operations: reserve and unreserve.
We have several point-like balls moving on a one-dimensional line. Every ball starts at some coordinate with its own velocity and mass.
The problem gives an array of positive integers called nums. We must find the contiguous subarray that is strictly increasing and has the largest possible sum. A subarray is contiguous, which means the elements must appear next to each other in the original array.
This problem asks whether it is possible to create the number 1 using the integers in the array, where each chosen numbe
The problem asks us to compute the maximum depth of an n-ary tree. An n-ary tree is a tree where each node can have zero or more children, unlike a binary tree where each node has at most two children.
This problem works with three relational database tables: Movies, Users, and MovieRating. The goal is to produce a result containing exactly two rows. The first row should contain the name of the user who rated the greatest number of movies.
The problem provides an array of exactly four digits. Using each digit exactly once, we must construct the latest possible valid 24-hour time in the format "HH:MM".
We are given two binary strings of equal length. Each character is either '0' or '1'. For every position, we compare the characters from the two strings. If the two characters are different, the resulting string gets '1' at that position.
We are given an array of at most 100 integers. Every value lies between 1 and 60. The task is simply to print the array in non-decreasing order.
We are given two integers, l and r. For every number n inside this interval, we build another number called its reflection. The reflection is created digit by digit. Every decimal digit d becomes 9 - d.
Let's dive deep and construct a thorough technical guide for LeetCode 1008, following your formatting rules. The problem asks us to construct a binary search tree (BST) from a given preorder traversal array.
The problem models the Android lock screen as a 3 x 3 grid containing digits 1 through 9. A valid unlock pattern is a sequence of distinct dots that follows a special movement rule. The first rule is straightforward: each dot can only be used once in a pattern.
The problem is asking us to simulate a flood fill operation on a 2D grid that represents an image. Each cell in the grid contains an integer representing a pixel color. You are given a starting pixel (sr, sc) and a target color color.
The problem gives us an array of k sorted linked lists. Each linked list is already sorted in ascending order, and our task is to combine all of them into one single sorted linked list. A linked list node contains a value and a pointer to the next node.
The problem gives us an array of non-negative integers called nums. We must choose two indices i and j such that 0 <= i <= j < n, then compute: The goal is to return the largest XOR value that can be produced from any pair in the array.
The problem gives us a sorted integer array nums, and asks whether we can divide every element into one or more subsequences that satisfy two conditions. First, each subsequence must consist of consecutive increasing integers.
This problem asks us to count the number of valid sequences for performing pickups and deliveries for n orders. Each ord
The problem asks us to calculate the sum of all odd-length subarrays of a given array of positive integers. A subarray is any contiguous sequence of elements from the original array.
The problem describes a corrupted set of integers. Originally, the set contained every number from 1 through n exactly once. Because of an error, one number was duplicated, which means one other number disappeared.
This problem asks us to process a sorted singly linked list and remove every value that appears more than once. The important distinction is that we are not keeping one copy of duplicated values. Instead, every node containing a duplicated value must be removed entirely.
The problem gives us an integer array nums, where each value represents the maximum distance we are allowed to jump forward from that position. You always start at index 0.
This problem models a chain reaction of falling dominoes. We are given a string where each character represents the initial state of a domino in a row. A domino can be in one of three states: - 'L' means the domino has been pushed to the left.
The problem gives us an array nums that is guaranteed to be a permutation of the integers from 0 to n - 1. A permutation means every value appears exactly once, and every value is within the valid index range of the array.
The problem gives an array of strings, and we must arrange them into a circular concatenation while preserving the original order of the strings. For every individual string, we are allowed to either keep it as-is or reverse it before joining it into the loop.
We are asked to compute the average sum of digits of a number _A_ when it is expressed in all bases from 2 up to _A_ - 1. The input is a single integer _A_ between 3 and 1000.
This problem gives us a string s and an integer k. We are allowed to repeatedly perform one operation: - Choose one of the first k characters of the string. - Remove that character from its current position. - Append it to the end of the string.
The problem gives us a binary matrix called grid, where each cell contains either 0 or 1. A value of 1 represents land, and a value of 0 represents water. An island is formed by connecting adjacent land cells in the four cardinal directions: up, down, left, and right.
The problem gives us an array of coin values, where each coin can be used at most once. We are asked to determine the maximum number of consecutive integer values that can be formed starting from 0.
We are asked to count how many distinct substrings of a given string t start with a string sbegin and end with another string send. The key detail is that substrings are considered different only by their content, not by their position in t.
This problem asks us to distribute candies to children standing in a line, based on their rating values. Each child must receive at least one candy, and any child with a higher rating than an adjacent neighbor must receive strictly more candies than that neighbor.
The problem gives us an integer num, and asks us to find two integers whose product equals either num + 1 or num + 2, su
We are asked to determine whether a set of non-zero integers forms a "friendly group," meaning every pair of numbers satisfies a divisibility relationship: one number divides the other.
We are simulating a simplified version of the party game "Boom" with multiple teams and cards. There are n teams, each with two players. Each player has an ability to explain words (a) and an ability to understand words (b).
This problem gives us two strings, s and t. The string t is created by taking all characters from s, shuffling them into a different order, and then inserting exactly one additional character somewhere in the string. Our task is to identify and return that extra character.
This problem asks us to repeatedly remove leaf nodes from a binary tree if their value matches a given target. A leaf no
The problem gives us an initially zero-filled array of size length. We are also given a list of update operations, where each update has the form: This means we must add inc to every element in the inclusive range from startIdx to endIdx.
We are given a map of _n_ cities in Berland, where the shortest distances between all pairs are already known. Conceptually, this means the country is fully connected via some unknown set of roads, and the distance matrix is already the all-pairs shortest path result.
We are given a valid binary search tree where every internal node has exactly two children. Each node stores a unique key. We are also given several query keys that are guaranteed not to appear in the tree. A normal BST search starts at the root.
This problem models a social network as an undirected graph. Each person is represented by an integer ID from 0 to n - 1
This problem asks us to generate random points uniformly inside a circle. The circle is defined by its radius and the coordinates of its center. Every call to randPoint() must return a point [x, y] such that the point lies either inside the circle or exactly on its boundary.
This problem asks us to identify all lucky numbers in a given matrix. A lucky number is defined as a value that satisfie
The problem is a combinatorial game involving three piles of stones with counts a, b, and c. In each move, you are allowed to pick two different non-empty piles and remove one stone from each, earning 1 point per move. The game ends when fewer than two piles have stones left.
The problem gives an array nums containing n distinct integers. Every number is supposed to come from the range [0, n], which means there are actually n + 1 possible values in total. Since the array only contains n numbers, exactly one value from that range is missing.
We are given two pieces of information about a person riding an elevator. The first input tells us which door the person used to enter, either the front door or the back door. The second input tells us which rail the person was holding, rail 1 or rail 2.
We are given a map of Berland with several control points and circular fences. Each knight occupies a control point. Fences separate the kingdom into regions, and a knight must cross fences to move between control points.
The problem asks us to determine how many words in the words array can be transformed into the target string s using a very specific type of expansion operation.
The problem asks us to find the "luckiest" substring of a given string of digits. By luckiest, we mean a substring that consists only of the digits 4 and 7, occurs in the string as many times as possible, and is the lexicographically smallest if there are ties.
We need to construct as many party days as possible using exactly n hobbits. Every day has a guest list. The construction must satisfy two conditions. First, every pair of different days must share at least one hobbit.
This problem is asking us to find the common elements between two lists of strings, but not just any common elements. We want the ones where the sum of their indices in both lists is the smallest.
This problem asks us to compare two binary trees based only on their leaf nodes. A leaf node is a node that has no left child and no right child.
The problem asks us to construct a lowercase string of length n such that the sum of the numeric values of its character
This problem gives us a two dimensional grid where each cell contains either 'X' or '.'. An 'X' represents part of a battleship, while '.' represents empty water. The important rule is that battleships are always placed in straight lines.
The problem gives four integer arrays, nums1, nums2, nums3, and nums4, each containing n elements. We must count how many index tuples (i, j, k, l) satisfy: The important detail is that we are counting tuples of indices, not unique value combinations.
The problem gives us a binary matrix where each cell is either land (1) or water (0). Land cells that touch vertically or horizontally belong to the same island. A grid is considered connected only when there is exactly one island in the entire grid.
The problem gives us a target array and asks us to build it starting from an array of all zeros. The only operation allowed is selecting any contiguous subarray and incrementing every element in that subarray by exactly one.
This problem gives us a list of people where each person is represented as a pair [h, k]. The value h represents the person's height. The value k represents how many people standing in front of this person must have a height greater than or equal to h.
The problem is asking us to swap the sex values of all employees in a Salary table. Each row in the table represents an employee with four columns: id, name, sex, and salary. The sex column is an ENUM with values 'm' for male and 'f' for female.
The problem asks us to determine how many longest strictly increasing subsequences exist in a given integer array. A subsequence is formed by deleting zero or more elements without changing the relative order of the remaining elements.
The problem asks us to identify books from a Books table that are considered unpopular based on their sales in the last year, relative to a fixed "today" date of 2019-06-23.
The problem gives a list of restaurant orders. Each order contains three pieces of information: - Customer name - Table number - Food item An order looks like this: The goal is to build a display table that summarizes how many times each food item was ordered at every table.
The problem asks us to determine which strings in words1 satisfy all character requirements imposed by every string in words2. A string b is considered a subset of another string a if every character in b appears in a at least as many times as it appears in b.
The problem gives an integer array nums and asks us to build a subsequence whose sum is strictly greater than the sum of
The problem asks us to determine the number of ways an encoded string can be decoded into letters, where 'A' maps to "1", 'B' maps to "2", up to 'Z' mapping to "26".
This problem asks us to take an integer array nums and perform a series of increment operations so that every element in the array becomes unique. An increment operation increases an element by exactly 1. The goal is to compute the minimum number of such moves required.
We have an undirected graph where every road has two thresholds attached to it. For a road with values (g, s), the road becomes safe only if the king gives at least g gold coins and at least s silver coins to the bandits.
This problem asks us to find the length of the longest contiguous substring in which every vowel appears an even number
In this game, two players, Alice and Bob, take turns removing stones from the beginning of a row. Each stone has an inte
This problem asks us to find the cheapest valid path through an array while respecting jump constraints and blocked positions. The array coins is 1-indexed in the problem statement, although programming languages will use 0-indexed arrays internally.
Each ticket is described by two positive integers. The first is the series number a, the second is the ticket number inside that series b.
This is a SQL database problem where we need to identify countries whose average call duration is strictly greater than
The problem asks us to find the maximum possible average value among all contiguous subarrays whose length is at least k. We are given an integer array nums and an integer k. A subarray is a contiguous portion of the array, meaning the elements must appear consecutively.
This problem asks us to find the nth number in a special sequence called "super ugly numbers". A super ugly number is a positive integer whose prime factors come only from the given array primes. That means every factorization of the number can use only primes from this list.
The problem asks us to find the length of the longest contiguous subarray that forms a valid mountain. A mountain is a sequence that strictly increases up to a peak, then strictly decreases afterward.
The problem gives an integer array nums where every value appears exactly twice except for one value that appears only once. The task is to find and return that unique value.
In this problem, we are given the positions of several black queens and exactly one white king on a standard 8 x 8 chessboard. The board uses 0-indexed coordinates, meaning every position is represented as [row, column], where both values range from 0 to 7.
This problem asks us to create a deep copy of an N-ary tree. An N-ary tree is a tree where each node can have zero or mo
The problem gives us a string called text and a list of dictionary words called words. We must find every substring inside text that exactly matches one of the words in the dictionary.
The problem gives us a string representing a mathematical expression containing integers, addition, subtraction, parentheses, and spaces. Our task is to evaluate the expression and return the final integer result. The expression is guaranteed to be valid.
This problem asks us to validate whether a partially filled Sudoku board follows the core Sudoku rules. The board is always a fixed 9 x 9 grid, and each cell contains either a digit from '1' to '9' or the character '.', which represents an empty cell.
A maximum tree is a special binary tree where every node contains a value greater than all values inside its subtree.
The problem gives us an integer array nums, and we need to find every unique triplet of numbers whose sum is exactly 0. A triplet consists of three different indices: - i != j - i != k - j != k The values themselves may be equal, but the indices must be different.
We are given three lines representing the three phrases of a poem. A valid haiku must contain exactly 5 vowel letters in the first phrase, 7 in the second, and 5 in the third. For this problem, syllables are simplified into vowel counts.
We are given a Tower of Hanoi variant with three pillars and n disks stacked on the first pillar. The disks are listed from bottom to top, and unlike the classical problem, several disks may have the same diameter. A move still transfers exactly one disk between pillars.
We are given a string representing a hockey team name and a list of forbidden substrings. Our task is to modify the original string so that any letter that is part of a forbidden substring can be replaced with another letter of our choosing.
This problem models a small social network with two database tables. The Friendship table stores friendship relationship
We are asked to count the number of valid ski bases after each road is added to an initially empty graph. The graph consists of junctions as vertices and roads as edges.
The problem asks us to extend the functionality of a standard iterator with an additional peek operation. A standard iterator provides two operations: next(), which returns the next element in the sequence and advances the pointer, and hasNext(), which tells whether there are…
We are asked to simulate a warehouse with a grid-like shelving system. Each shelf has m sections, and there are n shelves stacked from top to bottom. Every section can hold at most one box.
The problem asks us to calculate the minimum number of operations to convert one 24-hour time string, current, into anot
We have a connected weighted graph of cities. Runner i starts from city i and finishes in some other city. The finish city is chosen independently for every runner, and several runners may share the same destination.
The problem gives us a string s and an integer k. We are allowed to modify characters in the string, and our goal is to
The problem asks us to list all products that have been ordered in February 2020 with a total quantity of at least 100 u
This problem encodes a binary tree into a compact array of three-digit integers. Each integer stores three pieces of information: - The hundreds digit represents the depth of the node. - The tens digit represents the node's position within that depth level.
The problem gives us a database table named Activities with two columns: | Column | Meaning | | --- | --- | | selldate |
We are asked to simulate a turn-based game involving two players, Ciel and Hanako, who alternate taking coins from a common pile. The pile initially contains x 100-yen coins and y 10-yen coins. On each turn, the active player must remove exactly 220 yen.
The problem asks us to design an iterator that alternates between two input arrays, returning elements in a cyclic or zigzag order.
The problem gives us a numeric string s of even length and two operations that can be performed any number of times in a
The problem asks us to transform a valid IPv4 address into its "defanged" version. A defanged IP address is created by replacing every period character "." with the string "[.]". An IPv4 address consists of four numeric segments separated by periods. For example, "1.1.1.
This problem asks us to determine the minimum number of adjacent swaps needed to make exactly k ones appear consecutively in a binary array. The input array nums contains only 0 and 1. In one operation, we may swap two neighboring elements.
The problem asks us to determine whether a given positive integer num is a perfect square without using built in square root functions such as sqrt(). A perfect square is a number that can be written as: where k is an integer.
The problem asks us to repeatedly remove pairs of numbers from an array such that the sum of each selected pair equals a
The problem gives us a sorted integer array nums and a quadratic transformation function: We must apply this function to every element in the array and return the transformed values in sorted order.
This problem asks us to transform a given integer n into 0 using a specialized set of bit-flipping operations. Each oper
The problem asks us to rearrange an array so that it follows a strict alternating pattern: - nums[0] < nums[1] - nums[1] nums[2] - nums[2] < nums[3] - and so on. This pattern is called a wiggle sequence because the values repeatedly go up and down.
This problem models the spread of a virus on a two-dimensional grid. Each cell is either infected (1) or uninfected (0). The virus spreads every night from infected cells to adjacent uninfected cells in the four cardinal directions: up, down, left, and right.
The problem gives us a two-dimensional grid of integers called matrix. Each position in the grid contains a value, and we are asked to find the length of the longest strictly increasing path.
This problem asks whether it is possible to divide n people into exactly two groups such that no pair of people who dislike each other end up in the same group.
We are asked to compute the factorial of a given integer $n$. Factorial is the product of all positive integers up to $n$, so for $n = 3$, the factorial is $1 times 2 times 3 = 6$. The input is a single integer, and the output is the single integer result of this product.
The problem asks us to transform a Binary Search Tree, abbreviated as BST, into a sorted circular doubly linked list. The transformation must happen in place, meaning we are not allowed to create entirely new nodes for the linked list.
We are given a single integer n, and we need to construct the smallest positive integer whose number of divisors is exactly n.
The problem asks us to determine whether an integer array nums can be split into four non-empty subarrays with equal sums by selecting three indices i, j, and k that satisfy strict ordering constraints: 0 < i, i + 1 < j, j + 1 < k < n - 1.
The problem asks us to decide whether a positive integer $n$ can be represented as a sum of distinct integers, where each integer is either 1 or a prime number. If such a representation exists, we need to produce one that is lexicographically largest.
We have 2N - 1 boxes. Every box contains two numbers, the number of apples and the number of oranges inside it. We must choose exactly N boxes such that the chosen set contains at least half of all apples and at least half of all oranges across every box.
The problem describes a circular route containing n gas stations. At each station i, you can collect gas[i] units of fuel. Traveling from station i to station (i + 1) % n consumes cost[i] units of fuel.
We have a grid containing chips. Every chip stores one direction, left, right, up, or down. When we start a move from some chip, the process behaves like this: The current chip looks in the direction of its arrow.
Vasya is at the central station of a subway branch with two endpoints. Each endpoint corresponds to one of his girlfriends: Dasha or Masha. Trains to Dasha’s station arrive every a minutes and trains to Masha’s station arrive every b minutes.
We need to construct an array of n positive integers. The array must satisfy two conditions at the same time. The sum of squares of all elements must be at least x, while the ordinary sum of the elements must not exceed y. The task is not to optimize anything.
This problem models a shortest path search on a grid, but the moving object is not a single cell. Instead, the object is a snake that occupies exactly two adjacent cells.
Alice gives Bob a decimal number and asks him to rearrange its digits so that the resulting number is as small as possible, while still being a valid decimal number without leading zeroes.
This problem asks us to determine the minimum number of boats required to rescue all people, given two constraints. Each boat can carry at most two people, and the combined weight of the people in a boat cannot exceed the given weight limit.
The problem gives us two strings, order and s. The string order defines a custom character ordering. Unlike normal alphabetical ordering, the characters in order specify exactly how characters should be prioritized relative to one another.
Each person visiting the embassy must pass through three consecutive stages. The first stage has k1 identical windows, each service taking t1 time. The second stage has k2 windows with service time t2, and the third stage has k3 windows with service time t3.
The problem asks us to compute a "beauty" score for each element in an array nums, excluding the first and last elements. Specifically, for each index i in the range 1 <= i <= nums.length - 2, the beauty of nums[i] is determined by two conditions: 1.
The problem requires us to repeatedly remove consecutive nodes in a singly-linked list whose values sum to zero. The input is the head of a singly-linked list, where each node has an integer value in the range -1000 to 1000, and the list length ranges from 1 to 1000 nodes.
Gerald repeatedly removes either one candy or one stone. After every move, Mike looks at how many candies and stones Gerald has already eaten. If Gerald has eaten a candies and b stones so far, Mike awards $$f(a,b) = (xa + yb) bmod p$$ points.
The problem asks us to determine whether a given integer n is a power of two. A number is considered a power of two if it can be written in the form: where x is a non-negative integer.
The problem asks us to count how many contiguous, non-empty subarrays have a maximum element that lies within the inclusive range [left, right]. A subarray is a continuous segment of the original array. For every possible subarray, we look at its maximum value.
The problem is asking us to reconstruct a flight itinerary given a list of tickets. Each ticket is a one-way flight represented as a pair of airport codes [from, to]. The traveler always starts at "JFK", and the itinerary must use all tickets exactly once.
We are asked to count the numbers in a given interval $l, r$ that are simultaneously prime and expressible as the sum of two positive squares. In simpler terms, for each number in the interval, we need to check two independent properties.
The problem asks us to design a data structure that can efficiently answer majority element queries on subarrays of a given array.
The problem gives us two lowercase English strings, a and b. In a single operation, we may change any character in either string into any other lowercase English letter.
The problem asks us to find the optimal location for a service center on a 2D map such that the total Euclidean distance
We are asked to predict the next petrol station where Vanya will stop, given the stations he has already visited.
We have a fixed starting point, the handbag, and up to 24 scattered objects on a 2D plane. Lena always starts at the handbag, walks around collecting at most two objects, returns to the handbag, then repeats until every object is stored back.
The problem gives us the root of an arbitrary binary tree and asks us to find the size of the largest subtree that is also a valid Binary Search Tree, abbreviated as BST. A subtree is any node together with all of its descendants. This means we cannot selectively ignore children.
The problem asks us to simulate rainfall over a series of lakes and prevent flooding. Each day is represented by an inte
This problem asks us to implement a Skiplist from scratch, a probabilistic data structure that allows efficient search, insertion, and deletion operations.
We are tasked with counting how many pairs of soldiers in a detachment can form a reconnaissance unit. A unit consists of exactly two soldiers, and the difference in their heights cannot exceed a given threshold _d_.
The problem asks us to add two non-negative integers where each integer is given as a string instead of a numeric type. The goal is to return the resulting sum as another string. For example, if the inputs are "11" and "123", we should return "134".
In this problem, we are given an even number of people standing around a circle. Every person must participate in exactl
This guide is quite large and highly detailed under your required structure, especially with full worked examples, both
This problem is a variation of the classic House Robber dynamic programming problem, but instead of houses being arranged in a straight line, the houses form a binary tree.
The problem asks us to design an in-memory file system that simulates basic file system operations without interacting with the real filesystem.
This problem asks us to design a reusable data structure that can efficiently answer repeated shortest-distance queries between words in a fixed list of strings. We are given an array of words, wordsDict, during initialization.
The problem provides a database table named Activity, which stores information about processes executed on different machines.
We are asked to design a network of servers where each server is a node, and each connection between two servers is an undirected edge. There are n servers and we are allowed exactly m connections.
This problem asks us to evaluate boolean expressions stored in a database table. We are given two tables: Variables, whi
This problem asks us to determine whether two singly linked lists share a common node, and if they do, return the exact node where the intersection begins. The important detail is that an intersection is based on node reference, not node value.
We are given a three-dimensional space containing $n$ fixed points, representing the centers of energy spheres. There are $m$ levels, and in each level the player starts at a plane passing through the origin.
The problem presents a matrix isWater of size m x n where each cell is either water (1) or land (0). We are asked to assign heights to every cell in a way that satisfies three rules. First, all heights must be non-negative integers. Second, water cells must have a height of 0.
The problem gives us an n x n matrix called grid, where each cell contains a non-negative integer. The value grid[i][j] represents how many 1 x 1 x 1 cubes are stacked vertically at position (i, j). Each cube contributes surface area through its exposed faces.
The problem gives us a compressed string representation of a 2D coordinate. Originally, the coordinate looked something like "(1, 3)" or "(2, 0.5)", but all commas, spaces, and decimal points were removed. Our task is to reconstruct every possible valid original coordinate pair.
This problem involves a scenario with n passengers and n seats on an airplane. Each passenger has a designated seat, but the first passenger has lost their ticket and picks a seat randomly.
The problem gives us an array called groupSizes, where each index represents a person and each value represents the size
The problem gives an array nums containing exactly 2 n integers. We must divide these integers into two separate groups, each containing exactly n elements. After forming the two groups, we compute the sum of each group and measure the absolute difference between those sums.
We are asked to find the cheapest way for Gerald to buy three pieces of clothing that all match each other. Each clothing item has a price, and some pairs of items are marked as matching. The input gives us the total number of items, the prices, and a list of matching pairs.
This problem asks us to determine the minimum number of cameras required to monitor all nodes in a binary tree. Each camera placed on a node can monitor its parent, itself, and its immediate children.
The problem gives us a nested list structure where each element can either be: - A single integer - Another nested list containing additional integers or lists The goal is to compute a weighted sum of all integers, but unlike the standard "Nested List Weight Sum" problem, the…
The task is to reverse a number given as a string, taking care of signs and leading zeros. The input can be a very large integer, up to 10,000 digits, possibly with leading zeros.
The Sliding Puzzle problem presents a 2 x 3 board containing five numbered tiles from 1 to 5 and a blank space represented as 0.
We are given a queue of walruses, where index order goes from the tail toward the head. For every walrus at position i, we want to find the furthest position j i such that the walrus ahead is strictly younger, meaning a[j] < a[i]. If no such walrus exists, the answer is -1.
We are asked to maintain a set of points on a plane and support two operations: adding a point, and checking if a point lies inside the convex hull of the current set. The convex hull is the minimal convex polygon enclosing all points.
We need to tile a 4 × n board using ordinary dominoes. Each domino covers exactly two neighboring cells, either horizontally or vertically. The tiling must satisfy an extra condition.
Let's go through a complete, detailed technical solution guide for LeetCode 683 - K Empty Slots following your formatting instructions. The problem asks us to find the earliest day when there are exactly k bulbs turned off between two bulbs that are turned on.
The problem gives us a database table named Enrollments. Each row represents a student taking a course and receiving a grade for that course.
The problem gives us a string representation of a file system hierarchy. Every line represents either a directory or a file. The hierarchy is encoded using newline characters (n) to separate entries and tab characters (t) to indicate nesting depth.
The problem gives two non-negative integers, low and high, representing the inclusive bounds of an interval. We need to
The problem asks us to find the length of the longest subsequence that appears in both input strings. A subsequence is formed by deleting some characters from a string while keeping the remaining characters in the same relative order. The characters do not need to be contiguous.
The problem gives us a sorted array of positive integers, nums, and a target integer n. We are allowed to insert additional numbers into the array, called patches.
We have exactly seven heroes and three bosses. Each boss gives some amount of experience, and every hero assigned to that boss receives an equal share rounded down. If a boss gives x experience and its team has k heroes, then every hero in that team receives x // k.
The problem asks us to determine common free time intervals for a group of employees based on their schedules. Each employee has a list of non-overlapping intervals representing times when they are busy.
We are given an arithmetic expression built from terms involving a single variable a. Every term is one of two forms:
The problem asks us to determine the longest contiguous substring of s that can be transformed into the corresponding substring of t without exceeding a given budget, maxCost.
This problem asks us to find the smallest integer that is strictly greater than a given integer n, while using exactly the same digits as n. In other words, we are allowed to rearrange the digits of the number, but we cannot add or remove digits.
The problem asks us to compute the Hamming distance between two integers. The Hamming distance is defined as the number of bit positions where the two numbers differ in their binary representation. For example, consider x = 1 and y = 4.
We are asked to determine if a string of round brackets is balanced. A balanced sequence is one where every opening bracket «(» has a corresponding closing bracket «)», and brackets are properly nested.
We are given an undirected graph and need to decide whether it matches a very specific structure. The graph should contain exactly one simple cycle, and every other vertex must belong to a tree attached to some vertex on that cycle.
This problem asks us to convert a binary tree into a string using a very specific preorder traversal format. We must visit nodes in the order root → left subtree → right subtree, and represent the structure of the tree using parentheses.
We have a source string s1, which represents the headline of one newspaper. We may take as many copies of this headline as we want and concatenate them together.
The problem presents a weighted undirected graph where nodes represent junctions and edges represent roads with a given length. At each junction, there is a taxi that can carry a passenger up to a maximum distance and charges a fixed cost.
The problem asks us to find employees with the highest salary in each department from a company database. The input consists of two relational tables: Employee and Department. The Employee table contains employee details, including id, name, salary, and departmentId.
The problem gives us an m x n binary matrix where each cell contains either 0 or 1. We must count how many rectangular submatrices consist entirely of ones. A submatrix is any contiguous rectangular region inside the matrix.
We have a cycle of n cities arranged on a ring. Every pair of consecutive cities is already connected by the outer ring road, and city n is also connected back to city 1.
Each year the university sends a team of exactly n students to the finals. Some of those students may already have participated once before, and the rest are newcomers.
The problem describes a sequential candy distribution process across a row of people. We are given two integers, candies and numpeople. The goal is to simulate distributing candies in increasing order until all candies are exhausted.
We are given a permutation of jars. Jar i should finally stand at position i, but the current arrangement is shuffled. One operation allows us to choose any subset of at most five positions and permute the jars inside those positions arbitrarily.
The problem gives us a sequence of points on a 2D plane. These points are connected in order, and the final point connects back to the first point, forming a polygon. Our task is to determine whether that polygon is convex.
This problem asks us to determine how many strings in the words array are consistent with a given set of allowed charact
We have the classical river crossing setting, but with a precise safety rule that changes how transitions work.
This problem gives us access to a hidden function f(x, y) through an interface. We are not allowed to know or implement the formula directly. Instead, we can only call the function with positive integers x and y and observe the result.
This problem describes a frog moving through an undirected tree. A tree is a connected graph with no cycles, which means
The problem describes a two-player game involving a single heap of stones. There are n stones initially on the table, and players alternate turns removing stones. On each turn, a player may remove either 1, 2, or 3 stones. The player who removes the final stone wins the game.
The problem asks us to count the number of substrings in a string s such that every unique character in the substring occurs exactly count times. A substring is a contiguous segment of the string, so we are only considering consecutive characters.
This problem asks us to determine the minimum number of swaps needed to group all 1s in a binary array into one contiguous block. The block can appear anywhere in the array, as long as all 1s end up adjacent.
The problem requires us to implement a HashMap from scratch without using any built-in hash table libraries. In other words, we need to create a data structure that can store key-value pairs, allow efficient insertion, retrieval, and deletion of values based on their keys.
The problem gives us the root of a binary tree and an integer targetSum. We need to count how many downward paths in the tree have values that add up exactly to targetSum. A path can begin at any node and end at any node, as long as it always moves downward from parent to child.
We have six gems, each with one of six possible colors. The six gems must be placed onto the six faces of a cube. Two placements are considered identical if one can be rotated into the other. The input is simply a string of length six.
The problem gives us a binary array called flowerbed, where: - 0 represents an empty plot - 1 represents a plot that already contains a flower We are also given an integer n, which represents how many new flowers we want to plant.
This problem asks us to find the k points that are closest to the origin (0,0) from a list of 2D points. Each point is given as a pair [x, y], and the distance is defined using the Euclidean formula, sqrt(x^2 + y^2).
We want to cook soup using several ingredients that must appear in a fixed ratio. If the recipe says the proportions are a1, a2, ..., an, then the final soup must contain:
This problem gives us an m x n binary matrix where every cell contains either 0 or 1. For every cell in the matrix, we must compute the distance to the nearest cell containing 0. Distance is measured using Manhattan movement with four directions only: up, down, left, and right.
That is a long, structured technical document with multiple required sections, complete Python and Go implementations, w
The problem asks us to compute the total Hamming distance between every possible pair of integers in the array. The Hamming distance between two integers is defined as the number of bit positions where the two numbers differ.
Let's dive into a full, detailed technical solution guide for LeetCode 959 - Regions Cut By Slashes following your formatting rules. The problem gives us an n x n grid where each cell contains either a forward slash '/', a backslash '', or a blank space ' '.
The problem gives a binary string s, consisting only of the characters '0' and '1'. We must count how many substrings co
The problem asks us to count how many valid strings of length n can be formed using only the five lowercase vowels: - 'a' - 'e' - 'i' - 'o' - 'u' However, the strings are not arbitrary. Each vowel has strict rules about which vowels may appear immediately after it.
This problem asks us to design an iterator over a run-length encoded sequence instead of storing the full sequence explicitly.
The problem asks us to find the digit that appears at position n in an infinitely long sequence formed by concatenating all positive integers together.
We are given three points on the plane with integer coordinates. These three points already form a valid triangle, meaning they are not collinear.
This problem asks us to triangulate a convex polygon in such a way that the total triangulation score is minimized. We are given an array values, where each element represents the value assigned to a vertex of the polygon.
This problem asks us to compute the complement of a base-10 integer by flipping every bit in its binary representation. For instance, if the input n is 5, its binary form is 101. Flipping each 0 to 1 and each 1 to 0 produces 010, which equals 2 in decimal.
The problem gives us a sorted integer array where every value appears exactly twice, except for one value that appears only once. Our task is to find that unique value. The important detail is that the array is already sorted.
This problem is asking us to simulate the operation of a single-chef restaurant, where customers arrive at certain times and each customer has a specific preparation time for their order. The goal is to calculate the average waiting time for all customers.
We are given an rectangular grid representing a field, and within this field there are non-overlapping rectangles representing objects. Each rectangle occupies contiguous cells marked by '*', and all other cells are '.'.
The problem gives us the head of a singly linked list and asks us to return the middle node. A singly linked list is a sequence of nodes where each node contains a value and a pointer to the next node.
This problem asks us to count quadruples of indices (i, j, a, b) where substrings from two given strings are equal and the difference j - a is minimized. Specifically, i and j define a substring in firstString, while a and b define a substring in secondString.
The problem asks us to compute the ordinal day number of a year given a date in the format YYYY-MM-DD. For example, January 1st is always day 1, January 2nd is day 2, and December 31st is either day 365 or 366 depending on whether the year is a leap year.
The problem describes a simulation of squares falling onto the X-axis. Each square is represented by two values: - lefti, the X-coordinate of the square's left edge - sideLengthi, the side length of the square A square occupies the interval: When a square falls, it continues…
The problem asks us to compute the running sum, also called the prefix sum, of a one dimensional integer array. For ever
This problem asks us to generate all possible combinations of size k from the numbers in the inclusive range [1, n].
Nick wants to list all numbers of a given length n in base b, where digits range from 0 to b-1, but numbers cannot start with 0. Each page in his notepad holds exactly c numbers. We are asked to compute how many numbers appear on the last page he fills.
This problem asks us to determine whether a given string is a palindrome after applying two transformations: 1. Convert all uppercase letters to lowercase. 2. Remove all non-alphanumeric characters. An alphanumeric character is any English letter (a-z, A-Z) or digit (0-9).
The problem is about a network of computers represented as an undirected graph using an adjacency matrix. Each node represents a computer, and an edge between two nodes indicates a direct connection.
We maintain a dynamic set of points on a 2D plane. Three operations are supported.
The problem asks us to count how many distinct non-empty subsequences can be formed from a given string s. A subsequence is formed by deleting zero or more characters while keeping the remaining characters in their original order.
The problem requires determining the minimum amount of time to build all blocks using workers that can either build blocks or split into more workers. You are given a list blocks where blocks[i] indicates the time it takes for a worker to complete the i-th block.
The problem gives us an integer array arr and asks us to compute the average value after removing the smallest 5% and th
This problem gives us a collection of equations between variables, where each equation represents a division relationship.
This problem asks us to design a data structure that dynamically tracks intervals on the number line. The tracked intervals are represented as half-open intervals, meaning [left, right) includes every value x such that left <= x < right.
We are given a lowercase string that may contain accidental repeated keystrokes. Whenever two equal characters become adjacent, both characters must be deleted. After removing one pair, new adjacent equal pairs may appear, and those must also be removed.
In this problem, we are given a string that represents multiple possible words. Each position in the word may contain either a single fixed character or a set of alternative characters enclosed in curly braces.
We have a grid of digits. The pawn starts somewhere on the bottom row and moves upward one row at a time. At each step it may go diagonally left or diagonally right. Every visited cell contributes its digit to the total collected peas.
We have a 6 × 6 board containing the characters 0-9 and A-Z, each appearing exactly once. The target configuration is fixed: characters must appear in row-major order. The only allowed moves are cyclic shifts of complete rows or complete columns.
The problem is asking us to determine the earliest time at which a group of n people all become mutually acquainted, given a sequence of friendship formation events with timestamps.
This problem is a classic concurrency synchronization task. We are given a class FooBar with two methods, foo() and bar().
We are given a point $(x, y)$ on the 2D plane and an angle $k$ in degrees. The task is to rotate the point counter-clockwise around the origin by exactly $k$ degrees and print the coordinates of the new point.
We have n theorems, and each theorem has a proficiency value assigned to it. The exam contains exactly k different cards. Every card contains exactly floor(n / k) distinct theorems, and no theorem appears in more than one card.
The problem describes a staircase with n total steps. You start at the bottom and want to reach the top. At every move, you are allowed to climb either 1 step or 2 steps. The goal is to determine how many distinct sequences of moves can take you from the bottom to the top.
This problem gives us a database table named Queue that represents people waiting to board a bus. Each row contains a person's ID, name, weight, and boarding order. The turn column determines the exact sequence in which people attempt to board the bus.
We are given a laboratory grid of size n × n, representing a research station where some tiles are reactors and others are laboratories. One reactor is malfunctioning and will explode, causing toxic coolant to spread to neighboring labs.
The problem asks us to determine the length of the longest substring in a given string text where all characters are the same, and we are allowed to swap exactly two characters in the string.
This problem asks us to determine the weather type for each country during November 2019, based on the average weatherst
The problem gives us a non-negative integer num and asks us to compute how many operations are required to reduce it all the way down to 0.
The problem gives us n unique songs and asks us to build playlists of length goal. Songs may repeat, but two important constraints must always hold. First, every one of the n songs must appear at least once in the playlist.
This problem gives us the root of a Binary Search Tree, abbreviated as BST, along with two integers, low and high. We must compute the sum of all node values whose values lie inside the inclusive range [low, high].
This problem asks us to identify customers who consistently spend at least $100 in each of two consecutive months, June and July of 2020.
The problem is asking us to find all integers within a given range [low, high] that are stepping numbers. A stepping number is defined as a number in which the absolute difference between every pair of adjacent digits is exactly 1.
The problem asks us to generate every possible valid combination of n pairs of parentheses. A valid parentheses string is one where every opening parenthesis ( has a matching closing parenthesis ), and the parentheses are properly nested.
The problem describes a single-row keyboard where each key is positioned at a unique index from 0 to 25. You are given a string keyboard of length 26 that specifies the layout of all lowercase English letters.
We are given a sequence of four-digit years. Each year may contain mistakes, but we are allowed to repair it by changing at most one digit. After all repairs, the sequence must become non-decreasing, and every resulting year must stay between 1000 and 2011 inclusive.
The train has n wagons arranged in a line. The controller moves deterministically: every minute he walks one wagon in his current direction, and when he reaches either end he reverses direction. His path is completely fixed once we know the initial wagon and direction.
This problem asks us to find the length of the longest contiguous subarray that contains at most two distinct values. Each element in the fruits array represents the type of fruit produced by a tree.
The problem gives us n nodes labeled from 0 to n - 1. For every node i, we are told which node is its left child and which node is its right child through the arrays leftChild and rightChild. If leftChild[i] = x, then node x is the left child of node i.
In this problem, Alice repeatedly draws random numbers until her score reaches at least k. Every draw independently produces an integer between 1 and maxPts, inclusive, and each value is equally likely.
The problem gives us a collection of tokens, where each token has a numeric value. We also start with an initial amount of power and an initial score of 0. Every token can be used at most once, and each token can be played in exactly one of two ways.
The problem gives us a collection of unique points on a 2D plane. Our task is to determine the minimum possible area of any rectangle that can be formed using exactly four of these points as vertices.
The problem gives an array of intervals where each interval is represented as [start, end]. Each interval describes a continuous range on a number line. The task is to remove the minimum number of intervals so that the remaining intervals no longer overlap.
We are given a string that represents a valid XML-like structure. Every tag is either an opening tag like <a or a closing tag like </a, where the tag name is a single lowercase letter.
This problem asks us to reroot a binary tree at a given leaf node. Normally, a binary tree has a single root, and every node points downward to its children. In this problem, each node also contains a parent pointer, which allows traversal upward toward the root.
The problem gives us an integer array nums and an integer k. We are allowed to split the array into at most k contiguous, non-empty subarrays. For each subarray, we compute its average, then sum all of those averages together. Our goal is to maximize that total score.
The problem gives us a database table named Orders with two columns: Column Meaning --- --- ordernumber Unique identifier for an order customernumber Identifier for the customer who placed the order Each row represents a single order placed by a customer.
The problem asks us to find the length of the longest uncommon subsequence between two strings a and b. To understand the problem clearly, we first need to understand what an uncommon subsequence means.
The problem asks us to decompose a given string text into the largest possible number of contiguous substrings such that the sequence of substrings forms a palindromic pattern.
The problem requires us to implement a multithreaded class ZeroEvenOdd that coordinates three separate threads to print numbers in a specific sequence. Thread A prints 0s, Thread B prints even numbers, and Thread C prints odd numbers.
The problem asks us to count how many arithmetic subsequences exist inside a given integer array nums. An arithmetic sequence is a sequence where the difference between consecutive elements is constant. The sequence must contain at least three elements.
Here is a complete, detailed technical solution guide for LeetCode 1463 - Cherry Pickup II following your formatting req
The problem is asking us to determine the minimum number of semesters required to complete n courses given a set of prerequisite relationships.
Sure, let's build a full, detailed technical solution guide for LeetCode 1256 following your requested structure. The problem asks us to take a non-negative integer num and encode it as a string using a secret encoding rule.
The problem gives us a binary string s and an integer k. A binary string contains only the characters '0' and '1'. We mu
The problem gives us an Employee table containing three columns: Column Meaning --- --- id Unique employee identifier company Company name salary Employee salary We need to return the row or rows that correspond to the median salary for each company.
Here is a complete, detailed technical solution guide for LeetCode 1338, following your requested formatting and structu
This problem models how functions execute on a single-threaded CPU. Since the CPU is single-threaded, only one function can actively execute at any given moment. However, functions may call other functions, including recursive calls to themselves.
The problem asks us to replace every number in the input array with its rank when the array is sorted in ascending order.
The problem gives a collection of words on the left side of an equation and a single result word on the right side. Each distinct uppercase letter must be assigned a unique digit from 0 to 9.
The problem presents an array nums of length n that is a permutation of integers from 0 to n-1. A global inversion is defined as any pair (i, j) such that i < j and nums[i] nums[j], meaning the earlier element is larger than a later element anywhere in the array.
The problem describes a path traced on a two-dimensional grid. We begin at the origin (0, 0) and move according to the values in the distance array.
This problem asks us to simulate a baseball scoring system with a set of unusual rules. We begin with an empty score record and process a sequence of operations one by one. Each operation either adds a new score or modifies the history of previously recorded scores.
We are given a bracket string containing three kinds of characters: '(', ')', and '?'. Every '?' must eventually become either an opening or closing bracket. For each unknown position, the input provides two costs, one for replacing it with '(' and one for replacing it with ')'.
The problem asks us to design a mutable 2D range sum data structure. We are given a matrix of integers, and we must support two operations efficiently: 1. Update the value at a specific cell. 2. Compute the sum of all values inside a rectangular submatrix.
This problem gives us two strings, a and b, both having the same length. We are allowed to choose a split index and divi
Every student independently chooses one of the bathroom rooms uniformly at random. A room may contain several wash basins, so students entering that room are split into several queues.
This problem asks us to find the nearest node to the right of a given node u on the same level of a binary tree. A binar
The problem asks us to compute the number of ways to completely tile a 2 x n board using two types of tiles: a domino (2 x 1) and a tromino shape.
We are building a username registration system. Every incoming request contains a desired username. If that username has never appeared before, registration succeeds immediately and we print OK.
We are given two lowercase strings. The first string is exactly one character longer than the second. We want to find every position in the first string such that removing the character at that position makes the two strings identical.
This problem gives us an undirected weighted graph with n nodes labeled from 1 to n. Every edge connects two nodes and has a positive weight. The graph is guaranteed to be connected, meaning there is always at least one path between any pair of nodes.
We are given a rectangular slab divided into an grid of squares. Each square is either black or white in the final design, which is provided as input. The slab initially starts entirely white.
We are given up to one hundred thousand points on a 2D plane. For every unordered pair of points, we must compute the squared Euclidean distance between them and add all those values together.
This problem asks us to determine whether one string can be transformed into another using a specific recursive scrambling process. We are given two strings, s1 and s2, and they are guaranteed to have the same length. The scrambling process works recursively.
The problem presents a scenario where you are buying items in a shop, each with a given price stored in an array prices.
We are given several bags of cookies, where each bag contains some number of cookies. Olga wants to steal exactly one bag. After removing that bag, the total number of cookies left behind must be even so the two sisters can split them equally.
The problem gives us a binary array named nums, where every element is either 0 or 1. Our task is to determine the maximum number of consecutive 1s that appear anywhere in the array.
The problem asks us to simulate how a file system assigns folder names when duplicate names appear. We are given an array names, where names[i] represents the folder name requested at the i-th minute. The file system must ensure that every created folder has a unique name.
Each walrus builds a skyscraper whose height changes linearly over time. Walrus i starts with height a[i] and gains b[i] floors every minute, so at time t its height equals: $$hi(t) = ai + bi cdot t$$ For every query [l, r, t], we must find an index inside that interval whose…
This problem asks us to compute the post tax salary for every employee in the Salaries table. However, the tax rate is n
Lavrenty has a fixed amount of dough and several types of stuffing. Each stuffing type has a limited quantity and requires a certain amount of dough to make a bun, and each bun yields a profit.
This problem asks us to compute the total points for each football team after a series of matches in a tournament. The scoring rules are standard: a win gives 3 points, a draw gives 1 point, and a loss gives 0 points. The input consists of two tables: Teams and Matches.
This problem gives us a binary matrix where each cell contains either '0' or '1'. A '1' represents a black pixel, while a '0' represents a white pixel.
This problem requires determining how many candidates a company can hire as seniors and juniors under a fixed budget of $70000, following a strict priority: hire as many seniors as possible first, then use the remaining budget to hire juniors.
The problem gives us an n x n matrix called grid. Each cell grid[i][j] represents how many 1 x 1 x 1 cubes are stacked vertically at position (i, j) on a flat surface.
This problem asks us to compute the minimum number of days required to eat exactly n oranges, given three possible actions that can be performed each day. On any single day, we may choose one of the following operations: 1. Eat exactly one orange. 2.
In this problem, we are given a string text that must be displayed on a single line inside a screen with width w and hei
The problem gives us a set of allowed digits, stored as strings, and an integer n. We may construct any positive integer by repeatedly using digits from the given set. Each digit can be reused any number of times.
We are asked to calculate how many numbers from 1 to n are not divisible by any number in a given set of integers, called hit indicators. Each indicator is guaranteed to be coprime with the others, and there are k of them, each at most 1000.
The problem gives us a list of strings called words, where every string has the same length, and another string called target. We need to count how many different ways we can build target by selecting characters from the columns of words.
We are given a string made of two characters, C and P. Each character represents one object hanging on the wall. C means postcard, P means photo. Polycarpus removes objects from left to right. He cannot skip positions, and at any moment he may carry only one type of object.
The problem gives us a directed acyclic graph, usually abbreviated as a DAG. The graph contains n nodes labeled from 0 to n - 1. The graph is represented as an adjacency list, where graph[i] contains all nodes that can be reached directly from node i.
We are given a string written in the Borze encoding system. Every digit of a ternary number is represented by one of three patterns:
This problem asks us to parse a chemical formula represented as a string and return a canonical string that counts the number of each atom in the formula. Each atom starts with an uppercase letter and may have lowercase letters following it.
The elevator follows a completely deterministic cycle. It starts at floor 1 at time 0, climbs one floor per second until it reaches floor m, then immediately reverses direction and goes back down to floor 1, again moving one floor per second.
The problem asks us to determine whether a given string represents a valid IPv4 address, a valid IPv6 address, or neither.
This problem asks us to find the longest path in a binary tree where every node along the path has the same value. The key detail is that the path length is measured in edges, not nodes.
We have a lorry with capacity v. There are two kinds of boats.
This problem asks us to find the smallest number greater than or equal to n that satisfies two properties simultaneously: 1. The number must be a palindrome. 2. The number must be prime. A palindrome is a number that reads the same forward and backward.
We are given a simplified blackjack scenario where the first card is fixed: the queen of spades, which contributes 10 points. The player wants the sum of this card and a second card to equal a given number n, which ranges from 1 to 25.
The problem gives us a database table named RequestAccepted, where each row represents a successfully accepted friend request between two users. Every record contains a requesterid, an accepterid, and the date the request was accepted.
This problem asks us to find the largest possible island size in a binary grid after changing at most one 0 into 1.
The problem presents a company hierarchy encoded as a string. Each employee has a name, followed optionally by a colon and a comma-separated list of subordinates, ending with a dot. If an employee has no subordinates, the colon is omitted.
This problem asks us to compute the minimum number of moves required to gather all balls in a string of boxes into each individual box. Each box can either be empty ('0') or contain one ball ('1'). A single operation consists of moving a ball from one box to an adjacent box.
The problem asks us to determine the minimum number of operations required to transform a given password into a "strong" password according to three rules. A strong password must satisfy all of the following conditions: 1.
The problem asks us to merge two singly linked lists in a very specific way. We are given list1 and list2, with sizes n
The problem gives us a binary string, meaning the string contains only the characters '0' and '1'. We need to count how many substrings satisfy two conditions simultaneously: 1. The substring contains the same number of 0s and 1s. 2.
The problem asks us to design a data structure that continuously receives integers from a stream and can efficiently return the median of all numbers seen so far. The median is defined as the middle element in a sorted list.
This problem asks us to find all unique combinations of numbers from the candidates array such that the sum of the chosen numbers equals target. Unlike the original Combination Sum problem, each number can only be used once.
The problem gives us an integer array nums and a list of update operations called queries. Each query contains two values: - vali, the amount to add - indexi, the position in the array to update For every query, we must first update the array element: After applying the update…
We are given a sequence of books by Berlbury, each with a known height, arranged chronologically. The library wants to organize an exposition by selecting consecutive books such that the difference between the tallest and shortest book in the selection does not exceed a given…
The problem asks us to count how many contiguous subarrays contain exactly k odd numbers. We are given an integer array
The problem asks us to find all starting indices in a string s where a substring exists that is a concatenation of all the words in the array words exactly once, without any intervening characters. The words can appear in any order, and all words in words have the same length.
The problem gives us a numeric string num and an integer k. We must remove exactly k digits from the number so that the resulting integer is as small as possible. The key detail is that we are not allowed to reorder digits.
We are given two points in the plane, the ship's starting position and destination, together with a convex polygon representing an island. Moving through the sea costs 1 per unit distance. Moving through the interior of the island costs 2 per unit distance.
You are given two strings, a and b, where each string represents a binary number. A binary number contains only the characters '0' and '1'. The task is to add these two binary numbers together and return the result as another binary string.
The problem asks us to convert a binary matrix into a Quad-Tree representation. A Quad-Tree is a recursive tree structure where every internal node has exactly four children, each representing one quadrant of the current square region.
The problem asks whether it is possible to partition a deck of cards, represented as an integer array deck, into groups such that each group contains exactly x cards, all of which have the same integer value, and x is greater than 1.
The problem asks us to transform a binary tree into a flattened structure that behaves like a singly linked list. The transformation must happen in-place, meaning we are not supposed to create an entirely new tree or list structure.
The problem asks us to find the kth factor of a positive integer n, where the factors are ordered in ascending order. A
This problem asks us to analyze purchasing activity on an online shopping platform. We are given three database tables: Users, Orders, and Items.
This problem defines a special infinite sequence called the magical string. The string contains only the characters '1' and '2', and its defining property is very unusual: If you group consecutive identical digits together and record the length of each group, the resulting…
The problem asks us to convert a given 32-bit integer num into its hexadecimal representation as a string. Hexadecimal, or base-16, uses digits 0-9 and letters a-f to represent values 0-15.
The problem asks us to design two complementary operations for a binary tree: serialization and deserialization. Serialization converts a binary tree into a string representation that can be stored or transmitted.
This problem asks us to implement a thread-safe bounded blocking queue, which is a data structure that stores elements in a first-in-first-out (FIFO) order, but with the added constraint that multiple threads may be concurrently accessing it.
We are asked to analyze a straight-line race of length _s_ kilometers involving _n_ cars. Each car has a list of driving segments, and each segment specifies a constant speed and a duration. The car moves at that speed for the given time before switching to the next segment.
We are given a string s consisting of uppercase and lowercase Latin letters, and we are asked to analyze its substrings according to a specific property: a substring is "good" if the number of vowels it contains is at most twice the number of consonants.
The problem gives us two inputs: - An integer array nums - A list of queries, where each query is [xi, mi] For every query, we must find the maximum possible value of: subject to the condition: If no number in nums satisfies the condition nums[j] <= mi, then the answer for…
The problem asks us to modify a fixed-length integer array in-place by duplicating each zero and shifting the subsequent elements to the right.
The problem gives us a valid time string in the format "HH:MM" and asks us to construct the next chronological time using only the digits already present in the original time. The important detail is that digits may be reused any number of times.
This problem is a variant of the classic number guessing game with a twist: incorrect guesses cost money equal to the value guessed, and the goal is to minimize the maximum cost required to guarantee a win.
The problem asks us to delete a node with a specific value, key, from a Binary Search Tree, abbreviated as BST, and return the possibly updated root of the tree.
We have an array whose values are always at most 10000. Two operations must be processed online. The first operation adds a value d to every element inside a segment [l, r].
The problem gives the coordinates of two axis-aligned rectangles on a 2D plane. Each rectangle is represented using two corner points: - Bottom-left corner (x1, y1) - Top-right corner (x2, y2) For the first rectangle, the coordinates are: - (ax1, ay1) for the bottom-left -…
The problem is asking us to find the maximum difference between two elements in an array, with the condition that the smaller element appears before the larger element.
This problem asks us to simulate movement on a 2D plane according to a string of directional instructions. Each characte
The problem asks us to find the nth positive integer that is divisible by either a or b. A number is considered magical if at least one of the following is true: - It is divisible by a - It is divisible by b We are given three integers: - n, the position of the magical number…
The problem asks us to determine, for each given puzzle string, how many words from a given list words are valid for that puzzle. A word is valid for a puzzle if it contains the first character of the puzzle and all characters of the word are also present in the puzzle.
The problem asks us to determine whether an integer array forms a perfect sequence of consecutive numbers without gaps o
The problem gives us a 2D grid called picture, where each cell contains either: - 'B', representing a black pixel - 'W', representing a white pixel We need to count how many lonely black pixels exist in the matrix. A black pixel is considered lonely if: 1. It is black ('B') 2.
The problem gives us two integers, n and start. We must construct an array named nums of length n, where each element is
In this problem, we are given an integer array called satisfaction, where each element represents the satisfaction level
We are asked to generate a specific string of beads of length n according to a Martian’s unique notion of string equivalence. Each bead is either red (0) or blue (1).
The problem asks us to count how many ways we can split a string into two non-empty parts such that both parts contain the same number of distinct characters. A split occurs between two adjacent characters.
This problem asks us to construct a new string from the input string s such that every distinct character appears exactly once, while also ensuring that the final result is the smallest possible in lexicographical order.
We are given two real parameters chosen uniformly at random. The victim height p lies in the interval [0, a], and the wind parameter q lies in [-b, b]. For every pair (p, q), the anvil hits successfully if a certain quadratic equation has at least one real root.
We are given a tree with n vertices where each edge has a positive weight. Some edges are considered "lucky" if their weight consists only of the digits 4 and 7.
We are given a single lowercase string. Somewhere inside this string, there exists a hidden palindrome of odd length, but the palindrome was split into three consecutive parts:
We are given a tree, and two independent probability distributions over its vertices. One distribution chooses the starting vertex of a DFS, the other chooses the target vertex where the search stops.
A car starts at point A and moves along a straight road toward point B. The total distance is l meters, and the car always moves at a fixed speed v. Somewhere along the road, exactly d meters from the start, there is a traffic light.
The problem requires us to find the most frequently ordered product or products for each customer from a set of three tables: Customers, Orders, and Products. Each customerid may have multiple orders, and each order contains a productid.
We stack cubes one by one. Every cube is axis-aligned, and its projection on the ground is a square. Since the cubes are actual cubes, the side length is determined by the square base.
The problem asks us to design an autocomplete system that behaves similarly to a search engine suggestion feature. The system starts with a collection of historical sentences and the number of times each sentence has been typed before.
Each student points to exactly one other student, the person they consider their best friend. We may create a pair (a, b) only if either a chose b or b chose a. Every student can belong to at most one pair.
This problem asks us to collect the maximum possible number of cherries while making two trips across a square grid. The first trip starts at the top left corner (0, 0) and moves to the bottom right corner (n - 1, n - 1) using only right or down moves.
The problem asks us to identify all distinct integers that appear in at least two out of three given integer arrays. The input consists of three arrays nums1, nums2, and nums3, each containing integers in the range [1, 100].
This problem asks us to determine whether a rolling ball can stop exactly at a destination cell inside a maze. The maze is represented as a two-dimensional grid where: - 0 represents an empty space the ball can roll through - 1 represents a wall The important detail is that…
The problem asks us to implement a class RecentCounter that tracks the number of requests (or pings) made in the last 3000 milliseconds.
The problem asks us to design a special queue called an "Most Recently Used Queue", abbreviated as MRUQueue. Initially, the queue contains the integers from 1 to n in increasing order. The key operation is fetch(k), where k is 1-indexed. This operation does two things: 1.
The problem asks us to compute a clumsy factorial for a given positive integer n. A standard factorial multiplies all integers from n down to 1, but the clumsy factorial modifies the operations: instead of multiplying all numbers, it cycles through '', '/', '+', and '-' in…
The problem asks us to find the smallest positive integer x such that the product of all of its digits equals the given integer num.
Each input line describes the full path of one file inside a file system. A path looks like: The disk name is the root and is not considered a folder. Every component between the disk and the file is a folder. The last component is always a file.
The problem gives us a massive 1,000,000 x 1,000,000 grid. Each cell is identified by coordinates (x, y), and movement is allowed in four directions: up, down, left, and right. Some cells are blocked, meaning we cannot step onto them.
This problem gives us three inputs: - A sorted array nums - A lower bound lower - An upper bound upper The array contains unique integers, and every number in the array lies within the inclusive range [lower, upper].
We are given a string consisting of letters and digits, and we are asked to compute a special measure for every prefix called the _palindrome degree_.
The problem asks us to determine whether two strings are anagrams of each other. Two strings are considered anagrams if they contain exactly the same characters with exactly the same frequencies, but possibly in a different order.
We have an n × m chessboard. A billiard ball moves diagonally like a bishop, but unlike a bishop it reflects off the borders of the board. When it hits a vertical wall, the horizontal component of the direction flips. When it hits a horizontal wall, the vertical component flips.
We have a single row of hooks, numbered from 1 to n, which initially are all empty. Employees arrive and leave at specified times, and each arrival or departure is encoded by an employee ID: the first occurrence is an arrival, the second is the departure, and so on.
This problem asks us to determine, for every user in the marketplace system, whether the brand of the second item they sold matches their favorite brand. The database contains three tables: - The Users table stores user information, including their favoritebrand.
This problem is a variation of the classic House Robber dynamic programming problem, with one important twist: the houses are arranged in a circle rather than a straight line.
The problem gives us an integer array nums, and asks us to remove exactly one element from the array. After removing that element, the remaining elements shift left, which means their indices may change.
This problem asks us to query a database table named World and return a subset of countries that qualify as big countries.
The problem gives an array called temperatures, where each element represents the temperature recorded on a specific day. For every day, we must determine how many days in the future we need to wait until a warmer temperature appears.
The problem is asking us to design a system that tracks tweets by their timestamp and allows querying the number of tweets in specific time intervals, split according to a frequency (minute, hour, or day).
The turtle starts at coordinate 0 on a number line and initially faces the positive direction. Each command changes its state in one of two ways. If the command is F, the turtle moves one unit in the direction it is currently facing.
The problem asks us to count all the negative numbers in a 2D matrix grid where each row and column is sorted in non-inc
The problem is asking us to find the largest connected area of land in a 2D binary matrix. Each cell in the matrix represents either water (0) or land (1). A group of 1s forms an island if the 1s are connected 4-directionally (up, down, left, or right).
The problem asks us to count the number of ways we can partition a given array nums of positive integers into contiguous
The problem asks us to return every path from the root of a binary tree to each leaf node. A root-to-leaf path is formed by starting at the root node and continuously moving downward through child nodes until reaching a node that has no children.
This problem asks us to combine information from two database tables, Sales and Product, and produce a result containing the product name, the year of the sale, and the sale price for every sale record. The Sales table stores transactional information.
The problem provides a list of meeting intervals, where each interval is represented as [start, end]. Each pair describes the start time and end time of a meeting. The task is to determine whether a single person can attend every meeting without any scheduling conflicts.
In this problem, we are given an undirected graph represented as an adjacency matrix. Each node represents a computer in a network, and an edge between two nodes means those computers are directly connected. Some subset of nodes is initially infected with malware.
We are given three lowercase strings. We want to build a single string that contains all three as substrings, and we want this resulting string to be as short as possible.
We are given an array of integers and a fixed window size k. For each contiguous subarray (segment) of length k, we need to find the largest element that appears exactly once within that segment. If no element appears exactly once, we output "Nothing".
The problem gives us an integer array nums that may contain duplicate values. We must design a class that supports repeatedly selecting a random index for a given target value.
Here’s a full, detailed technical solution guide for LeetCode 756 - Pyramid Transition Matrix, following your formatting rules precisely.
The problem gives us a two dimensional grid of size m x n, where each cell contains a non-negative integer. We begin at the top-left corner of the grid, specifically at position (0, 0), and want to reach the bottom-right corner at position (m - 1, n - 1).
This problem asks us to compute the minimum number of moves a knight needs to travel from the origin [0, 0] to a target position [x, y] on an infinite chessboard.
We are given a snapshot of a tic-tac-toe board, represented as a 3×3 grid. Each cell is either empty (.), contains a cross (X), or contains a nought (0). The first player always places crosses, and the second player places noughts.
The problem is asking us to identify users who qualify as interview candidates based on their performance in LeetCode contests. We are provided two tables: Contests and Users.
You asked for the complete guide in a single response, but your previous message includes two separate LeetCode problems
We process students one by one. Each student either has a fixed house, represented by one of the letters G, H, R, S, or has ambiguous ancestry, represented by ?. A fixed student always goes into that house. A ? student behaves differently.
This problem asks us to find a contiguous subarray that contains only unique elements, meaning no value appears more than once inside that subarray. Among all such valid subarrays, we must return the maximum possible sum of its elements.
That’s a detailed reference guide with multiple required sections and full implementations. Before I start, please confi
The problem gives us an array jobDifficulty, where each element represents the difficulty of a job. The jobs must be com
This problem asks us to design a calendar system that supports adding time intervals and reporting the highest number of overlapping events seen so far. Each booking is represented as a half open interval [startTime, endTime).
We are asked to start with the number pair (1,1) and reach a pair where at least one of the numbers equals a given target n. At each step, we can add one number to the other to form a new pair. Concretely, if our current pair is (a, b), the next pair can be (a+b, b) or (a, a+b).
Edit The problem gives us a string s and asks for the minimum number of insertion operations required to transform the s
This problem asks us to find the minimum number of jumps required for a bug to move from position 0 to position x on a one dimensional number line. The bug follows several movement rules: - It may jump forward by exactly a units. - It may jump backward by exactly b units.
The problem asks us to find the length of the shortest contiguous subarray whose sum is at least k. We are given an integer array nums, which may contain both positive and negative numbers, and an integer k.
The problem describes an ant moving along a one-dimensional infinite line, starting at a boundary point, which we can co
It looks like your prompt accidentally concatenated two problems. I will answer the second one, LeetCode 173 - Binary Search Tree Iterator, following your required format.
The problem asks us to analyze sales data stored in a Sales table and extract the records corresponding to the first year each product was sold.
The problem asks whether a given undirected graph forms a valid tree. A tree is a special type of graph with two important properties: 1. The graph is fully connected, meaning every node can be reached from every other node. 2. The graph contains no cycles.
We are given a tree with n cities. Originally, city r1 is considered the capital, and for every other city we know its parent in the rooted tree. The value p[i] means that if we walk from the old capital toward city i, the last city before reaching i is p[i].
We have a row of n lamps, each initially either on or off. Each lamp has a number from 1 to n. We also have n keys numbered the same way. Pressing key i toggles every lamp whose number is divisible by i.
We are asked to construct a calendar from a list of city names. Each line of the calendar must contain exactly two city names separated by a given symbol, and all lines must have identical length. The calendar must use each city name exactly once.
The problem asks us to count how many permutations of the integers from 1 to n satisfy a special divisibility condition. A permutation is an arrangement of all numbers from 1 to n where every number appears exactly once.
This problem asks us to implement a read(buf, n) function using a restricted API called read4(buf4). The challenge is that we are not allowed to access the file directly, instead, we can only retrieve data in chunks of up to four characters at a time.
The problem gives us a string s of length n, where each character describes the relationship between two adjacent elements in a permutation. The permutation must contain every integer from 0 to n exactly once, so the permutation length is always n + 1.
We need to print a symmetric diamond made from numbers. The middle row contains numbers increasing from 0 up to n, then decreasing back to 0. Every row above and below follows the same pattern with a smaller maximum value.
The problem gives us a set of buckets where exactly one bucket contains poison. We need to determine which bucket is poisonous using the fewest number of pigs possible, under a strict time limit. Each pig can participate in multiple rounds of testing.
The problem gives us a collection of points on a 2D plane, where each point is represented as [x, y]. We need to determine whether there exists a vertical line, meaning a line parallel to the y-axis, such that reflecting every point across that line produces the exact same set…
The problem gives an integer array nums and an integer k. We must divide the array into exactly k non-empty contiguous subarrays. Among those subarrays, each one has its own sum, and the goal is to minimize the largest subarray sum.
The problem is asking us to identify all authors who have viewed at least one of their own articles. The input is a database table Views that contains information about articles, their authors, the viewers, and the dates on which the articles were viewed.
This problem provides two database tables, Candidate and Vote, and asks us to determine which candidate won the election. The Candidate table stores information about each candidate. Every candidate has a unique integer id and a corresponding name.
We are given a set of points in two-dimensional space, where each point can be thought of as a vector from the origin. Each vector has two coordinates, x and y, and we are allowed to independently flip the sign of each coordinate.
This problem gives us a row of seats represented by the array row, where each value is the ID of the person currently sitting in that seat. The row contains 2n people, meaning there are exactly n couples.
The problem gives us a string s that represents a license key. The string contains uppercase letters, lowercase letters, digits, and dashes. The existing dashes are only separators and do not necessarily represent the correct final grouping.
Here’s a complete, detailed technical solution guide for LeetCode 787, following your formatting instructions exactly. The problem asks us to find the cheapest flight cost from a source city src to a destination city dst while taking at most k stops along the way.
The problem gives us a string n representing a positive integer, and asks us to find the numerically closest palindrome that is not equal to the original number itself. A palindrome is a number that reads the same forward and backward. Examples include 121, 999, and 1331.
The problem asks us to determine whether an array contains a specific ordering pattern called a "132 pattern". A valid 132 pattern consists of three indices i, j, and k such that: - i < j < k - nums[i] < nums[k] < nums[j] The name "132" comes from the relative ordering of the…
The problem asks us to count all substrings in a string where every character inside the substring is identical. These are called homogenous substrings. A substring must be contiguous, meaning the characters must appear next to each other in the original string.
The problem describes a "strange printer" that can print sequences of the same character in one turn. The printer can overwrite any existing characters in the string.
The problem asks us to determine the maximum profit we can earn from at most two stock transactions given the daily prices of a stock in an array prices. A transaction is defined as buying once and selling once.
The problem asks us to determine the minimum total cost of cutting a wooden stick into pieces at specified positions. Th
The problem presents a binary matrix where each element is either 0 or 1. The goal is to transform this matrix into a well-isolated configuration, meaning that no 1 is 4-directionally connected to any other 1.
The problem gives us two arrays, values and labels, where each index represents a single item. The item at index i has a value values[i] and a category or group identifier labels[i].
This problem asks us to count how many contiguous subarrays of length k have an average value greater than or equal to a
The problem asks us to determine how many numbers in a given array nums have an even number of digits. The input is an a
The problem asks us to partition an array into the maximum possible number of contiguous chunks such that, after sorting each chunk independently and concatenating the results, the entire array becomes globally sorted. The key detail is that each chunk must remain contiguous.
The problem asks us to construct a binary tree from an input array arr of positive integers, where each integer represents a leaf node in an in-order traversal.
The houses and pipes form a directed graph with a very special structure. Every house can have at most one incoming pipe and at most one outgoing pipe. That restriction changes the graph from a general directed graph into a collection of independent chains.
This problem asks us to determine whether two strings become identical after processing backspace operations. Each string represents characters typed into a text editor, where the '' character means "delete the previous character", similar to pressing the backspace key on a…
The problem asks us to find all numbers within a given inclusive range [left, right] that satisfy the definition of a self-dividing number. A self-dividing number has two important properties: 1. Every digit inside the number must divide the number evenly. 2.
The problem asks us to implement a stack data structure while using only queue operations internally. A stack follows the Last-In-First-Out, or LIFO, principle. This means the most recently inserted element must be removed first.
This problem asks us to compute, for every node in a rooted tree, the smallest positive integer that does not appear in the subtree rooted at that node. We are given two arrays: parents describes the tree structure, and nums contains distinct genetic values assigned to each node.
This problem asks us to generate all unique combinations of exactly k distinct numbers chosen from the range 1 to 9 such that their sum equals n. Each number can only be used once in a combination. This means combinations like [1,1,5] are invalid because the number 1 is repeated.
This problem asks us to analyze a collection of user activity logs and determine how many users have a specific number of unique active minutes.
The problem gives us the root of a binary tree where every node stores an integer between 0 and 25. Each integer corresponds to a lowercase English letter: - 0 - 'a' - 1 - 'b' - ... - 25 - 'z' We need to construct strings that begin at a leaf node and move upward toward the root.
The problem asks us to design a data structure that continuously tracks the kth largest element in a stream of integers. We are given an integer k and an initial list of numbers called nums. After initialization, new numbers are added one at a time using the add method.
The problem is asking us to produce aggregated statistics on transactions, grouped by month and country, from the Transactions table.
We are given an integer array nums and an integer target. The goal is to find every unique quadruplet of numbers in the array whose sum equals the target value. A quadruplet consists of four elements: where all four indices are distinct.
This problem asks us to populate the next pointers of nodes in a perfect binary tree. A perfect binary tree has all levels completely filled; every internal node has exactly two children, and all leaves are at the same depth.
Each construction describes a staircase placed on a one-dimensional line of cells. A staircase starts at cell l, ends at cell r, and adds heights in an arithmetic progression.
The problem asks us to evaluate a boolean expression represented as a string. The expression can contain the literals 't' and 'f' for true and false, as well as three types of operators: logical NOT '!', logical AND '&', and logical OR ''. Each operator has a specific syntax: '!
The problem asks us to maximize the product of an array of non-negative integers after performing at most k increment op
We have two families of circles on the infinite plane.
The problem asks us to return the postorder traversal of a binary tree. A binary tree is made of nodes, where each node contains: - A value - A pointer to a left child - A pointer to a right child In postorder traversal, we must visit nodes in this exact order: 1.
This problem asks us to rotate a singly linked list to the right by k positions. A right rotation means that each node shifts one position toward the end of the list, and the last node wraps around to become the new head.
The problem is asking us to convert a given integer n from base 10 into another base k, then compute the sum of its digits in that base. For example, if n is 34 and k is 6, first we convert 34 into base 6, which yields 54, and then sum the digits: 5 + 4 = 9.
Yesterday the Gauls defeated n Roman soldiers, and n is guaranteed to be a prime number. Today they defeated m soldiers, where m n. We need to decide whether m is exactly the next prime number that comes immediately after n. The key detail is the phrase "next prime".
You are given a list of events, where each event is represented as: - startDay - endDay - value If you attend an event, you must attend the entire interval from startDay to endDay, inclusive.
The problem gives us a list of points on a 2D coordinate plane. Each point is represented as [x, y], where x and y are integer coordinates. Our task is to determine the maximum number of points that lie on the same straight line.
This problem asks us to compute a rolling seven day summary of restaurant revenue. The input table stores individual cus
We have an n × m grid. The king starts at the top-left cell (1,1) and must end there as well. Every other cell must be visited exactly once. Normal movement is allowed only between side-adjacent cells, but we may additionally install directed teleporters.
The problem asks us to determine the number of distinct binary trees that can be constructed from a given array of unique integers, where each integer is strictly greater than 1.
We are given an encrypted password represented as a binary string of length 80. The original password had exactly 8 decimal digits, and each digit was encoded into a block of 10 binary characters.
The problem asks us to determine whether a given numeric string remains valid and unchanged when rotated 180 degrees. This property is called being "strobogrammatic". A strobogrammatic number is formed only by digits that still represent valid digits after rotation.
The problem asks us to implement exponentiation manually, specifically computing: where x is a floating point number and n is an integer exponent. The input consists of two values: - x, the base - n, the exponent The output should be the result of raising x to the power n.
We are given a table where every row contains several string fields, and the first input line tells us the name of each column. Another line describes how the rows should be ordered. Each rule has the form COLUMNNAME ASC or COLUMNNAME DESC.
This problem asks us to modify a sorted integer array in-place so that every unique value appears at most twice. Since the array is already sorted in non-decreasing order, duplicate values will always appear next to each other, which is a very important property that makes an…
The problem provides a Delivery table that records food delivery orders. Each row represents a single order placed by a customer. Along with the order date, each customer also specifies a preferred delivery date.
This problem gives us a string s that contains only the characters 'L' and 'R'. The string is guaranteed to already be balanced overall, meaning the total number of 'L' characters equals the total number of 'R' characters.
This problem asks us to determine whether a singly linked list contains a cycle. A cycle exists when following the next pointers eventually leads back to a node that has already been visited, instead of reaching None. The input is the head node of a singly linked list.
We have a line of holes indexed from left to right. Every hole contains a jump length. If a ball is dropped into hole i, it immediately moves to i + a[i]. From there it jumps again using the value of the new hole, and this continues until the next jump leaves the array.
The problem asks us to generate a list of strings for all integers from 1 to n. For each number, we apply a set of divisibility rules: - If the number is divisible by both 3 and 5, we append "FizzBuzz". - If the number is divisible only by 3, we append "Fizz".
We are given two strings made of English letters. Both strings have the same length, but letters may appear in either uppercase or lowercase form. The task is to compare the two strings lexicographically while completely ignoring letter case.
We are asked to compute the minimum travel time for a car moving along a straight road from Berland to Bercouver. The road has length _l_, and at a distance _d_ from the start there is a speed sign that limits the car's instantaneous speed to _w_.
The problem gives us two integer arrays, nums1 and nums2, where every element is between 1 and 6. In one operation, we may choose any element from either array and change it to any value from 1 to 6.
This problem asks us to build a team of at most k engineers such that the team's performance is maximized. Each engineer
The problem asks us to determine whether a given string s can become a palindrome after removing at most k characters. A palindrome is a string that reads the same forwards and backwards, and a k-palindrome is one that can be turned into a palindrome with at most k deletions.
The task is to compute a kind of "distance" between a user-entered address and a list of potential addresses, according to a specific error function. The user enters a string s of length k. Then there are n potential addresses, each a string of arbitrary length.
This problem asks us to design a miniature peer-to-peer file sharing system. The file is divided into m chunks, where every chunk has an ID from 1 to m. Users can join the system while already owning some chunks, leave the system entirely, and request chunks from other users.
The problem presents two database tables: NPV and Queries. The NPV table contains historical net present value (NPV) dat
The problem asks us to group strings that belong to the same cyclic shifting sequence. Two strings belong to the same group if one can be transformed into the other by repeatedly shifting every character forward or backward in the alphabet, with wraparound between 'z' and 'a'.
The problem asks us to compute a simple mathematical transformation on the digits of an integer. Given an integer n, we
This problem asks us to construct an array of length n where every element is a positive integer, the difference between adjacent elements is at most 1, and the total sum of the array does not exceed maxSum.
We have an array of integers, and we want every element to become strictly larger than the one before it. The only operation allowed is choosing a single element and increasing it by exactly d. Every use of this operation counts as one move.
The problem provides an array of positive integers and asks us to perform operations to transform it so that two conditions are satisfied: the first element must be 1, and the absolute difference between adjacent elements cannot exceed 1.
This problem asks us to compute the number of distinct combinations of coins that can sum to a target amount. We are given an array coins, where each value represents a coin denomination, and an integer amount, which represents the target sum we want to form.
We are asked to simulate a fair random selection of one knight out of n using a coin that can only produce two outcomes. Each coin flip has an equal probability of landing heads or tails.
This problem is asking us to determine the minimum number of vertical arrows required to burst all balloons represented as intervals along the x-axis. Each balloon is defined by its start and end x-coordinates, [xstart, xend].
The problem gives an integer array nums where every value appears exactly twice, except for two numbers that appear only once. The task is to find those two unique numbers and return them in any order.
The problem asks us to compute the sum of two integers without using the arithmetic operators + and -. Instead of relying on normal arithmetic, we must use bit manipulation to simulate how addition works at the binary level. The input consists of two integers, a and b.
This problem asks us to build the lexicographically largest possible number of length k using digits taken from two arrays, nums1 and nums2. Each array represents a sequence of digits from a number.
The problem asks us to convert a positive integer into the column naming format used by Microsoft Excel. Excel labels columns alphabetically instead of numerically. The sequence begins as: The input, columnNumber, represents a 1-based column index.
The problem asks us to determine if two strings of equal length, s1 and s2, can be made identical by performing at most one string swap on exactly one of the strings. A string swap allows exchanging any two characters at different or same indices.
The problem asks us to find the minimum number of moves required to reach a specific position on an infinite number line starting from position 0. Each move i allows you to move exactly i steps, either to the left or the right.
The problem gives us two sorted integer arrays, nums1 and nums2, and asks us to return the k pairs with the smallest sums. A pair is formed by taking one element from nums1 and one element from nums2.
The problem asks us to find all integers within a given range [low, high] whose digits are sequential, meaning that each
This problem asks us to identify all sellers who did not make any sales during the year 2020. We are given three databas
We are asked to organize a company party such that no group contains both a manager and their subordinate, directly or indirectly. The input describes each employee's immediate manager: a number from 1 to n, or -1 if they have no manager.
This problem asks us to identify a special number in an array called a lucky integer. A lucky integer is defined as an i
The problem asks us to find the length of the longest strictly increasing subsequence in an array of integers. A subsequence is formed by deleting zero or more elements from the array without changing the order of the remaining elements.
The problem provides a database table named Sessions with two columns: | Column | Meaning | | --- | --- | | sessionid |
Edit The problem gives us an array hours, where each element represents the number of hours an employee worked on a specific day. A day is classified as tiring if the employee worked strictly more than 8 hours. Otherwise, the day is considered non-tiring.
We are given a sequence of points on the 2D integer grid: a starting point M0 and a sequence of n points A0, A1, …, An-1, where n is always odd. We then define a new sequence M1, M2, … where each Mi is the reflection of Mi-1 over one of the Ai points.
The problem asks us to design a class that processes stock prices one day at a time and, for every new price, returns the stock span for that day.
This problem is asking us to determine the maximum number of contiguous chunks into which we can split an array such that sorting each chunk individually and concatenating them results in a fully sorted array.
The problem asks us to find the Lowest Common Ancestor, usually abbreviated as LCA, of two nodes in a binary tree. A binary tree consists of nodes where each node may have a left child and a right child.
The Task Scheduler problem asks us to determine the minimum number of CPU intervals required to complete a list of tasks with a cooling constraint. Each task is represented by an uppercase letter A-Z. The CPU can execute one task per interval or remain idle.
The problem gives us a binary matrix, meaning every cell contains either 0 or 1. We are allowed to rearrange the columns of the matrix in any order we want.
The problem defines a special value called the rotation function for an array. Given an integer array nums of length n, we rotate the array clockwise by k positions to create a new array arrk.
This problem revolves around identifying which subtasks of a task were not executed. We are given two tables: Tasks and Executed.
The problem asks us to build a basketball team with the maximum possible total score, while ensuring that the team does not contain any conflicts.
Jack repeats a cyclic sequence consisting of three possible actions. L means a left-foot step, R means a right-foot step, and X means standing still for one beat. The sergeant expects the infinite alternating pattern:
This problem asks us to minimize the total number of ship trips required to deliver boxes from storage to ports, while r
We are asked to examine permutations of numbers from 1 to n and focus on "lucky numbers" - integers that contain only the digits 4 and 7.
The problem is asking us to simulate a single-tab browser with a history mechanism. You start on a homepage, and from there, you can visit new URLs, backtrack a certain number of steps, or forward a certain number of steps.
We start with a distinct boxes and b distinct items. Since boxes may stay empty and every item independently chooses one of the boxes, the number of possible placements is simply:
The problem presents three stones placed on distinct positions along a one-dimensional X-axis, represented by integers a, b, and c.
The problem gives an array called height, where each element represents the height of a vertical bar in an elevation map. Every bar has width 1. After rainfall, water may become trapped between taller bars. The task is to compute the total amount of water that can be trapped.
The problem gives us two binary search trees, root1 and root2. A binary search tree, commonly abbreviated as BST, has the important property that for every node: - All values in the left subtree are smaller than the node's value - All values in the right subtree are larger…
The problem gives us a database table named Cinema with two columns: Column Meaning --- --- seatid Unique identifier for a seat free Whether the seat is available, where 1 means free and 0 means occupied We need to find all seats that are part of at least one consecutive…
This problem gives us a directed graph with n nodes labeled from 0 to n - 1. Each directed edge [a, b] means there is a one way path from node a to node b.
The task is to take a positive integer represented in binary and determine how many steps it takes to reduce it to 1 using a simple iterative process. In each step, if the number is odd, we increment it by 1, and if it is even, we divide it by 2.
The problem is asking us to compute the time at which each car in a line of cars collides with the car immediately in front of it, or return -1 if it never collides. Each car is represented by its position and speed.
The problem gives us a singly linked list where every node contains a unique integer value. We are also given an array nums, and every value in nums is guaranteed to appear somewhere in the linked list.
We are given a string representing the genome of an organism, where each character is one of the first K capital letters. Adjacent genes contribute to the total “risk of disease” according to a given K × K matrix of non-negative integers.
This problem asks us to identify bank accounts whose monthly income exceeds a predefined limit for at least two consecutive months. We are given two tables: The Accounts table stores the maximum allowed monthly income for each account.
We need to construct a binary prefix code.
The problem gives us an array called heights, where each value represents the height of a student standing in a line. The school wants the students arranged in non-decreasing order, meaning heights should appear from smallest to largest, allowing duplicates.
We are asked to represent a positive integer given in binary as a sum of powers of two, with the option of using negative powers, such that the total number of terms is minimized. Formally, we want to write the number as a sum of expressions of the form +2^x or -2^x.
The problem gives us a list of query strings and a target camel case pattern. For each query, we must determine whether the query can be formed by inserting only lowercase English letters into the pattern.
The problem provides access to a single API, rand7(), which returns a uniformly random integer from 1 to 7. The task is to implement another function, rand10(), which must return a uniformly random integer from 1 to 10.
We are given a set of small arrays and a sequence of indexes indicating how to concatenate them into one larger array. Once the large array is built in this way, the goal is to find the maximum sum of a contiguous subarray.
We are given a small array of integers and need to find the smallest value that is strictly larger than the minimum element in the array.
This problem asks us to find the maximum sum of any two distinct numbers in an integer array nums such that their sum is strictly less than a given integer k.
This problem asks us to perform a postorder traversal on an n-ary tree. In a binary tree, each node has at most two children. In an n-ary tree, each node can have any number of children. Every node contains a value and a list of child nodes. A postorder traversal means: 1.
This problem gives us every node of an N-ary tree in an arbitrary order, and asks us to determine which node is the root
This problem asks us to simulate a game of Tic-Tac-Toe on a 3 x 3 grid given a sequence of moves. Each move specifies th
The problem provides a database table named Activity. Each row represents one login session for a player on a specific date.
The problem asks us to determine how many items in a list satisfy a given rule. Each item is represented as a list of three strings: its type, color, and name. The rule is given as two strings: ruleKey and ruleValue.
We have a tank that wants to move from point A to point B along the straight segment connecting them. The tank moves with constant speed $v$, which we must choose as small as possible. There are $n$ enemy tanks placed on the plane.
We are given an undirected graph. Each edge represents a segment drawn between two points. We may erase exactly one edge, and after removing it we want the remaining graph to become bipartite.
The problem asks us to traverse a jagged 2D integer array nums diagonally. Specifically, elements are accessed by their
This problem asks us to determine the maximum frequency of an element in an integer array nums after performing at most k increment operations. Each operation allows you to choose an element and increase it by 1.
The problem asks us to calculate aggregate monetary information for each product based on invoices. We have two tables:
The problem presents a grid where each cell represents a street segment with a specific orientation, denoted by a number
We are asked to help King Copa maximize his expected number of hits in a shooting gallery. The gallery is represented as a 2D plane, and each target appears at a specific point exactly at a specific time and disappears immediately afterward.
This problem asks us to determine whether a robot moving on an infinite two dimensional plane will remain within some bounded region if it repeats a sequence of instructions forever. The robot starts at coordinate (0, 0) facing north.
This problem asks us to add two polynomials represented as singly linked lists. Each node in the linked list represents a single term of a polynomial, with a coefficient and a power.
We are asked to compute the n-th Tribonacci number, but only its value modulo 26. The sequence starts with: $$t0 = 0,quad t1 = 0,quad t2 = 1$$ and every later value is formed by summing the previous three: $$ti = t{i-1} + t{i-2} + t{i-3}$$ The input contains a single integer n…
The problem gives us a sorted array of distinct integers and a target value. Our task is to determine where the target belongs in the array. If the target already exists, we return its index.
We are asked to verify a watering schedule for flowers over a set of consecutive holiday days. Each day must be watered exactly once. The schedule specifies, for each of several people, the range of days they are assigned to water the flowers.
The problem gives us an integer array nums and asks us to move every 0 element to the end of the array while preserving the relative order of all non-zero elements. The phrase "relative order" is extremely important.
This problem asks us to simulate a dynamic permutation of integers and answer position queries as elements continuously
The problem describes a robot moving on a rectangular m x n grid. The robot starts at the top left corner of the grid and wants to reach the bottom right corner. At every step, the robot is only allowed to move either one cell to the right or one cell downward.
The problem asks us to transform a binary tree into a formatted 2D string matrix representation. Each node must appear in a specific row and column based on its position in the tree. The output is not simply a traversal order, it is a visual layout of the tree structure.
This problem asks us to compute the daily cancellation rate for taxi trips over a fixed date range, specifically from "2013-10-01" to "2013-10-03".
We are given a weighted tree rooted at hall 1, where Gerald starts. The treasure is hidden uniformly at random in one of the other halls. Gerald only discovers the treasure when he first enters the correct hall.
This is an interactive problem where we must identify a hidden six-letter word from a given list of candidate words. We are not allowed to directly inspect the secret word. Instead, we can interact with the provided Master API by calling master.guess(word).
The problem asks us to implement a circular queue, a fixed-size queue where the end wraps back to the start to efficiently reuse empty space.
We are given a grid of size n × m, representing a board where Gennady has placed 28 domino chips. Each domino occupies exactly two adjacent squares, and the squares of the same domino are marked with the same letter, while different dominoes have different letters.
The problem presents an array arr of length n where each element is defined by the formula arr[i] = 2 i + 1. This genera
The problem gives us a line of washing machines, where each machine contains some number of dresses. In one move, we are allowed to choose any subset of machines, and every chosen machine may pass exactly one dress to one of its adjacent machines simultaneously.
This problem asks us to find the longest contiguous subarray of 1s in a binary array nums after deleting exactly one ele
The problem describes a room containing n light bulbs, where every bulb starts in the on state. There are four buttons available, and each button flips a specific subset of bulbs. Flipping means changing on to off or off to on.
We are given eight distinct points on the plane. The task is to split them into two disjoint groups of four points each. One group must form a square. The other group must form a rectangle.
The problem asks us to count how many strings of length n satisfy three conditions simultaneously: 1. The string must be lexicographically greater than or equal to s1 2. The string must be lexicographically less than or equal to s2 3.
We are given a string s composed of lowercase letters, with length n. The task is to determine if we can rearrange the letters so that for every prime p less than or equal to n, all positions in the string that are multiples of p can contain the same character.
The problem requires designing a data structure that can efficiently maintain and retrieve the first unique integer in a
The problem requires calculating the sum of a subrange of all possible contiguous subarray sums of a given array nums.
The problem gives us several classes, where each class is represented as [passi, totali]. The value passi tells us how many students currently pass the exam, while totali tells us the total number of students in that class.
This problem asks us to determine whether an entire input string s matches a wildcard pattern p. The pattern supports two special wildcard characters: - '?' matches exactly one character. - '' matches any sequence of characters, including an empty sequence.
The problem gives us an undirected graph with n original nodes. Each edge has an associated subdivision count, meaning the edge is replaced by a chain of intermediate nodes. For an edge [u, v, cnt], the original direct connection between u and v no longer exists as a single edge.
We are given Masha's phone number as a string of digits. From this number, she generates another phone number digit by digit. The first digit of the new number can be any digit from 0 to 9.
The problem asks us to perform a level order traversal of an n-ary tree. In other words, we need to return the values of the tree nodes grouped by their depth. The root node represents level 0, its immediate children are level 1, their children are level 2, and so on.
The problem requires transforming a rope of balloons into a "colorful" rope, which means no two consecutive balloons sha
The problem gives us an array of strings called words and an integer k. Each string represents a word, and words may appear multiple times in the array. Our task is to return the k most frequent words. The result must follow two ordering rules.
In this problem, we are simulating navigation inside a file system. The system starts at the main folder, and we are giv
The problem asks us to split a string into substrings such that every substring is a palindrome. Among all valid palindrome partitions, we must return the minimum number of cuts required. A cut divides the string into two parts.
The problem asks us to find the length of the longest contiguous substring in a string s that contains at most two distinct characters. A substring is a continuous sequence of characters inside the original string.
This problem asks us to design a thread-safe traffic light system for a road intersection where cars arrive concurrently
We are given an integer array and need to count how many contiguous subarrays are "magical". A subarray is magical when its minimum value is equal to its maximum value. A minimum and maximum can only be equal if every element inside the subarray is the same.
This problem asks us to determine how many different valid ways a numeric string can be decoded into letters using the mapping: - "1" → 'A' - "2" → 'B' - ... - "26" → 'Z' The input is a string s containing only digits.
We are asked to estimate the minimum number of presents the Hedgehog will receive over the next N days. He has two rules governing present reception: every holiday he receives a gift, and he cannot go more than K days without receiving one.
The problem asks us to find the lexicographically largest contiguous subarray of length k from a given array of distinct integers. Two arrays are compared lexicographically.
The problem gives us an n x n binary matrix called grid. Each cell contains either 0 or 1. We are allowed to perform operations where we swap two adjacent rows. The goal is to transform the matrix into a valid configuration using the minimum number of adjacent row swaps.
The problem asks us to make a string good, which means no two distinct characters in the string have the same frequency.
We have n walruses arranged in a circle. The presenter starts with m chips and distributes them in order. Walrus 1 receives 1 chip, walrus 2 receives 2 chips, and so on up to walrus n, after which the cycle repeats again from walrus 1.
Each lawn contains a distinct positive integer. Two lawns are considered connected if their numbers can appear together in some primitive Pythagorean triple.
The problem gives us two arrays: - boxes, where each value represents the height of a box - warehouse, where each value represents the height of a room in the warehouse Every box has width 1, and every warehouse room also has width 1.
This problem asks us to simulate one move in the classic Minesweeper game. We are given a two dimensional grid representing the current game board, along with the coordinates of a user click. Our task is to update the board according to the rules of Minesweeper.
The problem asks us to partition a binary array arr into three non-empty contiguous segments such that each segment, when interpreted as a binary number (allowing leading zeros), is equal to the other two segments.
The problem describes a scenario where we have a limited number of opportunities to invest in projects before an IPO.
This problem asks us to determine whether a binary tree contains at least one valid root-to-leaf path whose node values add up to a given target sum. A binary tree consists of nodes where each node can have a left child and a right child. Each node also stores an integer value.
This problem describes a hierarchical relationship between geographic regions. Each list in regions represents a parent-child relationship where the first element is the parent region and every remaining element in the list is directly contained within that parent.
That is a long, detailed reference document with multiple required sections, complete implementations, worked traces, and test coverage. I can provide the full guide in a single response.
The problem gives us a binary string s, which means the string contains only the characters '0' and '1'. In one operation, we are allowed to flip a character, meaning we can change '0' into '1' or '1' into '0'.
The problem asks us to construct a permutation of the integers from 1 to n such that the sequence of absolute differences between adjacent elements contains exactly k distinct values.
We are given several lowercase words and must shorten only the ones that are considered "too long". A word is too long if its length is greater than 10. The shortening rule is very specific.
The problem gives us two database tables, Visits and Transactions. The Visits table records every time a customer visite
This problem asks us to determine whether a robot, starting at the origin (0, 0) on a 2D plane, returns to the origin after executing a sequence of moves. Each move is represented by a character in a string: 'R' moves right, 'L' moves left, 'U' moves up, and 'D' moves down.
The problem asks us to rotate a given integer array nums to the right by k steps. In other words, each element of the array should be shifted k positions forward, and the elements that "fall off" the end wrap around to the front of the array.
The problem asks us to simulate a hiring process with a fixed salary budget of $70,000, where candidates are classified as either "Senior" or "Junior" and each has a unique salary.
This problem asks us to generate every possible subset of a given array of unique integers. A subset is any selection of elements from the array, including the empty subset and the subset containing every element.
We are given several strings of digits, all with the same length. We may choose one permutation of digit positions and apply it to every string. After rearranging the digits according to that shared permutation, each string becomes a new integer, possibly with leading zeroes.
This problem asks us to identify the project or projects that have the largest number of employees assigned to them. We are given two database tables. The Project table stores relationships between projects and employees.
We start with an initial string and pass it around a circle of n people. Every person is allowed to modify the string in only one of two ways: 1. Remove exactly one character from the end. 2. Add exactly one character to the end. A person may also choose to do nothing.
The problem gives us a list of email addresses and asks how many unique destinations actually receive emails after applying Gmail-like normalization rules.
Here is a complete, detailed technical solution guide for LeetCode 1386 - Cinema Seat Allocation in a single, comprehens
We are given a list of people standing in a fixed left-to-right order on a sinking ship. Every person has a name and a role. The evacuation order depends entirely on the role priority. Rats leave first. Women and children share the next priority level. Men leave after them.
We have a garden represented as a one-dimensional array of sections, each with a fixed height. Petya can create artificial rain above exactly one section, and water will flow to neighboring sections as long as their height is less than or equal to the section the water comes…
The problem asks us to count how many pairs of substrings, one taken from s and one taken from t, differ by exactly one
We start with a set of distinct lattice points on the plane. We may add more points, and the final set must satisfy a geometric condition for every pair of points. Take any two points.
Let's go step by step and create a complete technical solution guide for LeetCode 708 - Insert into a Sorted Circular Linked List following your formatting requirements. The problem asks us to insert a value into a sorted circular linked list such that the list remains sorted.
Here is the complete, detailed technical solution guide for LeetCode 1648 - Sell Diminishing-Valued Colored Balls follow
This problem asks us to determine whether the given array nums is both: 1. A valid shortest supersequence of all arrays in sequences 2. The only possible shortest supersequence A supersequence is a sequence that contains every sequence in sequences as a subsequence.
We have n towers on a 2D plane. Every tower must belong to exactly one of two generals. For each general, the cost he demands is the maximum Manhattan distance between any two towers assigned to him. The king only pays the larger of the two costs.
The problem asks us to implement integer division without using the standard arithmetic operators for multiplication, division, or modulo. We are given two integers, dividend and divisor, and we must compute the quotient obtained by dividing the dividend by the divisor.
The problem is asking us to implement an image smoother, a filter that modifies each cell in a 2D grayscale image based on the average of itself and its surrounding cells in a 3 x 3 window.
The problem gives us a string s containing only the characters 'a', 'b', and 'c'. We may repeatedly perform a deletion operation on the string.
The problem is asking us to identify the center node of a star graph. A star graph is a very specific type of graph with n nodes: there is one node called the center, and it is connected to all other n-1 nodes.
The problem is asking us to design a data structure that can efficiently answer queries about the leader of an election at any given time. We are given two integer arrays, persons and times.
This problem asks us to determine the minimum number of days required to make a given number of bouquets from a garden o
The problem asks us to identify all large groups in a given string and return their positions as intervals. A group is a sequence of consecutive identical characters.
The problem can be restated as simulating bullets flying along the positive _Z_ axis at given coordinates on a 2D shooting plane (_XOY_). Each bullet may hit one of multiple rectangular targets floating at distinct heights along the _Z_ axis.
The problem asks us to build a complete Sudoku solver. We are given a partially filled 9 x 9 grid where each cell contains either a digit from '1' to '9' or the character '.', which represents an empty space.
The problem asks for the minimum sum of any falling path through an n x n integer matrix. A falling path is defined as a sequence of elements starting from any element in the first row and moving row by row to the last row.
We are given a string of 2n digits. The digits are processed strictly from left to right. At every step, either Homer or Marge takes the current leftmost digit and appends it to their own number. By the end, both players must have taken exactly n digits.
This problem gives us a sorted array of integers named nums and a target integer named target. The array is sorted in ascending order, meaning every element is smaller than the elements that come after it. We must determine whether the target value exists in the array.
This problem asks us to construct a special ordering of all integers from 0 to 2^n - 1. The ordering must satisfy the properties of a circular Gray code sequence.
This problem asks us to return a specific row from Pascal's Triangle. The input, rowIndex, represents the zero-based index of the row we want to generate.
We are given a black-and-white photo represented as a grid of size n × m where each cell is either '1' for a white pixel or '0' for a black pixel.
The problem gives us a string s containing only lowercase English letters. We repeatedly look for two adjacent characters that are equal, remove both of them, and continue this process until no such adjacent duplicate pair exists.
We are given a string of lowercase letters, and the task is to find the length of the longest substring that occurs at least twice in the string. The repeated occurrences may overlap. The input is a single string of at most 100 characters.
The problem gives us an array of lowercase English words. We need to find two different words such that they do not share any common letters, then return the maximum possible product of their lengths. More formally, for every pair of indices (i, j) where i !
We are given a directed graph where nodes represent crossroads and edges represent one-way tramlines. The engine house is at node 1. Every node has at least one outgoing edge, so the tram is never trapped.
The problem asks us to find the longest "nice" substring within a given string s. A substring is considered nice if for every letter it contains, both its lowercase and uppercase forms appear. For example, "aAa" is nice because it contains 'a' and 'A'.
The problem gives us an array nums with exactly 2n elements. The structure of the array is guaranteed to follow a very specific pattern: This means the first half of the array contains all the x values, and the second half contains all the y values.
Each input line describes a geometric object in the plane. The equation
This problem asks us to identify students who are "quiet" across all exams they participated in. A student is considered
This problem is asking us to determine how to write a string s on multiple lines when each character has a specific pixel width and no line can exceed 100 pixels.
This problem asks us to compute several descriptive statistics-minimum, maximum, mean, median, and mode-from a very large sample of integers ranging from 0 to 255.
The task is to generate a multiplication table for numbers in a positional numeral system with a base k. Unlike the decimal system, the digits in this system range from 0 up to k-1. Petya wants to see products of numbers from 1 to k-1 expressed in this system.
The problem asks us to determine whether one string can be transformed into another string using repeated left rotations. A single shift operation removes the first character of the string and appends it to the end.
LeetCode 1349 is long and detailed enough that a complete high quality guide with all required sections, full prose expl
This problem asks whether two sequences of integers, pushed and popped, could represent valid operations on a stack.
The problem gives us a two dimensional grid representing a gold mine. Each cell contains either a positive amount of gold or 0, which means the cell is empty. We are allowed to start from any cell that contains gold and move through the grid collecting gold along the way.
The problem asks us to count the number of subarrays within a given integer array arr whose sums are odd. A subarray is defined as a contiguous segment of the original array. The input is a list of integers, and we need to return the count of subarrays whose total sum is odd.
We are asked to count numbers in given ranges that Volodya would call beautiful. A number is beautiful if it is divisible by each of its nonzero digits. For example, 128 is beautiful because 128 is divisible by 1, 2, and 8.
This problem asks us to maximize the number of satisfied customers in a bookstore over n minutes. We are given two arrays: customers, which indicates how many customers arrive each minute, and grumpy, which indicates whether the bookstore owner is grumpy (1) or not grumpy (0)…
The problem gives us a 2D grid called rooms, where each cell represents one of three possible states. A value of -1 represents a wall or obstacle that cannot be passed through. A value of 0 represents a gate.
The problem gives us a list of words and a target pattern string. We need to determine which words follow the exact same structural character relationship as the pattern.
The problem is asking us to find a "fixed point" in a sorted array of distinct integers. A fixed point is an index i such that the value at that index equals the index itself, i.e., arr[i] == i.
We are given three points in the plane. Each point is the midpoint of one side of an unknown strictly convex quadrilateral, and all four sides of that quadrilateral have equal length.
This problem asks us to find the k strongest values in an array according to a custom definition of "strength". The firs
The problem gives us a binary number represented as a string s. Our task is to determine how many operations are require
We need to count integers inside the interval $[a,b]$ whose smallest divisor greater than $1$ is exactly $k$. For a number $x$, saying that its smallest divisor is $k$ means two things happen simultaneously: 1. $x$ is divisible by $k$. 2.
The problem gives us the head of a singly linked list and asks us to return a random node value such that every node in the list has exactly the same probability of being selected. The class has two operations: 1. Solution(head) initializes the object with the linked list. 2.
LeetCode 428, LeetCode Serialize and Deserialize N-ary Tree, asks us to design a reversible encoding system for an N-ary tree. The problem is not asking for a specific serialization format.
We are given a line of mushrooms, each with a weight, initially sorted in non-decreasing order. Every minute, new mushrooms grow between every pair of neighboring mushrooms, and the weight of each new mushroom equals the sum of the two neighboring mushrooms.
The problem asks whether it is possible to make two strings equal by performing exactly one swap operation on the first string s. A swap operation means choosing two different indices i and j in s and exchanging the characters at those positions.
The problem asks us to repeatedly replace a number with the sum of its digits until the number becomes a single-digit number. The input is a number n that can be extremely large, up to 10 million digits.
We are given an integer array flowers, where each element represents the beauty value of a flower in a line. We may remove any flowers we want, while preserving the relative order of the remaining flowers. After removals, the remaining sequence must form a valid garden.
The game records a sequence of rounds. In each round, one player either gains or loses some number of points. At the end, the player with the highest total score should win.
The problem asks whether a given undirected graph is bipartite. In simpler terms, we are given a graph represented as an adjacency list graph, where graph[u] lists all nodes directly connected to node u.
This problem gives us a list of points on a 2D plane, where each point is represented as [x, y]. We need to find the widest vertical area that contains no points strictly inside it. A vertical area is defined by two vertical lines.
Edit This problem gives us a permutation array arr containing integers from 1 to n, where n is the size of the binary st
We are given a tree with _n_ nodes, described by _n-1_ edges. A tree is a connected acyclic graph, so there is exactly one path between any two vertices. Bob can delete any subset of edges, possibly none, and then he looks at the sizes of the resulting connected components.
We are given a rectangular grid representing a sheet of graph paper with _n_ rows and _m_ columns. Some of the squares are shaded, marked by *, and the rest are unshaded, marked by ..
The problem asks us to design a data structure called RandomizedCollection that behaves like a multiset. Unlike a normal set, duplicate values are allowed.
We have a large circle, the plate, with radius R. Inside it there is another circle, the Golden Honduras, with radius r. The Honduras circle is tangent to the plate from the inside, so its center is exactly R - r units away from the plate center.
The problem asks us to determine the minimum number of moves required to identify a critical floor in a building using a limited number of eggs.
The problem gives us an integer array nums and defines the "value" of the array as the sum of the absolute differences b
The problem presents an array arr where all elements are equal except for one element that is strictly larger than the o
We are given several closed intervals on the number line. A nail placed at an integer coordinate covers every segment that contains that coordinate, including endpoints.
The problem asks us to convert a non-negative integer into its English words representation. Instead of returning digits such as 12345, we must produce a properly formatted English phrase such as "Twelve Thousand Three Hundred Forty Five".
We are given a country with cities connected by roads such that the road network forms a tree: there is exactly one simple path between any two cities. Each road has equal length in terms of counting the number of edges.
This problem is asking us to find classes in a school database that have at least five students enrolled. The input is a table named Courses with two columns: student and class. Each row represents one student being enrolled in one class.
This problem asks us to find the Lowest Common Ancestor, or LCA, of multiple nodes in a binary tree. Unlike the classic
This problem asks us to design a data structure that continuously processes a stream of integers and supports querying t
This problem asks us to verify whether a list of words is sorted according to a custom alphabet order, instead of the normal English alphabetical order. In normal lexicographical ordering, characters are compared from left to right using the standard alphabet.
The problem asks us to find the length of the longest contiguous subarray in an integer array nums such that the absolut
The problem gives us an array of positive integers called sticks, where each value represents the length of a stick. We are allowed to repeatedly connect any two sticks together.
The problem is asking us to find the maximum sum of a contiguous subarray in a given integer array, with the additional twist that we are allowed to delete at most one element from that subarray.
This problem asks us to count all subarrays of a binary array nums that contain more 1s than 0s. In other words, for any contiguous slice of the array, if the number of 1s exceeds the number of 0s, it should be counted.
We are given a circular arrangement of hills around a capital. Each hill has a height and a watchman who can send signals via fire. The core question is: how many pairs of watchmen can see each other’s signals?
We are given a row of mice located along a horizontal line at coordinate y = Y0 and a row of cheese pieces along another horizontal line at y = Y1. Each mouse can run directly towards any piece of cheese, and all mice run at the same speed.
We need to place the integers from 1 to n * m into an n × m grid so that every pair of side-adjacent cells differs by at least 2. Two cells are adjacent only if they share an edge, diagonal neighbors do not matter.
The problem describes a sequence of n bulbs, all initially turned off. We perform n rounds of operations on these bulbs. In the first round, every bulb is turned on. In the second round, every second bulb is toggled, meaning on bulbs become off and off bulbs become on.
This problem asks us to count how many valid ways exist to assign hats to people under two constraints: 1. Every person
The problem gives us a collection of stones placed on a 2D grid. Each stone occupies a unique coordinate (x, y). A stone can be removed only if there is at least one other stone that shares either the same row or the same column.
The problem gives us two arrays, aliceSizes and bobSizes, representing the candy boxes owned by Alice and Bob. Each element in the arrays is the number of candies in a particular box.
The problem gives us the root node of a binary tree and asks whether the tree is "uni-valued". A binary tree is considered uni-valued if every node in the tree contains exactly the same integer value.
We are given a set of cups, each containing some amount of juice, and we want to determine whether the volumes could result from the pages pouring juice from one cup to another exactly once, or not at all.
We are asked to implement a simple memory manager for a linear memory array of size _m_. Each memory cell can either be free or occupied by a block. We are to process a sequence of operations: alloc n, erase x, and defragment.
The problem gives us a string s and a target character c. For every index in the string, we must compute the distance to the nearest occurrence of c. The distance between two indices is defined as: where i is the current index and j is the index of some occurrence of c.
The problem gives us a string s containing lowercase English letters and balanced parentheses. Our task is to repeatedly reverse the substrings enclosed inside every matching pair of parentheses, starting from the innermost pair first.
Scrooge signs papers by moving a pen along a polyline. The signature starts at the first point, then draws straight segments between consecutive points until the last point is reached.
The problem asks whether it is possible to reorder an even-length array arr of integers such that every element can be paired with another element that is exactly double its value.
The problem asks us to identify transactions that are potentially invalid based on two rules. First, any transaction with an amount greater than $1000 is invalid.
LeetCode 212, LeetCode Word Search II, asks us to find every word from a given dictionary that can be formed inside a 2D character grid. Each word must be built by moving one cell at a time horizontally or vertically.
The problem gives two geometric shapes: 1. A circle, defined by: - radius - center coordinates (xCenter, yCenter) 2. An axis-aligned rectangle, defined by: - bottom-left corner (x1, y1) - top-right corner (x2, y2) The goal is to determine whether the circle and rectangle overlap.
The task is to decompose a given integer n into its prime factors and print them in non-decreasing order, with each prime repeated according to its multiplicity. Essentially, if a number is a product of primes like $n = 2^2 cdot 3^1 cdot 5^2$, the output should be 2 2 3 5 5.
The problem asks us to construct an array containing exactly n unique integers whose total sum is equal to 0. The keywor
The problem gives us an integer num, and we are allowed to perform a digit replacement operation twice, independently. I
The problem asks us to reconstruct digits from a jumbled string of English letters representing numbers from 0 to 9.
This problem asks us to find a unique number in an array where every other number appears exactly three times. In simpler terms, if you imagine counting all the numbers in the array, every number except one will show up three times, and our task is to identify the number that…
The problem is a two-player game played on a binary tree, where each player colors nodes starting from an initial chosen node. Player 1 picks a node x and colors it red, while Player 2 picks a different node y and colors it blue.
We have two arrays representing the order of students in a line. Array a is the desired arrangement, and array b is the current arrangement. In one operation we may swap two neighboring students. We must output any sequence of adjacent swaps that transforms b into a.
The problem asks us to design a data structure that behaves like a key-value store, but with an important twist: the same key can have multiple values over time.
This problem asks us to select exactly three items from a store while satisfying both an index ordering condition and a
The problem gives an array nums containing 2n integers. Our task is to divide these integers into exactly n pairs. For each pair (ai, bi), we take the smaller value, min(ai, bi). After computing the minimum value from every pair, we sum all of those minimums together.
The problem asks us to find the largest value in each row of a binary tree, where a row is defined as all nodes at the same depth. The input is the root of a binary tree, which may contain up to 10,000 nodes, and each node's value is a signed 32-bit integer.
The problem asks us to construct the shortest possible string that contains both str1 and str2 as subsequences. A subsequence does not require characters to appear contiguously. Instead, the characters only need to appear in the same relative order.
The problem asks us to find the kth largest element in an unsorted integer array. The important detail is that we are looking for the element that would appear in the kth position if the array were sorted in descending order. We are not looking for the kth distinct value.
The problem asks us to find the length of the longest substring in which every character appears at least k times. A substring is a contiguous sequence of characters inside the original string.
The task is to find the next time on a 24-hour digital clock that reads as a palindrome. The input is a string formatted as "HH:MM", representing hours and minutes in 24-hour notation.
The problem asks us to find the minimum difference between values of any two nodes in a Binary Search Tree (BST). In other words, given a BST, we need to calculate the smallest absolute difference a - b where a and b are values of two distinct nodes.
We have a line with a series of discrete platforms. Each platform has a fixed length l and is separated from the next by a fixed distance m - l, so platform k occupies the segment from (k-1)m to (k-1)m + l.
The problem asks us to add two numbers that are represented in negabinary, which is base -2. Unlike standard binary (base 2), each bit in negabinary contributes a value multiplied by powers of -2.
We are given two uppercase strings, s and t. We may transform s using three operations: 1. Insert a character at any position. 2. Delete a character from any position. 3. Replace one character with another. Every operation costs exactly one move.
The problem asks us to compute, for every element in the array, how many elements to its right are strictly smaller than it. Given an array nums, we must return another array counts of the same length.
The problem gives us a mathematical expression as a string containing integers and arithmetic operators (+, -, ). We are asked to compute all possible results that can be obtained by inserting parentheses in every valid way.
This problem describes a group of people where some people are known to be richer than others. The richer array represents directed relationships between people. If richer[i] = [a, b], it means person a definitely has more money than person b.
Jack starts at position 0 on a number line. His jumps have fixed lengths: the first jump must have length 1, the second jump length 2, the third jump length 3, and so on. For every jump he may choose either direction.
The problem is asking us to transform the array arr1 into a strictly increasing array using the minimum number of operations. Each operation allows replacing an element of arr1 with an element from arr2.
The problem asks us to find the smallest positive integer that does not appear in an unsorted integer array. The key detail is that we only care about positive integers starting from 1.
The problem asks us to identify a redundant edge in a directed graph that originated as a rooted tree. A rooted tree has a single root with all other nodes having exactly one parent.
We have a rectangular grid. Each cell contains either a terrain type represented by a lowercase letter, the start cell S, or the target cell T. We may move in four directions between side-adjacent cells. Every move costs exactly one minute. The path must start at S and end at T.
We have n independent logs, each with length m. On a move, a player chooses one existing log and splits it into several equal pieces. If a log of length x is split into t equal parts, then t 1, t must divide x, and every resulting part must have length at least k.
The problem asks us to find the most experienced employees in each project from two database tables, Project and Employee. The Project table contains a mapping of employeeids to projectids, indicating which employees work on which projects.
The problem asks us to find the maximum average value among all contiguous subarrays of a fixed length k. In other words, given an integer array nums, we must examine every possible subarray whose size is exactly k, compute its average, and return the largest average found.
The problem is asking whether it is possible to convert one string s into another string t using at most k moves, follow
The problem gives us a special alphabet board and asks us to generate the shortest sequence of moves needed to spell a target string. The board contains lowercase English letters arranged in rows: We always begin at position (0, 0), which corresponds to the character 'a'.
The problem gives a list of envelopes, where each envelope is represented as a pair [w, h]. The value w is the width and h is the height. An envelope can fit inside another envelope only if both dimensions are strictly smaller.
The problem asks us to create a dessert using one ice cream base and zero or more toppings, with the additional restriction that each topping type can be used at most twice.
LeetCode 669, Trim a Binary Search Tree, asks us to modify a binary search tree so that every remaining node has a value within the inclusive range [low, high].
The problem gives us a 2D integer matrix called grid and an integer x. In one operation, we may either add x to a cell or subtract x from a cell. Our goal is to make every value in the grid equal using the minimum number of operations.
In this problem, every rabbit tells us how many other rabbits share its color. The input array answers contains these responses. If a rabbit says x, that means there are exactly x + 1 rabbits of that color group in total, including itself.
The problem describes a voting based ranking system where every voter ranks all teams from best to worst. Each vote is r
We are given a collection of n cities connected by n-1 roads. This means the current network forms a forest: a set of trees, because in graph terms, a connected tree with n nodes has exactly n-1 edges.
We are given a weighted tree with n cities and n - 1 roads. Every ordered pair of distinct cities represents one military campaign, so there are n (n - 1) total trips. For a trip from city u to city v, the army travels along the unique path between them.
The problem asks us to design a data structure that can efficiently answer multiple rectangular sum queries on a fixed 2D matrix.
This is a SQL database problem where we need to identify the employees who belong to the top three unique salaries within each department.
We are asked to find the minimum distance Shapur must travel to visit all cities at least once. The cities are connected in a tree structure, meaning there are exactly $n-1$ bidirectional roads and a unique path between any two cities.
This problem asks us to compare the average salary of each department against the company's overall average salary for each month. We are given two tables: Salary and Employee.
This problem gives us a collection of 3D cuboids, where each cuboid is represented by three dimensions: width, length, a
You are given a sequence of roads that must be traveled in order. Each road has a distance, and you travel at a fixed sp
We start with an undirected graph that may be disconnected and may contain cycles. We are allowed to repeatedly merge two vertices into one. Merging decreases the number of vertices by one, while the number of edges stays unchanged.
This problem asks us to design a simplified file system that supports two operations: 1. Creating a new path with an associated integer value. 2. Retrieving the value stored at a path. A path behaves similarly to a directory structure in a real operating system.
The problem gives us an integer array nums in which the largest value is guaranteed to be unique. Our task is to determine whether this largest number is at least twice as large as every other number in the array.
We have an undirected graph where each vertex stores some positive integer, the number of mushrooms on that lawn. For every known edge, we are given two values: the gcd of the two endpoint values and the lcm of the two endpoint values.
We are given three integers, A, B, and n. The task is to find an integer value X such that:
We are given an array of positive integers. For every position, we remove that element and compute the arithmetic mean of the remaining numbers. We must find all positions where the removed value itself is exactly equal to that mean. Suppose the array is [1, 2, 3, 4, 5].
The problem asks us to determine whether a given matrix satisfies the Toeplitz property. A matrix is considered Toeplitz if every diagonal running from the top-left corner toward the bottom-right corner contains identical values.
The problem asks us to clean and aggregate sales data from a Sales table. Each row contains a unique saleid, a productna
The problem gives us two integer arrays, arr1 and arr2, both having the same length. We must compute the maximum possible value of the following expression across every pair of indices (i, j): The task is not asking for the indices themselves, only the largest achievable value.
The problem asks us to maximize the total points we can collect from a row of cards, where each card has a point value.
The problem gives us an integer array nums and asks us to count how many pairs of indices (i, j) satisfy two conditions: 1. nums[i] == nums[j] 2. i < j Such pairs are called "good pairs".
Here’s a fully detailed, reference-style solution guide for LeetCode 1609 - Even Odd Tree, following all your formatting
We are given the name of a user on a social network and a sequence of activity messages between users. Every activity contributes a certain number of points between the two people involved.
We are given a rectangular board where some cells are missing. Every remaining cell is already colored either black or white in a chessboard pattern. The task is to cover all existing cells using straight triminoes of size 1 × 3 or 3 × 1.
The problem asks us to move n chips located at various positions along a one-dimensional line so that they all end up at the same position. Each chip can be moved either by 2 units at zero cost or by 1 unit at a cost of 1.
This problem asks us to count how many subsets of crimes satisfy two constraints simultaneously: 1. The total number of members used is at most n 2.
We are given the full results of a football tournament where every pair of teams plays exactly one match. For each match, we know how many goals each side scored. Using the tournament rules, we must determine which teams advance to the knockout stage.
We are asked to implement a simplified autocomplete function. The input consists of a string s, which represents the text the user has typed so far, followed by a list of previously visited pages.
In Baden, the unit conversion rules are unusual. One inch equals 3 centimeters, and one foot contains 12 inches. We are given a length in centimeters and must express it as feet and inches. The tricky part is the rounding rule.
This problem provides a database table named Users with two columns: userid and name. The userid column uniquely identifies each user, while the name column stores a person's name using a mixture of uppercase and lowercase English letters.
The problem asks us to simulate a process of pouring champagne into a pyramid of glasses. Each glass can hold exactly one cup of champagne, and any excess from a glass flows evenly to the two glasses immediately below it.
The problem gives us an integer array nums and an integer k. For every element in the array, we are allowed to modify it once by adding any integer value in the range [-k, k].
This problem gives us a single database table named Employee. Each row represents one employee and contains four pieces of information: Column Meaning --- --- id Unique identifier for the employee name Employee name salary Employee salary managerId The id of that employee's…
The problem asks us to determine whether a given integer becomes a different valid number after being rotated by 180 degrees. Not every digit remains valid after rotation.
The problem asks us to insert HTML-style bold tags into a string whenever a substring matches any word from a given dictionary of words.
The problem asks us to find the length of the longest consecutive sequence path in a binary tree, where a consecutive sequence path is defined as a path in which the values increase by exactly one from parent to child.
The problem gives us the head of a singly linked list where every node contains either 0 or 1. These values together rep
The problem asks us to take a valid parentheses string seq and split it into two disjoint subsequences A and B such that each subsequence is itself a valid parentheses string (VPS).
The problem gives us a string that contains encoded patterns of the form k[encodedstring]. The integer k tells us how many times the substring inside the brackets should be repeated. Our task is to fully decode the string and return the expanded result.
The problem asks us to count the number of strings of length n composed only of vowels a, e, i, o, u such that each stri
The problem gives us an array of non-negative integers called nums. We need to count how many index pairs (i, j) satisfy the following conditions: - i < j - nums[i] + rev(nums[j]) == nums[j] + rev(nums[i]) Here, rev(x) means reversing the digits of the integer.
We are given a string of characters consisting of lowercase letters and digits. The task is to count the number of ordered pairs of positions in the string where the characters at those positions are identical.
The problem gives us a string s that contains only the characters 'a' and 'b'. In one operation, we are allowed to remov
The problem gives us two database tables, Users and Transactions, and asks us to compute the final account balance for every user after applying all recorded transactions. The Users table contains the starting credit balance for each user.
The problem asks us to determine the minimum depth of a binary tree. In other words, given the root of a binary tree, we want to find the shortest path from the root node down to the nearest leaf node.
The problem is asking us to rearrange a list of barcodes so that no two adjacent barcodes are the same. The input is an array barcodes of integers where each integer represents a type of barcode. The output should be a rearranged array that satisfies the adjacency constraint.
The problem asks us to count how many distinct phone numbers of length n can be generated by moving a chess knight across a numeric keypad. The keypad layout looks like this: A knight moves in an L-shape. From any current digit, it can jump only to specific other digits.
We are given two non-negative integers, A and B. We need to construct two other non-negative integers, X and Y, such that: - their sum equals A - their bitwise xor equals B Among all valid pairs, we must output the one with the smallest possible X.
The problem gives us an integer array nums, a value called target, and an index called start. We need to find an index i where nums[i] == target and the distance between i and start is as small as possible.
The problem asks us to convert every uppercase English letter in a string into its lowercase equivalent. Any character that is already lowercase, or is not an alphabetic character at all, should remain unchanged. The input is a string s containing printable ASCII characters.
The problem asks for the length of the longest arithmetic subsequence in a given array arr such that the difference between consecutive elements equals a specified integer difference. A subsequence can skip elements but must preserve the original order.
The problem requires classifying triangles based on the lengths of their three sides, represented by the columns A, B, a
The problem defines an "ugly number" as any positive integer that is divisible by at least one of the three given integers a, b, or c.
The problem asks us to design a data structure that can repeatedly return a random integer from the range [0, n - 1], while excluding all integers that appear in a blacklist. Every valid number must have exactly the same probability of being chosen.
The problem asks us to design our own HashSet implementation without using any built in hash table libraries such as Python's set or Go's built in map type as the primary solution idea. A HashSet is a data structure that stores unique values.
This problem asks us to find the mode(s) in a Binary Search Tree (BST). A mode is the value that appears most frequently in the tree. Since duplicates are allowed in this BST definition, a value may occur multiple times. The input is the root node of a BST.
The problem gives us a source string s and a list of candidate words called dictionary. We must determine which dictionary word can be formed by deleting characters from s without changing the relative order of the remaining characters.
The task gives spreadsheet cell coordinates written in one of two formats, and for every coordinate we must convert it into the other format.
In this problem, we are standing at a fixed position on a 2D plane, represented by location = [posx, posy]. Around us, there are multiple points, each with integer coordinates. We are allowed to rotate in place, but we cannot move.
Here is a comprehensive solution guide for LeetCode 1340 following your requested format. The problem presents an array
The problem asks us to determine how many starting indices in an array allow reaching the last element by performing a series of jumps defined as either odd-numbered or even-numbered.
The problem asks us to determine the maximum number of times the word "balloon" can be formed using the characters from a given string text. Each character in text can be used at most once per occurrence of the word.
The problem asks us to find the Lowest Common Ancestor, commonly abbreviated as LCA, of two nodes in a Binary Search Tree (BST). A lowest common ancestor of two nodes p and q is the deepest node in the tree that has both p and q as descendants.
The office is represented as a rectangular grid. Every uppercase letter represents part of a desk, and all cells with the same letter belong to the same rectangular desk. The president’s desk color is given as c.
The problem asks us to multiply two non-negative integers where each integer is provided as a string instead of a numeric type. The result must also be returned as a string.
We are given all pairwise unions of some hidden disjoint sets. The original sets themselves are not shown. Suppose the hidden sets are $S1, S2, dots, Sn$. For every pair $i neq j$, we are given the set $Si cup Sj$.
We have an n × m grid. Every cell must contain one of four corner-shaped pipe pieces. Each piece connects exactly two adjacent sides of the cell.
This problem asks us to generate all possible synonymous variations of a given sentence based on a list of equivalent wo
This problem asks us to determine the minimum number of character replacements needed to transform string t into an anagram of string s. Two strings are anagrams if they contain exactly the same characters with the same frequencies, regardless of order.
The problem asks us to split a positive integer n into the sum of at least two positive integers, then maximize the product of those integers. In other words, we are not allowed to keep the number as-is.
The problem is asking us to determine if a set of equations between single-letter variables can all be satisfied simultaneously. Each equation is either of the form "xi==yi" or "xi!=yi", where xi and yi are lowercase letters from 'a' to 'z'.
The problem asks us to group the nodes of a binary tree by their vertical columns and return those groups from left to right.
We are asked to determine the maximum number of banana pieces Maria can produce from a birthday cake decorated with n non-overlapping circular banana pieces using exactly k straight-line cuts. Each circle has a center coordinate (x, y) and a radius r.
We are asked to count the number of simple cycles in an undirected graph. A simple cycle is a closed loop where no vertex or edge is repeated.
The problem asks us to generate every unique palindrome that can be formed by rearranging the characters of a given string. A palindrome is a string that reads the same forward and backward. For example, "abba" and "racecar" are palindromes.
The problem asks us to find the maximum number of times a string str2 can be obtained from another string str1 when both are repeated multiple times. Specifically, str1 is the string s1 repeated n1 times, and str2 is the string s2 repeated n2 times.
This problem asks us to count how many pairs of rectangles are interchangeable. Each rectangle is represented by two integers, width and height. Two rectangles are considered interchangeable if their width-to-height ratios are exactly the same.
This problem is asking us to find the convex hull of a set of points on a 2D plane. The input trees is a list of coordinates where each coordinate [xi, yi] represents the location of a tree in the garden.
This problem asks us to remove duplicate values from a sorted singly linked list so that every distinct value appears exactly once.
The problem gives an integer array nums and asks us to find a contiguous subarray that has the largest possible sum. A subarray means the elements must appear next to each other in the original array. We are not allowed to rearrange elements or skip positions arbitrarily.
This problem asks us to determine the minimum number of buses required to travel from a given starting bus stop (source) to a destination bus stop (target) given a set of bus routes.
This problem asks us to split a collection of steel rods into two groups such that both groups have exactly the same total height. Among all possible equal-height pairs, we want the maximum achievable height. Each rod can be used in one of three ways: 1.
We are given a binary grid. Cells containing '1' form usable tiles, while '0' cells are blocked. We want the longest cycle made entirely from '1' cells. The cycle must behave like a simple closed curve on the grid.
The problem asks us to check whether all 1s in a given binary array nums are separated by at least k positions. In other
The problem gives us an m x n matrix where every cell initially contains 0. We need to design a data structure that supports two operations efficiently: 1.
The problem gives us a connected, weighted, undirected graph with n vertices and a list of edges. Each edge is represented as [u, v, weight], meaning there is a bidirectional connection between vertices u and v with the given cost.
This problem asks us to calculate the total storage volume occupied by products inside each warehouse. We are given two database tables: The Warehouse table tells us which products are stored in each warehouse and how many units of each product exist there.
Marina can pick any subset of flowers from the field. Each flower has a certain number of petals, and she will pluck all petals from all chosen flowers one by one. The phrases alternate between "Loves" and "Doesn't love", starting from "Loves" on the first petal.
We are given the value of the largest coin denomination, n. We must build a sequence of distinct coin values such that every larger coin is divisible by every smaller coin. Among all valid sequences, we want the one containing the maximum possible number of coins.
The problem gives us a two dimensional matrix of integers with m rows and n columns. We must modify the matrix in place so that whenever a cell contains 0, every element in that cell's entire row and entire column also becomes 0.
The problem is asking whether an array of integers nums can be divided into exactly k subsets such that each subset has the same sum. The input consists of the integer array nums and the integer k.
The problem gives an array nums consisting of positive integers. We must find the longest prefix of the array such that, after removing exactly one element from that prefix, every remaining number appears the same number of times.
This problem asks us to manipulate a binary tree so that its pre-order traversal matches a given sequence called voyage.
The problem gives an array stones representing positions of stones on the X-axis, where each position is unique. The goal is to move the endpoint stones-the stones at the smallest and largest positions-so that eventually all stones occupy consecutive positions on the X-axis.
The problem asks us to generate a valid Gray code sequence for a given number of bits, n. A Gray code sequence is a special ordering of numbers where every adjacent pair differs by exactly one bit in binary form.
The problem asks whether a given integer n can be expressed as the sum of distinct powers of three. A power of three is any number of the form: where x is a non-negative integer. That means the valid powers are: - 3^0 = 1 - 3^1 = 3 - 3^2 = 9 - 3^3 = 27 - and so on.
This problem asks us to implement our own linked list data structure from scratch without using any built in linked list library.
We are asked to simulate a two-player game involving numbers represented as nine-digit strings. The first player selects a string s1 that represents a number not exceeding a given value a. The second player responds with a string s2 representing a number not exceeding b.
The problem gives us an integer array nums and a target integer x. In one operation, we are allowed to remove either the leftmost element or the rightmost element from the array. Whenever we remove a value, we subtract it from x.
We are given the heights of soldiers standing in a circle. Two soldiers can form a reconnaissance unit if they stand next to each other and their height difference is as small as possible among all neighboring pairs.
The problem requires us to add one to a non-negative integer that is represented as a singly-linked list. Each node in the list contains a single digit, and the head of the list corresponds to the most significant digit.
This problem asks us to examine the smallest number in the input array and determine whether the sum of its digits is odd or even. We are given an integer array nums. The task consists of three clear steps: 1. Find the minimum integer in the array. 2.
We are given a set of C-style #define macros and an expression, and we are asked to determine whether the expression becomes "suspicious" after macro substitution.
We are given a rectangular picture represented as an A × B grid of letters. The task is to determine how many ways we can cut this picture into smaller rectangular pieces such that each piece is unique up to rotations, and to identify the smallest possible piece size among…
The problem asks us to determine how many unique Morse code transformations exist among a list of words. Each lowercase English letter maps to a specific Morse code representation.
The problem asks us to arrange a sequence of books on a bookshelf with multiple shelves while minimizing the total height of the bookshelf. Each book is described by its thickness and height, and the books must be placed in the given order.
Thumbelina wants to cross a swamp by riding a single frog. The swamp contains hills numbered from 1 to n, where hill i is exactly i meters away from the island. A frog with jump length d lands on hills d, 2d, 3d, ...
We are given a connected undirected graph representing a road network. Every junction is a vertex, every road is an edge, and there is at most one edge between any pair of vertices.
The problem is asking us to compute a score for a string of balanced parentheses according to a set of rules. A balanced parentheses string is one in which every opening parenthesis '(' has a corresponding closing parenthesis ')' and the pairs are properly nested.
The problem is asking us to design a specialized queue that allows insertions and removals not only at the front and bac
This problem asks us to connect all given points on a 2D plane with the minimum possible total cost. Each point is repre
The problem gives us a grid called heights, where each cell contains an integer representing elevation. We start at the top-left corner (0, 0) and want to reach the bottom-right corner (rows - 1, columns - 1).
We are given a string template consisting of lowercase letters and question marks. The final string must satisfy three conditions simultaneously. First, it must be a palindrome, so characters mirrored around the center must match.
Valera has a shopping list containing a number of fruits, possibly with duplicates if he wants more than one of the same type. At the market, the stall has _n_ types of fruits and _n_ price tags, but the tags are not yet attached to the fruits.
The problem asks us to design a data structure that supports weighted random sampling. You are given an array w, where each element represents the weight of an index. Instead of choosing every index with equal probability, the selection probability depends on its weight.
The problem asks us to compute the top five average score for each student given a list of [ID, score] pairs. Each ID represents a unique student, and score represents a single score they received.
The problem gives us a deck of unique integer cards and asks us to arrange the deck so that a very specific reveal process produces the cards in increasing order. The reveal process works like this: 1. Reveal the top card and remove it from the deck. 2.
The input describes an undirected tree with n cities. Since the graph is a tree, there are exactly n - 1 edges and there is a unique simple path between every pair of cities. The problem asks us to examine every possible connected subset of cities.
We are asked to plan snow purchases over n days from m companies, ensuring we buy exactly W cubic meters each day. Each company produces a fixed daily amount w[i], but the cost of all snow from that company decreases linearly: c[i] on day 1, c[i] - a[i] on day 2, and so on.
We receive a single string containing uppercase and lowercase English letters. The task is to transform this string according to three rules. First, every vowel must be removed. The vowels in this problem are A, O, Y, E, U, I in both uppercase and lowercase forms.
The problem gives us a strictly increasing array called target and an integer n. We are also given access to a stream of integers starting from 1 and ending at n. The numbers arrive in order, and once a number is skipped, we cannot go back to it.
Each ray enters the box through one hole on the left side and exits through one hole on the right side. The order of holes on both sides matters.
We have a grid representing a basement. Some cells are walls, some are walkable, and exactly one walkable cell is the exit. A robot starts in an unknown walkable cell. We are given a fixed sequence of movement commands such as L, R, U, D.
The problem gives us two strings, word1 and word2. We must choose a non-empty subsequence from each string, concatenate them together, and form a palindrome. Our goal is to maximize the length of that palindrome. A subsequence does not require characters to be contiguous.
We are asked to compute the maximum number of “tentacles” that can traverse a cylindrical grid from the leftmost column to the rightmost column, given capacities for every corridor connecting adjacent cells.
The problem asks us to construct a maximum binary tree from a given integer array nums with unique elements. A maximum binary tree is a binary tree where each node is the maximum element of the subarray it represents.
We are given an undirected connected graph representing roads between junctions. The mayor wants to remove some roads so that the remaining graph becomes a tree, meaning it stays connected and contains exactly n - 1 edges.
This problem asks us to find, for each interval in a list, the "right interval" that starts at or after the end of the current interval and is the closest such interval in terms of starting time.
This problem gives us a directed graph represented as an adjacency list. Each node represents a state, and each directed edge represents a possible transition from one node to another. The input graph[i] contains all nodes that can be reached directly from node i.
We are given an array of positive integers, and we want to sort it in non-decreasing order. The twist is that we can only swap elements if at least one of the two numbers involved is lucky. A lucky number is defined as a number containing only the digits 4 and 7.
We start with an empty string. The first player picks any single letter that appears somewhere inside at least one dictionary word. After that, players alternately extend the current string by adding exactly one character either to the front or to the back.
We have two sets of points on the plane. Red points may be used as triangle vertices, blue points are obstacles. No three points are collinear, which removes all degeneracies involving points on triangle borders.
The problem gives us a directed graph with n nodes labeled from 0 to n - 1. Every edge in the graph has a color, either red or blue.
The problem asks us to determine the number of ways to paint a grid of size n × 3 using exactly three colors-Red, Yellow
The problem asks us to calculate the sum of beauty for all substrings of a given string s. The beauty of a substring is defined as the difference between the highest frequency and the lowest frequency of any character that occurs in that substring.
Here is a complete, detailed technical solution guide for LeetCode 1673 - Find the Most Competitive Subsequence followin
The problem asks us to determine the number of similarity groups within a list of strings. Two strings are defined as similar if they are either identical or if we can swap exactly two letters in one string to make it equal to the other.
This problem asks us to compute the diameter of an N-ary tree. An N-ary tree is a tree where each node can have any numb
The problem is asking us to find the maximum path sum in a binary tree. A path is defined as any sequence of nodes connected by edges, where each node is included at most once. The path does not need to start at the root or end at a leaf.
The problem is asking us to determine whether two sentences can be made identical by inserting a contiguous sequence of words (possibly empty) into one of them. Each sentence is a string of words separated by single spaces.
This problem asks whether a linked list appears as a continuous downward path inside a binary tree. The path does not ne
The problem asks us to design a data structure that simulates a royal inheritance system. There is a king at the top of the family tree, and over time people can be born or die. We must support three operations efficiently: 1. Add a child to an existing person. 2.
This problem models a directed weighted graph. Each node represents a computer in the network, and each directed edge represents the time required for a signal to travel from one node to another.
This problem asks us to compute the transpose of a given two dimensional matrix. A transpose operation flips a matrix across its main diagonal.
The problem gives us a sorted integer array nums, arranged in non-decreasing order. For every index i, we must compute t
The problem gives us a string s and three parallel arrays: - indices[i] tells us where a replacement might happen - sources[i] is the substring we expect to find at that index - targets[i] is the string we should replace it with if the match is valid For each operation, we…
In this problem, we are given an integer n, and we must count how many numbers in the range [1, n] are considered confusing numbers. A number becomes a confusing number if, after rotating every digit by 180 degrees, the resulting number is both: 1. Valid after rotation 2.
This problem asks us to remove the minimum number of columns from a list of equal length strings so that the resulting array of strings becomes lexicographically sorted.
The problem gives us a two-dimensional binary grid where each cell contains either 0 or 1. A value of 1 represents the home location of a friend, while 0 represents an empty cell.
This problem asks us to determine whether a circular array contains a valid cycle under a specific movement rule. Each element in the array represents how far we move from the current index.
We are given a chess position containing exactly four pieces on a standard 8×8 board. White has two rooks and one king, black has only a king. The position is already legal, meaning no two pieces share a square and the two kings are not adjacent.
This problem is asking us to calculate a retention metric from a table of player activities. Specifically, we need to determine the fraction of players who log in on the day immediately following their first login.
The problem is asking us to identify all playback sessions during which no advertisements were shown. We are given two tables: Playback and Ads. The Playback table lists sessions for each customer, with the start and end times of each session.
The problem gives us an encoded string s that represents a potentially enormous decoded string. The decoding process works incrementally from left to right. When we encounter a letter, we append it directly to the decoded tape.
The problem gives us an 8 x 8 chessboard represented as a matrix of characters. Each cell contains one of four possible values: - 'R' represents the white rook - 'p' represents a black pawn - 'B' represents a white bishop - '.
The problem asks whether we can divide the given array nums into two non-empty groups such that both groups have the same average. Suppose the array is split into subsets A and B.
The problem gives us a database table named Person with two columns: Column Description --- --- id A unique integer identifier for each row email The email address associated with that row The goal is to find all email addresses that appear more than once in the table.
The problem gives us two strings, start and result, both consisting only of the characters 'L', 'R', and 'X'. We are allowed to transform the start string using only two kinds of moves: - Replace "XL" with "LX" - Replace "RX" with "XR" The goal is to determine whether it is…
We are given four stick lengths, and we must choose exactly three of them. Depending on the relationship between those three lengths, there are three possible outcomes.
We are asked to find an integer point on a straight line described by the equation Ax + By + C = 0. The inputs are three integers, A, B, and C, which define the slope and position of the line.
We are given a rectangular garden represented as an n×m grid. Each cell contains either grass, which does not require mowing, or weeds, which do. We start at the top-left corner of the garden, always on grass, and initially facing right.
The problem is asking us to find the length of the longest good palindromic subsequence in a given string s. A good pali
The problem is asking us to identify the largest integer in an array that occurs exactly once. In other words, among all integers that are unique (appear only one time), we need to find the maximum. If no such integer exists, the output should be -1.
This problem asks us to rank integers according to a special metric called the "power value". The power value of a numbe
This problem asks us to select a subset of words that can be constructed using a limited supply of letters, such that th
This problem asks us to determine the distance between two nodes in a binary tree, given their values p and q. The distance is defined as the number of edges in the shortest path connecting the two nodes.
The problem asks us to find the minimum size of a set of integers, nums, such that each given interval [starti, endi] contains at least two integers from nums.
This problem asks whether we can satisfy a set of customer requests using repeated integers from the array nums. Each customer wants a certain quantity of numbers, given by quantity[i]. The important restriction is that every number given to a single customer must be identical.
The problem asks us to reverse only the English letters in a string while keeping all non-letter characters fixed in their original positions. In other words, letters move, but symbols, digits, and punctuation marks stay exactly where they started.
We are asked to simulate a queue of customers, where each customer has a known service time. The cashier can serve two people simultaneously, and the time to serve two people at once is the maximum of their individual times.
The problem gives us a binary array nums, where every element is either 0 or 1. We want to find the longest contiguous subarray containing only 1s after we are allowed to flip at most one 0 into a 1.
This problem asks us to determine whether a given binary tree satisfies the rules of a Binary Search Tree, commonly abbreviated as BST. A binary tree consists of nodes where each node contains a value and pointers to a left child and a right child.
We are given a keyboard laid out in an n by m grid. Each key contains either a lowercase Latin letter or a special "Shift" key represented by S.
The problem asks us to find the length of the longest substring that is considered "beautiful" according to two strict conditions. First, the substring must contain all five vowels, 'a', 'e', 'i', 'o', and 'u', at least once.
The problem asks us to manipulate a special binary string to produce the lexicographically largest possible string.
We start on an infinite chessboard at square (0, 0). A knight moves using the usual chess move, two cells in one direction and one in the perpendicular direction. Some squares are removed from the board, and the knight is never allowed to stand on them.
The problem asks us to count how many unique pairs of integers in the array have an absolute difference equal to k. A pair is considered valid if: - The two elements come from different indices. - The absolute difference between the two values is exactly k.
LeetCode 10, Regular Expression Matching, asks us to determine whether an entire input string s matches a pattern p. The pattern supports two special regular expression characters: - .
The problem asks us to determine whether a given binary string s contains all binary representations of integers from 1 to n as substrings.
The problem asks us to extract all consecutive records from a Stadium table where the people count is at least 100, and the consecutive streak has a length of three or more. Each row has a unique id and a corresponding visitdate.
We are given a database table called Traffic that records different user activities on different dates. Each row contains a userid, an activity type, and an activitydate.
This problem asks us to modify the structure of an N-ary tree by moving one subtree under another node. Every node contains a unique value, and each node may have any number of children.
The problem asks us to determine whether an array contains a continuous subarray whose sum is a multiple of k, while also satisfying an important constraint: the subarray must contain at least two elements.
We are given a set of wooden bars, each with a positive integer length. Vasya wants to build towers by stacking bars of the same length. Each tower must consist of bars that are identical in length, but different towers can have different lengths.
The problem asks us to count how many integers in the range 0 <= x < 10^n contain no repeated digits. For example, when n = 2, the valid range is: This means we consider every number from 0 to 99. Among these numbers, we only count those whose digits are all unique.
We are given a messy textual representation of a sequence. The string may contain positive integers, commas, spaces, and the special token .... Spaces may appear in the wrong places or appear multiple times. The task is purely formatting.
We are given the dimensions of an uploaded photo, height h and width w. We want to cut out a smaller rectangle from it. The cut rectangle must satisfy three conditions.
We are given several unsigned integer datatypes, each defined by its bit length. A datatype with a bits can store every integer from 0 up to 2^a - 1. We want to know whether there exists some integer x and two datatypes with sizes a[i] < a[j] such that: 1.
The problem asks us to generate strings of length n using only the characters 'a', 'b', and 'c', with one important rest
This problem gives us an undirected tree with n nodes labeled from 0 to n - 1. The input edges describes the connections between nodes, and because the graph is guaranteed to be a tree, several important properties immediately hold true.
This problem asks us to determine the minimum number of moves required to make all elements in an integer array equal, where a move consists of incrementing or decrementing a single element by 1.
The problem asks us to determine whether a given collection of cards can be rearranged into groups of consecutive numbers, each of size groupSize.
We are given a sequence of binary strings, all of equal length, and we need to split them into two subsequences in a way that minimizes the sum of the lengths of their compressed forms.
The problem gives us a circular array called code and an integer k. We must produce a new array where each element is replaced according to the value of k. If k 0, each element becomes the sum of the next k elements in the circular array.
The problem asks us to determine whether a given string can be constructed by repeatedly inserting the substring "abc" into an initially empty string. The operation is very specific. At any point, we may take the current string and insert "abc" at any position.
This problem asks us to identify all buyers who purchased the product named S8 but never purchased the product named iPhone. We are given two tables: The Product table stores information about products. Each row contains a unique productid, the product name, and its unit price.
We need to compute the $n$-th hexagonal number. The sequence is defined by the formula $$hn = 2n^2 - n$$ The input contains a single integer $n$, and the output is the value produced by this formula.
This problem asks us to determine the minimum ship capacity required to transport all packages within a fixed number of days. We are given an array called weights, where weights[i] represents the weight of the i-th package on a conveyor belt.
The problem asks us to count the number of 3 x 3 magic squares inside a larger grid. A magic square is defined as a square where all numbers are distinct integers from 1 to 9 and the sum of each row, column, and the two diagonals is the same.
The problem gives a date string in a human readable format such as "20th Oct 2052" and asks us to convert it into the st
We have a rectangular grid with n columns and m rows. Two lasers point at two different cells. Both lasers always move together, meaning their relative offset never changes. If one laser moves by (dx, dy), the other must move by exactly the same vector.
We are given a polynomial already factorized into linear terms: $$p(x) = (x + a1)(x + a2)dots(x + an)$$ The task is to expand this product and print the polynomial in the usual descending-power form: $$x^n + b1x^{n-1} + dots + bn$$ The tricky part is not the expansion itself.
This problem asks us to determine whether any permutation of s1 appears as a contiguous substring inside s2. A permutation means the characters are rearranged, but the frequency of each character remains the same. For example, the permutations of "ab" are "ab" and "ba".
The problem gives us a rooted tree with n nodes labeled from 0 to n - 1. Instead of providing the tree as adjacency lists or edges, the input is represented using a parent array. For every node i, parent[i] tells us which node is its direct parent.
This problem asks us to find the largest absolute difference between the values of two nodes in a binary tree, under one important condition: one node must be an ancestor of the other.
The problem gives us a triangular array of integers and asks us to compute the minimum path sum from the top row to the bottom row. The input is a two-dimensional array called triangle, where: - The first row contains exactly one number.
This problem describes a two-player impartial game played on a string consisting only of '+' and '-' characters. A valid move consists of selecting any pair of consecutive "++" characters and flipping them into "--". The players alternate turns.
We are given a one-dimensional map of a village where all the houses lie along the _x_-axis. Each house is square, specified by its center coordinate and its side length. No two houses overlap, though they may touch edges.
The problem asks us to process a string s that contains bracketed keys like (name) or (age) and replace them with corres
The problem asks us to count how many integers in the range [1, n] contain at least one repeated digit. A repeated digit means that some digit appears more than once in the number. For example, 11 has a repeated 1, 100 has repeated 0, and 121 has repeated 1.
The problem asks us to find the maximum width ramp in an array of integers. A ramp is defined as a pair of indices (i, j) such that: - i < j - nums[i] <= nums[j] The width of the ramp is simply the distance between the two indices: Our goal is to compute the largest possible…
The problem gives us a circular array nums, and for every element, we must find the next greater element. The phrase next greater element means the first value encountered while moving forward in the array that is strictly larger than the current number.
The problem asks us to count the number of good triplets in an integer array arr based on three difference constraints a, b, and c.
The problem asks us to compute the maximal network rank for a set of cities connected by bidirectional roads. Each city can be thought of as a node in a graph, and each road as an undirected edge.
The problem gives us an integer array arr and asks us to replace every element with the greatest value that appears some
This problem asks us to maximize the number of distinct candy types Alice can eat while respecting a strict limit on how many candies she is allowed to consume. We are given an integer array candyType, where each element represents the type of a candy.
This problem asks whether it is possible to construct the string ransomNote using characters taken from the string magazine. Each character in magazine can only be used once, which means character frequency matters.
The problem gives us all adjacent pairs from an unknown array nums, and our task is to reconstruct the original array. Suppose the original array was: The adjacent pairs would be: The important detail is that the pairs can appear in any order and can also be reversed.
The problem is asking us to take a table Department that records revenue per department per month in a vertical format and convert it into a horizontal format, often called a "pivot" table. In the input, each row contains a department id, a revenue value, and a month.
This problem asks us to retrieve the second highest distinct salary from the Employee table. The key word here is distinct. We are not looking for the second row after sorting salaries, we are looking for the second unique salary value.
The problem asks us to find the third distinct maximum number in an integer array. The key word is distinct. Duplicate values should only be counted once when determining rankings. For example, in the array [2,2,3,1], the distinct values are {3,2,1}.
In this problem, we are given a binary matrix where each cell contains either 0 or 1. A value of 1 represents land, while 0 represents water. An island is formed by connecting adjacent land cells in the four cardinal directions: up, down, left, and right.
The problem is asking us to determine, for each user in a social media application, how many followers they have. The input is a table called Followers with two columns: userid and followerid. Each row represents a relationship where followerid follows userid.
The problem asks us to find the longest substring that is both a prefix and a suffix of the given string s, while also e
The problem defines a special kind of binary tree called a Fibonacci tree. The structure is recursive: - order(0) is an empty tree. - order(1) is a single node. - order(n) has: - a root node, - a left subtree equal to order(n - 2), - a right subtree equal to order(n - 1).
We are given a string s. For every substring x of s, we look at all positions where x appears inside s. Suppose these occurrences start at positions: $$p1 < p2 < dots < pk$$ The function F(s, x) counts how many contiguous segments we can choose from this ordered list of…
The problem requires simulating the transformation of an array over consecutive days according to a simple local rule. Each day, every element of the array (except the first and last) is compared with its immediate neighbors.
We are given a set of atoms with known atomic numbers and a target set of atoms we want to produce using fusion. Each fusion operation combines exactly two atoms into one, and the resulting atom’s atomic number is the sum of the two original numbers. We cannot split atoms.
The problem gives us a binary string s, meaning the string contains only the characters '0' and '1'. We must split this
The problem asks us to find the length of the longest string in the array that is not a subsequence of any other string in the same array. A subsequence is formed by deleting zero or more characters from a string without changing the order of the remaining characters.
This problem gives us a special type of array called a mountain array. A mountain array strictly increases until it reaches a single peak element, then strictly decreases afterward. For example: increases up to 5, then decreases.
This problem requires designing a simulation for an exam room seating arrangement. We have n seats in a single row, labeled from 0 to n - 1. Students enter one by one, and each student chooses a seat such that the distance to the closest occupied seat is maximized.
This problem asks us to compare two software version numbers represented as strings. Each version consists of one or more numeric revisions separated by dots (.). For example, "1.2.10" contains three revisions: 1, 2, and 10.
The problem asks us to determine how many times a binary string becomes prefix-aligned during a series of bit flips. We
The problem asks us to repeatedly transform a number by summing its digits until only a single digit remains. The final single digit should then be returned. For example, if the input is 38, we first compute 3 + 8 = 11.
This problem asks us to count how many root to leaf paths in a binary tree are "pseudo-palindromic". A palindrome is a s
In this problem, we are given a circular bus route with n stops numbered from 0 to n - 1. The array distance describes the distance between neighboring stops. Specifically, distance[i] represents the distance from stop i to stop (i + 1) % n.
We have an L-shaped corridor with widths a and b on the two branches. A rectangular coffin of fixed length l must be moved through the corner.
Bob is at a sale with a collection of old TVs, each with a price. Some TVs are free, some have positive prices, and some even have negative prices, which means the owner is willing to pay Bob to take them away.
We are asked to count the number of times a digital clock shows a moment where at least k digits change simultaneously while Vasya is watching it. The clock is not necessarily 24-hour or 60-minute - it has h hours and m minutes, where both are arbitrary integers up to 10^9.
This problem asks us to process a stream of non-negative integers and continuously maintain a compact representation of all numbers seen so far. Instead of storing every individual number separately, we want to group consecutive numbers into disjoint intervals.
We are given the sequence of goals scored during a football match. Every line after the first contains the name of the team that scored one goal. The task is to determine which team scored more goals overall.
The problem asks us to sort a singly linked list using the insertion sort algorithm. Unlike array sorting problems where elements can be accessed directly by index, linked lists require sequential traversal, so insertion operations behave differently and require careful…
We have several accumulators, each storing some amount of energy. We are allowed to move energy between them, but every transfer wastes a fixed percentage. If we send x units from one accumulator, the sender loses all x, while the receiver only gains x (100 - k) / 100.
We are filling an n × m grid with brackets. Every cell contains either "(" or ")". The grid is called valid if every monotone path from the top-left corner to the bottom-right corner forms a correct bracket sequence. A monotone path only moves right or down.
The problem gives us a string called text and two target words, first and second. We must find every occurrence where the words appear consecutively in the exact order: For every such occurrence, we return the value of third.
This problem asks us to identify students who are enrolled in university departments that no longer exist in the departm
The problem asks us to sort every diagonal of a matrix independently in ascending order. A diagonal is defined as a sequence of cells that begins either from the first row or the first column, then continues by repeatedly moving one row down and one column to the right.
We are given two strings. The first string is the newspaper headline, and the second string is the anonymous letter Vasya wants to build from it.
This problem simulates a lemonade stand where each lemonade costs 5, 20 bill. The challenge is to provide exact change to every customer as they pay. The input is a list of integers bills, where each element represents the bill a customer gives.
The problem requires us to transform an array nums of even length n into a complementary array. An array is complementar
The problem asks us to count how many index triplets (i, j, k) satisfy two conditions: - The indices must follow the order i < j < k - The values at those indices must sum to the given target Formally, we want: The important detail is that we are counting tuples of indices…
In this problem, we are given a binary array called bits. The array represents a sequence of encoded characters using the following rules: - A one-bit character is represented by a single 0 - A two-bit character is represented by either 10 or 11 The array is guaranteed to end…
We need to build a circular sequence of colors for n eggs. There are exactly seven available colors: R, O, Y, G, B, I, V Two conditions must hold simultaneously. First, every color must appear at least once.
We have two armies, each containing n soldiers. The value of n is always even. The game lasts exactly three turns: 1. Valera attacks Arcady. 2. Arcady attacks Valera. 3. Valera attacks Arcady again.
The problem is asking us to solve a linear equation containing a single variable 'x' and integer coefficients, expressed as a string. The equation may include addition '+', subtraction '-', and the equality operator '='.
We are given the coordinates of three vertices of some regular polygon. The polygon itself is unknown: we do not know how many sides it has, where its center is, or which vertices the three points correspond to.
Each quarry contributes a consecutive range of heap sizes to a standard Nim game.
The children stand in a circle numbered from 1 to n. Child 1 starts with the ball. The first throw moves the ball forward by 1 position, the second throw moves it forward by 2 positions, the third throw by 3 positions, and so on.
This problem asks us to determine whether a singly linked list contains a cycle and, if it does, return the exact node where that cycle begins. A linked list is normally a sequence of nodes where each node points to the next one, eventually ending with null.
The problem asks whether it is possible to split a binary tree into two separate trees such that both resulting trees have the same sum of node values. The split must happen by removing exactly one edge from the original tree. A binary tree is given through its root node.
The problem is asking us to find the minimum cost to connect two groups of points, where the first group has size1 point
The problem asks us to determine how many complete rows of a coin staircase can be formed using exactly n coins. A staircase arrangement follows a very specific structure: - The first row contains 1 coin - The second row contains 2 coins - The third row contains 3 coins -…
The operation described in the problem is exactly the process of generating the previous lexicographical permutation of a string. Starting from the current string, each operation transforms it into the lexicographically largest string that is still smaller than the current one.
The problem describes a classic interactive guessing game. A hidden number called pick is chosen somewhere in the range from 1 to n, inclusive. We are not allowed to access pick directly.
This problem is essentially a simulation of the Candy Crush game, where we need to repeatedly crush candies in a grid until the board reaches a stable state. The input is an m x n matrix of integers representing different types of candies, and 0 represents empty cells.
The problem provides two traversal orders of the same binary tree: - inorder, which follows the order: left subtree, root, right subtree - postorder, which follows the order: left subtree, right subtree, root We must reconstruct the original binary tree and return its root node.
This problem asks us to compute the XOR sum of every possible pairwise bitwise AND between two arrays. More formally, for every pair (i, j): - Take arr1[i] AND arr2[j] - Add that result to a conceptual list - Compute the XOR of all values in that list The challenge is that…
The problem asks us to determine whether the binary representation of a given positive integer n has alternating bits. In other words, for the binary digits of n, no two consecutive bits should be the same.
This problem asks us to calculate the Click-Through Rate, abbreviated as CTR, for every advertisement stored in the Ads
This problem asks us to find the maximum possible sum of a contiguous subarray when we are allowed to delete at most one element from that chosen subarray. The important detail is that the resulting subarray must still contain at least one element after deletion.
This guide will focus on the optimal monotonic queue solution, which achieves linear time complexity and is necessary to
The problem requires transforming an n x n binary matrix in two steps: first flipping it horizontally, then inverting it.
The problem asks us to compute the power of a string. The power is defined as the length of the longest contiguous subst
The task is to analyze a program written in the esoteric HQ9+ language and determine whether executing it will produce any visible output. The program is provided as a single string consisting of printable ASCII characters.
The problem is a variant of the classic "Sokoban" puzzle. You are given a grid representing a warehouse, where a player
The problem asks us to perform in place string compression on an array of characters. The input is not a string object, but a mutable array named chars, where each element is a single character. The compression rule is based on groups of consecutive repeated characters.
The problem gives us an array candiesCount where candiesCount[i] represents how many candies exist for candy type i.
We are given a stripe consisting of n squares, each containing an integer. The task is to cut this stripe into three contiguous, non-empty segments such that the sum of numbers in each segment is identical. The output is the number of valid ways to perform these cuts.
This problem asks us to swap the seat IDs of every two consecutive students in a classroom. The input is a table called Seat with two columns: id, which is a unique integer representing the seat number, and student, which is a string representing the student's name.
The problem gives us a numeric string num and an integer k. Each operation allows us to swap two adjacent digits. We may
This problem asks us to determine whether a given Tic-Tac-Toe board configuration could occur during a real game that follows all official rules. The input is a 3 x 3 board represented as an array of three strings.
The problem asks us to find the length of the longest contiguous substring that forms a valid parentheses sequence. A valid parentheses sequence is one where every opening parenthesis '(' has a corresponding closing parenthesis ')', and the pairs are correctly nested.
The problem gives us two sentences, where each sentence is represented as an array of words. We are also given a list of word pairs that define which words are considered similar. Our task is to determine whether the two sentences are similar according to the following rules: 1.
The problem asks whether string s can be formed from string t by deleting some characters from t without changing the order of the remaining characters. A subsequence does not require characters to be adjacent. The only requirement is that the relative ordering stays the same.
The problem asks us to assign a rank to every element in a matrix while preserving ordering relationships inside rows and columns. For any two elements that share the same row or the same column: - If one value is smaller, its rank must also be smaller.
This problem asks us to determine whether a string s3 can be formed by interleaving two other strings, s1 and s2. An interleaving means we combine characters from s1 and s2 while preserving the relative order of characters within each original string.
We are given a string consisting solely of opening and closing parentheses. The task is to determine the maximum length of a subsequence that forms a valid, or regular, bracket sequence.
We are asked to enumerate all ways to split n toys into piles, starting from a single pile containing all toys. The toys are numbered from 1 to n, and the order within a pile or between piles does not matter for uniqueness beyond the actual grouping.
The problem asks us to generate a string of length n such that each character in the string occurs an odd number of time
This problem asks us to combine information from two database tables and return the result in a specific format. We are
The problem gives us several types of boxes, where each box type contains two values: - The number of boxes available for that type - The number of units inside each box of that type We also have a truck that can carry at most truckSize boxes total, regardless of type.
The problem gives us two binary search trees, root1 and root2, along with an integer target. We must determine whether there exists one node from the first tree and one node from the second tree such that their values add up exactly to target.
This problem models a squirrel collecting nuts in a 2D garden grid. The garden has a fixed tree position, a starting squirrel position, and multiple nuts scattered around the grid.
The problem gives us a two dimensional grid where each cell contains either '1' or '0'. A cell containing '1' represents land, while a cell containing '0' represents water. An island is defined as a group of connected land cells.
The problem asks us to design a data structure that supports two operations on an integer array: 1. Reset the array back to its original order. 2. Return a randomly shuffled version of the array.
The problem gives us a sorted list containing every integer from 1 to n. We repeatedly eliminate numbers in alternating directions until only one number remains.
This problem gives us a binary string consisting only of '0' and '1'. We are allowed to repeatedly apply two transformation rules: - Replace "00" with "10" - Replace "10" with "01" We may perform these operations as many times as we want, in any order, and the goal is to…
We are given positions of queens on a chessboard. A queen attacks in eight directions: left, right, up, down, and the four diagonals. A queen does not attack every queen in a direction, only the first one encountered along that ray.
We are given a small grid of lowercase letters. A cell survives only if its letter is unique both inside its row and inside its column. If the same character appears somewhere else in the same row, that cell is removed.
In this problem, we are given the root node of a binary tree, and we need to compute the average value of all nodes at each depth level of the tree. A binary tree is organized into levels.
We are given two integers, k and l. The task is to determine whether l can be written as an exact power of k. In other words, we need to check whether there exists a non-negative integer n such that: $l = k^n$ If such an n exists, we print "YES" and also print the importance…
The problem gives us an m x n multiplication table where each cell contains the product of its row index and column index, using 1-based indexing. That means the value at position (i, j) is simply i j.
This problem asks us to find the maximum average value among all subtrees of a given binary tree. A subtree is defined as any node along with all of its descendants. The average of a subtree is the sum of its node values divided by the number of nodes in that subtree.
The problem asks us to compute the median for every contiguous subarray, or "window", of size k as that window slides from left to right across the input array. For each position of the window, we consider exactly k elements.
This problem asks us to retrieve the three most recent orders for every customer from a database. If a customer has fewe
The problem gives us a binary tree where every node value has been contaminated and replaced with -1. However, we know t
This problem requires designing a specialized stack data structure, called CustomStack, that not only supports the usual
The problem asks us to count the number of valid teams of three soldiers from a line of n soldiers, where each soldier h
The problem asks us to find the optimal location to build a house on a grid so that the total travel distance to all existing buildings is minimized. The input is an m x n grid of integers where each cell is either empty land (0), a building (1), or an obstacle (2).
The problem asks us to count the number of good integers within a given range [1, n]. An integer is considered good if each of its digits, when rotated 180 degrees, forms another valid digit and the resulting number is different from the original.
This problem asks us to transform an initial array arr of zeros into a target array nums using a minimal number of operations. There are two allowed operations: incrementing any single element of the array by 1, or doubling all elements of the array.
The problem gives us an integer array nums of length n, where every value is guaranteed to be in the range [1, n]. Each number appears either once or twice. Our task is to return all numbers that appear exactly twice.
The problem gives us a list of unique strings called words. We must find every ordered pair of indices (i, j) such that: - i != j - concatenating words[i] + words[j] forms a palindrome A palindrome is a string that reads the same forward and backward.
This problem asks us to build the lexicographically largest possible string by repeatedly choosing characters from the front of two given strings.
We are asked to reconstruct the missing result of a round-robin tournament. There are _n_ participants, and every participant plays against every other participant exactly once.
The office floor is represented as an n × m grid. Each cell is either free, written as 0, or blocked by furniture, written as 1. We want to place one rectangular table whose sides stay aligned with the grid. Every cell covered by the rectangle must be free.
The problem gives us an integer array arr and two integers, m and k. We need to determine whether there exists a contiguous subarray of length m that repeats consecutively at least k times.
The plane is colored using concentric rings centered at the origin. Every ring between two consecutive integer distances alternates color. The borders themselves, meaning all points whose distance from the origin is an integer, are always black.
The problem asks us to find pairs of songs whose total duration is divisible by 60. Specifically, we are given an array time where each element represents the length of a song in seconds. We need to count all unique pairs (i, j) where i < j and (time[i] + time[j]) % 60 == 0.
The problem asks whether a given string s can become a palindrome after deleting at most one character. A palindrome is a string that reads the same forwards and backwards.
The problem asks us to compute the n-th Fibonacci number, but with a slight twist in indexing: the sequence starts with f₀ = 1, f₁ = 1. Every subsequent number is the sum of the previous two.
This problem asks us to maximize profit from stock trading over a sequence of days, where each transaction incurs a fixed fee. The input array prices represents the stock price on each day, and fee represents the transaction fee charged for every completed buy-sell transaction.
This problem asks us to simulate a simple programming language with a single integer variable X that starts at 0. We are given a list of string operations, each of which either increments (++X or X++) or decrements (--X or X--) the value of X by 1.
The problem asks us to determine the sign of the product of all numbers in an integer array, without necessarily computing the actual product itself.
The problem asks us to generate every strobogrammatic number of a given length n. A strobogrammatic number is a number that still appears valid after being rotated 180 degrees. Not every digit works under rotation.
The problem gives us an array called salary, where each element represents the salary of an employee. Every salary value
The problem gives an array of intervals, where each interval is represented as [start, end]. Each interval describes a continuous range of values from start to end, inclusive. The goal is to combine all intervals that overlap into a single larger interval.
The problem asks us to determine the minimum eating speed k for Koko such that she can finish all piles of bananas within a given number of hours h.
We are given three circles on the plane. Each circle represents a stadium, with a center point and a radius. We need to find a point from which all three stadiums are seen under the same angle.
We are asked to place as many standard dominoes as possible on a rectangular board of size _M_ by _N_. Each domino covers exactly two adjacent squares, and dominoes cannot overlap or extend outside the board.
The problem provides a database table named Employee that stores information about which departments employees belong to.
The problem asks us to replace words in a sentence using a set of predefined root words. A root is a shorter word that can serve as a prefix for a longer derivative word.
The problem describes a cinema hall in Berland with K rows and K seats per row, where K is always odd. Customers come in groups of size M and request consecutive seats.
Solomon stands on the fortress wall at position 0. To his right there may exist a chain of ice blocks occupying positions 1, 2, .... Initially there are no blocks at all. The battlefield is a line of length n. At battlefield position i, there may be a demon with strength a[i].
The problem asks us to count how many subtrees in a binary tree are "uni-value" subtrees. A uni-value subtree is a subtree in which every node has the same value. A subtree consists of a node together with all of its descendants.
We have a line of archers, each with some health. A fireball can only be thrown at positions 2 ... n-1. If we throw at position i, then:
This problem asks us to compute how many Grand Slam tennis titles each player has won across all years recorded in the database. We are given two tables: The Players table contains information about tennis players.
We are given a shuffled array that originally came from concatenating several permutations. Each permutation may have a different size. After concatenation, all numbers were mixed together, so the original grouping disappeared.
We have a queue of animals waiting for the doctor. Animal i must visit the doctor exactly a[i] times before leaving forever. Whenever an animal is examined, one of two things happens. If it still needs more visits, it immediately moves to the back of the queue.
The problem is asking us to determine whether a given array nums could be the result of taking a sorted array in non-decreasing order and then rotating it by some number of positions. A non-decreasing array is one where each element is greater than or equal to the previous one.
This problem asks us to design a simplified version of a social media platform similar to Twitter. The system must support four main operations: 1. Users can post tweets. 2. Users can follow other users. 3. Users can unfollow other users. 4.
We are given several force vectors acting on a body in three-dimensional space. Each vector has three components: its effect along the x-axis, y-axis, and z-axis. A body is in equilibrium only if the total force acting on it is zero in every direction.
We start with a rectangle drawn on grid paper. Only the border matters, not the interior. Players repeatedly draw a strictly smaller rectangle inside the previous one.
Here’s a full technical solution guide following your formatting rules for LeetCode 629: The problem asks us to determine the number of arrays of length n containing numbers from 1 to n such that there are exactly k inverse pairs.
The problem is asking whether a given string s can be rearranged to form exactly k non-empty palindrome strings using al
The problem gives us the root node of a binary tree and asks us to compute the sum of all left leaves in the tree. A binary tree node may have up to two children, a left child and a right child.
The problem asks us to count how many nodes exist in a complete binary tree. A complete binary tree has a very specific structure. Every level except possibly the last one is completely filled, and the nodes on the final level appear as far left as possible.
The problem asks us to generate all structurally unique binary search trees (BSTs) that contain exactly n nodes labeled from 1 to n.
This problem gives us a rectangular cake with height h and width w. We are also given two arrays: - horizontalCuts, whic
The problem asks us to design a calendar system that supports booking events without allowing overlapping intervals. Each event is represented as a half-open interval [startTime, endTime). This means the event includes startTime, but does not include endTime.
The problem asks us to compute every possible bitwise OR value that can be formed from all non-empty contiguous subarrays of the given array arr, then return how many distinct values exist. A subarray is any contiguous slice of the array.
We are given a weighted undirected graph with up to 15 vertices and up to 2000 edges. Each edge has a positive weight, and there may be multiple edges connecting the same pair of vertices or edges that loop back to the same vertex.
We are given an undirected graph where vertices represent people and edges represent relationships. For every query (u, v), we must decide whether there exists a simple path from u to v whose length is odd. The keyword here is "simple". We are not allowed to revisit vertices.
We have bus stops placed on a line from 0 to n. Gerald starts at stop 0 and wants to reach stop n. Each bus is described by an interval [s, t]. Gerald may board that bus at any stop from s through t - 1, but once he rides it, he must stay on until stop t.
The problem asks us to design a custom data structure that behaves like a map from strings to integers, while also supporting efficient prefix-based sum queries. There are two operations: 1. insert(key, val) This operation stores a string key with an integer value.
This problem asks us to determine the maximum overlap between two binary square matrices (img1 and img2) when one matrix is translated over the other. Each matrix contains only 0s and 1s, where 1 represents a filled pixel and 0 represents an empty pixel.
The problem asks us to determine whether a given integer array forms a valid mountain array. A mountain array has a very specific structure. The values must first strictly increase until they reach a single peak, then strictly decrease after the peak.
We are asked to classify a single non-negative integer based on how it relates to its digits. Specifically, each digit of the number is considered as a potential divisor.
The problem gives us the root of a Binary Search Tree, abbreviated as BST, and asks us to transform it into a Greater Sum Tree. In a Binary Search Tree, every node follows an important ordering rule: - All values in the left subtree are smaller than the current node.
The problem gives us a strictly increasing array of positive integers, arr, and an integer k. The array does not necessa
We are given three musical notes, and the task is to classify the chord they form as either major, minor, or "strange". Notes are represented in the twelve-tone chromatic scale: C, C, D, D, E, F, F, G, G, A, B, H, and the scale is cyclic, so after H comes C again.
The problem asks us to determine the minimum number of semesters required to complete n courses when there are prerequis
This problem asks us to combine two binary trees into a single merged tree. Each tree consists of nodes where every node contains a value and pointers to a left and right child.
The problem gives us a positive integer n and asks for the minimum number of operations required to transform it into 1.
The problem gives us an integer array arr and an integer k. We are allowed to partition the array into contiguous subarrays where each subarray has length at most k.
We start with an n × n pattern consisting of black cells (*) and white cells (.). This pattern acts like a template.
The problem asks us to compute the minimum absolute difference in subarrays of a given integer array nums for multiple q
The problem is asking us to sort an array of unique integers using only pancake flips. A pancake flip is defined as reversing a prefix of the array from index 0 to index k-1 for some integer k between 1 and the length of the array.
A deck of cards contains R red cards and B black cards.
Two players A and B are playing a variant of Nim.
Solution to Project Euler Problem 992.
The problem describes a row of houses, where each house contains some amount of money. You are acting as a robber who wants to maximize the total amount stolen, but there is one important restriction, you cannot rob two adjacent houses.
The problem gives us a database table named Weather that stores daily temperature records. Each row contains three fields: a unique id, a recordDate, and the temperature recorded on that date.
An L-expression is defined as any one of the following: - a natural number; - the symbol A; - the symbol Z; - the symbol
A contiguous range of positive integers is called a divisible range if all the integers in the range can be arranged in
An ascent of a column j in a matrix occurs if the value of column j is smaller than the value of column j + 1 in all row
The prime factors of 13195 are 5, 7, 13 and 29.
nThere are n distinct piles of stones, each of size n-1.
The problem asks us to reverse the binary representation of a 32-bit integer. Instead of reversing the decimal digits of a number, we reverse the order of its individual bits.
The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Let sigma(n) be the sum of the divisors of n.
The sum of the squares of the first ten natural numbers is, The square of the sum of the first ten natural numbers is, H
The four adjacent digits in the 1000-digit number that have the greatest product are 9 times 9 times 8 times 9 = 5832.
This problem asks us to process a text file named file.txt and print only the phone numbers that match one of two valid formats. The valid formats are: and In both formats, every x represents a single digit from 0 to 9.
- A Sierpiński graph of order-1 (S1) is an equilateral triangle.
The Fibonacci sequence (fi) is the unique sequence such that - f0=0 - f1=1 - f{i+1}=fi+f{i-1} Similarly, there is a uniq
This problem asks us to print exactly the 10th line from a text file named file.txt. The file contains multiple lines of plain text, and each line is separated by a newline character. The input is not provided as function arguments like many algorithm problems.
A square of side length b<1 is rolling around the inside of a larger square of side length 1, always touching the larger
The numbers from 1 to 12 can be arranged into a 3 times 4 matrix in either row-major or column-major order: By swapping
Solution to Project Euler Problem 980.
There is a plane on which all points are initially white, except three red points and two blue points.
A multiplicative function f(x) is a function over positive integers satisfying f(1)=1 and f(a b)=f(a) f(b) for any two c
The problem gives us an integer array called nums and another integer called target. Our task is to find two different elements in the array whose sum equals target, then return their indices.
Three friends attempt to collectively choose one of n options, labeled 1,dots,n, based upon their individual preferences
We define the unlucky prime of a number n, denoted u(n), as the smallest prime number p such that the remainder of n div
Let operatorname{smpf}(n) be the smallest prime factor of n.
A peerless tree is a tree with no edge between two vertices of the same degree.
The problem gives us an array of positive integers called nums and a positive integer called target. We need to find the smallest possible length of a contiguous subarray whose sum is greater than or equal to target.
Solution to Project Euler Problem 975.
Two players play a game with two piles of stones.
Solution to Project Euler Problem 983.
For a positive integer n we define tau(n) to be the count of the divisors of n.
Each new term in the Fibonacci sequence is generated by adding the previous two terms.
The problem gives us the root node of a binary tree and asks us to determine which nodes are visible when looking at the tree from the right side.
A unitary divisor of a positive integer n is a divisor d of n such that gcdleft(d,frac{n}{d}right)=1.
If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9.
Let B(n) be the smallest number larger than n that can be formed by rearranging digits of n, or 0 if no such number exis
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
A bouncing point moves counterclockwise along a circle with circumference 1 with jumps of constant length l lt 1, until
Solution to Project Euler Problem 985.
This problem asks us to determine whether a target value exists inside a rotated sorted array that may contain duplicate values. The original array was sorted in non-decreasing order, meaning values are arranged from smallest to largest, and duplicates are allowed.
We are given a string and want to remove the minimum number of characters so that the resulting string no longer contains "pie" or "map" as a substring.
Let theta=sqrt{-2}. Define T to be the set of numbers of the form a+btheta, where a and b are integers and either agt 0,
Solution to Project Euler Problem 988.
The following game is a classic example of Combinatorial Game Theory: Two players start with a strip of n white squares
The problem asks us to transform a string into a zigzag pattern across a fixed number of rows, then read the characters row by row to produce the final result.
The function s(n) is defined recursively for positive integers by s(1) = 1 and s(n+1) = big(s(n) - 1big)^3 +2 for ngeq 1
Define M(n) to be the minimum number of matchsticks needed to represent the number n.
A round number is a number that ends with one or more zeros in a given base.
Let {x} denote the fractional part of a real number x.nnDefine fN(x) to be the minimal value of {nx} for integer n satis
A composition of n is a sequence of positive integers which sum to n.
The problem gives a Roman numeral string and asks us to convert it into its corresponding integer value. Roman numerals use seven symbols: Symbol Value --- --- I 1 V 5 X 10 L 50 C 100 D 500 M 1000 Most Roman numerals follow a simple additive rule.
There are xyz dice arranged in an x times y times z box such that touching faces have the same value.
Solution to Project Euler Problem 973.
Given a right-angled triangle with integer sides, the smaller angle formed by the two medians drawn on the the two perpe
A clock sequence is a periodic sequence of positive integers that can be broken into contiguous segments such that the s
Two players play a game with at least two piles of stones.
There are N seats in a row.
Solution to Project Euler Problem 974.
A game is played with three piles of stones and two players.
For every integer n 1, the family of functions f{n,a,b} is defined by for integers a, b, x with 0 < a < n, 0 le b < n, 0
Let '' denote the bitwise AND operation.
For a positive integer n construct a graph using all the divisors of n as the vertices.
A Pythagorean triplet is a set of three natural numbers, a lt b lt c, for which, For example, 3^2 + 4^2 = 9 + 16 = 25 =
An L-expression is defined as any one of the following: - a natural number; - the symbol A; - the symbol Z; - the symbol
2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
This problem asks us to transpose the contents of a text file. Transposing means converting rows into columns and columns into rows. Every word in the input file is separated by a single space, and each row contains the same number of columns. The input is a file named file.txt.
We define nu2(n) to be the largest integer r such that 2^r divides n.
An irrational number x can be uniquely expressed as a continued fraction [a0; a1,a2,a3,dots]: where a0 is an integer and
The problem asks us to determine whether two strings, s and t, are isomorphic. Two strings are considered isomorphic if there exists a one-to-one mapping between characters in s and characters in t such that replacing every character in s according to this mapping produces t.
An integer partition of a number n is a way of writing n as a sum of positive integers.
A Young diagram is a finite collection of (equally-sized) squares in a grid-like arrangement of rows and columns, such t
The problem asks us to remove every node from a singly linked list whose value is equal to a given integer val. After all matching nodes are removed, we must return the head of the modified linked list.
A Young diagram is a finite collection of (equally-sized) squares in a grid-like arrangement of rows and columns, such t
Solution to Project Euler Problem 996.
A driller drills for water.
The problem asks us to compute the largest difference between two consecutive elements after sorting the array. At first glance, this may seem straightforward: sort the array, then scan through adjacent pairs to find the maximum difference.
Consider the following recurrence relation: Note that a0 is the golden ratio.
Solution to Project Euler Problem 979.
nA positive integer n is considered B-trivisible if the sum of all different prime factors of n which are not larger tha
nStarting at zero, a kangaroo hops along the real number line in the positive direction.
The minimum bounding square of a triangle is the smallest square that can be drawn which fully covers the triangle.
Consider a stack of bottles of wine.
This problem asks us to write a shell script that reads a text file named words.txt and computes how many times each word appears. After counting the occurrences, the script must print every unique word together with its frequency, sorted in descending order of frequency.
The problem gives us two non empty singly linked lists. Each linked list represents a non negative integer, but the digits are stored in reverse order. That means the head node contains the least significant digit.
The problem asks us to find the longest substring of a given string that is also a palindrome. A palindrome is a sequence that reads the same forward and backward. The substring must be contiguous, which means the characters must appear next to each other in the original string.
Let B(n) be the smallest number larger than n that can be formed by rearranging digits of n, or 0 if no such number exis
Bozo sort, not to be confused with the slightly less efficient bogo sort, consists out of checking if the input sequence
Let's call a pair of positive integers p, q (p lt q) reciprocal, if there is a positive integer rlt p such that r equals
This problem provides a database table named Person with two columns: The id column is unique because it is the primary key. The email column may contain duplicate values, meaning multiple rows can share the same email address.
A palindromic number reads the same both ways.
Solution to Project Euler Problem 976.
The Euclidean algorithm can be used to find the greatest common divisor of two positive integers.
This problem asks us to determine a valid order in which courses can be completed given prerequisite relationships between them. You are given numCourses, which represents the total number of courses labeled from 0 to numCourses - 1.
Solution to Project Euler Problem 994.
For each prime p and each positive integer n define two polynomials: Let S(p) be the smallest positive integer s such th
Let Sn be an integer sequence produced with the following pseudo-random number generator: Let A(i, j) be the minimum of
The problem gives an array called height, where each element represents the height of a vertical line drawn on a coordinate plane. The line at index i starts at (i, 0) and ends at (i, height[i]).
The problem asks us to reverse the digits of a signed 32-bit integer. Given an integer x, we must return a new integer whose digits appear in reverse order while preserving the sign. For example, if the input is 123, reversing the digits produces 321.
The sequence sn is defined by s1 = 102022661 and sn = s{n-1}^2 bmod {998388889} for n 1.
Given is an integer sided triangle ABC with BC le AC le AB.
A permutation pi of 1, dots, n can be represented in one-line notation as pi(1),ldots,pi(n) .
The sequence an is defined by a1=1, and then recursively for ngeq1: The first ten terms are 1, 2, -5, 4, 17, -10, -17, 8
Solution to Project Euler Problem 990.
Solution to Project Euler Problem 986.
Given nge 2 bowls arranged in a circle, mge 2 balls are distributed amongst them.
Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.
For the year 2025 Given positive integers a and b, the concatenation ab we call a 2025-number if ab = (a+b)^2.
This problem models course dependencies as a directed graph. Each course is represented as a node, and each prerequisite relationship is represented as a directed edge. If prerequisites[i] = [a, b], that means course b must be completed before course a.
The problem gives us two arrays of coin values. The first array represents coins in Rudolf's left pocket, and the second array represents coins in his right pocket.
An infant's toy consists of n cups, labelled C1,dots,Cn in increasing order of size.
For a positive integer n and digits d, we define F(n, d) as the number of the divisors of n whose last digits equal d.
Solution to Project Euler Problem 982.
Consider the problem of determining a secret number from a set 1, ..., N by repeatedly choosing a number y and asking "I
Solution to Project Euler Problem 984.
We are given an array of non-negative integers. In one operation, we choose an index $i$ such that $2 le i le n-1$, and
The problem asks us to reverse a singly linked list. A singly linked list is a sequence of nodes where each node stores a value and a pointer to the next node in the sequence. The input, head, represents the first node of the linked list.
This problem is based on (but not identical to) the scoring for the card game Cribbage.
Let P(n) be the number of permutations of 1,2,3,ldots,2n such that: 1.
Let G(n) denote the largest possible area of an n-gona polygon with n sides contained in the region (x, y) in Bbb R^2: x
The problem gives two individually sorted integer arrays, nums1 and nums2, with lengths m and n. The task is to compute the median of the combined sorted sequence formed by merging both arrays. The median is the middle value of a sorted sequence.
Write Fn for the n-th Fibonacci number, with F1 = F2 = 1 and F{n+1} = Fn + F{n-1}.
The classical eight queens puzzle is the well known problem of placing eight chess queens on an 8 times 8 chessboard so
A full k-ary tree is a tree with a single root node, such that every node is either a leaf or has exactly k ordered chil
Claire Voyant is a teacher playing a game with a class of students.
Given any integer n gt 1 a binary factor tree T(n) is defined to be: - A tree with the single node n when n is prime.
Solution to Project Euler Problem 987.
The problem gives a string s and asks for the length of the longest substring that contains no repeated characters. A substring is a continuous section of the string. This detail matters because characters must remain adjacent.
For coprime positive integers a and b, let C{a,b} be the curve defined by: where t varies between 0 and 2pi.
We are given d days and a target total number of study hours, sumTime. For every day, Peter must study at least minTime[i] hours and at most maxTime[i] hours. The task is to construct any valid schedule whose total sum is exactly sumTime.
Solution to Project Euler Problem 993.
For a given integer R consider all primitive Pythagorean triangles that can fit inside, without touching, a circle with
Solution to Project Euler Problem 977.
nThe hyperbolic plane can be represented by the open unit disc, namely the set of points (x, y) in Bbb R^2 with x^2 + y^
The problem asks us to design and implement a Trie, also called a Prefix Tree. A Trie is a specialized tree structure for storing strings in a way that makes prefix-based operations very efficient.
nDefinennas the sum of 2^a3^b5^c7^d11^e over all quintuples of non-negative integers (a, b, c, d, e) such that the sum o
nStarting at zero, a kangaroo hops along the real number line in the positive direction.
Three epistemologists, known as A, B, and C, are in a room, each wearing a hat with a number on it.
In a standard 52 card deck of playing cards, a set of 4 cards is a Badugi if it contains 4 cards with no pairs and no tw
The problem asks us to count how many prime numbers exist that are strictly smaller than a given integer n. A prime number is a positive integer greater than 1 that has exactly two divisors: 1 and itself. Examples of prime numbers include 2, 3, 5, 7, and 11.
This problem asks us to determine whether a given positive integer n is a happy number. A happy number is defined through a repeated transformation process. Starting with the original number, we repeatedly replace it with the sum of the squares of its digits.
The problem asks us to determine whether a given integer reads the same forward and backward. Such numbers are called palindromes. For example, the number 121 is a palindrome because reversing its digits still produces 121.
nLet I(a, b, c) be the largest possible area of intersection between a triangle of side lengths a, b, c and a circle whi
Starting with one piece of integer-sized rectangle paper, two players make moves in turn.
A permutation pi of 1, dots, n can be represented in one-line notation as pi(1),ldots,pi(n) .
Given is an integer sided triangle ABC with BC le AC le AB.nk is the angular bisector of angle ACB.nm is the tangent at
Define the sequence a(n) as the number of adjacent pairs of ones in the binary expansion of n (possibly overlapping).
A group of k(k-1) / 2 + 1 children play a game of k rounds.nAt the beginning, they are all seated on chairs arranged in
We call a triangle fortunate if it has integral sides and at least one of its vertices has the property that the distanc
The problem asks us to implement a simplified version of the C/C++ atoi function, which converts a string into a 32-bit signed integer. The conversion process is strict and follows several rules in a specific order.
A round clock only has three hands: hour, minute, second.
Solution to Project Euler Problem 981.
nThis game starts with a positive integer.
Let p be a prime of the form 5k-4 and define fp(x) = left(x^k+xright) bmod p.nnLet C(p) be the number of values 0 le x l
Solution to Project Euler Problem 978.
A frog is placed on the number line.
The problem asks us to count how many bits are set to 1 in the binary representation of a positive integer n. A set bit is simply a bit whose value is 1. For example, the number 11 in binary is: This binary representation contains three 1 bits, so the answer is 3.
Solution to Project Euler Problem 991.
NOTE: This problem is related to Problem 882.
The problem asks us to compute the bitwise AND of every integer in the inclusive range [left, right]. For example, if left = 5 and right = 7, the numbers in the range are: Applying bitwise AND across all values: So the answer is 4.
Let sn be the n-th positive integer that does not contain three consecutive ones in its binary representation.
We shall call a positive integer A an "Alexandrian integer", if there exist integers p, q, r such that: and For example,
The McCarthy 91 function is defined as follows: We can generalize this definition by abstracting away the constants into
The sequence Sn is defined by S0 = 290797 and Sn = S{n - 1}^2 bmod 50515093 for n 0.
It is well known that if the square root of a natural number is not an integer, then it is irrational.
Let f(n) represent the number of ways one can fill a 3 times 3 times n tower with blocks of 2 times 1 times 1.
A program written in the programming language Fractran consists of a list of fractions.
Define Q(n) to be the smallest number that occurs in exactly n Pythagorean triples (a,b,c) where a lt b lt c.
Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for examp
A game is played with two piles of stones and two players.
The hyperexponentiation or tetration of a number a by a positive integer b, denoted by amathbin{uparrow uparrow}b or ^b
One variant of N.G. de Bruijn's silver dollar game can be described as follows: On a strip of squares a number of coins
Let S(A) represent the sum of elements in set A of size n.
A natural number, N, that can be written as the sum and product of a given set of at least two natural numbers, a1, a2,
A geoboard (of order N) is a square board with equally-spaced pins protruding from the surface, representing an integer
Let tk be the tribonacci numbers defined as: quad t0 = t1 = 0; quad t2 = 1; quad tk = t{k-1} + t{k-2} + t{k-3} quad text
In the hexadecimal number system numbers are represented using 16 different digits: The hexadecimal number mathrm{AF} wh
Consider the sequence n^2+3 with n ge 1.
For a polygon P, let t(P) be the number of ways in which P can be tiled using rhombi and squares with edge length 1.
A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when conn
A dynamical polynomial is a monicleading coefficient is 1 polynomial f(x) with integer coefficients such that f(x) divid
A touch-screen device can be unlocked with a "password" consisting of a sequence of two or more distinct spots that the
Here are the records from a busy telephone system with one million users: | RecNr | Caller | Called | |:----------:|:---
A monic polynomial is a single-variable polynomial in which the coefficient of highest degree is equal to 1.
A cubic Bézier curve is defined by four points: P0, P1, P2, and P3.
For a positive number n, define S(n) as the sum of the integers x, for which 1 lt x lt n and x^3 equiv 1 bmod n.
Consider the isosceles triangle with base length, b = 16, and legs, L = 17.
Alice walks on a lattice grid.
This problem uses half open interval notation where [a,b) represents a le x < b.
A hexagonal orchard of order n is a triangular lattice made up of points within a regular hexagon with side n.
Let ABCD be a convex quadrilateral, with diagonals AC and BD.
A certain type of chandelier contains a circular ring of n evenly spaced candleholders.
For every integer n1, the family of functions f{n,a,b} is defined by f{n,a,b}(x)equiv a x + b mod n for a,b,x integer an
Barbara is a mathematician and a basketball player.
In the card game poker, a hand consists of five cards and are ranked, from lowest to highest, in the following way: - Hi
Define f(0)=1 and f(n) to be the number of ways to write n as a sum of powers of 2 where no power occurs more than twice
Build a triangle from all positive integers in the following way: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
The cube, 41063625 (345^3), can be permuted to produce two other cubes: 56623104 (384^3) and 66430125 (405^3).
A segment is uniquely defined by its two endpoints.
The Knights of the Order of Fibonacci are preparing a grand feast for their king.
An unbiased coin is tossed repeatedly until two consecutive heads are obtained.
A Harshad or Niven number is a number that is divisible by the sum of its digits.
Dr. One and Dr. Zero are playing the following partisan game. The game begins with one 1, two 2's, three 3's, ..., n n's
For an odd prime p, define f(p) = leftlfloorfrac{2^{(2^p)}}{p}rightrfloorbmod{2^p} For example, when p=3, lfloor 2^8/3rf
A symmetrical convex grid polygon is a polygon such that: - All its vertices have integer coordinates.
A square piece of paper with integer dimensions N times N is placed with a corner at the origin and two of its sides alo
The positive integers, x, y, and z, are consecutive terms of an arithmetic progression.
For any integer n, consider the three functions and their combination We call (x, y, z) a golden triple of order k if x,
Let us define a balanced sculpture of order n as follows: - A polyominoAn arrangement of identical squares connected thr
Consider a rectangle made up of W times H square cells each with area 1.
The Fibonacci numbers fn, n ge 0 are defined recursively as fn = f{n-1} + f{n-2} with base cases f0 = 0 and f1 = 1.
A positive integer matrix is a matrix whose elements are all positive integers.
Let S(n,m) = sumphi(n times i) for 1 leq i leq m.
Let S(A) represent the sum of elements in set A of size n.
The game Number Mind is a variant of the well known game Master Mind.
Any positive real number x can be decomposed into integer and fractional parts lfloor x rfloor + x, where lfloor x rfloo
For a positive integer n, let sigma2(n) be the sum of the squares of its divisors.
Let ai be the sequence defined by ai=153^i bmod 10000019 for i ge 1.
A prime number p is called a Panaitopol prime if p = dfrac{x^4 - y^4}{x^3 + y^3} for some positive integers x and y.
Let Bbb R^2 be the set of pairs of real numbers (x, y).
You are given a unique investment opportunity.
The Gauss Factorial of a number n is defined as the product of all positive numbers leq n that are relatively prime to n
Many numbers can be expressed as the sum of a square and a cube.
For any triangle T in the plane, it can be shown that there is a unique ellipse with largest area that is completely ins
A modified Collatz sequence of integers is obtained from a starting value a1 in the following way: a{n+1} = frac {an} 3
A definition for an ellipse is: Given a circle c with centre M and radius r and a point G such that d(G,M) lt r, the loc
The 5-digit number, 16807=7^5, is also a fifth power.
Let S(n) = sum a + b + c over all triples (a, b, c) such that: - a, b and c are prime numbers.
Define A(n) to be the 10 decimal digits from the nth digit onward.
A Slider is a chess piece that can move one square left or right.
Construct triangle ABC such that: - Vertices A, B and C are lattice points inside or on the circle of radius r centered
Consider the following algorithm for sorting a list: - 1.
Define f(0)=1 and f(n) to be the number of different ways n can be expressed as a sum of integer powers of 2 using each
Find the number of integers 1 lt n lt 10^7, for which n and n + 1 have the same number of positive divisors.
Consider the natural numbers having at least 5 prime factors, which don't have to be distinct.
For any integer n0 and prime number p, define nup(n) as the greatest integer r such that p^r divides n.
Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2, 3, 4.
In the classic "Crossing Ladders" problem, we are given the lengths x and y of two ladders resting on the opposite walls
How many 20 digit numbers n (without any leading zero) exist such that no three consecutive digits of n have a sum great
Find the smallest x + y + z with integers x gt y gt z gt 0 such that x + y, x - y, x + z, x - z, y + z, y - z are all pe
We create an array of points Pn in a two dimensional plane using the following random number generator: s0=290797 s{n+1}
Despite the popularity of Hilbert's infinite hotel, Hilbert decided to try managing extremely large finite hotels, inste
A cyclic number with n digits has a very interesting property: When it is multiplied by 1, 2, 3, 4, dots, n, all the pro
For an integer n, we define the square prime factors of n to be the primes whose square divides n.
Let n be an integer and S(n) be the set of factors of n.
An unbiased single 4-sided die is thrown and its value, T, is noted.
In the following equation x, y, and n are positive integers.
Having three black objects B and one white object W they can be grouped in 7 ways like this: | | | | | | | | |--------|-
A partition of n is a set of positive integers for which the sum equals n.
We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and hor
We use xoplus y for the bitwise XOR of x and y.
Tom has built a random generator that is connected to a row of n light bulbs.
Start from an ordered list of all integers from 1 to n.
Two positive numbers A and B are said to be connected (denoted by "A leftrightarrow B") if one of these conditions holds
We define a pseudo-geometric sequence to be a finite sequence a0, a1, dotsc, an of positive integers, satisfying the fol
Two players play a game with two piles of stones.
You are given a pizza (perfect circle) that has been cut into m cdot n equal pieces and you want to have exactly one top
If we take 47, reverse and add, 47 + 74 = 121, which is palindromic.
The flipping game is a two player game played on an N by N square board.
Let H(n) be the number of distinct integer sided equiangular convex hexagons with perimeter not exceeding n.
The number 145 is well known for the property that the sum of the factorial of its digits is equal to 145: Perhaps less
The number sequence game starts with a sequence S of N numbers written on a line.
Consider the following Diophantine equation: where x, y and z are positive integers.
It is possible to write five as a sum in exactly six different ways: How many different ways can one hundred be written
Let x be a real number.
The (a,b,m)-sequence, where 0 leq a,b lt m, is defined as $begin{align} g(0)&=a g(1)&=b g(n)&= big(g(n-1) + g(n-2)big) b
In the game, Monopoly, the standard board is set up in the following way: !0084monopolyboard.png A player starts on the
A triomino is a shape consisting of three squares joined via the edges.
The harmonic series 1 + frac 1 2 + frac 1 3 + frac 1 4 + cdots is well known to be divergent.
When we calculate 8^n modulo 11 for n=0 to 9 we get: 1, 8, 9, 6, 4, 10, 3, 2, 5, 7.
An integer of the form p^q q^p with prime numbers p neq q is called a hybrid-integer.
Euler's totient function, phi(n) [sometimes called the phi function], is used to determine the number of positive number
We are trying to find a hidden number selected from the set of integers 1, 2, dots, n by asking questions.
A horizontal row comprising of 2n + 1 squares has n red counters placed at one end and n blue counters at the other end,
A pythagorean triangle with catheti a and b and hypotenuse c is characterized by the well-known equation a^2+b^2=c^2.
It is easily proved that no equilateral triangle exists with integral length sides and integral area.
A game is played with many identical, round coins on a flat table.
ABC is an integral sided triangle with sides a le b le c.
Create a sequence of numbers using the "Blum Blum Shub" pseudo-random number generator: Concatenate these numbers s0s1s2
Given is the function f(x) = lfloor 2^{30.403243784 - x^2}rfloor times 10^{-9} (lfloor rfloor is the floor-function), th
Let N(i) be the smallest integer n such that n! is divisible by (i!)^{1234567890} Let S(u)=sum N(i) for 10 le i le u.
Let p(t) denote the (t+1)th prime number.
Let g(n) denote the number of ways a positive integer n can be represented in the form: where x and y are integers.
A window into a matrix is a contiguous sub matrix.
By replacing the 1st digit of the 2-digit number 3, it turns out that six of the nine possible values: 13, 23, 43, 53, 7
For every positive number n we define the function mathop{streak}(n)=k as the smallest positive integer k such that n+k
In laser physics, a "white cell" is a mirror system that acts as a delay line for the laser beam.
Let P(n) be the set of the first n positive integers 1, 2, dots, n.
Consider equations of the form: a^2 + b^2 = N, 0 le a le b, a, b and N integer.
k defects are randomly distributed amongst n integrated-circuit chips produced by a factory (any number of defects may b
Euler's totient function, phi(n) [sometimes called the phi function], is defined as the number of positive integers not
If we calculate a^2 bmod 6 for 0 leq a leq 5 we get: 0,1,4,3,4,1.
Let g(a, n, b, m) be the smallest non-negative solution x to the system: x = a bmod n x = b bmod m if such a solution ex
A series of three rooms are connected to each other by automatic doors.
Leonhard Euler was born on 15 April 1707.
Two positive integers x and y (x y) can generate a sequence in the following manner: - ax = y is the first term, - a{z+1
The function f is defined for all positive integers as follows: It can be proven that f(n) is integer for all values of
How many 18-digit numbers n (without leading zeros) are there such that no digit occurs more than three times in n?
An m times n maze is an m times n rectangular grid with walls placed between grid cells such that there is exactly one p
Consider a unit circlecircle with radius 1 C0 on the plane that does not enclose the origin.
In a sliding game a counter may slide horizontally or vertically into an empty space.
NOTE: This problem is a significantly more challenging version of Problem 81.
Let S(n) be the sum of all contiguous integer-substrings that can be formed from the integer n.
For a positive integer n, let s(n) be the integer obtained by shifting the leftmost digit of the decimal representation
Define d(n,k) to be the number of ways to write n as a product of k ordered integers Further define D(N,K) to be the sum
Define s(n) to be the smallest number that has a digit sum of n.
For a positive integer n, d(n) is defined to be the sum of the digits of n.
Solution to Project Euler Problem 710.
Fermat's Last Theorem states that no three positive integers a, b, c satisfy the equation for any integer value of n gre
The smallest positive integer n for which the numbers n^2 + 1, n^2 + 3, n^2 + 7, n^2 + 9, n^2 + 13, and n^2 + 27 are con
We use xoplus y for the bitwise XOR of x and y.
Let varphi(n) be Euler's totient function.
A dominating number is a positive integer that has more than half of its digits equal.
Let A be an affine plane over a radically integral local field F with residual characteristic p.
The proper divisors of a number are all the divisors excluding the number itself.
A number consisting entirely of ones is called a repunit.
Take a sequence of length n.
Given a set, L, of unique lines, let M(L) be the number of lines in the set and let S(L) be the sum over every line of t
Three circles of equal radius are placed inside a larger circle such that each pair of circles is tangent to one another
In a tournament there are n teams and each team plays each other team twice.
The Fibonacci sequence is defined by the recurrence relation: Fn = F{n - 1} + F{n - 2}, where F1 = 1 and F2 = 1.
Let T(r) be the number of integer quadruplets x, y, z, t such that x^2 + y^2 + z^2 + t^2 le r^2.
Let g(n) be the number of undirected graphs with n nodes satisfying the following properties: - The graph is connected a
Phil the confectioner is making a new batch of chocolate covered candy.
In the following equation x, y, and n are positive integers.
Let us consider mixtures of three substances: A, B and C.
A composite number can be factored many different ways.
Let an be the largest real root of a polynomial g(x) = x^3 - 2^n cdot x^2 + n.
Consider a honey bee's honeycomb where each cell is a perfect regular hexagon with side length 1.
Coprime Nim is just like ordinary normal play Nim, but the players may only remove a number of stones from a pile that i
We call the convex area enclosed by two circles a lenticular hole if: - The centres of both circles are on lattice point
For any N, let f(N) be the last twelve hexadecimal digits before the trailing zeroes in N!.
NOTE: This problem is a more challenging version of Problem 81.
A number where one digit is the sum of the other digits is called a digit sum number or DS-number for short.
There are exactly ten ways of selecting three from five, 12345: 123, 124, 125, 134, 135, 145, 234, 235, 245, and 345 In
A prime is drawn uniformly from all primes not exceeding N.
For every positive integer n the Fibonacci sequence modulo n is periodic.
Let f be a function from a finite set S to itself.
Define G(N) = sumS operatorname{lcm}(S) where S ranges through all subsets of 1, dots, N and operatorname{lcm} denotes t
Two players play a game with a deck of cards which contains s suits with each suit containing n cards numbered from 1 to
Two players play a game.
Given an integer sided triangle ABC: Let I be the incenter of ABC.
There are several ways to write the number dfrac{1}{2} as a sum of square reciprocals using distinct integers.
Let Sk be the set containing 2 and 5 and the first k primes that end in 7.
The Carmichael function lambda(n) is defined as the smallest positive integer m such that a^m = 1 modulo n for all integ
It is possible to show that the square root of two can be expressed as an infinite continued fraction.
For a positive integer n define T(n) to be the number of strictly larger integers which can be formed by permuting the d
The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form 2^{
Let q be a prime and A ge B 0 be two integers with the following properties: - A and B have no prime factor in common, t
The sequence 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201, dots is defined by T1 = T2 = T3 = 1 and Tn = T{n -
Given two positive integers a,b, Alex and Bianca play a game in ab rounds.
The quadtree encoding allows us to describe a 2^N times 2^N black and white image as a sequence of bits (0 and 1).
An integer partition of a number n is a way of writing n as a sum of positive integers.
n runners in very different training states want to compete in a race.
Define f(n) as the sum of the factorials of the digits of n.
A particular school offers cash rewards to children with good attendance and punctuality.
4n people stand in a circle with their heads down.
The divisors of 12 are: 1,2,3,4,6 and 12.
Let us define a geometric triangle as an integer sided triangle with sides a le b le c so that its sides form a geometri
Consider the alphabet A made out of the letters of the word "text{project}": A=text c,text e,text j,text o,text p,text r
The following diagram shows a billiard table of a special quadrilateral shape.
A positive integer n is considered cube-full, if for every prime p that divides n, so does p^3.
Consider the following binary quadratic form: A positive integer q has a primitive representation if there exist positiv
The first number n for which phi(n)=13! is 6227180929.
Three distinct points are plotted at random on a Cartesian plane, for which -1000 le x, y le 1000, such that a triangle
Assuming that two points are chosen randomly (with uniform distribution) within a rectangle, it is possible to determine
A non-decreasing sequence of integers an can be generated from any positive real value theta by the following procedure:
Consider graphs built with the units A: and B: , where the units are glued along the vertical edges as in the graph .
Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given poi
Let D0 be the two-letter string "Fa".
A positive integer, n, is factorised into prime factors.
Two players are playing a game, alternating turns.
Let d(n) be the number of divisors of n.
The kernel of a polygon is defined by the set of points from which the entire polygon's boundary is visible.
We shall define a pythagorean polygon to be a convex polygon with the following properties: - there are at least three v
A row of five grey square tiles is to have a number of its tiles replaced with coloured oblong tiles chosen from red (le
We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and hor
Consider the values of log2(8), log4(64) and log3(27).
There are N seats in a row.
"What? Where? When?" is a TV game show in which a team of experts attempt to answer questions.
A Gaussian integer is a number z = a + bi where a, b are integers and i^2 = -1.
Two positive integers a and b are 2-friendly when gcd(a,b) = 2^t, t gt 0.
Consider all lattice points (a,b,c) with 0 le a,b,c le N.
A gozinta chain for n is a sequence 1,a,b,dots,n where each element properly divides the next.
A riffle shuffle is executed as follows: a deck of cards is split into two equal halves, with the top half taken in the
For integers a and b, we define D(a, b) as the domain enclosed by the parabola y = x^2 and the line y = acdot x + b: D(a
In a 3 times 2 cross-hatched grid, a total of 37 different rectangles could be situated within that grid as indicated in
Card Stacking is a game on a computer starting with an array of N cards labelled 1,2,ldots,N.
Define F(n) to be the number of integers x≤n that can be written in the form x=a^2b^3, where a and b are integers not ne
A set of lattice points S is called a titanic set if there exists a line passing through exactly two points in S.
Suppliers 'A' and 'B' provided the following numbers of products for the luxury hamper market: | Product | 'A' | 'B' | |
Let C(x, y) be a circle passing through the points (x, y), (x, y + 1), (x + 1, y) and (x + 1, y + 1).
Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.
Define F(m,n) as the number of n-tuples of positive integers for which the product of the elements doesn't exceed m.
Two players play a game with a number of piles of stones, alternating turns.
Let (p1 p2 ldots pk) denote the permutation of the set {1, ..., k} that maps pimapsto i.
A triplicate number is a positive integer such that, after repeatedly removing three consecutive identical digits from i
Let G(a, b) be the smallest non-negative integer n for which operatorname{mathbf{gcd}}Greatest common divisor(n^3 + b, (
We define two sequences S = S(1), S(2), ..., S(n) and S2 = S2(1), S2(2), ..., S2(n): S(k) = (pk)^k bmod 10007 where pk i
Two players play a game with two piles of stones, alternating turns.
All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as 2^i
Adam plays the following game with his birthday cake.
The square root of 2 can be written as an infinite continued fraction.
Consider a function f(k) defined for all positive integers k0.
Consider the number 6.
Looking at the table below, it is easy to verify that the maximum possible sum of adjacent numbers in any direction (hor
The Golomb's self-describing sequence (G(n)) is the only nondecreasing sequence of natural numbers such that n appears e
For every n ge 1 the prime-counting function pi(n) is equal to the number of primes not exceeding n.
Let a, b and c be positive numbers.
Each new term in the Fibonacci sequence is generated by adding the previous two terms.
An even positive integer N will be called admissible, if it is a power of 2 or its distinct prime factors are consecutiv
Let p = p1 p2 p3 cdots be an infinite sequence of random digits, selected from 0,1,2,3,4,5,6,7,8,9 with equal probabilit
Let us call a positive integer k a square-pivot, if there is a pair of integers m gt 0 and n ge k, such that the sum of
Gauss famously proved that every positive integer can be expressed as the sum of three triangular numbers (including 0 a
The primes 3, 7, 109, and 673, are quite remarkable.
A matrix M is called idempotent if M^2 = M.
For an integer n ge 4, we define the lower prime square root of n, denoted by operatorname{lps}(n), as the largest prime
Tom (the cat) and Jerry (the mouse) are playing on a simple graph G.
Consider the Gaussian integer i-1.
A 4 times 4 grid is filled with digits d, 0 le d le 9.
It can be shown that the polynomial n^4 + 4n^3 + 2n^2 + 5n is a multiple of 6 for every integer n.
For every positive integer n the Fibonacci sequence modulo n is periodic.
An infinite number of people (numbered 1, 2, 3, etc.) are lined up to get a room at Hilbert's newest infinite hotel.
Gary and Sally play a game using gold and silver coins arranged into a number of vertical stacks, alternating turns.
A k-bounded partition of a positive integer N is a way of writing N as a sum of positive integers not exceeding k.
The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is 28.
Let n be a positive integer.
Nim is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any hea
When (1+sqrt 7) is raised to an integral power, n, we always get a number of the form (a+bsqrt 7).
Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
The well-known Rubik's Cube puzzle has many fascinating mathematical properties.
Consider a horizontal frictionless tube with length L millimetres, and a diameter of 20 millimetres.
Consider the infinite repeating sequence of digits: 1234321234321234321...
For a positive integer n, let sigma(n) be the sum of all divisors of n.
(x,y) is called a nested radical pair if x and y are non-zero integers such that dfrac{x}{y} is not a cube of a rational
Let x1, x2, dots, xn be a sequence of length n such that: - x1 = 2 - for all 1 lt i le n: x{i - 1} lt xi - for all i and
Susan has a prime frog.
Alice is playing a game with n cards numbered 1 to n.
Let T(n) be the nth triangle number, so T(n) = dfrac{n(n + 1)}{2}.
Su Doku (Japanese meaning number place) is the name given to a popular puzzle concept.
Consider a row of n dice all showing 1.
The logical-OR of two bits is 0 if both bits are 0, otherwise it is 1.
For every real number a gt 1 is given the sequence ga by: g{a}(x)=1 for x lt a g{a}(x)=g{a}(x-1)+ga(x-a) for x ge a G(n)
ABC is an integer sided triangle with incenter I and perimeter p.
Let f(N) be the number of points with integer coordinates that are on a circle passing through (0,0), (N,0),(0,N), and (
Let (a, b, c) represent the three sides of a right angle triangle with integral length sides.
Let C be the circle with radius r, x^2 + y^2 = r^2.
A rectangular sheet of grid paper with integer dimensions w times h is given.
The regular star polygon p/q, for coprime integers p,q with p gt 2q gt 0, is a polygon formed from p edges of equal leng
A mountain range consists of a line of mountains with slopes of exactly 45^circ, and heights governed by the prime numbe
Let n be a positive integer.
A sequence is defined as: - gk = 1, for 0 le k le 1999 - gk = g{k-2000} + g{k - 1999}, for k ge 2000.
For any positive integer n, the nth weak Goodstein sequence g1, g2, g3, dots is defined as: - g1 = n - for k gt 1, gk is
A row measuring seven units in length has red blocks with a minimum length of three units placed on it, such that any tw
Tom has built a random generator that is connected to a row of n light bulbs.
Consider a wtimes h grid.
Odd and Even are playing a game with N cookies.
Let d(i,b) be the digit sum of the number i in base b.
Let sigma(n) be the sum of all the divisors of the positive integer n, for example: sigma(10) = 1+2+5+10 = 18.
For a positive integer, n, define g(n) to be the maximum perfect square that divides n.
A group of chefs (numbered 1, 2, etc) participate in a turn-based strategic cooking competition.
2^N binary digits can be placed in a circle so that all the N-digit clockwise subsequences are distinct.
Let f(n) be the largest prime factor of n.
Circles A and B are tangent to each other and to line L at three distinct points.
For any positive integer k, a finite sequence ai of fractions xi/yi is defined by: a1 = 1/k and ai = (x{i - 1} + 1) / (y
Let displaystyle S(n)=sumlimits{k=0}^{n}binom{n}{k}k^n.
A number consisting entirely of ones is called a repunit.
A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next low
Let n be a positive integer and let En be the set of n-tuples of strictly positive integers.
Let ABCD be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows: A(a, 0), B(0, b),
Find the unique positive integer whose square has the form 1234567890, where each “” is a single digit.
Jack has three plates in front of him.
Let C(n) be the number of squarefree integers of the form x^2 + 1 such that 1 le x le n.
On a chess board, a centaur moves like a king or a knight.
Let F(r, c, n) be the number of ways to colour a rectangular grid with r rows and c columns using at most n colours such
Consider a three dimensional grid of cubes.
Two players share an unbiased coin and take it in turns to play The Race.
A root or zero of a polynomial P(x) is a solution to the equation P(x) = 0.
As we all know the equation x^2=-1 has no solutions for real x.
What is the length of the shortest pipe, of internal radius pu{50 mm}, that can fully contain 21 balls of radii pu{30 mm
Given the set 1,2,dots,n, we define f(n, k) as the number of its k-element subsets with an odd sum of elements.
In a triangular array of positive and negative integers, we wish to find a sub-triangle such that the sum of the numbers
The complexity of an ntimes n binary matrix is the number of distinct rows and columns.
For a prime p let S(p) = (sum (p-k)!) bmod (p) for 1 le k le 5.
An integer is called B-smooth if none of its prime factors is greater than B.
Define two functions on lattice points: r(x,y) = (x+1,2y) s(x,y) = (2x,y+1) A path to equality of length n for a pair (a
You probably know the game Fifteen Puzzle.
Two cars are on a circular track of total length 2n, facing the same direction, initially distance n apart.
Let P(m,n) be the number of distinct terms in an mtimes n multiplication table.
A positive number is pandigital in base b if it contains all digits from 0 to b - 1 at least once when written in base b
Given the positive integers, x, y, and z, are consecutive terms of an arithmetic progression, the least value of the pos
Let a1, a2, dots, an be an integer sequence of length n such that: - a1 = 6 - for all 1 le i lt n: phi(ai) lt phi(a{i +
A Pythagorean triangle is called supernatural if two of its three sides are consecutive integers.
Let y0, y1, y2, dots be a sequence of random unsigned 32-bit integers (i.e.
Consider the following set of dice with nonstandard pips: Die A: 1 4 4 4 4 4 Die B: 2 2 2 5 5 5 Die C: 3 3 3 3 3 6 A gam
The sum of the kth powers of the first n positive integers can be expressed as a polynomial of degree k+1 with rational
The game of Mahjong is played with tiles belonging to s suits.
A deck of cards numbered from 1 to n is shuffled randomly such that each permutation is equally likely.
The lambda-calculus is a universal model of computation at the core of functional programming languages.
Let r be the real root of the equation x^3 = x^2 + 1.
Let F(N) be the maximum number of lattice points in an axis-aligned Ntimes N square that the graph of a single strictly
Define m = M(n, d) to be the smallest positive integer such that when m^2 is written in base n it includes the base n di
An integer s is called a superinteger of another integer n if the digits of n form a subsequenceA subsequence is a seque
A robot moves in a series of one-fifth circular arcs (72^circ), with a free choice of a clockwise or an anticlockwise ar
Using a combination of grey square tiles and oblong tiles chosen from: red tiles (measuring two units), green tiles (mea
The four right-angled triangles with sides (9,12,15), (12,16,20), (5,12,13) and (12,35,37) all have one of the shorter s
The most naive way of computing n^{15} requires fourteen multiplications: But using a "binary" method you can compute it
A positive integer is called square root smooth if all of its prime factors are strictly less than its square root.
After buying a Gerver Sofa from the Moving Sofa Company, Jack wants to buy a matching cocktail table from the same compa
A standard deck of 52 playing cards, which consists of thirteen ranks (Ace, Two, ..., Ten, King, Queen and Jack) each in
A Fibonacci tree is a binary tree recursively defined as: - T(0) is the empty tree.
For integers m, n (0 leq n lt m), let L(m, n) be an m times m grid with the top-right n times n grid removed.
A firecracker explodes at a height of pu{100 m} above level ground.
On the Euclidean plane, an ant travels from point A(0, 1) to point B(d, 1) for an integer d.
By replacing each of the letters in the word CARE with 1, 2, 9, and 6 respectively, we form a square number: 1296 = 36^2
An equilateral triangle with integer side length n ge 3 is divided into n^2 equilateral triangles with side length 1 as
Comparing two numbers written in index form like 2^{11} and 3^7 is not difficult, as any calculator would confirm that 2
Let A and B be bit strings (sequences of 0's and 1's).
Consider the infinite integer sequence S starting with: S = 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 8, 4, 9, 1, 10, 11, 5
A circle C of circumference c centimetres has a smaller circle S of circumference s centimetres lying off-centre within
Rand48 is a pseudorandom number generator used by some programming languages.
Writing down the numbers which have a digit sum of 10 in ascending order, we get: 19, 28, 37, 46,55,64,73,82,91,109, 118
An infinitely long cylinder has its curved surface fully covered with different coloured but otherwise identical rectang
The triangle triangle ABC is inscribed in an ellipse with equation frac {x^2} {a^2} + frac {y^2} {b^2} = 1, 0 lt 2b lt a
A depot uses n drones to disperse packages containing essential supplies along a long straight road.
For a positive integer d, let f(d) be the number created by sorting the digits of d in ascending order, removing any zer
For any positive integer n the function operatorname{nextprime}(n) returns the smallest prime p such that p gt n.
Mamma Triangolo baked a triangular pizza.
There are 16 positive integers that do not have a zero in their digits and that have a digital sum equal to 5, namely: 5
A laborious ant walks randomly on a 5 times 5 grid.
The divisors of 6 are 1,2,3 and 6.
A row of n squares contains a frog in the leftmost square.
Consider numbers t(n) of the form t(n) = 2n^2 - 1 with n gt 1.
Define operatorname{Co}(n) to be the maximal possible sum of a set of mutually co-prime elements from 1,2,dots,n.
Numbers of the form n^{15}+1 are composite for every integer n gt 1.
A random generator produces a sequence of symbols drawn from the set {I, V, X, L, C, D, M, }.
Let H(n) denote the number of sets of positive integers such that the least common multiple of the integers in the set e
For a positive integer n, let f(n) be the sum of the squares of the digits (in base 10) of n, e.g.
Sam and Tom are trying a game of (partially) covering a given line segment of length L by taking turns in placing unit s
A composite is a number containing at least two prime factors.
Four points with integer coordinates are selected: A(a, 0), B(b, 0), C(0, c) and D(0, d), with 0 lt a lt b and 0 lt c lt
There are some prime values, p, for which there exists a positive integer, n, such that the expression n^3 + n^2p is a p
Consider an infinite row of boxes.
A snowflake of order n is formed by overlaying an equilateral triangle (rotated by 180 degrees) onto each equilateral tr
There is a method that is used by Bell ringers to generate all variations of the order that bells are rung.
For an integer M, we define R(M) as the sum of 1/(p cdot q) for all the integer pairs p and q which satisfy all of these
Let S be the set consisting of the four letters texttt{A'},texttt{E'},texttt{F'},texttt{R'}.
In a very simplified form, we can consider proteins as strings consisting of hydrophobic (H) and polar (P) elements, e.g
Let ra, rb and rc be the radii of three circles that are mutually and externally tangent to each other.
Alice and Bob have enjoyed playing Nim every day.
Given an n-tuple of numbers another n-tuple is created where each element of the new n-tuple is chosen randomly from the
Albert chooses a positive integer k, then two real numbers a, b are randomly chosen in the interval [0,1] with uniform d
The RSA encryption is based on the following procedure: Generate two distinct primes p and q.
Let Sm = (x1, x2, dots , xm) be the m-tuple of positive real numbers with x1 + x2 + cdots + xm = m for which Pm = x1 cdo
For two integers n,e gt 1, we define an (n,e)-MPS (Mirror Power Sequence) to be an infinite sequence of integers (ai){ig
The SET® card game is played with a pack of 81 distinct cards.
A positive fraction whose numerator is less than its denominator is called a proper fraction.
On planet J, a year lasts for D days.
If a triple of positive integers (a, b, c) satisfies a^2+b^2=c^2, it is called a Pythagorean triple.
Recall that a graph is a collection of vertices and edges connecting the vertices, and that two vertices connected by an
Bob plays a single-player game of chance using two standard 6-sided dice and twelve cards numbered 1 to 12.
A gambler decides to participate in a special lottery.
It is possible to write ten as the sum of primes in exactly five different ways: What is the first value which can be wr
When wrapping several cubes in paper, it is more efficient to wrap them all together than to wrap each one individually.
Let f(n) be the largest positive integer x less than 10^9 such that the last 9 digits of n^x form the number x (includin
In a room N chairs are placed around a round table.
The coefficients in the expansion of (x+1)^k are called binomial coefficients.
Consider the consecutive primes p1 = 19 and p2 = 23.
Consider the following variant of "The Chase" game.
Taking three different letters from the 26 letters of the alphabet, character strings of length three can be formed.
In the context of formal languages, any finite sequence of letters of a given alphabet Sigma is called a word over Sigma
We define an S-number to be a natural number, n, that is a perfect square and its square root can be obtained by splitti
Starting from zero the natural numbers are written down in base 10 like this: Consider the digit d=1.
Peter is playing a solitaire game on an infinite checkerboard, each square of which can hold an unlimited number of toke
Find the number of non-empty subsets of 1^1, 2^2, 3^3,dots, 250250^{250250}, the sum of whose elements is divisible by 2
A googol (10^{100}) is a massive number: one followed by one-hundred zeros; 100^{100} is almost unimaginably large: one
70 coloured balls are placed in an urn, 10 for each of the seven rainbow colours.
An arrangement of coins in one or more rows with the bottom row being a block without gaps and every coin in a higher ro
The positive integral solutions of the equation x^y=y^x are (2,4), (4,2) and (k,k) for all k 0.
Consider the numbers 15, 16 and 18: 15=3times 5 and 3+5=8.
The blancmange curve is the set of points (x, y) such that 0 le x le 1 and y = sum limits{n = 0}^{infty} {dfrac{s(2^n x)
A number is p-smooth if it has no prime factors larger than p.
Let Si be an integer sequence produced with the following pseudo-random number generator: - S0 = 290797 - S{i+1} = Si ^2
On the parabola y = x^2/k, three points A(a, a^2/k), B(b, b^2/k) and C(c, c^2/k) are chosen.
A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in
Given is an integer sided triangle ABC with sides a le b le c.
Let's call an integer sided triangle with exactly one angle of 60 degrees a 60-degree triangle.
Consider the set S of all possible products of n positive integers not exceeding m, that is S= x1x2cdots xn mid 1 le x1,
!0315clocks.gif Sam and Max are asked to transform two digital clocks into two "digital root" clocks.
Let n be a positive integer.
A bag contains one red disc and one blue disc.
For each integer p gt 1 coprime to 10 there is a positive divisibility multiplier m lt p which preserves divisibility by
A unit fraction contains 1 in the numerator.
Three mirrors are arranged in the shape of an equilateral triangle, with their reflective surfaces pointing inwards.
Consider the right angled triangle with sides a=7, b=24 and c=25.
A best approximation to a real number x for the denominator bound d is a rational number frac r s (in reduced form) with
Inside a rope of length n, n - 1 points are placed with distance 1 from each other and from the endpoints.
For positive integers n and m, we define two polynomials Fn(x) = x^n and Gm(x) = (x-1)^m.
Let pn be the nth prime: 2, 3, 5, 7, 11, dots, and let r be the remainder when (pn - 1)^n + (pn + 1)^n is divided by pn^
Consider the set S(r) of points (x,y) with integer coordinates satisfying |x| + |y| le r.
A positive integer n is powerful if p^2 is a divisor of n for every prime factor p in n.
Let S(k) be the sum of three or more distinct positive integers having the following properties: - No value exceeds k.
Let f(n) = n^2 - 3n - 1.
Consider the sequence of real numbers an defined by the starting value a0 and the recurrence displaystyle a{n+1}=an-frac
Let g(m) be the integer defined by the following double sum of products of binomial coefficients: You are given that g(1
Let us call an integer sided triangle with sides a le b le c barely acute if the sides satisfy a^2 + b^2 = c^2 + 1.
Consider n coins arranged in a circle where each coin shows heads or tails.
We call an integer sided triangle n-pandigital if it contains one angle of 120 degrees and, when the sides of the triang
"And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of
A standard 52-card deck comprises 13 ranks in four suits.
For a non-negative integer k, define where sigmak(n) = sum{d mid n} d^k is the sum of the k-th powers of the positive di
For 0 le x lt 1, define di(x) to be the ith digit after the binary point of the binary representation of x.
The Möbius function, denoted mu(n), is defined as: - mu(n) = (-1)^{omega(n)} if n is squarefree (where omega(n) is the n
A Pythagorean triple consists of three positive integers a, b and c satisfying a^2+b^2=c^2.
An axis-aligned cuboid, specified by parameters (x0, y0, z0), (dx, dy, dz), consists of all points (X,Y,Z) such that x0
Let phi be Euler's totient function, i.e.
A list initially contains the numbers 2, 3, dots, n.
This problem combines the game of Nim with the Towers of Hanoi.
A printing shop runs 16 batches (jobs) every week and each batch requires a sheet of special colour-proofing paper of si
Define where oplus, vee, wedge are the bitwise XOR, OR and AND operator respectively.
A set, S, of integers is called 123-separable if S, 2S and 3S are disjoint.
An ellipse E(a, b) is given at its initial position by equation: frac {x^2} {a^2} + frac {(y - b)^2} {b^2} = 1 The ellip
Let f(n) be the largest prime factor of n and displaystyle F(n) = sum{i=2}^n f(i).
Whenever Peter feels bored, he places some bowls, containing one bean each, in a circle.
We use xoplus y for the bitwise XOR of x and y.
For some fixed rho in [0, 1], we begin a sum s at 0 and repeatedly apply a process: With probability rho, we add 1 to s,
A 10-substring of a number is a substring of its digits that sum to 10.
Siegbert and Jo take turns playing a game with a heap of N pebbles: 1.
Let p(n) be the number of ways to write n as the sum of powers of two, ignoring order.
Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for I
Every triangle has a circumscribed circle that goes through the three vertices.
The following is a function defined for all positive rational values of x.
Let N and K be two positive integers.
Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are
The function f is defined for all positive integers as follows: - f(1)=1 - f(3)=3 - f(2n)=f(n) - f(4n + 1)=2f(2n + 1) -
In this problem we consider triangles drawn on a hexagonal lattice, where each lattice point in the plane has six neighb
Let D(n) be the n-th positive integer that has the sum of its digits a prime.
In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbe
Consider the triangle with sides 6, 8, and 10.
Triangle numbers Tk are integers of the form frac{k(k+1)} 2.
Consider the number 54.
The eight divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
A sequence of rooted trees Tn is constructed such that Tn has n nodes numbered 1 to n.
For positive real numbers a,b, an atimes b torus is a rectangle of width a and height b, with left and right sides ident
Let T(n, m) be the number of m-tuples of positive integers such that the sum of any two neighbouring elements of the tup
Let n be a positive integer.
Let S(A) represent the sum of elements in set A of size n.
It is possible to find positive integers A and B such that given any triangular number, Tn, then ATn +B is always a tria
Given positive integers a le b le c le d, it may be possible to form quadrilaterals with edge lengths a,b,c,d (in any or
Two players play a game with a single pile of stones of initial size n.
A small child has a “number caterpillar” consisting of N jigsaw pieces, each with one number on it, which, when connecte
The numbers 545, 5995 and 15151 are the three smallest palindromes divisible by 109.
We define 123-numbers as follows: - 1 is the smallest 123-number.
The Torpids are rowing races held annually in Oxford, following some curious rules: - A division consists of n boats (ty
In the 5 by 5 matrix below, the minimal path sum from the top left to the bottom right, by only moving to the right and
Let S(n) be the sum of all positive integers m not exceeding n having the following property: a^{m + 4} equiv a pmod m f
Given any positive integer n, we can construct a new integer by inserting plus signs between some of the digits of the b
Define function P(n, k) = 1 if n can be written as the sum of k prime numbers (with repetitions allowed), and P(n, k) =
Consider the number 142857.
Oscar and Eric play the following game.
A standard 52 card deck comprises thirteen ranks in four suits.
Alice and Bob are taking turns playing a game consisting of c different coins on a chessboard of size n by n.
Given an integer n, n geq 3, let B=mathrm{false},mathrm{true} and let B^n be the set of sequences of n values from B.
We shall define a sqube to be a number of the form, p^2 q^3, where p and q are distinct primes.
A positive integer, n, is divided by d and the quotient and remainder are q and r respectively.
Consider all permutations of 1, 2, ldots N, listed in lexicographic order.
Consider the number 48.
The number of divisors of 120 is 16.
Consider all the words which can be formed by selecting letters, in any order, from the phrase: thereisasyetinsufficient
The binomial coefficients displaystyle binom n k can be arranged in triangular form, Pascal's triangle, like this: | | |
Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots
For any prime p the number N(p, q) is defined by N(p, q) = sum{n = 0}^q Tn cdot p^n with Tn generated by the following r
By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top
A gozinta chain for n is a sequence 1,a,b,dots,n where each element properly divides the next.
The radical of n, operatorname{rad}(n), is the product of the distinct prime factors of n.
We define a simber to be a positive integer in which any odd digit, if present, occurs an odd number of times, and any e
A company specialises in producing large rectangular metal sheets, starting from unit square metal plates.
Let sk be the number of 1’s when writing the numbers from 0 to k in binary.
The largest integer le 100 that is only divisible by both the primes 2 and 3 is 96, as 96=32times 3=2^5 times 3.
Consider the set Ir of points (x,y) with integer co-ordinates in the interior of the circle with radius r, centered at t
Starting with 1 gram of gold you play a game.
Let T(n) be the number of tours over a 4 times n playing board such that: - The tour starts in the top left corner.
Let fk(n) be the sum of the kth powers of the first n positive integers.
n families, each with four members, a father, a mother, a son and a daughter, were invited to a restaurant.
Consider the fraction, dfrac n d, where n and d are positive integers.
Let fn(k) = e^{k/n} - 1, for all non-negative integers k.
5-smooth numbers are numbers whose largest prime factor doesn't exceed 5.
Let dn(x) be the nth decimal digit of the fractional part of x, or 0 if the fractional part has fewer than n digits.
Frodo and Sam need to travel 100 leagues due East from point A to point B.
All square roots are periodic when written as continued fractions and can be written in the form: For example, let us co
In this problem oplus is used to represent the bitwise exclusive or of two numbers.
Each one of the 25 sheep in a flock must be tested for a rare virus, known to affect 2 of the sheep population.
Two friends A and B are great fans of Chess.
A positive integer with k (decimal) digits is called balanced if its first lceil k/2 rceil digits sum to the same value
We define the rounded-square-root of a positive integer n as the square root of n rounded to the nearest integer.
We stack n plates into k non-empty piles where each pile is a different size.
Consider all the triangles having: - All their vertices on lattice pointsInteger coordinates.
The above is an example of a cryptic kakuro (also known as cross sums, or even sums cross) puzzle, with its final soluti
Consider the following algorithm for sorting a list: - 1.
Let tn be the tribonacci numbers defined as: t0 = t1 = 0; t2 = 1; tn = t{n-1} + t{n-2} + t{n-3} for n ge 3 and let rn =
Some positive integers n have the property that the sum [n + operatorname{reverse}(n)] consists entirely of odd (decimal
Consider the following configuration of 64 triangles: We wish to colour the interior of each triangle with one of three
In the following equation x, y, and n are positive integers.
Anton and Bertrand love to play three pile Nim.
We want to tile a board of length n and height 1 completely, with either 1 times 2 blocks or 1 times 1 blocks with a sin
2^7=128 is the first power of two whose leading digits are "12".
Consider a single game of Ramvok: Let t represent the maximum number of turns the game lasts.
An n times n grid of squares contains n^2 ants, one ant per square.
Let r be the remainder when (a - 1)^n + (a + 1)^n is divided by a^2.
For a non-negative integer k, the triple (p,q,r) of positive integers is called a k-shifted Pythagorean triple if (p, q,
Let's call S the (infinite) string that is made by concatenating the consecutive positive integers (starting from 1) wri
A group of p people decide to sit down at a round table and play a lottery-ticket trading game.
It is a common recreational problem to make a target number using a selection of other numbers.
Three points, P1, P2 and P3, are randomly selected within a unit square.
The following undirected network consists of seven vertices and twelve edges with a total weight of 243.
The look and say sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...
Let varphi be the golden ratio: varphi=frac{1+sqrt{5}}{2}.
In the context of formal languages, any finite sequence of letters of a given alphabet Sigma is called a word over Sigma
There are 1111 ways in which five 6-sided dice (sides numbered 1 to 6) can be rolled so that the top three sum to 15.
Consider a two dimensional grid of squares.
Given a fixed real number c, define a random sequence (Xn){nge 0} by the following random process: - X0 = c (with probab
Consider the region constrained by 1 le x and 0 le y le 1/x.
Let R(M, N) be the number of lattice points (x, y) which satisfy MltxleN, MltyleN and largeleftlfloorfrac{y^2}{x^2}right
N disks are placed in a row, indexed 1 to N from left to right.
For a positive integer n gt 1, let p(n) be the smallest prime dividing n, and let alpha(n) be its p-adic order, i.e.
A 30 times 30 grid of squares contains 900 fleas, initially one flea per square.
Given the values of integers 1 < a1 < a2 < dots < an, consider the linear combination q1 a1+q2 a2 + dots + qn an=b, usin
Define: xn = (1248^n bmod 32323) - 16161 yn = (8421^n bmod 30103) - 15051 Pn = (x1, y1), (x2, y2), dots, (xn, yn) For ex
Given a character string s, we define L(k,s) to be the length of the longest substring of s which appears at least k tim
There is a grid of length and width 50515093 points.
A sliding block puzzle is a puzzle where pieces are confined to a grid and by sliding the pieces a final configuration i
Consider the problem of building a wall out of 2 times 1 and 3 times 1 bricks (text{horizontal} times text{vertical} dim
The inversion count of a sequence of digits is the smallest number of adjacent pairs that must be swapped to sort the se
By counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles: Although the
A pack of cards contains 4n cards with four identical cards of each value.
If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next te
Consider a positive integer sequence S = (s1, s2, dots, sn).
A Hilbert number is any positive integer of the form 4k+1 for integer kgeq 0.
Given n equally spaced points on a circle, we define an n-star polygon as an n-gon having those n points as vertices.
Let An be the smallest positive integer satisfying An bmod pi = i for all 1 le i le n, where pi is the i-th prime.
A positive integer N is stealthy, if there exist positive integers a, b, c, d such that ab = cd = N and a+b = c+d+1.
Let D(m,n)=displaystylesum{dmid m}sum{k=1}^nsigma0(kd) where d runs through all divisors of m and sigma0(n) is the numbe
Larry and Robin play a memory game involving a sequence of random numbers between 1 and 10, inclusive, that are called o
Let F5(n) be the number of strings s such that: - s consists only of '0's and '1's, - s has length at most n, and - s co
A long long time ago in a galaxy far far away, the Wimwians, inhabitants of planet WimWi, discovered an unmanned drone t
Let Seq(n,k) be the number of positive-integer sequences ai{1 le i le n} of length n such that: - n is divisible by ai f
A binary matrix is a matrix consisting entirely of 0s and 1s.
Let ak, bk, and ck represent the three solutions (real or complex numbers) to the equation frac 1 x = (frac k x)^2(k+x^2
A simple quadrilateral is a polygon that has four distinct vertices, has no straight angles and does not self-intersect.
Consider the number 45656.
The smallest number m such that 10 divides m! is m=5.
Let E(x0, y0) be the number of steps it takes to determine the greatest common divisor of x0 and y0 with Euclid's algori
Let b(n) be the largest power of 2 that divides n.
We call a positive integer double pandigital if it uses all the digits 0 to 9 exactly twice (with no leading zero).
We define the mathcal{I} operator as the function and mathcal{I}-expressions as arithmetic expressions built only from v
Let H be the hyperbola defined by the equation 12x^2 + 7xy - 12y^2 = 625.
Define fk(n) = sum{i=0}^n fk(lfloorfrac i k rfloor) where fk(0) = 1 and lfloor x rfloor denotes the floor function.
Using only a six-sided fair dice and a five-sided fair dice, we would like to emulate an n-sided fair dice.
5-smooth numbers are numbers whose largest prime factor doesn't exceed 5.
We define the Matrix Sum of a matrix as the maximum possible sum of matrix elements such that none of the selected eleme
Let S(n, k, b) represent the number of valid solutions to x1 + x2 + cdots + xk le n, where 0 le xm le b^m for all 1 le m
Let ABC be a triangle with all interior angles being less than 120 degrees.
This game has a box of N unfair coins and N fair coins.
A list initially contains the numbers 2, 3, dots, n.
For a positive number n, define C(n) as the number of the integers x, for which 1 lt x lt n and x^3 equiv 1 bmod n.
The arithmetic derivative is defined by - p^prime = 1 for any prime p - (ab)^prime = a^prime b + ab^prime for all intege
Consider the number 50.
Consider the following Diophantine equation: where x, y and z are positive integers.
Given an odd prime p, put the numbers 1,...,p-1 into frac{p-1}{2} pairs such that each number appears exactly once.
A polygon is a flat shape consisting of straight line segments that are joined to form a closed chain or circuit.
Every divisor d of a number n has a complementary divisor n/d.
Any positive integer can be written as a product of prime powers: p1^{a1} times p2^{a2} times cdots times pk^{ak}, where
Consider the number 3600.
Consider the divisors of 30: 1,2,3,5,6,10,15,30.
Let's call two numbers friend numbers if their representation in base 10 has at least one common digit.
Consider writing a natural number as product of powers of natural numbers with given exponents, additionally requiring d
Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for examp
Consider the triangles with integer sides a, b and c with a le b le c.
For any N, let f(N) be the last five digits before the trailing zeroes in N!.
We define a block to be a rectangle with a height of 1 and an integer-valued length.
A position in chess is an (orientated) arrangement of chess pieces placed on a chessboard of given size.
Upside Down is a modification of the famous Pythagorean equation: A solution (x,y,z) to this equation with x,y and z pos
Define g(n, m) to be the largest integer k such that 2^k divides binom{n}m.
Oregon licence plates consist of three letters followed by a three digit number (each digit can be from [0..9]).
The Collatz sequence is defined as: a{i+1} = left large{frac {ai} 2 atop 3 ai+1} {text{if }aitext{ is even} atop text{if
There are 5 ways to tile a regular dodecagon of side 1 with regular polygons of side 1.
For an integer n, we define the square prime factors of n to be the primes whose square divides n.
A list of size n is a sequence of n natural numbers.
Given n and k two positive integers we begin with an urn that contains kn white balls.
The nth harmonic number Hn is defined as the sum of the multiplicative inverses of the first n positive integers, and ca
For some positive integers k, there exists an integer partition of the form 4^t = 2^t + k, where 4^t, 2^t, and k are all
If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken
For any two strings of digits, A and B, we define F{A, B} to be the sequence (A,B,AB,BAB,ABBAB,dots) in which each term
Let's call a lattice point (x, y) inadmissible if x, y and x+y are all positive perfect squares.
How many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter tha
Let f(N) be the smallest positive integer that is not coprime to any positive integer n le N whose least significant dig
Every day for the past n days Even Stevens brings home his groceries in a plastic bag.
A graph is made up of vertices and coloured edges.
A moon could be described by the sphere C(r) with centre (0,0,0) and radius r.
Members of a species of bacteria occur in two different types: alpha and beta.
A certain type of tile comes in three different sizes - 1 times 1, 1 times 2, and 1 times 3 - and in four different colo
We call a natural number a duodigit if its decimal representation uses no more than two different digits.
Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hanoi," which consists of three upright rods an
Consider positive integer solutions to a^2+b^2+c^2 = 3abc For example, (1,5,13) is a solution.
Let Aq(n) be the number of subsets, B, of the set 1, 2, ..., q cdot n that satisfy two conditions: 1) B has exactly n el
A Hamming number is a positive number which has no prime factor larger than 5.
The number 7 is special, because 7 is 111 written in base 2, and 11 written in base 6 (i.e.
Consider the diophantine equation frac 1 a + frac 1 b = frac p {10^n} with a, b, p, n positive integers and a le b.
Let pi(x) be the prime counting function, i.e.
Two players, Anton and Bernhard, are playing the following game.
A set of disks numbered 1 through 100 are placed in a line in random order.
At Euler University, each of the n students (numbered from 1 to n) occupies a bed in the dormitory and uses a desk in th
NOTE: This is a more difficult version of Problem 114.
In Plato's heaven, there exist an infinite number of bowls in a straight line.
A sequence of integers S = si is called an n-sequence if it has n elements and each element si satisfies 1 leq si leq n.
Let f(n) be the number of 6-tuples (x1,x2,x3,x4,x5,x6) such that: - All xi are integers with 0 leq xi < n - gcd(x1^2+x2^
Let F(n) be the number of connected graphs with blue edges (directed) and red edges (undirected) containing: - two verti
The radical of n, operatorname{rad}(n), is the product of distinct prime factors of n.
It turns out that pu{12 cm} is the smallest length of wire that can be bent to form an integer sided right angle triangl
Let: begin{array}{ll} x(0)&=0 x(1)&=1 x(2k)&=(3x(k)+2x(lfloor frac k 2 rfloor)) text{ mod } 2^{60} text{ for } k ge 1 te
A triplet of positive integers (a, b, c) is called a Cardano Triplet if it satisfies the condition: For example, (2,1,5)
Given two points (x1, y1, z1) and (x2, y2, z2) in three dimensional space, the Manhattan distance between those points i
Ea is an ellipse with an equation of the form x^2 + 4y^2 = 4a^2.
Let d(k) be the sum of all divisors of k.
By using each of the digits from the set, 1, 2, 3, 4, exactly once, and making use of the four arithmetic operations (+,
A certain type of flexible tile comes in three different sizes - 1 times 1, 1 times 2, and 1 times 3 - and in k differen
We can easily verify that none of the entries in the first seven rows of Pascal's triangle are divisible by 7: | | | | |
ABCD is a convex, integer sided quadrilateral with 1 le AB lt BC lt CD lt AD.
For every integer n1, the family of functions f{n,a,b} is defined by f{n,a,b}(x)equiv a x + b mod n for a,b,x integer an
Let T(m, n) be the number of the binomial coefficients ^iCn that are divisible by 10 for n le i lt m (i, m and n are pos
6174 is a remarkable number; if we sort its digits in increasing order and subtract that number from the number you get
For a positive integer n, define f(n) as the least positive multiple of n that, written in base 10, uses only digits le
An electric circuit uses exclusively identical capacitors of the same value C.
For a positive integer k, define d(k) as the sum of the digits of k in its usual decimal representation.
Given is the function f(a,n)=lfloor (lceil sqrt a rceil + sqrt a)^n rfloor.
It was quite an ordinary day when a mysterious alien vessel appeared as if from nowhere.
The Thue-Morse sequence Tn is a binary sequence satisfying: - T0 = 0 - T{2n} = Tn - T{2n + 1} = 1 - Tn The first several
A regular hexagon table of side length N is divided into equilateral triangles of side length 1.
The binomial coefficient displaystyle binom {10} 3 = 120.
The Chase is a game played with two dice and an even number of players.
It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different orde
Alice and Bob are playing a modified game of Nim called Scatterstone Nim, with Alice going first, alternating turns with
Consider a wire of length 1 unit between two posts.
For a number written in Roman numerals to be considered valid there are basic rules which must be followed.
Consider the real number sqrt 2 + sqrt 3.
A window into a matrix is a contiguous sub matrix.
A positive integer, n, is a near power sum if there exists a positive integer, k, such that the sum of the kth powers of
A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting at "12 o'clock" and numbering th
The following equation represents the continuous topography of a mountainous region, giving the elevationheight above se
A positive integer n is called squarefree, if no square of a prime divides n, thus 1, 2, 3, 5, 6, 7, 10, 11 are squarefr
The binomial coefficient displaystyle{binom{10^{18}}{10^9}} is a number with more than 9 billion (9times 10^9) digits.
We define a permutation as an operation that rearranges the order of the elements 1, 2, 3, ..., n.
For a positive integer n create a graph using its divisors as vertices.
Let P{a,b} denote a path in a atimes b lattice grid with following properties: - The path begins at (0,0) and ends at (a
For a positive integer n, the function g(n) is defined as For example, g(4) = -gcd left(4,1^2right) + gcd left(4,2^2righ
Let f(n) be the number of couples (x, y) with x and y positive integers, x le y and the least common multiple of x and y
Let Sn be the regular n-sided polygon – or shape – whose vertices vk (k = 1, 2, dots, n) have coordinates: Each Sn is to
Let N be a positive integer and let N be split into k equal parts, r = N/k, so that N = r + r + cdots + r.
Consider an equilateral triangle in which straight lines are drawn from each vertex to the middle of the opposite side,
A permutation of 2,3,ldots,n is a rearrangement of these numbers.
We use xoplus y to be the bitwise XOR of x and y.
Consider a directed graph made from an orthogonal lattice of Htimes W nodes.
Let f(n) be the number of divisors of 2n^2 that are no greater than n.
Both 169 and 961 are the square of a prime.
The minimum number of cubes to cover every visible face on a cuboid measuring 3 times 2 times 1 is twenty-two.
Amidakuji (Japanese: 阿弥陀籤) is a method for producing a random permutation of a set of objects.
Let B(n) = displaystyle prod{k=0}^n {n choose k}, a product of binomial coefficients.
Consider the equation 17^pa+19^pb+23^pc = n where a, b, c and p are positive integers, i.e.
Define the sequence a1, a2, a3, dots as: - a1 = 1 - a{n+1} = 6an^2 + 10an + 3 for n ge 1.
Alice enlists the help of some friends to generate a random number, using a single unfair coin.
Starting with three numbers a, b, c, at each step do one of the three operations: - change a to 2(b + c) - a; - change b
The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) eac
Consider the term small sqrt{x+sqrt{y}+sqrt{z}} that is representing a nested square root.
Let S = 2, 3, 5, dots, 4999 be the set of prime numbers less than 5000.
Julie proposes the following wager to her sister Louise.
Considering 4-digit primes containing repeated digits it is clear that they cannot all be the same: 1111 is divisible by
A secret integer t is selected at random within the range 1 le t le n.
We say that a d-digit positive number (no leading zeros) is a one-child number if exactly one of its sub-strings is divi
A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.
A number consisting entirely of ones is called a repunit.
For an n-tuple of integers t = (a1, dots, an), let (x1, dots, xn) be the solutions of the polynomial equation x^n + a1 x
Given a non-square integer d, any real x can be approximated arbitrarily close by quadratic integers a+bsqrt{d}, where a
We wish to tile a rectangle whose length is twice its width.
Using all of the digits 1 through 9 and concatenating them freely to form decimal integers, different sets can be formed
A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle.
Let an be a sequence recursively defined by:quad a1=1,quaddisplaystyle an=biggl(sum{k=1}^{n-1}kcdot akbiggr)bmod n.
Christopher Robin and Pooh Bear love the game of Poohsticks so much that they invented a new version which allows them t
An ant moves on a regular grid of squares that are coloured either black or white.
Consider the infinite polynomial series AF(x) = x F1 + x^2 F2 + x^3 F3 + dots, where Fk is the kth term in the Fibonacci
Alice and Bob play the game Nim Square.
A standard envelope shape is a convex figure consisting of an isosceles triangle (the flap) placed on top of a rectangle
Secret Santa is a process that allows n people to give each other presents, so that each person gives a single present a
Given is the arithmetic-geometric sequence u(k) = (900-3k)r^{k - 1}.
A line segment of length 2n-3 is randomly split into n segments of integer length (n ge 3).
A triangle is cut into four pieces by two straight lines, each starting at one vertex and ending on the opposite edge.
Let p(n) represent the number of different ways in which n coins can be separated into piles.
We are trying to find a hidden number selected from the set of integers 1, 2, dots, n by asking questions.
Turan has the electrical water heating system outside his house in a shed.
It can be verified that there are 23 positive integers less than 1000 that are divisible by at least four distinct prime
Consider the triangle with sides sqrt 5, sqrt {65} and sqrt {68}.
A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules:
For fixed integers a, b, c, define the crazy function F(n) as follows: F(n) = n - c for all n gt b F(n) = F(a + F(a + F(
A and B play a game. A has originally 1 gram of gold and B has an unlimited amount. Each round goes as follows: - A choo
The first 15 Fibonacci numbers are: 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610.
Let a0, a1, dots be an integer sequence defined by: - a0 = 1; - for n ge 1, an is the sum of the digits of all preceding
Take the number 6 and multiply it by each of 1273 and 9854: By concatenating these products we get the 1 to 9 pandigital
A k-input binary truth table is a map from k input bits (binary digits, 0 [false] or 1 [true]) to 1 output bit.
Let W(p,q,r) be the number of words that can be formed using the letter A p times, the letter B q times and the letter C
Denote the average of k numbers x1, ..., xk by bar{x} = frac{1}{k} sumi xi.
Pentagonal numbers are generated by the formula: Pn = tfrac 12n(3n-1) giving the sequence: Some pentagonal numbers can b
We shall call a fraction that cannot be cancelled down a resilient fraction.
Consider an n-player game played in consecutive pairs: Round 1 takes place between players 1 and 2, round 2 takes place
The moon has been opened up, and land can be obtained for free, but there is a catch.
For any set A of numbers, let operatorname{sum}(A) be the sum of the elements of A.
If a,b are two nonnegative integers with decimal representations a=(dots a2a1a0) and b=(dots b2b1b0) respectively, then
Consider the Fibonacci sequence 1,2,3,5,8,13,21,ldots.
An infinite sequence of real numbers a(n) is defined for all integers n as follows: For example, a(0) = dfrac{1}{1!} + d
G(N)=sum{j=1}^Nsum{i=1}^j gcd(i,j).
A common security method used for online banking is to ask the user for three random characters from a passcode.
Let f5(n) be the largest integer x for which 5^x divides n.
A sequence is created by starting with a positive integer n and incrementing by (n+m) at the m^{th} step.
12n musicians participate at a music festival.
Consider the infinite polynomial series AG(x) = x G1 + x^2 G2 + x^3 G3 + cdots, where Gk is the kth term of the second o
Starting from a positive integer n, at each step we subtract from n the largest perfect cube not exceeding n, until n be
The number 512 is interesting because it is equal to the sum of its digits raised to some power: 5 + 1 + 2 = 8, and 8^3
A unitary divisor d of a number n is a divisor of n that has the property gcd(d, n/d) = 1.
Jeff eats a pie in an unusual way.
A spider, S, sits in one corner of a cuboid room, measuring 6 by 5 by 3, and a fly, F, sits in the opposite corner.
Let f(n) be the number of ways an ntimes n square grid can be coloured, each cell either black or white, such that each
Two friends, a runner and a swimmer, are playing a sporting game: The swimmer is swimming within a circular pool while t
The Pythagorean tree is a fractal generated by the following procedure: Start with a unit square.
The function operatorname{mathbf{lcm}}(a,b) denotes the least common multiple of a and b.
The palindromic number 595 is interesting because it can be written as the sum of consecutive squares: 6^2 + 7^2 + 8^2 +
Consider the following process that can be applied recursively to any positive integer n: - if n = 1 do nothing and the
This problem involves an iterative procedure that begins with a circle of nge 3 integers.
Let a, b and c be the sides of an integer sided triangle with one angle of 120 degrees, a le b le c and b-a le 100.
Consider the fraction, dfrac n d, where n and d are positive integers.
The 3-digit number 376 in the decimal numbering system is an example of numbers with the special property that its squar
Alice plays the following game, she starts with a list of integers L and on each step she can either: - remove two eleme
Dave is doing his homework on the balcony and, preparing a presentation about Pythagorean triangles, has just cut out a
Consider the number 15.
Recall the blancmange function from Problem 226: T(x) = sumlimits{n = 0}^inftydfrac{s(2^nx)}{2^n}, where s(x) is the dis
Let omega(n) denote the number of distinct prime divisors of a positive integer n.
For two positive integers a and b, the Ulam sequence U(a,b) is defined by U(a,b)1 = a, U(a,b)2 = b and for k gt 2, U(a,b
Let n be a natural number and p1^{alpha1}cdot p2^{alpha2}cdots pk^{alphak} its prime factorisation.
N times N disks are placed on a square game board.
Let us call an integer sided triangle with sides a le b le c barely obtuse if the sides satisfy a^2 + b^2 = c^2 - 1.
For a positive integer n we define q(n) to be the number of solutions to: where 0 leq ai, bi lt n.
Let R(a, b, c) be the maximum area covered by three non-overlapping circles inside a triangle with edge lengths a, b and
Given a circle C and an integer n 1, we perform the following operations.
Let g(n) be a sequence defined as follows: g(4) = 13, g(n) = g(n-1) + gcd(n, g(n-1)) for n gt 4.
How many integers 0 le n lt 10^{18} have the property that the sum of the digits of n equals the sum of digits of 137n?
Peter moves in a hallway with N + 1 doors consecutively numbered from 0 through N.
A number consisting entirely of ones is called a repunit.
Consider the fraction, dfrac n d, where n and d are positive integers.
A 3-smooth number is an integer which has no prime factor larger than 3.
A number chain is created by continuously adding the square of the digits in a number to form a new number until it has
Consider quadratic Diophantine equations of the form: For example, when D=13, the minimal solution in x is 649^2 - 13 ti
There are 3 buckets labelled S (small) of 3 litres, M (medium) of 5 litres and L (large) of 8 litres.
The points P(x1, y1) and Q(x2, y2) are plotted at integer co-ordinates and are joined to the origin, O(0,0), to form tri
defhtmltext1{style{font-family:inherit;}{text{1}}} For non-negative integers m, n, the Ackermann function A(m,n) is defi
Fred the farmer arranges to have a new storage silo installed on his farm and having an obsession for all things square
Let d(p, n, 0) be the multiplicative inverse of n modulo prime p, defined as n times d(p, n, 0) = 1 bmod p.
Let W(n,k) be the number of ways in which n can be written as the product of k distinct positive integers.
A train is used to transport four carriages in the order: ABCD.
For a positive integer n, define f(n) to be the number of non-empty substrings of n that are divisible by 3.
The number 209 can be expressed as a^2 + 3ab + b^2 in two distinct ways: qquad 209 = 8^2 + 3cdot 8cdot 5 + 5^2 qquad 209
An integer is called eleven-free if its decimal expansion does not contain any substring representing a power of 11 exce
Given an irrational number alpha, let Salpha(n) be the sequence Salpha(n)=lfloor {alpha cdot n} rfloor - lfloor {alpha c
For a set of positive integers a, a+1, a+2, dots , b, let C(a,b) be the number of non-empty subsets in which the product
A square is drawn around a circle as shown in the diagram below on the left.
There are n stones in a pond, numbered 1 to n.
Let S(n) be the number of pairs (a,b) of distinct divisors of n such that a divides b.
A lattice cube is a cube in which all vertices have integer coordinates.
N trolls are in a hole that is DN cm deep.
A bracelet is made by connecting at least three numbered beads in a circle.
Gary and Sally play a game using gold and silver coins arranged into a number of vertical stacks, alternating turns.
Each of the six faces on a cube has a different digit (0 to 9) written on it; the same is done to a second cube.
Consider a circle where 2n distinct points have been marked on its circumference.
Given the representation of a continued fraction alpha is a real number with continued fraction representation: alpha =
Given two unequal positive integers a and b, we define a self-describing sequence consisting of alternating runs of as a
The Fibonacci sequence is defined by the recurrence relation: Fn = F{n - 1} + F{n - 2}, where F1 = 1 and F2 = 1.
145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.
A permutation is an ordered arrangement of objects.
By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top
Consider all integer combinations of a^b for 2 le a le 5 and 2 le b le 5: If they are then placed in numerical order, wi
The following iterative sequence is defined for the set of positive integers: - n to n/2 (n is even) - n to 3n + 1 (n is
The total number of prime factors of n, counted with multiplicity, is denoted Omega(n).
Work out the first ten digits of the sum of the following one-hundred 50-digit numbers.
We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once.
Given a prime p and a positive integer n lt p, let R(p, n) be the multiplicative order of p modulo n!.
Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits: As 1 = 1^4 is
Left and Right play a game with a word consisting of L's and R's, alternating turns.
n! means n times (n - 1) times cdots times 3 times 2 times 1.
The number 3797 has an interesting property.
If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 lette
It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a s
Left and Right play a game with a number of words, each consisting of L's and R's, alternating turns.
Two players play a game using a deck of 2n cards: n red and n black.
In the classical game of Nim two players take turns removing stones from piles.
The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime
If p is the perimeter of a right angle triangle with integral length sides, a, b, c, there are exactly three solutions f
Given a set E of positive integers, an element x of E is called an element divisor (elevisor) of E if x divides another
Pentagonal numbers are generated by the formula, Pn=n(3n-1)/2.
A band of pirates has come into a hoard of treasure, and must decide how to distribute it amongst themselves.
The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order,
Take the number 192 and multiply it by each of 1, 2, and 3: By concatenating each product we get the 1 to 9 pandigital,
Euler discovered the remarkable quadratic formula: n^2 + n + 41 It turns out that the formula will produce 40 primes for
de Bruijn has a digital combination lock with k buttons numbered 0 to k-1 where k le 10.
We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once; for example, th
Using names.txt (right click and 'Save Link/Target As...'), a 46K text file containing over five-thousand first names, b
Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
The fraction 49/98 is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly
A perfect number is a number for which the sum of its proper divisors is exactly equal to the number.
2^{15} = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.
The nth term of the sequence of triangle numbers is given by, tn = frac12n(n+1); so the first ten triangle numbers are:
A unit fraction contains 1 in the numerator.
The series, 1^1 + 2^2 + 3^3 + cdots + 10^{10} = 10405071317.
A positive integer is called heptaphobic if it is not divisible by seven and no number divisible by seven can be produce
Given a natural number q, let p = 2^q - 1 be the q-th Mersenne number.
In the 20 times 20 grid below, four numbers along a diagonal line have been marked in red.
Triangle, pentagonal, and hexagonal numbers are generated by the following formulae: | | | | | | |------------|-----|---
An irrational decimal fraction is created by concatenating the positive integers: It can be seen that the 12th digit of
In the United Kingdom the currency is made up of pound (£) and pence (p).
We use xoplus y for the bitwise XOR of x and y.
The decimal number, 585 = 10010010012 (binary), is palindromic in both bases.
The sequence of triangle numbers is generated by adding the natural numbers.
You are given the following information, but you may prefer to do some research for yourself.
Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows: 21 22
The prime 41, can be written as the sum of six consecutive primes: This is the longest sum of consecutive primes that ad
A sequence (an){n ge 0} starts with a0 = 3 and for each n ge 0, - if an is a triangle numberA triangle number is a numbe
Starting in the top left corner of a 2 times 2 grid, and only being able to move to the right and down, there are exactl
The first two consecutive numbers to have two distinct prime factors are: The first three consecutive numbers to have th
Lowering Mochi `stream<T>`, stream definitions, `on`-handlers, agent records, and `intent` methods, plus the M:N work-stealing scheduler over minicoro fibers that runs them.
Inventory of the third-party and home-grown components the C runtime can stand on: GC (BDWGC, MMTk, Perceus), allocator (mimalloc, scudo), coroutines (minicoro), I/O (libuv, libxev), strings, hash tables, JSON/YAML/CSV, HTTP, LLM, FFI.
Risks (semantic, build, supply chain, performance, ergonomic), explicit alternatives considered (LLVM IR, WASM, Rust, JIT, C++, Zig), kill switches that demote the transpiler back to optional, comparable industrial precedent.
The five guiding principles behind the Mochi-to-C transpiler (spec-first, boring C, no ABI surprises, portability over performance, verifiable output), plus the runtime shape and a sample C output.
Every Mochi construct the MEP-45 codegen must lower: value core, function core, collection core, ADT core, query DSL, stream/agent core, logic, AI/FFI, tests, modules, error model, concurrency semantics.
Survey of transpilers and AOT compilers that emit C or behave like a C-target system: Nim, Crystal, Vala, OCaml, Roc, Koka, MLton, Cosmopolitan, zig cc, Cython, ATS, Soufflé. Twelve distilled lessons.
The C target itself: C23 features used, compiler matrix (clang, gcc, msvc, zig cc, cosmocc, tcc), tier-1/2/3 architectures and OSes, ABI per arch, libc matrix, sanitisers, reproducibility, hardening, style guide for emitted C.
Type-system lowering details: generics/monomorphisation, records, sum types with niche optimisation, closures with fat pointer, strings with SSO, lists, maps with Swiss-table, sets, time/duration, error values with built-in code table.
Testing strategy: differential testing against vm3, BG corpus, fuzzing, sanitiser matrix (ASan/UBSan/TSan/MSan/LeakSan), property tests, reproducibility check, 16 phased CI gates.
Build pipeline: `mochi build` command surface, output layout, amalgamated runtime, cross-compilation via bundled zig cc, APE via cosmocc, WASM via wasi-sdk, content-addressed caching, reproducibility.
Codegen pipeline, why a C IR, name mangling rules, type-lowering table, value representation with `mochi_value` boxed type, expression lowering, statement lowering, for-loop lowering, try/catch via setjmp, Maranget pattern matching, modules, amalgamation.
Lowering the Mochi query DSL (LINQ-style from/where/select/join/group by/order/limit/union/intersect/except) to C with arena allocation, operator fusion, and load/save adapters.
One paragraph each, Phase 1 vs Phase 2.
Gatekeeper, notarytool, SmartScreen, Authenticode, and the real cost of shipping a desktop binary.
A conceptual essay on where Mochi MEP-42 should sit on the curve from "compile slowly, run fast" (LLVM -O3) through "compile and run at medium speed" (Cranelift, B3) to "compile instantly, run okay" (copy-and-patch, Sparkplug).
Statically linked, position-independent, ASLR-friendly, and no dynamic loader required.
One-paragraph recommendation, plus reasoning, for which naive-emission technique Mochi MEP-42 should adopt as the first cut.
Comparison table and Phase 1 / Phase 2 recommendation for Mochi's native code-generation backend.
The two competing approaches to debugging Wasm, and why DWARF won inside Chrome.
Cross-cutting patterns from twelve production AOT pipelines, with a recommendation for which one Mochi should learn from most.
The published material a Mochi engineer should keep open while implementing MEP-42 phase 1. Cooper/Torczon 3rd ed for the canonical theory, Nystrom for a hands-on bytecode compiler walkthrough, Appel for the verified-compiler-curious, plus 2024-2026 course materials covering…
The "honorable mentions" beyond LLVM/Cranelift/MIR/QBE.
What it takes to write a valid ELF/Mach-O/COFF from a backend's raw bytes.
Fat Mach-O wrapping arm64 + x86_64, the lipo tool, and the end of x86_64 support.
The Apple object/executable format: macOS, iOS, iPadOS, tvOS, watchOS, visionOS.
Microsoft's native debug format, the sidecar PDB file, and the pain of producing one on Linux.
A single binary that is simultaneously a valid ELF, Mach-O, PE, and BSD a.out.
The .wasm container, plus the component model binary and WAT text format.
The most actively researched mid-tier compiler backend of the 2020s. ISLE for instruction selection, proof-carrying code for Wasm-sandbox memory accesses, and VeriISLE for verified instruction-lowering rules.
The Windows executable and object format.
Single-page roll-up of every (target ISA x OS) combination Mochi MEP-42 could ship, with status and engineering complexity recommendations.
Apple's Swift in two modes: full LLVM-driven AOT for app platforms, and a stripped Embedded Swift mode for microcontrollers and freestanding binaries.
The universal Unix object/executable format: Linux, *BSD, Solaris, Haiku, embedded.
Python-syntax systems language built on MLIR, with both AOT and JIT pipelines, on the path to 1.0 in H1 2026.
The Glasgow Haskell Compiler's native code generator: hand-written, x86_64 / aarch64 / risc-v, alongside an LLVM alternative.
Meta's typed Python subset that can be AOT-aware, the Cinder JIT, and CPython 3.13+'s experimental copy-and-patch JIT.
Multi-Level IR, dialect framework, the foundation under Mojo, IREE, and OpenXLA.
The Triton dialect (OpenAI, GPU codegen), the IREE dialect (Google, ML compiler), Mojo's MLIR-based KGEN compiler, and the broader trend of language frontends as MLIR dialects. The "if we wanted to be ambitious" backend story for Mochi.
Recent foundational work from POPL and its co-located CPP (Certified Programs and Proofs) workshop on verified compilation, secure calling conventions, and packet-filter codegen.
The cross-platform binary debug format, its current standard, and where the next version is going.
"Mochi → Wasm → wasmtime compile → native": skip the backend, use the wasm ecosystem.
"How small can a useful native backend be?"
The lowest-cognitive-load path: emit C, let GCC/Clang do the rest.
Recent work from the flagship PL conference on lightweight backends, baseline JIT design, superoptimization for stack bytecode, and library composition that sidesteps heavyweight compilation.
Direct syscalls on Linux, why this is impossible on macOS, and the APE alternative.
A 14k-LOC SSA compiler backend by Quentin Carbonneaux that targets x86-64, arm64, and riscv64 from a textual SSA IR. The "70% of LLVM in 10% of the code" pitch. The natural fallback if writing our own emitter feels too risky.
JIT-first scientific language adding a real AOT path via juliac and a trimming-based static binary mechanism.
Statically-typed Go-influenced language that defaults to emitting C, with experimental native and LLVM backends and a multi-mode memory model.
cgo, c-shared, c-archive: keep vm3 in Go and call into it from native code.
Tiny systems language built on QBE, deliberately constrained, BSD/Linux-only, manual memory.
Static, GC-by-default systems language that emits C (or C++, JavaScript, LLVM) and inherits the host toolchain's optimiser.
One binary, six operating systems, two ISAs. The APE format and cosmocc toolchain.
The default Linux libc, and the reasons "fully static linking" is officially unsupported.
GCC as a shared library, used by Emacs native compilation, GCC Rust, GDC, Cython.
Go's internal assembler exported as a library, already used by Mochi's vm2jit.
Stencil-based binary stitching from OOPSLA 2021, now shipping in CPython 3.13/3.14.
Mike Pall's preprocessor-driven assembler with runtime patching; LuaJIT's secret weapon.
Rui Ueyama's ~10k LOC C compiler that emits x86-64 GAS assembly directly from a recursive-descent parser. The clearest published example of a single-pass codegen pipeline that produces correct code with no IR, no SSA, no register allocator.
Survey of niche or specialized architectures that MEP-42 should be aware of but probably defer.
The general-purpose 64-bit RISC-V baseline plus vector and recent extensions.
Wasm 3.0 (2025) as the stable target, plus WASI Preview 2/3 and the component model.
A converged design pattern across three independent implementations: per-opcode template emission, a virtual operand stack with lazy register promotion, no IR, no global optimization. The current state of the art in "fast and simple" Wasm code generation.
Windows 11 on Snapdragon X / X2 Elite, Surface Pro X lineage, and ARM64EC interop.
The static-friendly C library that makes "build once, ship anywhere" actually work on Linux.
The general pattern that Sparkplug, Liftoff, JSC Baseline, and the HotSpot template interpreter all instantiate. Per-op native template, fixed register convention, stub calls for slow paths, optional inline caches as patchable code regions.
Microsoft's production successor to CoreRT: trimmed CoreCLR plus RyuJIT-as-AOT, shipping single-file native binaries.
Systems language rebuilding its compiler around its own native backends, with LLVM demoted to an optional path.
Closed-world AOT compilation that turns a managed JVM application into a self-contained native executable.
Ruby-syntax statically-typed language with global type inference, LLVM backend, and Boehm GC.
Tree-walk to tuples to native assembly. The textbook recipe from the Dragon Book and Cooper/Torczon, still the right starting point when you want correctness before performance.
Emitting a self-contained executable from the compiler, no external linker required.
Stencil-driven code generation that harvests pre-compiled opcode snippets at build time and stitches them into native code at runtime via relocation patches. No IR, no register allocator, no instruction selector. The technique CPython 3.13+ ships in production.
Bytecode Alliance's Rust-native SSA backend, ISLE-driven, ~10x faster compile than LLVM.
The closed-source Mach-O linker that ships in Xcode 15 and later.
The workhorse SSA infrastructure, version 20 era, evaluated for MEP-42.
Quentin Carbonneaux's deliberately tiny SSA backend, "70% of LLVM in 10% of the code."
The historical Linux workhorse and its dying sibling.
Vladimir Makarov's lightweight JIT+AOT, fast compile times, lazy basic-block versioning.
The 4-register fast-call convention used by every Windows-on-AMD64 binary.
The base 64-bit Arm calling standard, with Apple and Microsoft deltas called out.
The default 64-bit Unix ABI: Linux, macOS, FreeBSD, OpenBSD, NetBSD, illumos.
A single-pass, IR-free transpiler from Ignition bytecode to native machine code, designed to add a fast tier between an interpreter and an optimizer with minimal engineering cost.
Apple WebKit's original template-style baseline compiler, the longest-running production baseline JIT for a dynamic language, and the design template for every modern four-tier VM.
The fastest production ELF linker, single-author, MIT licensed.
One binary, four object formats, cross-link from any host.
Six capabilities (iso, trn, ref, val, box, tag) make actor-based concurrency data-race-free at compile time. Production at WallarooLabs and Microsoft; foundation for Verona's region work.
Foundational separation-logic verification of Rust's safe / unsafe core, and its 2024-2026 RefinedRust successor.
The DoD's evolving acquisition posture on memory-safe languages, the SWFT framework, and aerospace coding standards.
The state of Rust's borrow checker as the Polonius "alpha" lands behind a nightly feature gate, and what a non-ownership language can still steal from it.
MarkUs and quarantine-style UAF prevention
The voluntary US federal pledge that has set the de-facto industry baseline for memory-safety roadmaps.
Arm Morello
Intel LAM (Linear Address Masking) and the x86 top-byte-tag story
The NSA's formal language-level guidance, the named-language list, and the joint CISA reissue.
The 2024-2026 industry picture on temporal memory safety, kernel mitigations, and the convergence of UAF defences.
A WebAssembly extension that replaces linear memory with segments and handles. Handles are unforgeable, typed pointers carrying bounds and provenance — closely modelled on CHERI capabilities but pure software.
Arm PAC + BTI
A deductive verifier for safe Rust that compiles to Why3 and discharges to off-the-shelf SMT solvers.
WebKit's retreating-wavefront concurrent garbage collector. Marks objects while JS runs, throttles allocation when it falls behind, and uses logical versioning to skip clearing bitmaps.
Stacked Borrows, the 2025 Tree Borrows replacement, and the trajectory of NLL / two-phase formalisation.
Single ownership with deterministic destruction, multiple simultaneous borrows allowed, runtime borrow-count enforcement. No GC, no compile-time borrow checker, no lifetime variables. Concurrency uses `uni T` (unique values) for safe inter-process transfer.
Microsoft Research's SMT-backed Rust verifier, the leading tool for verified systems software at scale in 2025-2026.
The bootstrapping verified ML compiler, its 2024 PLDI agenda, and the FM-JIT verified-JIT effort that builds on it.
A purely software in-process sandbox for the V8 JS heap. Ban raw pointers, replace with offsets into a 1 TB sandbox region and indices into out-of-sandbox pointer tables. About 1% perf cost, enabled by default in Chrome 123. Every modern JIT is moving this direction.
Linearity attached to function arrows, not to types. Backwards-compatible: ordinary code continues to type-check unchanged. Experimental since GHC 9.0; still labelled experimental in 9.12 / 9.15 (2024–2026).
Chrome's two cooperating collectors. Orinoco runs V8's young-generation JS heap in parallel; Oilpan is Blink's traced C++ GC, recently hosted inside V8 as a library, learning to do generational collection with conservative stack scanning.
Orthogonal modes layered onto OCaml's type system. Mode is a property of a value, separate from its type, tracked through inference. Production in Jane Street; open-sourced as OxCaml in 2025.
Arm MTE
The White House Office of the National Cyber Director's memory-safety report and the C/C++-adverse federal stance.
The biggest real-world deployment of Perceus reference counting, layered with Morphic alias analysis and "seamless slices" so functional code rarely allocates.
CHERIoT
The canonical "secure allocator inside a managed runtime." Scudo is Android's hardened native heap, used for ART's non-managed allocations (JIT code, off-heap buffers, JNI). Pairs with ARM MTE on Armv9 hardware for hardware-checked tagging.
Swift 6.x has shipped a complete move-only/noncopyable system layered on top of ARC. Parameter conventions are explicit; the law of exclusivity (Swift's variant from 2017) supplies the static aliasing discipline.
What every shipping JIT must do on day 1 to be production-grade: never have a code page that is both writable and executable to the same thread at the same time.
A research framework that cleanly separates GC plans from policies, plus the LXR collector that proves a stop-the-world RC+mark-region design can beat industrial concurrent GCs on tail latency.
Java's flagship low-latency collector. Sub-millisecond pauses on multi-TB heaps via colored 64-bit pointers and concurrent everything.
Rust verification by functional translation: compile Rust to a pure lambda calculus, verify there.
The canonical industry data point on memory-safety vulnerability prevalence, and every follow-up through 2026.
The CG-track answer to "what's beyond a single 4 GB linear memory?" Multiple memories shipped in Wasm 3.0 (Sep 2025). Memory64 shipped at the same time. A formal "segmented memory" proposal in the MSWasm vein has not yet entered the CG track but is influencing design.
The CRA's 2026-2027 enforcement timeline and the implicit pressure toward memory-safe languages.
Capability tracking in the type system. Each value's type may carry a capture set listing which capabilities it could reference. Foundation for capability-based effects, separation checking (System Capybara), and ownership for resources.
Mutable value semantics with subscript-based projection borrowing, no lifetime variables, and the Law of Exclusivity enforced at call sites.
Speculative-execution attacks haven't gone away. As of May 2026, every shipping JIT either implements index masking + bounds-check hardening, or relies on process-level Site Isolation, or both.
Intel CET Shadow Stack + IBT
Scudo & friends: hardened allocators in production
Strict linear types as the load-bearing primitive for memory and protocol safety, plus capability-based effect control. Spec-and-compiler-small enough to read in a weekend.
Stenberg's curl CVE data, the Hyper / Rust experiment, and the unresolved 2026 question of whether memory-safe languages are the whole answer.
MEP-41 Research Substrate: Memory Safety Advances 2023–May 2026
The production-grade formally verified C compiler, the CompCertO / Owlang line, and why it matters for JIT verification.
Native GC primitives in a portable bytecode. Ratified in Wasm 3.0 (Sep 2025), shipped in all major browsers by Dec 2024. Dart, Kotlin, OCaml, Java/Scala/Scheme can now compile to Wasm without bundling a GC.
Cerise, the CHERI-C Coq memory model, and capability-safety logical relations as the formal model behind handle-based runtimes.
MTE in a Managed Runtime
The direct intellectual ancestor of vm3's handle design: a per-allocation generation counter, a per-reference remembered generation, and a check on every dereference. Same idea, different layer.
Google's published 2022-2026 data on memory-safety progress — Android's 76% → <20% trajectory, the CVE-2025-48530 near-miss, the V8 sandbox.
Chrome Security's published per-quarter memory-safety data for 2024-2026, the JSON / PNG / fonts Rust rollouts, and the V8 sandbox.
"Garbage-free" precise reference counting with reuse — in-place updates without locks, statically inserted at compile time.
A pragmatic language by Wouter van Oortmerssen that elides 95% of refcount ops at compile time through flow-typed lifetime analysis. Cycles handled by a cleanup at program exit.
CHERI (Next Generation)
A Rust-like ownership system with simpler call-site syntax, taped onto a Python-shaped surface, sitting on MLIR, headed to open source in fall 2026.
Microsoft Research's experimental concurrent-ownership language. Ownership is over **regions** (groups of objects) instead of individual objects. Cowns (concurrent owners) serialise access; behaviours schedule work over multiple cowns atomically.
AWS's bit-precise bounded model checker for Rust, deployed in CI on Firecracker and the standard library.
Iris-based separation logics with space credits, tracing GC, and the 2025 IrisFit + Nextgen-Modality lines.
A direct ancestor of vm3's 32-bit slab index. V8 squeezes 64-bit pointers down to 32-bit offsets within a per-isolate 4 GB virtual region (the "cage"). Cut V8's heap by 43%, Chrome renderer memory by 20%.
This book describes the Zig programming language as of Zig 0.16.
A
These are short answers or sketches for selected exercises from the book. Many exercises have several reasonable solutions. The important part is clarity and correctness.
This appendix records the kinds of changes that matter when moving older Zig code to Zig 0.16. It is a practical checklist, not a full release history.
Zig tries to report errors at the point where the program becomes invalid. Many messages are direct: a value has the wrong type, a variable is unused, an error was ignored, or...
One of Zig's central goals is direct interoperability with C. Zig can call C code, compile C code, link system libraries, export functions to C, and translate C headers.
Zig builds programs with Zig code. The build script is named:
The Zig standard library is imported as:
Builtin functions are part of the language. Their names begin with @.
This appendix lists the common operators by use. When an expression is not obvious, use parentheses.
This appendix summarizes the core syntax of Zig 0.16. It is a compact reference, not a tutorial.
This appendix is a quick map of Zig syntax. It is not a grammar. The full Zig grammar is part of the official language reference. Zig 0.16 also keeps the language small enough...
The programs in this chapter are small, but they have the shape of larger Zig programs.
This section combines the ideas from the previous sections into one program structure.
A program is finished only when another machine can build and run it reliably.
Exercise 19-1. Write a program that overflows a u8 using +=. Run it in Debug mode and observe the panic.
Threads are a mechanism, not a design.
A benchmark measures the cost of a program or operation. The result is useful only if the measurement is repeatable and the work being measured is clearly defined.
FFI means foreign function interface. It is the boundary where Zig calls code written in another language, or where another language calls Zig.
A semaphore is a counter used for synchronization.
This section collects the chapter exercises into one working set.
Exercise 16-1. Write a C file containing this function:
Tests should be close to the code they check. Zig makes this easy with test blocks.
A normal struct is laid out for efficient access.
Creating a thread is expensive.
Zig can build the same source code in different optimization modes.
A C program and a Zig program can share data only when both sides agree on layout.
These exercises use the material from Chapters 15.1 through 15.8. Most are small. The goal is to practice reading and writing build.zig files until the structure becomes familiar.
Zig can build for a different target than the machine running the compiler.
A Zig package may depend on another package.
Examples are small programs kept inside the project.
This section collects the exercises for Chapter 14.
Zig has built-in support for tests.
The standard library gives access to process information through std.process.
This section collects the exercises for the chapter.
Zig 0.16 changes how I/O code is written in the standard library.
The standard library gives access to files and directories through std.fs.
Input and output can fail.
A library is code meant to be used by another program. In Zig, a library can be built from the same kind of source files as an executable. The difference is in how the build...
Most memory in a program behaves normally.
Zig once had async functions as a language feature.
A program is made from object code.
C can call Zig when the Zig function is exported with a C-compatible ABI.
Zig has tests in the language.
Zig uses format strings to turn values into text.
A program usually has three standard streams:
The exercises in this section are meant to make allocation habits precise. Each one should be written as a complete program unless stated otherwise.
Allocation creates a responsibility.
A fixed-buffer allocator uses memory that already exists.
An arena allocator is used when many allocations have the same lifetime.
An integer type can store only a fixed range of values.
An HTTP client opens a network connection, sends a request, receives a response, and writes the response body.
A mutex protects shared data.
A Zig target may use a C library, or it may use none.
Calling a C function is only half the job. The linker must also find the code for that function.
A build option is a value passed to build.zig from the command line.
A hash map stores key-value pairs.
Reading or writing one byte at a time is expensive.
The general-purpose allocator is used for ordinary heap allocation.
Exercise 11-1. Write a generic min function for values that support <.
Generic functions in Zig are specialized at compile time.
Zig has no built-in interface keyword.
A line filter reads text, changes or selects some lines, and writes the result. Many Unix programs have this shape.
A pointer has a type.
A mutex makes one thread wait while another thread uses shared data.
The architecture part of a target tells Zig what kind of processor the program will run on.
@cImport does two jobs.
A build file describes steps.
std.ArrayList is a growable array.
Writing bytes is the opposite of reading them.
The page allocator gets memory from the operating system.
Zig does not have a formal trait or interface system.
1. Write a program that computes several Fibonacci numbers with comptime. Print the results at runtime.
A file copier is a useful small program. It opens one file for reading, opens another file for writing, then copies bytes from the first to the second.
Zig inserts safety checks for operations that are valid only under certain conditions. These checks are present in safe build modes. They catch mistakes at the point where the...
A mutex is a lock for shared data.
The operating system part of a target tells Zig what kind of system the program will run on.
Writing every C declaration by hand is tedious. Zig can read C headers directly.
The file build.zig is a Zig program.
The std.mem module contains operations on memory, slices, bytes, and basic data movement. Much of Zig programming eventually passes through std.mem.
A file is read as bytes.
An allocator is a value that knows how to allocate and free memory.
A generic struct is a function that returns a type.
A Zig function can return a type.
Many programs begin the same way: they read command-line arguments, decide what the user requested, then execute an operation.
Zig gives the programmer direct access to memory, integers, pointers, and machine operations. This makes many programs simple and efficient. It also makes some operations...
Reflection means inspecting a type from inside the program.
A thread is an independent flow of execution.
One of Zig's main design goals is cross compilation.
A normal loop runs while the program runs.
One of Zig's design goals is direct interoperability with C.
Large programs are rarely built with a single compiler command.
In Zig, a type is a value.
Exercise 9-17. Declare an optional integer.
The std.debug module contains utilities for debugging programs. The most commonly used function is print, which writes formatted output to standard error.
Optionals and errors both describe a value that may not be produced.
Most programs spend their time moving bytes.
A function parameter can be marked comptime.
Pointers are often optional.
A program uses memory to store values.
A function in Zig can take types as parameters.
An optional value cannot be used as the payload type until it is unwrapped.
A Zig program runs in two stages.
null is the value used when an optional has no payload.
1. Write a function parseUpper that accepts a byte and returns the uppercase letter value. Return error.InvalidUppercase if the byte is not between 'A' and 'Z'.
Sometimes a value may or may not exist.
Most functions should not decide what an error means.
A program that opens a file must close it. A program that allocates memory must free it. A program that locks a mutex must unlock it.
catch handles an error union.
As of the current public Zig release line, the version is Zig 0.16, not Zig 1.16. Zig has not reached 1.0 yet. In this appendix, read “Zig 1.16” as “Zig 0.16” unless...
try is a shortcut for a common operation:
You now know the core ideas behind Zig.
Zig has not reached 1.0 yet. The current release line discussed in this book is Zig 0.16, not Zig 1.16.
Git stores snapshots of files.
IR means intermediate representation.
A function returns an error the same way it returns a normal value: with return.
Zig changes quickly because it is still moving toward 1.0.
Zig is still before 1.0, so the language and standard library can change between releases. Code written for one Zig version may not compile on another version without edits.
A game engine is a program structure for running interactive simulations.
An error union combines a normal value with an error set.
Programs fail for many reasons. A file may not exist. Memory allocation may fail. Input may be malformed. A network connection may close unexpectedly.
These exercises close the chapter on structs, enums, and unions. They are meant to make the data model concrete.
Performance ideas become clearer when you see them inside real code.
High performance concurrent design means using several threads or tasks without making the program slower, more fragile, or harder to reason about.
A struct is not only a list of fields. It also has a layout in memory.
A shell is a program that reads commands and runs other programs.
A data race happens when two threads access the same memory at the same time, at least one access writes, and there is no proper synchronization.
Wrapping a C library means building a Zig layer around it.
A normal struct is laid out for ordinary program use. The compiler may add padding between fields so that each field has a suitable address.
A union is a value that may hold one of several types.
An enum is a type whose values come from a fixed set of names.
Zig has no class syntax.
Fields are selected with the dot operator.
A struct is a type made from named fields.
This section closes the chapter. The exercises are small programs. Write them by hand. Compile them. Change them. Run them again.
Zig strings are bytes. UTF-8 is one way to interpret those bytes as text.
A Zig string is a sequence of bytes.
Some arrays have a special value after the last ordinary element. This value is called a sentinel.
A bytecode VM is a small machine inside your program.
Packaging means preparing your program so other people can download it, install it, run it, and trust what they are running.
A self-hosted compiler is a compiler written in the language it compiles.
An abstraction is a way to hide detail behind a simpler interface.
A CLI tool is a command-line program.
A queue is a data structure for passing work from one part of a program to another.
C headers are the bridge between Zig and C.
Benchmarking measures how fast code runs.
A slice is a view into an array.
An array literal creates an array value.
An array is a sequence of values of the same type.
1. Write a function increment that takes i32 and adds 1 to the pointed-to value.
A pointer is useful only while the value it points to still exists.
Memory has addresses. Types also have alignment.
Zig changes quickly before 1.0. That includes the build system.
Pointer arithmetic means forming a new pointer by moving from one element to another.
Packaging means preparing your program so another person can download it, install it, and run it.
A build does not only compile files. It can also install the results into a predictable output directory.
A slice is a pointer and a length.
A many-item pointer points to the first item in a sequence.
A single-item pointer points to one value.
A variable has a value. In Zig, a variable may also have an address.
1. Write a function max3 that returns the largest of three integers.
A program grows gradually.
Large programs are divided into smaller files. Zig uses @import to include declarations from another file.
Logging means recording what a program is doing.
Networking means communicating with another program through a network.
A Zig program may be split across many files. Declarations can be made visible outside a file with pub.
An HTTP client is code that sends a request to a web server and reads the response.
When a function is called, values are passed from the caller to the function parameters.
Functions communicate through parameters and return values.
JSON is a text format for storing structured data.
A function groups statements into a single operation. Functions are the basic unit of organization in a Zig program.
Compression means making data smaller.
This section collects the exercises for the chapter. They are meant to be small programs. Each one should compile and run.
defer runs a statement when control leaves the current block.
A process is a running program.
break leaves a loop.
A for loop visits the elements of an array, slice, or range.
An environment variable is a named value provided to a program by the operating system.
A while loop repeats while a condition is true.
A switch chooses one branch from several alternatives.
In Zig, if is an expression. It can produce a value.
Random numbers are used when a program needs variation.
A block is a sequence of statements inside braces.
This chapter introduced the basic forms of values and declarations: names, constants, variables, integer types, floating-point types, booleans, bytes, inferred types, and...
A data structure is never just a container.
Zig does not perform most numeric conversions automatically.
A declaration may name its type explicitly.
A collection is a type that stores many values.
This chapter covered the builtins you will see most often as a beginner.
@embedFile reads a file at compile time and embeds its bytes into the final program.
Zig does not have a separate character type for ordinary strings.
A boolean value is either true or false.
Zig has floating-point types for numbers with fractional parts.
Zig has signed and unsigned integer types.
Zig has two kinds of variable declarations:
A Zig program is made from declarations.
The programs in this chapter are small. They are meant to be changed, broken, rebuilt, and studied.
A useful program reads text and counts something.
Programs often need input.
A program can compute a value before it prints it.
The function std.debug.print writes text.
A program works with values. In Zig, values are usually stored in constants or variables.
There are two common ways to run a small Zig program.
The first program in Zig is small.
The first program in Zig is small.
@compileError stops compilation with a custom error message.
@panic stops the program immediately with a message.
Zig code should be explicit, simple, and easy to inspect. The goal is not cleverness. The goal is code that another programmer can read, verify, and maintain.
Contributing to Zig means helping the language, compiler, standard library, documentation, tests, or tooling improve.
A lexer is the first stage of many programming language tools.
A cross-platform Zig program should not pretend that every operating system behaves the same way. Windows, Linux, macOS, WebAssembly, and embedded targets have different...
A compiler translates one form of code into another form.
Benchmarking means measuring how fast code runs.
A TCP server is a program that waits for clients to connect.
An event loop is code that waits for events, then runs the right piece of work for each event.
ABI means Application Binary Interface.
Memory debugging means finding mistakes in how a program uses memory.
Zig has built-in support for tests.
Programs often need to work with time.
@floatCast converts one floating-point value to another floating-point type.
This appendix lists useful Zig projects to read after you know the basics.
A programming language is never only its syntax.
A database stores data so it can be saved, searched, updated, and loaded again later.
A portable API is an interface that works across more than one platform.
A virtual machine is a program that runs another program.
Branch prediction is a CPU optimization.
A network protocol is a rulebook for how programs talk over a network.
await means: wait until an asynchronous operation has finished, then continue with its result.
Zig can call C, but C can also call Zig.
GDB and LLDB are debuggers.
Zig can build the same program in different optimization modes.
Parsing means turning text into data.
A tree is a collection made of nodes.
An allocation-friendly API makes memory behavior clear to the caller.
@truncate converts an integer to a smaller integer type by keeping only the low bits.
Zig is designed to work closely with C. You can call C from Zig, call Zig from C, compile C code with Zig, and link Zig programs against existing C libraries.
LLVM is a compiler infrastructure project.
A memory allocator is code that gives memory to the rest of a program.
Cross-target debugging means debugging a program built for a different machine, operating system, or CPU architecture than the one you are sitting at.
A parser reads text and turns it into structure.
Copying data is sometimes necessary, but unnecessary copying is one of the easiest ways to waste time and memory.
Endianness means the order used to store the bytes of a multi-byte value.
Async code lets a program start an operation now and receive the result later.
A mixed Zig and C project contains source files from both languages.
An assertion is a check that must be true while the program runs.
Linking is the step where the compiler connects your program with the code it depends on.
Formatting means turning values into text.
A ring buffer is a fixed-size queue that reuses its storage.
Memory lifetime means:
@intCast converts one integer value to another integer type.
comptime is one of Zig’s strongest features, but it should not be used everywhere.
When Zig compiles a program, it can build the program in different modes.
Code generation is the compiler stage that turns analyzed program meaning into target code.
A thread pool is a group of worker threads that wait for jobs.
ARM is a CPU architecture family used in phones, tablets, laptops, routers, Raspberry Pi boards, microcontrollers, servers, and many embedded devices. When you write Zig for...
A game engine core is the small central layer that runs the game.
Allocations are one of the most common causes of slow programs.
A binary file format stores data as bytes with a specific structure.
A condition variable lets one thread sleep until another thread says that something has changed.
Importing a C header lets Zig understand a C API. Linking gives the final program the actual compiled code.
A stack trace shows how your program reached a failure.
Cross compilation means building a program for a different machine than the one you are using.
A file lives inside a directory.
A bit set is a compact collection of yes-or-no values.
A memory leak happens when a program allocates memory and then loses the ability to free it.
@memcpy copies bytes from one memory region to another.
Metaprogramming means writing code that helps create, inspect, or specialize other code.
Most memory in Zig is ordinary memory.
A clear explanation of counting subarrays where the leftmost element is not larger than any other element, using a monotonic stack.
A clear explanation of finding the complement of a number by XORing with a bitmask of the same bit length.
A clear explanation of finding the shortest path from top-left to bottom-right in a binary matrix using BFS.
A clear explanation of removing outermost parentheses from each primitive decomposition by tracking nesting depth.
A clear explanation of maximizing the sightseeing score by tracking the best left value seen so far in a single pass.
A clear explanation of counting song pairs whose total duration is divisible by 60 using remainder frequency counting.
A clear explanation of summing root-to-leaf binary numbers in a binary tree using DFS with accumulated values.
A clear explanation of finding the longest string that divides both strings using the GCD of their lengths.
A clear explanation of finding the smallest index where arr[i] equals i using binary search on a sorted distinct array.
A clear explanation of finding the most experienced employee(s) for each project using a window function or correlated subquery.
A clear explanation of computing the clumsy factorial by simulating cyclic operators with a stack.
A clear explanation of generating all strings from a brace expansion pattern in lexicographic order using backtracking.
A clear explanation of finding the best seller(s) by total price using GROUP BY, SUM, and a subquery for the maximum.
A clear explanation of determining if a source can reach a target in a very large grid with blocked cells using BFS with a cell count limit.
A clear explanation of computing statistical measures (minimum, maximum, mean, median, mode) from a frequency count array.
A clear explanation of minimizing total rounding error when rounding prices to meet a target sum using a greedy approach.
A clear explanation of finding the longest word chain where each word is formed by inserting one letter into the previous word, using dynamic programming.
A clear explanation of finding the minimum and maximum moves to make three stones consecutive by analyzing gap cases.
A clear explanation of finding the minimum number of subsequences of source needed to form target using greedy two-pointer scanning.
A clear explanation of finding the k-th missing number in a sorted array using binary search on the missing count.
A clear explanation of finding the maximum number of rows that can be made all-equal by flipping columns, using row pattern normalization.
A clear explanation of finding all words that follow a two-word sequence in a text string.
A clear explanation of finding characters that appear in all words using minimum frequency counts.
A clear explanation of reconstructing a binary tree from a depth-encoded preorder traversal string using a stack.
A clear explanation of determining if a robot stays in a bounded circle by checking position and direction after one instruction cycle.
A clear explanation of finding a target in a mountain array using three binary searches on the interface API.
A clear explanation of finding the maximum ancestor-node difference in a binary tree by tracking min and max along each root-to-leaf path.
A clear explanation of finding minimum rotations to make all tops or bottoms equal using a greedy candidate check.
A clear explanation of verifying that all paths from a source node lead to a destination using DFS with cycle detection.
A clear explanation of finding two non-overlapping subarrays with maximum combined sum using prefix sums and running maximums.
A clear explanation of rearranging barcodes so no two adjacent barcodes are equal using a greedy max-heap approach.
A clear explanation of checking if three points form a boomerang (non-collinear) using the cross product.
A clear explanation of computing the average years of experience per project using JOIN and AVG aggregation.
A clear explanation of validating a string by repeatedly removing 'abc' substrings using a stack.
A clear explanation of finding minimum and maximum moves to make stones consecutive using a sliding window.
A clear explanation of counting all distinct non-empty sequences from a set of letter tiles using backtracking with frequency counting.
A clear explanation of simulating stone smashing to find the last remaining weight using a max heap.
A clear explanation of retrieving product names and their sale years using a JOIN between Sales and Product tables.
A clear explanation of pruning tree nodes where all root-to-leaf paths through them have sum less than a limit, using post-order DFS.
A clear explanation of checking if an array can be split into three contiguous parts with equal sum using a greedy two-pass approach.
A clear explanation of finding the lexicographically smallest equivalent string using Union-Find with canonical representatives.
A clear explanation of counting confusing numbers up to n using digit backtracking with rotation validation.
A clear explanation of finding the longest substring that appears at least twice using binary search on length with rolling hash.
A clear explanation of checking if a number becomes a different valid number when rotated 180 degrees.
A clear explanation of greedily assigning bikes to workers based on Manhattan distance, prioritizing by distance then worker then bike index.
A clear explanation of minimizing the last stone weight by splitting stones into two groups using 0/1 knapsack dynamic programming.
A clear explanation of finding the longest arithmetic subsequence in an array using dynamic programming with difference hash maps.
A clear explanation of counting numbers up to n with at least one repeated digit using digit DP and combinatorics.
A clear explanation of maximizing array sum after exactly k negations using a greedy strategy.
A clear explanation of checking divisibility of binary prefixes by 5 using running remainder tracking.
A clear explanation of selecting the maximum sum subset under item and label count constraints using a greedy approach.
A clear explanation of finding the lexicographically smallest subsequence with all distinct characters using a greedy stack approach.
A clear explanation of finding customers who purchased every product in the catalog using GROUP BY and HAVING with COUNT DISTINCT.
A clear explanation of finding the minimum number of video clips to cover a time range using a greedy interval covering approach.
A clear explanation of finding actor-director pairs with at least three collaborations using GROUP BY and HAVING.
A clear explanation of checking if the digit sum of the array minimum is odd or even.
A clear explanation of duplicating zeros in-place in an array without using extra space by working backwards.
A clear explanation of counting the occurrences of a specific digit in all numbers from 1 to n using digit dynamic programming.
A clear explanation of why Alice wins the divisor game if and only if n is even, proven by mathematical induction.
A clear explanation of finding the longest duplicate substring using binary search on length combined with Rabin-Karp rolling hash.
A clear explanation of checking camelCase pattern matching by verifying uppercase consistency with a two-pointer approach.
A clear explanation of finding the fraction of players retained the day after their first login using self-join and window functions.
A clear explanation of finding the first year each product was sold using a self-join or window function.
A clear explanation of eliminating adjacent duplicate character pairs from a string using a stack.
A clear explanation of checking whether a binary string contains all binary representations of integers from 1 to n.
A clear explanation of finding the smallest repunit divisible by K by tracking remainders to detect cycles.
A clear explanation of assigning 4 flower types to garden nodes with no adjacent conflicts using greedy graph coloring.
A clear explanation of counting submatrices with a given sum using 2D prefix sums combined with the subarray sum equals k technique.
A clear explanation of finding all index pairs where a word from the list appears in a text string using a trie.
A clear explanation of reconstructing a BST from its preorder traversal using value range bounds.
A clear explanation of finding books with fewer than 10 sales in the last year that were not sold in the last year using LEFT JOIN and GROUP BY.
A clear explanation of finding the lexicographically largest permutation smaller than the given array using at most one swap.
A clear explanation of simulating lamp illumination on a grid using hash maps for rows, columns, and diagonals.
A clear explanation of counting substrings of length k with all unique characters using a sliding window.
A clear explanation of generating all strings from a brace expansion expression using recursive parsing and set union/concatenation.
A clear explanation of finding buyers who bought an iPhone but not an iPad using JOIN and NOT IN filtering.
A clear explanation of maximizing array sum by partitioning into subarrays of at most k elements, each filled with their maximum value, using dynamic programming.
A clear explanation of determining if a car can transport all passengers using a difference array for passenger count tracking.
A clear explanation of finding the minimum ship capacity to deliver all packages within D days using binary search.
A clear explanation of finding the project with the most employees using GROUP BY, COUNT, and a subquery for the maximum.
A clear explanation of coloring the border of a connected component in a grid using BFS.
A clear explanation of finding the maximum sum of two numbers less than k using a two-pointer approach on a sorted array.
A clear explanation of computing total quantity sold per product using GROUP BY and SUM aggregation.
A clear explanation of finding the longest subarray of ones by flipping at most k zeros using a sliding window.
A clear explanation of adding two non-positive integers represented as arrays of digits.
A clear explanation of finding the shortest string containing both input strings as subsequences using LCS dynamic programming.
A clear explanation of efficiently querying a stream of characters against a word list using an Aho-Corasick trie.
A clear explanation of finding products sold only in the first quarter of 2019 using GROUP BY with date range conditions.
A clear explanation of finding the minimum total Manhattan distance to assign bikes to workers using bitmask dynamic programming.
A clear explanation of counting students not in the expected height order by comparing the array to its sorted version.
A clear explanation of counting land cells unreachable from the grid border using BFS from boundary land cells.
A clear explanation of converting a non-negative integer to its base negative-two representation.
A clear explanation of converting a BST to a greater sum tree by accumulating values in reverse inorder traversal.
A clear explanation of sorting matrix cells by Chebyshev distance from a given center cell using BFS.
A clear explanation of minimizing total travel cost for two-city scheduling using a greedy refund approach after sending everyone to city A.
A clear explanation of maximizing uncrossed connecting lines between two arrays using longest common subsequence dynamic programming.
A clear explanation of finding the next greater value for each node in a linked list using a monotonic stack.
A clear explanation of maximizing satisfied customers by choosing the best window for the owner to not be grumpy using a sliding window.
A clear explanation of minimizing the total score of triangulating a polygon using interval dynamic programming.
A clear explanation of computing each student's top-5 average score using sorting and grouping.
Zig gives you direct control over memory. That control is useful, but it also means you must follow clear rules.
Semantic analysis is the compiler stage that checks what a program means.
A debug build is a build made for finding mistakes.
WebAssembly, often shortened to Wasm, is a portable binary instruction format. It lets you compile code once and run it inside different hosts, such as web browsers, servers,...
Memory safety means using memory only while it is valid, only through the right type, and only inside the allowed range.
Allocation can fail.
@bitCast reinterprets the bits of one value as another type.
Static dispatch means the compiler decides which code to call before the program runs.
Type reflection means asking questions about a type while Zig is compiling the program.
Data-oriented design means you organize a program around the data it processes.
Efficient text processing means working with text without doing unnecessary allocation, copying, or decoding.
SIMD means Single Instruction, Multiple Data.
Embedded development means writing software for small computers inside devices.
Zig’s standard library is imported with:
A custom allocator is an allocator you design for a specific memory policy.
@ptrCast converts one pointer type into another pointer type.
A dangling pointer is a pointer that refers to memory that is no longer valid.
A generic data structure is a data structure that works with more than one element type.
Type coercion means Zig converts a value from one type to another when the conversion is safe and well-defined.
A state machine is a simple way to describe a program that moves between fixed states.
A dynamic string is text whose length is not fixed ahead of time.
A build option is a value passed from the command line into build.zig.
Queues and stacks are two simple ways to organize a collection of items.
The page allocator asks the operating system for memory directly.
@typeInfo asks the compiler for structured information about a type.
Reflection means inspecting a type as data.
An error union is a value that can contain either:
A union stores one active field at a time.
Zig string data is usually stored as UTF-8 bytes.
Zig programs use memory in different places. The two most important places are the stack and the heap.
Terminal programming means writing programs that interact with the command line as more than simple text output.
A static file server is a program that reads files from a directory and sends them to a browser over HTTP.
@Type builds a type from compile-time type information.
Generating code at compile time means using Zig code to create specialized program behavior before the final executable is built.
anytype means the function parameter can accept many different types.
A tagged union is a type that can store one value from several possible shapes.
A sentinel-terminated array is an array with a special value at the end.
An opaque type is a type whose internal structure is hidden.
One of Zig’s strengths is compiler diagnostics. Zig tries to explain problems precisely instead of silently accepting dangerous behavior.
Zig does not have a separate built-in mutable String type.
An enum is a type whose value must be one item from a fixed list.
A linked list is a collection where each item points to the next item.
A C struct groups several fields into one value.
Fuzz testing means testing a program with many generated inputs.
A fixed buffer allocator gives memory from a buffer you already own.
@alignOf asks the Zig compiler for the required memory alignment of a type.
A compile-time loop is a loop that runs while Zig is compiling your program.
Parsing is the part of the compiler that reads source code and turns it into structure.
A Zig project can depend on other Zig packages.
macOS is one of Zig’s main desktop targets. You can use Zig on macOS to write command-line tools, development utilities, servers, libraries, and cross-platform applications.
Writing a file means sending bytes from your program to the operating system so they can be stored on disk.
An atomic operation is a small operation that can safely happen while several threads are running.
Alignment is a rule about where a value may be placed in memory.
Errors are for expected failures.
A plugin architecture lets a program be extended without rewriting the whole program.
Modern CPUs are fast, but memory is much slower.
Systems programming means working directly with the operating system.
Most Zig programs start with small error sets:
A generic function is a function that works with many types instead of only one type.
A JSON parser reads JSON text and turns it into data your program can use.
const std = @import"std";
A vector is a fixed-size group of values of the same type.
An anonymous struct is a struct type without a name.
A string literal is text written directly in your source code.
Pointer arithmetic means moving a pointer forward or backward through memory.
An error API is the part of your function signature that tells callers how failure works.
An exported function is a function made visible outside the current Zig program.
Calling a C function from Zig has three parts.
An arena allocator is an allocator that frees many allocations at once.
A StringHashMap is a hash map where the key is a string.
A table-driven test checks many input cases with one test loop.
@sizeOf asks the Zig compiler how many bytes a type needs in memory.
Inline branching means Zig chooses a branch during compilation, not during runtime.
An optional pointer is a pointer that may have no value.
Propagating an error means passing it to the caller instead of handling it immediately.
A calling convention defines how functions communicate at the machine level.
A compile error means Zig refused to build your program.
A slice is a view into memory.
A pointer stores the address of a value in memory.
A normal Zig struct is designed for ordinary data modeling.
When people talk about Zig compiler internals, they often mention stage2.
An inline function is a function where the compiler may place the function’s code directly at the call site instead of performing a normal function call.
Some parts of a program should never run.
When you first open the Zig source code repository, it can feel overwhelming.
Linux is one of the most natural platforms for Zig. Many Zig programs are built, tested, and deployed on Linux because Linux is common in servers, containers, embedded...
Reflection means a program can inspect information about types while the program is being compiled or running.
When a program feels slow, your first job is not optimization.
A system call is a request from your program to the operating system.
A mutex is a lock for shared data.
@cImport is Zig’s built-in way to import C declarations from header files.
Reading a file means asking the operating system for bytes stored on disk.
A unit test checks one small piece of code in isolation.
A Zig build is made from steps.
A HashMap is a data structure for storing values by key.
The general purpose allocator is Zig’s standard allocator for ordinary heap allocation.
One of the first Zig builtins you will learn is @import.
In the previous section, you learned that Zig can execute code during compilation.
errdefer is a cleanup tool.
An anonymous function is a function without a permanent name.
Many programs need to clean something up after using it.
Optional unwrapping means taking the value out of an optional.
A slice is a view into a sequence of values.
A slice is a view into a sequence of values.
A struct field can have a default value.
catch handles an error at the place where it happens.
Functions are values.
Zig has normal loops that run when the program runs.
Zig 0.16.0 was released on April 14, 2026. The release contains 8 months of work, with changes from 244 contributors across 1,183 commits. The largest themes are the new I/O...
The Zig compiler is not only a compiler for the Zig language. It is also the center of the Zig toolchain.
In this project, we will build a small command-line calculator.
Windows is one of Zig’s main supported platforms. You can write Zig programs on Windows, build Windows executables, call Windows system APIs, link with C libraries, and...
Zig has a formatting system built into the standard library. You have already used it many times through std.debug.print.
Performance is one of the main reasons people choose Zig.
A memory mapped file is a file that the operating system places into your program's address space.
A thread is a separate path of execution inside one program.
Zig works unusually well with C because it treats C as a first-class part of systems programming.
Zig has a built-in test system. You do not need a separate testing library to start writing tests.
Zig has a built-in build system.
std is Zig’s standard library.
An ArrayList is one of the most important data structures in Zig.
Memory is one of the most important ideas in Zig.
Zig has special built-in functions whose names start with @.
comptime means “compile time.”
An optional type is a type that can hold either a value or no value.
A struct method is a function that belongs to a struct.
A multidimensional array is an array whose elements are also arrays.
A many item pointer is a pointer that can move across several values of the same type.
try is the most common way to handle errors in Zig.
Recursion is a technique where a function calls itself.
A struct is a type that groups several values together.
An array literal is the syntax you use to write array values directly in source code.
A pointer is a value that stores the address of another value.
An error union type means:
Many functions need to produce more than one piece of information.
Loops repeat code. But sometimes you do not want a loop to finish in the normal way.
An array is a group of values stored next to each other.
Memory is where a program keeps its data while it runs.
An error set is a group of possible error names.
Zig has blocks.
Functions often need to produce results.
A keyword is a word that has special meaning in Zig.
Most programming languages need a way to handle failure.
A while loop repeats while a condition is true.
Programs often need to repeat work.
Names are part of the program.
Function parameters are the inputs of a function.
Functions are reusable blocks of code.
An if statement is good for general conditions:
Comments are notes for humans who read the code.
A string is text.
Initialization means giving a value to something when it is created.
In Zig, undefined means “this value has not been initialized.”
Type inference means Zig can figure out a type from the value you write.
A boolean is a value that can be only one of two things:
Floating point numbers are numbers with fractional parts.
Integers are whole numbers.
Every value in a Zig program has a type.
Sooner or later, every Zig beginner meets the compiler.
You can write Zig code in any text editor.
So far, we have used single-file programs.
When beginners hear the word “compiler,” they often think of one job:
In the previous section, we used this command:
Now we will write and run a complete Zig program.
Before writing larger Zig programs, you need to understand an important fact:
Before we write more Zig code, we need the Zig compiler.
Zig exists because low-level programming is still important, but the old tools have painful tradeoffs.
Programs need to make choices.
A program stores values so it can use them later. In Zig, you store values with two main keywords:
Zig is a programming language for writing programs that are fast, clear, and close to the machine.
Zig is a programming language for writing fast, small, reliable programs.
Zig is a programming language for writing fast, small, reliable programs.
A set is a collection of objects called elements.
Modern number theory continues to evolve rapidly.
The Langlands program is one of the largest and most influential research programs in modern mathematics.
An elliptic curve over $\mathbb{Q}$ may be written in Weierstrass form
The Riemann zeta function is one of the central objects in mathematics.
Fermat's Last Theorem states that there are no positive integers
Number theory contains some of the oldest and deepest unsolved problems in mathematics.
Arithmetic statistics studies the distribution of arithmetic objects inside large families.
Prime numbers are deterministic objects, but many aspects of their distribution resemble random behavior.
The Riemann zeta function is defined for $\operatorname{Re}s>1$ by
Sieve methods are extremely effective for estimating how many integers avoid small prime factors. They have produced major results about:
The Twin Prime Conjecture states that infinitely many primes satisfy
A probabilistic algorithm uses random choices during its execution. In number theory, this is often a practical advantage rather than a weakness.
The Prime Number Theorem for arithmetic progressions states that for
The classical Langlands program relates:
Write the two integers as
One of the central problems of analytic number theory is understanding how primes distribute among residue classes.
A primality test determines whether an integer is prime.
Modular curves parameterize elliptic curves and connect modular forms with arithmetic geometry.
Fourier analysis decomposes functions into harmonic frequencies.
Classical sieve methods estimate how many integers survive congruence restrictions. The large sieve approaches these problems from a different direction.
One of the deepest ideas in algebraic number theory is that prime numbers possess hidden symmetry inside field extensions.
Let
Brun's sieve introduced the idea of estimating sifted sets through truncated inclusion-exclusion. However, Brun's method often produced bounds that were technically difficult...
A positive integer is called $y$-smooth if all of its prime factors are at most $y$.
The Riemann zeta function
Sieve methods are techniques for counting integers that remain after removing residue classes modulo primes.
In the ordinary integers, every nonzero integer factors uniquely into prime numbers.
Number theory often studies exact statements about individual integers. For example, one may ask whether a given integer is prime, squarefree, smooth, or representable as a...
The Langlands program predicts that many different arithmetic objects are connected by systematic transfers.
In additive number theory, ordinary asymptotic density is often too weak to control additive behavior.
Classical number theory studies arithmetic globally over fields such as
A zero-knowledge proof allows one party to convince another that a statement is true without revealing why it is true.
Galois groups encode the symmetries of algebraic equations and field extensions.
A central question in additive number theory asks whether every integer can be represented as a sum of elements from a fixed set.
The real numbers arise by completing the rational numbers using the ordinary absolute value. The $p$-adic numbers arise by completing the rational numbers using the $p$-adic...
Modern public-key cryptography relies heavily on two computational assumptions:
The Langlands program is one of the most ambitious and influential theories in modern mathematics.
Exponential sums are among the central tools of analytic number theory.
In ordinary analysis, the absolute value
Ordinary integers satisfy several remarkable properties simultaneously:
Study empirical properties of prime numbers through computation.
Lattice cryptography is a family of cryptographic systems based on the presumed hardness of computational problems on high-dimensional lattices.
Many problems in additive number theory ask whether an integer can be represented in the form
Classical modular form theory begins with analytic functions satisfying symmetry conditions.
Let $R$ be a commutative ring. An ideal $I\subseteq R$ is called principal if there exists an element $\alpha\in R$ such that
Waring's problem asks whether every sufficiently large positive integer can be written as a sum of a bounded number of fixed powers.
Pairing-based cryptography uses special maps defined on elliptic curve groups. A pairing is a function
Modular forms are functions on the upper half-plane satisfying symmetry conditions under the modular group
The discriminant is one of the most important invariants of a number field. It measures how the arithmetic of the field differs from ordinary rational arithmetic.
Goldbach-type problems ask whether integers can be represented as sums of primes. They are among the oldest and most famous problems in additive number theory.
Elliptic curve cryptography is a public-key cryptographic framework based on the arithmetic of elliptic curves over finite fields.
For centuries, elliptic curves and modular forms were studied as separate objects.
Let $K$ be a number field and let
In ordinary integers, every ideal is generated by a single element:
Secure communication requires two parties to share secret information. In classical symmetric cryptography, both parties must already possess the same secret key before...
Additive number theory studies arithmetic structure through addition of integers and subsets of integers.
An elliptic curve is simultaneously:
1. Prove that the sum of two even integers is even.
The classical Riemann Hypothesis concerns the zeros of the Riemann zeta function
In ordinary integers, every number factors uniquely into primes. In many rings of algebraic integers, this property fails.
One of the central properties of the ordinary integers is unique factorization.
Let $K$ be a number field of degree
Classical cryptography uses a shared secret key. Both sender and receiver must know the same secret information in advance.
The modular group acts on the upper half-plane by fractional linear transformations:
A central theme in analytic number theory is determining when an $L$-function is nonzero at a particular point.
Modern number theory relies heavily on computation. Two broad computational paradigms dominate the subject:
An arithmetic progression is a sequence of the form
Modular forms already possess symmetry under the modular group. Yet a deeper arithmetic structure emerges through another family of operators: the Hecke operators.
In ordinary arithmetic, the integers
The Riemann zeta function
Elliptic curves occupy a central position in modern number theory, arithmetic geometry, and cryptography.
Modular forms satisfy strong symmetry conditions under the modular group. Among them, cusp forms form the deepest and most arithmetic subclass.
Modular forms are highly structured analytic functions with deep arithmetic properties. Although their definitions involve complex analysis and group actions, modular forms...
Among all modular forms, Eisenstein series are the most explicit and computationally accessible.
Dirichlet characters behave analogously to exponential functions in Fourier analysis. Just as complex exponentials separate frequencies, characters separate residue classes...
A number field is a finite extension of the rational numbers. Concretely, it is a field $K$ satisfying
Modular forms are among the central objects of modern number theory.
A lattice is a discrete additive subgroup of Euclidean space. More concretely, let
| Period | Development |
The Riemann zeta function studies prime numbers globally, without distinguishing congruence classes. However, many arithmetic questions concern primes satisfying conditions such as
The modular group acts on the upper half-plane by fractional linear transformations:
An algebraic number is a complex number that satisfies some nonzero polynomial equation with rational coefficients. Thus $\alpha\in\mathbb{C}$ is algebraic if there exists a...
| Definition | Location |
The Riemann zeta function was introduced through the series
Integer factorization asks for the prime decomposition of a positive integer. Given
Euler products are one of the central ideas of analytic number theory. They express infinite sums over integers as infinite products over primes.
An arithmetic function $fn$ can be encoded into an infinite series of the form
Modular forms begin with the action of certain matrix groups on the complex upper half-plane.
The ordinary integers
| Theorem | Location |
A prime number is an integer greater than $1$ whose only positive divisors are
One of the central goals of algebraic number theory is to classify field extensions of a number field
One of the deepest ideas in analytic number theory is that the zeros of the zeta function determine the distribution of prime numbers.
Diophantine approximation studies how closely real numbers can be approximated by rational numbers.
Arithmetic functions often fluctuate strongly from one integer to the next.
Many arithmetic functions are defined through sums over divisors. For example,
Recall that a Pell equation has the form
The Riemann zeta function has nontrivial zeros inside the critical strip
Modern computational number theory depends fundamentally on efficient arithmetic with large integers.
Global class field theory studies finite abelian extensions of number fields such as
Arithmetic functions can be added and multiplied pointwise, but number theory has another product that is better adapted to divisibility.
The convergents of a continued fraction are the rational numbers obtained by truncating the expansion at finite stages.
The zeros of the Riemann zeta function are the complex numbers $s$ satisfying
| Symbol | Meaning |
One of the central ideas of modern analysis is that functions may be decomposed spectrally into elementary pieces.
One of the central discoveries of algebraic number theory is that unique factorization may fail in rings of algebraic integers.
The defining series of the zeta function,
Many important numbers are irrational:
An arithmetic function is a function defined on the positive integers. Such a function
The Liouville function is an arithmetic function denoted by
The defining series of the Riemann zeta function is
One of the oldest themes in number theory is reciprocity: the phenomenon that solvability conditions for one prime are controlled by arithmetic involving another prime.
Finite continued fractions correspond exactly to rational numbers. When the Euclidean algorithm never terminates, the continued fraction becomes infinite.
Functoriality is the unifying mechanism of the Langlands program. It predicts systematic relationships between automorphic representations attached to different algebraic groups.
Euler's totient function is an arithmetic function denoted by
The defining series of the Riemann zeta function is
A central goal of algebraic number theory is to understand field extensions of a given base field, especially extensions of the rational numbers
A finite continued fraction is an expression of the form
A group $G$ is abelian if
One of the central objects of analytic number theory is the Riemann zeta function. It connects infinite series, prime numbers, complex analysis, and arithmetic structure into...
The Euclidean algorithm is one of the oldest and most important algorithms in mathematics. It computes the greatest common divisor of two integers using repeated division.
The rational numbers may be studied through their completions:
The Möbius function is an arithmetic function denoted by
The Langlands program is a broad collection of conjectures connecting number theory, representation theory, harmonic analysis, and algebraic geometry. Its central idea is that...
A pair of primes
Quadratic residue theory is not only a theoretical subject. It also plays a major role in computational number theory, cryptography, primality testing, and algorithm design.
A central problem in number theory is determining whether an equation possesses rational or integral solutions.
Number theory is one of the oldest parts of mathematics, but modern number theory is not a single ancient subject carried forward unchanged. It is a layered discipline....
Divisor functions measure the positive divisors of an integer. They are among the first examples of arithmetic functions, because their values depend directly on the prime...
Modular arithmetic is not only a theoretical language for divisibility. It is also one of the main tools of computation with integers.
Let
Number theory studies arithmetic simultaneously at two levels:
Quadratic reciprocity describes when one prime is a square modulo another prime. A natural question is whether similar laws exist for higher powers.
One of the central ideas of number theory is that congruences modulo powers of a prime often approximate genuine arithmetic solutions.
Modular arithmetic often requires computing powers such as
Computation has become an essential part of number theory. Classical arithmetic relied mainly on symbolic reasoning and hand calculations. Modern arithmetic combines rigorous...
The Prime Number Theorem describes the average distribution of primes up to a large number $x$:
Classically, number theory studied special analytic functions such as modular forms. These functions satisfy strong symmetry conditions under actions of arithmetic groups.
The rational numbers form a field rich enough for arithmetic, yet insufficient for many limiting processes.
The Chinese remainder theorem describes when several congruence conditions can be combined into one congruence. Its cleanest form occurs when the moduli are pairwise coprime.
The Prime Number Theorem states that
Representation theory studies abstract algebraic objects by expressing them as linear transformations of vector spaces.
The real numbers arise by completing the rational numbers with respect to the ordinary absolute value. This completion produces a field suited to Euclidean geometry and...
Gauss sums arise from combining multiplicative and additive structures modulo a prime. They form one of the fundamental tools of analytic and algebraic number theory.
Category theory studies mathematical structures through objects and maps between them. Instead of looking only at what objects are made of, it studies how they relate to other...
One of the central problems in arithmetic geometry is understanding the number of solutions of polynomial equations over finite fields.
The ordinary absolute value on the real numbers measures magnitude:
The theory of quadratic residues asks a fundamental question:
A system of congruences asks for an integer satisfying several congruence conditions simultaneously.
In ordinary arithmetic, division by a nonzero number means multiplication by its reciprocal. Modular arithmetic is more delicate. A residue class may or may not have a...
A linear congruence is a congruence of the form
Arithmetic modulo $n$ is arithmetic performed on residue classes modulo $n$. Instead of distinguishing all integers separately, we identify integers that have the same...
Congruence modulo $n$ groups integers according to their remainders after division by $n$. If two integers have the same remainder, they are congruent modulo $n$.
Ordinary equality compares integers exactly. In many arithmetic problems, however, only the remainder after division matters.
The infinitude of primes guarantees that primes continue indefinitely, but it says nothing about how frequently primes occur.
Euclid proved that there are infinitely many primes by contradiction. Euler discovered a very different proof based on infinite series and products.
Euclid's proof of the infinitude of primes is one of the earliest examples of a general argument in number theory. It does not depend on computation, experimentation, or...
Prime numbers are the building blocks of the positive integers. Once unique prime factorization is known, a natural question arises: are there only finitely many primes, or do...
An arithmetic function is a function whose domain is the positive integers. It assigns a value to each integer
Unique prime factorization says that every integer $n>1$ can be written as a product of primes. The canonical prime decomposition is the ordered and exponentiated version of...
A vector space over a field $F$ is a set $V$ equipped with addition and scalar multiplication satisfying the usual algebraic rules.
The logarithmic integral is the function
Classical topology studies geometric spaces using invariants such as homology and cohomology. Over the complex numbers, algebraic varieties can often be viewed as topological...
One of the central ideas of algebraic number theory is that prime numbers may behave differently after passing to a larger field.
The Prime Number Theorem describes the asymptotic distribution of prime numbers. It states that
Measure theory extends the ideas of length, area, volume, and integration to more general settings. In number theory, measure appears in probability, harmonic analysis,...
Euler criterion gives an efficient way to decide whether an integer is a square modulo an odd prime. Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$....
Arithmetic geometry often studies families of algebraic curves varying over arithmetic bases. The most important base is
One of the most important classes of number fields arises from the solutions of the equation
The prime counting function
The Legendre symbol
Let $p$ be an odd prime and let $a\in\mathbb{Z}$. The Legendre symbol is defined by
A quadratic congruence is a congruence involving a square. The basic form is
Topology studies continuity, convergence, connectedness, and geometric structure in an abstract setting. In number theory, topology appears naturally in real analysis, complex...
An algebraic curve is a geometric object whose dimension is one. Curves are among the oldest and most important objects in number theory and algebraic geometry.
The familiar fields
One of the oldest questions in number theory asks how prime numbers are distributed among the positive integers. Since primes become less frequent as numbers grow larger,...
The fundamental theorem of arithmetic states that every integer $n>1$ can be written as a product of prime numbers, and that this product is unique up to the order of the factors.
A Diophantine equation is first an arithmetic object. It asks for solutions in integers or rational numbers. But every polynomial equation also defines a geometric object.
Two integers $a$ and $b$, not both zero, are called coprime if their greatest common divisor is $1$:
Let $a$ and $b$ be integers. An integer of the form
The real numbers $\mathbb{R}$ extend the rational numbers $\mathbb{Q}$ by filling gaps such as
Geometry is not only concerned with spaces themselves, but also with maps between spaces. In algebraic geometry and arithmetic geometry, these maps are called morphisms.
A polynomial equation may possess several roots related by hidden algebraic symmetries. Consider
In analytic number theory, one often studies sums of the form
A central problem in number theory is to study solutions of polynomial equations whose coordinates belong to a specified number system. Two important cases are:
The Euclidean algorithm computes the greatest common divisor of two integers. The extended Euclidean algorithm does more. It also expresses the gcd as an integer linear...
The greatest common divisor of two integers can be found by listing divisors, but this method becomes inefficient for large numbers. For example, finding
An exponential Diophantine equation is a Diophantine equation in which one or more unknowns appear as exponents. Typical examples include
A Catalan-type equation is a Diophantine equation involving powers whose values differ by a small amount. The classical example is
Let $a$ and $b$ be nonzero integers. An integer $m$ is called a common multiple of $a$ and $b$ if
Analytic number theory studies infinite sums, products, and integrals. Before such expressions can be manipulated safely, one must understand the meaning of convergence.
One of the oldest questions in number theory asks which integers can be written as sums of squares. Typical examples are
Let $a$ and $b$ be integers, not both zero. An integer $d$ is called a common divisor of $a$ and $b$ if
Abstract algebra studies sets equipped with operations. In number theory, these structures organize arithmetic behavior.
Euler products arise when an infinite series has coefficients controlled by multiplication. The simplest and most important example is the zeta series
A Pell equation is a Diophantine equation of the form
The division algorithm is one of the basic structural facts about the integers. It says that any integer can be divided by a positive integer with a unique quotient and remainder.
Classical algebraic geometry studies varieties defined by polynomial equations. This theory works well over algebraically closed fields, especially over $\mathbb{C}$. However,...
A central problem in algebra is to determine where a polynomial factors completely into linear terms. Consider the polynomial
An infinite product has the form
A Pythagorean triple is a triple of positive integers
A positive integer $n>1$ is called composite if it is not prime.
A mathematical proof is a logically complete argument establishing the truth of a statement from accepted assumptions, definitions, and previously proved results.
Prime numbers are the fundamental building blocks of arithmetic.
A set is a collection of objects, called its elements. If $x$ is an element of a set $A$, we write $x \in A$. If $x$ is not an element of $A$, we write $x \notin A$.
A clear explanation of the Long Pressed Name problem using a two-pointer scan.
A clear explanation of counting good starting indices using next-jump preprocessing and dynamic programming.
A clear explanation of the Rectangle Area II problem using sweep line and merged active y-intervals.
A clear explanation of counting subarrays whose sum is divisible by k using prefix sums and remainder frequencies.
A clear explanation of returning the k closest points to the origin using squared distance and sorting.
A clear explanation of the Maximize Distance to Closest Person problem using gaps between occupied seats.
A clear explanation of the Shifting Letters problem using suffix sums and modulo arithmetic.
A clear explanation of comparing rational numbers written as decimal strings with optional repeating parts.
A clear explanation of the Shortest Path Visiting All Nodes problem using multi-source BFS and bitmask state compression.
A clear explanation of minimizing malware spread by analyzing connected components with Union Find.
A clear explanation of matching a binary tree preorder traversal by greedily flipping nodes.
A clear explanation of the Hand of Straights problem using sorting, frequency counting, and greedy grouping.
A clear explanation of counting index triplets with duplicate values using frequency counts and combinatorics.
A clear explanation of designing an iterator over a run-length encoded sequence without expanding it.
A clear explanation of generating all powerful integers using bounded powers and a set.
A clear explanation of placing even numbers at even indices and odd numbers at odd indices using two pointers.
A clear explanation of the Longest Mountain in Array problem using peak detection and two-pointer expansion.
A clear explanation of finding the lexicographically smallest string after queue operations using rotation and sorting.
A clear explanation of sorting an array using prefix reversals by repeatedly placing the largest remaining value.
A clear explanation of merging consecutive stone piles with minimum cost using interval dynamic programming.
A clear explanation of finding the minimum banana-eating speed using binary search on the answer.
A clear explanation of making a parentheses string valid using greedy counting.
A clear explanation of the Backspace String Compare problem using stack simulation and an O(1) space two-pointer scan.
A clear explanation of solving Reveal Cards In Increasing Order using sorting and queue simulation over indices.
A counting solution for computing how many directed friend requests are allowed by age rules.
A clear explanation of counting distinct bitwise OR results from all non-empty subarrays using rolling sets.
A clear explanation of counting how many pawns a rook can capture by scanning four directions on a chessboard.
A clear explanation of placing the minimum number of cameras in a binary tree using postorder DFS.
A clear explanation of simulating robot movement on an infinite grid using direction vectors and obstacle lookup.
A clear explanation of solving Largest Time for Given Digits by checking all permutations of four digits.
A clear explanation of the Guess the Word interactive problem using candidate filtering and minimax-style guessing.
A clear explanation of counting valid music playlists using dynamic programming over playlist length and unique songs used.
A string simulation solution for converting each word in a sentence into Goat Latin.
A clear explanation of generating all n-digit numbers whose adjacent digits differ by k.
A clear explanation of inserting a value into a maximum binary tree by following the right spine.
A clear explanation of finding the longest Fibonacci-like subsequence using dynamic programming and value-to-index lookup.
A clear explanation of the Split Array into Fibonacci Sequence problem using backtracking, leading-zero checks, and 32-bit integer limits.
A clear explanation of maintaining a complete binary tree inserter using level-order indexing.
A dynamic programming solution for counting binary trees where every non-leaf node is the product of its children.
A clear explanation of solving Bag of Tokens using sorting, greedy choices, and two pointers.
A clear explanation of rearranging a binary search tree into an increasing right-only tree using inorder traversal.
A clear explanation of implementing a spellchecker with exact, case-insensitive, and vowel-error matching.
A clear explanation of identifying the town judge using trust indegree and outdegree counts.
A clear explanation of finding the maximum circular subarray sum using Kadane's algorithm.
A clear explanation of the Keys and Rooms problem using graph traversal from room 0.
A clear explanation of comparing two binary trees by collecting their leaf value sequences with DFS.
A hash set solution for finding the smallest number that can be hidden from all front-facing cards.
A clear explanation of solving Most Stones Removed with Same Row or Column using connected components and union-find.
A clear explanation of checking whether an array is monotonic using one pass and direction flags.
A clear explanation of checking whether every node in a binary tree has the same value.
A clear explanation of reversing only English letters while keeping all non-letter characters fixed.
A clear explanation of counting unique permutations where every adjacent pair sums to a perfect square using backtracking.
A clear explanation of the Magic Squares In Grid problem using fixed-size subgrid validation.
A two-pass solution for computing the shortest distance from each index to the nearest occurrence of a target character.
A clear explanation of minimizing refueling stops using a greedy max heap over reachable stations.
A clear explanation of expressing a target using the fewest operators with repeated uses of x.
A clear explanation of solving Validate Stack Sequences by simulating stack push and pop operations.
A clear explanation of designing a stack that pops the most frequent value, breaking ties by most recent insertion.
A clear explanation of finding universal words by merging character frequency requirements from words2.
A clear explanation of finding the closest shorthand RGB color by rounding each color channel to the nearest repeated hexadecimal pair.
A clear explanation of checking whether every global inversion is also a local inversion using distance constraints.
A clear explanation of splitting a linked list into k consecutive parts with sizes as equal as possible.
A clear explanation of making all bits equal to 1 using greedy left-to-right flips and a sliding window flip parity.
A suffix-removal solution for finding the shortest reference string that can encode every word.
A clear explanation of maximizing the advantage of one array over another using sorting, greedy matching, and two pointers.
A clear explanation of the Similar String Groups problem using graph connectivity and union-find.
A clear explanation of finding the minimum-area rectangle from points when the rectangle may be rotated.
A clear explanation of finding the smallest left partition using prefix maximums and suffix minimums.
A clear explanation of generating all full binary trees with n nodes using recursion and memoization.
A clear explanation of solving Minimum Increment to Make Array Unique by sorting and greedily assigning the next available value.
A clear explanation of simulating overflow in a champagne glass pyramid using dynamic programming.
A clear explanation of solving Delete Columns to Make Sorted by checking each column independently.
A clear explanation of finding the smallest rotation with maximum score using a difference array.
A clear explanation of minimizing the largest adjacent gas-station distance using binary search on the answer.
A clear explanation of finding the leftmost pivot index using prefix sums and a running left sum.
A clear explanation of finding the minimum time for all oranges to rot using multi-source BFS.
A clear explanation of checking whether two binary tree nodes are cousins using BFS with parent tracking.
A hash map and string parsing solution for finding the most frequent non-banned word in a paragraph.
A clear explanation of solving the 2 x 3 sliding puzzle using breadth-first search over board states.
A clear explanation of restoring a Candy Crush board to a stable state using repeated marking, crushing, and gravity simulation.
A clear explanation of finding every path from node 0 to node n - 1 in a directed acyclic graph using DFS and backtracking.
A clear explanation of checking whether the digits of a number can be reordered to form a power of two using digit frequency signatures.
A clear explanation of counting subarrays with exactly k distinct integers using the at-most-k sliding window trick.
A clear explanation of counting special-equivalent string groups by building canonical signatures from even and odd positions.
Count axis-aligned rectangles whose four corners are 1 using column-pair frequency counting.
A dynamic programming solution for finding the shortest instruction sequence that drives a race car to the target position.
A clear explanation of solving Find the Shortest Superstring using pairwise overlaps and bitmask dynamic programming.
A clear explanation of checking whether one string can become another by repeated left rotations.
A clear explanation of evaluating arithmetic expressions with parentheses, precedence, and integer division.
A clear explanation of removing line comments and block comments from source code using a state machine.
A clear explanation of finding the maximum distance between adjacent set bits in a binary representation.
A clear explanation of finding the minimum operations by working backward from target to startValue.
Simulate virus containment by repeatedly quarantining the most dangerous infected region and spreading the remaining regions.
A clear explanation of computing the exposed surface area of stacked cubes by adding tower area and subtracting shared faces.
A clear explanation of solving DI String Match using a greedy two-pointer construction.
A hash set and linked list traversal solution for counting consecutive components whose values appear in nums.
A clear explanation of transposing a matrix by swapping row and column indices.
A clear explanation of counting how many stones are jewels using a hash set for fast membership checks.
A clear explanation of counting contiguous subarrays whose maximum value lies inside a given inclusive range.
A clear explanation of merging accounts that share emails using union find and sorted email groups.
A clear explanation of Pow(x, n) using binary exponentiation to compute powers in logarithmic time.
Find the shortest word that contains all required license plate letters using frequency counting.
A clear explanation of summing subsequence widths by sorting and counting each element as a maximum and minimum.
A clear explanation of solving Valid Mountain Array by walking up the increasing slope and then down the decreasing slope.
A clear explanation of checking equality and inequality constraints using union-find.
A clear explanation of finding the smallest prime palindrome greater than or equal to n by generating odd-length palindromes and testing primality.
An enumeration solution for reconstructing all valid coordinate pairs after commas, spaces, and decimal points were removed.
A clear explanation of counting uni-value subtrees using post-order DFS.
A clear explanation of the Rising Temperature SQL problem using a self join and date comparison.
A clear explanation of simplifying algebraic expressions by parsing, substituting variables, and combining polynomial terms.
A clear explanation of finding the longest buildable word using sorting and a hash set.
Find whether the maximum element is at least twice every other element using a single linear scan.
A clear explanation of validating whether a Tic-Tac-Toe board can occur in a legal game.
A clear explanation of Group Anagrams using a hash map keyed by each word's sorted character signature.
A clear explanation of finding words that match a pattern using bijective character mapping.
A clear explanation of solving Distinct Subsequences II using dynamic programming and last occurrence tracking.
A clear explanation of finding how many integers have exactly k trailing zeroes in their factorial.
A clear explanation of splitting a permutation into the maximum number of chunks using prefix maximums.
Find the minimum cost to reach the top of the staircase using dynamic programming.
A clear explanation of finding the kth smallest pair distance using sorting, binary search on the answer, and a two-pointer count.
A clear explanation of grouping strings by their shifting sequence using normalized hash keys.
A clear explanation of the Delete Duplicate Emails SQL problem using DELETE with a self join.
A clear explanation of Rotate Image using in-place matrix transpose and row reversal.
A clear explanation of counting strobogrammatic numbers in a string range using recursive generation and range filtering.
A clear explanation of generating all strobogrammatic numbers of length n using recursion from the inside out.
A clear explanation of Permutations II using sorting, depth-first search, and duplicate-skipping backtracking.
A clear explanation of the Tenth Line shell problem using awk, sed, head, and tail.
A clear explanation of the Transpose File shell problem using awk to transform rows into columns.
Support fast prefix and suffix queries by indexing every prefix-suffix combination with the largest word index.
A clear explanation of finding the longest common contiguous subarray using dynamic programming.
A clear explanation of Permutations using depth-first search and backtracking.
A clear explanation of the Strobogrammatic Number problem using digit rotation rules and two pointers.
A clear explanation of adding an integer to an array-form number using digit-by-digit simulation.
A BFS solution for finding the minimum number of buses needed to travel from a source stop to a target stop.
A clear explanation of finding the smallest subtree that contains all deepest nodes using bottom-up DFS.
A clear explanation of counting how many words are subsequences of a string using waiting queues.
A clear explanation of splitting an array into the maximum number of chunks so sorting each chunk gives the fully sorted array.
Use binary search to find the smallest character strictly greater than the target with wraparound handling.
A clear explanation of determining whether the last character must be a one-bit character using greedy parsing.
A clear explanation of the Valid Phone Numbers shell problem using grep and regular expressions.
A clear explanation of the Word Frequency shell problem using Unix text-processing tools.
A clear explanation of Jump Game II using a greedy range expansion approach to find the minimum number of jumps.
A clear explanation of Minimum Factorization using greedy digit factors from 9 down to 2.
A clear explanation of the Shortest Word Distance III problem, including the special case where both target words are the same.
A clear explanation of Wildcard Matching using dynamic programming over string and pattern prefixes.
A clear explanation of the Shortest Word Distance II problem using preprocessing and two pointers.
A clear explanation of Multiply Strings using grade-school multiplication with digit arrays.
A clear explanation of Maximum Distance in Arrays using sorted endpoints and a greedy scan.
A clear explanation of the Shortest Word Distance problem using one pass and the latest seen indices of both words.
A clear explanation of rearranging a string so that selected characters follow a custom order.
A clear explanation of rearranging characters so no two adjacent characters are equal using a greedy max heap.
Find the time needed for a signal to reach all nodes in a directed weighted graph using Dijkstra's algorithm.
A clear explanation of designing a stack that supports push, pop, top, peekMax, and popMax.
A clear explanation of the Trapping Rain Water problem using left and right boundaries, then an optimized two-pointer solution.
A clear SQL guide for solving Combine Two Tables using LEFT JOIN.
A clear explanation of Add One Row to Tree using tree traversal and careful subtree reconnection.
A clear explanation of the First Missing Positive problem using in-place index placement to achieve O(n) time and O(1) extra space.
A clear explanation of Design Circular Queue using a fixed array, a front pointer, and a size counter.
A clear explanation of checking whether every top-left to bottom-right diagonal in a matrix has the same value.
Find the nearest leaf to a target node by converting the tree into an undirected graph and running breadth-first search.
A clear explanation of counting tilings of a 2 x n board using dominoes and L-shaped trominoes with dynamic programming.
A clear explanation of designing a range module that can add, query, and remove half-open intervals.
A clear explanation of Task Scheduler using frequency counting and the greedy block formula.
A clear explanation of solving Minimum Area Rectangle using diagonal point pairs and constant-time point lookup.
A clear explanation of reconstructing a binary tree from preorder and postorder traversals using recursion and index ranges.
Maximize cherries collected on a round trip by converting the problem into two simultaneous forward paths and solving with dynamic programming.
A clear explanation of maximizing stock trading profit with unlimited transactions and a fixed transaction fee using dynamic programming.
A clear explanation of finding the maximum width ramp using a monotonic decreasing stack.
A clear explanation of the Push Dominoes problem using force propagation and a two-pass scan.
A clear explanation of minimizing swaps so every couple sits together using greedy position tracking.
Transform the problem into House Robber dynamic programming by grouping equal values into total points.
A clear explanation of deciding whether escape is possible by comparing Manhattan distances to the target.
A clear explanation of checking whether card counts share a common group size using the greatest common divisor.
A clear explanation of finding the lexicographically smallest leaf-to-root string in a binary tree using DFS.
A clear explanation of finding the minimum moves to collect all keys in a grid using BFS with key bitmasks.
A postorder DFS solution for removing every binary tree subtree that does not contain a 1.
A clear explanation of finding the element repeated N times using a hash set.
A clear explanation of finding one candy box swap that makes Alice and Bob have equal total candies.
A clear explanation of solving Range Sum of BST using DFS with binary search tree pruning.
A clear explanation of counting good numbers after rotating every digit by 180 degrees.
Find how many days each temperature must wait for a warmer future day using a monotonic stack.
A clear explanation of counting contiguous subarrays whose product is less than k using a sliding window.
A clear explanation of finding the largest plus sign in a mined grid using four directional dynamic programming scans.
A clear explanation of the New 21 Game problem using probability dynamic programming and a sliding window sum.
Search for a target value in a binary search tree and return the subtree rooted at the matching node.
A clear explanation of Cat and Mouse using game states, reverse BFS, and topological propagation.
Simulate falling squares on a number line and track the maximum stack height after each placement.
A dynamic programming and prefix sum solution for partitioning an array into adjacent groups with maximum total average.
A clear explanation of vertical tree traversal using coordinates, DFS, sorting, and column grouping.
A clear explanation of finding all binary tree nodes at distance k from a target node by treating the tree as an undirected graph.
A clear explanation of using dynamic programming to minimize the ASCII cost of deletions needed to make two strings equal.
A clear explanation of deleting the minimum number of columns so every remaining row is individually sorted.
Find the largest number less than or equal to n whose digits are monotone increasing using a greedy digit adjustment.
A clear explanation of partitioning a string into the maximum number of parts so each character appears in at most one part.
A clear explanation of finding the minimum worst-case number of moves using dynamic programming over eggs and moves.
A clear explanation of the Rectangle Overlap problem using axis projections and positive intersection area.
A clear explanation of solving Reorder Data in Log Files using custom sorting and stable handling of digit logs.
A clear explanation of finding the cheapest flight route with at most k stops using bounded Bellman-Ford relaxation.
A clear explanation of sorting an array without built-in sorting using merge sort.
A dynamic programming and prime factorization solution for finding the minimum operations needed to produce n characters.
Decide whether an array can be divided into k non-empty subsets with equal sums using backtracking and pruning.
A queue-based simulation for predicting which party wins after senators ban opponents in turn order.
A clear explanation of counting distinct island shapes under rotation and reflection using normalization and geometric transformations.
A clear explanation of finding intersections between two sorted disjoint interval lists using two pointers.
A geometry solution for finding the largest triangle area by checking every triplet of points with the cross product formula.
A clear explanation of finding the shortest non-empty subarray with sum at least k using prefix sums and a monotonic deque.
A clear explanation of counting numbers whose binary representation has a prime number of set bits.
A clear explanation of solving Stamping The Sequence using reverse simulation and BFS-style processing.
A clear explanation of counting regions formed by slashes using union find over four triangles per cell.
A clear explanation of checking whether people can be split into two groups using graph coloring and bipartite graph detection.
Check sentence similarity with transitive word relationships using union-find.
A clear explanation of the Image Overlap problem using translation vectors and frequency counting.
A clear explanation of finding the kth smallest fraction from a sorted array using a min-heap.
A clear explanation of Online Election using preprocessing and binary search over vote times.
Find the shortest contiguous subarray with the same degree as the whole array using frequency counts and first occurrence indices.
A trie-based solution for replacing each derivative word with the shortest matching root.
Count substrings with equal consecutive groups of 0s and 1s using run lengths.
A center expansion solution for counting every palindromic substring in a string.
Find the largest connected island area in a binary grid using depth-first search.
A clear explanation of selecting a uniformly random integer while excluding blacklisted values using remapping and hashing.
A clear explanation of maximizing a binary matrix score using greedy row and column flips.
A clear explanation of cutting trees in increasing height order using repeated BFS on a grid.
A hash map solution for accumulating visit counts across domains and all of their parent subdomains.
A greedy interval scheduling solution for finding the longest chain of valid pairs.
Count unique island shapes in a binary grid using DFS and relative coordinates.
A clear explanation of making a special binary string lexicographically largest using recursive decomposition and sorting.
Evaluate a Lisp-like expression with integers, variables, let bindings, addition, multiplication, and lexical scope.
A clear explanation of solving Knight Dialer using dynamic programming over the phone keypad graph.
A clear explanation of maintaining the sum of even numbers after each array update.
A clear explanation of generating grid coordinates in an outward clockwise spiral using simulation.
A clear explanation of checking whether a binary tree is complete using level-order traversal.
A clear explanation of minimizing the array range after adding either +k or -k to every element.
A clear explanation of checking whether an undirected graph can be split into two independent sets using graph coloring.
A clear explanation of the Sum of Distances in Tree problem using tree DP, subtree sizes, and rerooting.
A clear explanation of converting uppercase ASCII letters to lowercase by scanning the string once.
A math and bit manipulation solution for deciding whether Alice wins the XOR removal game.
A clear explanation of Lemonade Change using greedy simulation and bill counting.
A counting and math solution for finding the duplicated number and the missing number in a corrupted set.
A clear explanation of finding the longest strictly increasing contiguous subarray using a single scan.
Check whether every adjacent bit in a positive integer's binary representation is different.
A clear explanation of mapping each element in one array to a matching index in its anagram using a hash map.
Simulate asteroid collisions using a stack that keeps the surviving asteroids in order.
A clear explanation of solving Shortest Bridge using DFS to mark one island and BFS to expand toward the other island.
A clear explanation of finding uncommon words by counting word frequencies across both sentences.
A clear explanation of constructing a string with exact counts of a and b while avoiding three equal consecutive characters.
A binary search solution for finding the maximum average of any contiguous subarray with length at least k.
A clear explanation of simulating prison cell transitions efficiently using cycle detection.
Find the k most frequent words using frequency counting and custom sorting by count and lexicographical order.
A clear explanation of counting how many longest strictly increasing subsequences exist using dynamic programming.
A clear explanation of counting possible bulb states after pressing four toggle buttons exactly presses times.
Find the minimum number of stickers needed to form a target string using top-down dynamic programming with memoization.
A sliding window solution for finding the maximum average among all contiguous subarrays of fixed length k.
A clear explanation of Snakes and Ladders using breadth-first search over board squares.
Compute the total importance of an employee and all direct and indirect subordinates using a hash map and depth-first search.
A trie-based design for returning the top three historical sentences for a typed prefix.
A clear explanation of generating all strings formed by independently changing each letter to lowercase or uppercase.
A clear explanation of the Find And Replace in String problem using simultaneous replacement, source matching, and a replacement map.
A clear explanation of finding the second minimum value in a special binary tree using DFS.
A clear explanation of inserting a value into a sorted circular linked list while preserving the circular sorted order.
A two-pointer group comparison solution for counting how many words can be stretched to match a target string.
A clear explanation of checking whether one swap in a string can make it equal to another string.
A clear explanation of finding common free time by merging all employee busy intervals and returning the gaps.
Check whether two word arrays are sentence-similar using a hash set of symmetric similar word pairs.
A clear explanation of computing the projection areas of stacked cubes from top, front, and side views.
A clear explanation of solving Number of Recent Calls using a queue as a sliding time window.
A clear explanation of finding the cheapest way to cover all travel days using dynamic programming.
A clear explanation of minimizing an array score after each value can move by at most k.
A clear explanation of solving Tallest Billboard using dynamic programming over height differences.
A clear explanation of finding the minimum difference between any two nodes in a BST using inorder traversal.
A clear explanation of the Flipping an Image problem using row reversal, bit inversion, and an in-place two-pointer method.
Find three non-overlapping subarrays of length k with maximum total sum and return the lexicographically smallest starting indices.
A clear explanation of maximizing an integer by swapping at most two digits once.
An array-based circular buffer solution for implementing a fixed-size double-ended queue.
A string parsing solution for reducing a linear equation into coefficient and constant terms.
A clear explanation of trimming a BST so that all remaining node values lie inside a given inclusive range.
Compute the probability that a knight remains on an n x n chessboard after exactly k random moves using dynamic programming.
A dynamic programming solution for counting decodings of a digit string with wildcard characters.
Find the longest path in a binary tree where every node on the path has the same value using depth-first search.
A SQL guide for filtering movies with odd IDs and non-boring descriptions, then sorting by rating.
A clear explanation of implementing a linked list from scratch using nodes, a dummy head, and a size counter.
A clear explanation of marking matching substrings and merging overlapping bold ranges.
A clear explanation of Mirror Reflection using room unfolding, least common multiples, and parity.
A probability dynamic programming solution for computing whether soup A empties before soup B, with an early return for large input.
A clear explanation of solving Beautiful Array using divide and conquer with odd and even transformations.
Recolor the connected component containing the starting pixel using depth-first search.
A clear explanation of counting ordered triples whose bitwise AND is zero using pairwise AND counts.
A clear explanation of finding the kth smallest value in an m by n multiplication table using binary search on answer.
A DFS and memoization solution for finding the minimum cost to satisfy item needs using individual prices and reusable special offers.
A SQL guide for finding the largest number that appears exactly once in a table.
Find the minimum number of times one string must be repeated so another string becomes a substring.
Find the directed edge to remove so a graph becomes a rooted tree again, handling both cycles and nodes with two parents.
A breadth-first search solution for computing the average value of nodes at each level of a binary tree.
A clear explanation of constructing an array with exactly k distinct adjacent differences using a greedy pattern.
A SQL guide for pivoting rows into columns using ranking and conditional aggregation.
A stack-based solution for computing exclusive execution time from nested start and end logs.
A recursive tree traversal guide for merging two binary trees node by node.
A clear explanation of counting reachable original and subdivided nodes using Dijkstra's shortest path algorithm.
A clear explanation of summing subarray minimums using a monotonic stack and contribution counting.
A clear explanation of transforming a binary board into a chessboard using feasibility checks and minimum row and column swaps.
A clear explanation of the Masking Personal Information problem using string parsing and format-specific masking rules.
Find the extra edge in an undirected graph that creates a cycle using Union-Find.
A clear explanation of computing all root-to-leaf path sums from a compact three-digit binary tree encoding.
A string-marking guide for adding bold tags around all matched words while merging overlapping and adjacent bold regions.
A design solution for storing timestamped logs and retrieving IDs by inclusive time range at a chosen granularity.
A clear explanation of solving interval intersection constraints using greedy sorting and minimal point selection.
A clear explanation of designing a hash map without using built-in hash table libraries.
A clear explanation of hiring exactly k workers with minimum total cost using wage-to-quality ratios, sorting, and a max heap.
A greedy solution for increasing building heights as much as possible while preserving every skyline view.
Track the maximum number of overlapping calendar events using a sweep line difference map.
A clear explanation of designing a time-based key-value store using a hash map and binary search.
A clear explanation of solving Minimum Falling Path Sum using dynamic programming over matrix rows.
Find the earliest day when two turned-on bulbs have exactly k turned-off bulbs between them using a sliding window over bloom days.
A clear explanation of checking whether an array can become non-decreasing by modifying at most one element.
A dynamic programming and combinatorics solution for counting permutations with no fixed positions.
Simulate a baseball scoring system using a stack to process operations and compute the final score.
A clear explanation of minimizing rescue boats using sorting, greedy choice, and two pointers.
A clear explanation of finding the minimum possible number of rabbits using counting and greedy grouping.
A clear explanation of counting super-palindromes by generating palindromic roots and checking their squares.
A clear explanation of the Positions of Large Groups problem using a simple two-pointer scan.
A SQL guide for comparing each department's monthly average salary against the company's monthly average salary.
A two-pointer and number theory solution for checking whether an integer can be written as the sum of two square numbers.
A clear explanation of minimizing printer turns using interval dynamic programming.
A clear explanation of designing a hash set without using built-in hash table libraries.
A clear explanation of solving Pyramid Transition Matrix using backtracking and memoization over pyramid rows.
A clear explanation of scoring a balanced parentheses string using depth counting.
A simple simulation solution for counting how many 100-pixel lines are needed to write a string.
Allow double bookings but reject triple bookings using overlap interval tracking.
Find the next valid 24-hour time using only the digits from the current time.
A clear explanation of solving Binary Subarrays With Sum using prefix sums and a frequency map.
A clear explanation of finding the kth character in a decoded string without building the full decoded string.
A clear explanation of counting all paths from start to end that visit every non-obstacle square exactly once using backtracking.
A SQL guide for finding users who both follow someone and have followers, then counting how many followers they have.
A clear explanation of sorting an array by parity using a two-pointer partition method.
A clear explanation of checking whether a binary tree can be split into two equal-sum trees by removing one edge.
A clear explanation of checking whether one point can reach another by working backward with modulo.
A heap-based solution for finding the smallest range that contains at least one number from each sorted list.
A clear explanation of the Consecutive Numbers Sum problem using arithmetic series formulas and divisibility analysis.
A clear explanation of searching for a target in a sorted array using binary search.
A clear explanation of simulating water droplets over an elevation map by checking left first, then right.
A clear explanation of simulating an exam room by maintaining occupied seats in sorted order.
A dynamic programming solution for deciding whether an array can be split into two non-empty groups with the same average.
Count distinct non-empty palindromic subsequences using interval dynamic programming and duplicate handling.
Check whether a string can become a palindrome after deleting at most one character using two pointers.
A clear explanation of solving Unique Email Addresses using string normalization and a hash set.
A clear explanation of balancing coins in a binary tree using postorder DFS and subtree coin balance.
A clear explanation of counting profitable crime schemes using 0/1 knapsack dynamic programming with members and profit states.
A SQL guide for finding the minimum distance between any two unique points on the X-axis.
A clear explanation of finding the longest contiguous subarray with equal numbers of 0 and 1 using prefix sums and a hash map.
A clear explanation of deleting the minimum number of columns so rows become lexicographically sorted.
A clear explanation of checking whether an array can be reordered into pairs where one number is double the other.
A clear explanation of Fruit Into Baskets using a sliding window with at most two distinct fruit types.
A clear explanation of computing the maximum width of a binary tree using level-order traversal and complete-tree indices.
A clear explanation of finding the kth symbol in the grammar sequence using recursion and the parent-child relationship.
A clear explanation of finding the longest dictionary word obtainable as a subsequence using two pointers and sorting rules.
A SQL guide for finding the minimum Euclidean distance between any two points in a 2D plane.
A design solution for a small Excel-like spreadsheet that supports set, get, and dynamic sum formulas.
A clear digit dynamic programming solution for counting numbers whose binary representation does not contain consecutive ones.
A set-based solution for counting how many different Morse code transformations appear among a list of words.
A clear explanation of finding the minimum number of swaps needed to transform one anagram string into another using BFS.
A clear explanation of reaching a target on a number line using cumulative sums and parity.
Implement a calendar that accepts a booking only when it does not overlap with any existing booking.
A clear explanation of maintaining the kth largest element in a stream using a fixed-size min heap.
Determine whether four numbers can be combined with arithmetic operations and parentheses to produce 24.
A clear explanation of solving Minimize Malware Spread II by removing each infected node and simulating the final malware spread.
A clear explanation of finding the longest subarray whose adjacent comparisons alternate between greater-than and less-than.
A clear explanation of finding the nth magical number using binary search, greatest common divisor, least common multiple, and inclusion-exclusion.
A clear explanation of Count Unique Characters of All Substrings using contribution counting with previous and next occurrences.
A clear explanation of checking whether words are sorted according to a custom alien alphabet order.
A clear explanation of counting valid DI permutations using dynamic programming and prefix sums.
A clear explanation of averaging neighboring pixels in a matrix using direct simulation.
A clear explanation of finding the nth positive integer that does not contain the digit 9 using base-9 conversion.
A clear explanation of Distribute Candies using a set to count candy types and a simple limit argument.
A clear explanation of detecting a subarray whose sum is a multiple of k using prefix sums and modular arithmetic.
A clear explanation of finding the minimum time to reach the bottom-right cell using a priority queue and minimax path reasoning.
A two-pointer guide for counting triplets that can form valid triangles after sorting the side lengths.
A clear explanation of deciding whether a sorted array can be split into consecutive subsequences of length at least three.
A SQL guide for checking whether three side lengths can form a valid triangle using the triangle inequality.
A clear explanation of calculating the fraction of players who logged in again the day after their first login.
A clear hash map solution for finding common strings with the smallest index sum.
A clear explanation of finding the k closest elements to a target using binary search and a sliding window.
A hash map guide for grouping file paths by identical file content and returning only duplicate groups.
A clear explanation of finding the longest increasing or decreasing consecutive path in a binary tree using DFS.
A SQL guide for classifying binary tree nodes as Root, Inner, or Leaf based on parent-child relationships.
A clear explanation of determining whether a robot returns to the origin after executing movement instructions.
A clear math solution for counting the maximum values after repeated top-left matrix increment operations.
A clear explanation of Winning Candidate using SQL aggregation to count votes and return the candidate with the most votes.
A clear explanation of splitting an array into four equal-sum parts using prefix sums and set-based search.
A clear explanation of finding the longest uncommon subsequence among many strings using subsequence checks.
A clear explanation of finding the longest uncommon subsequence between two strings using simple case analysis.
A SQL guide for finding salespeople who never had an order related to the company named RED.
A greedy heap solution for taking the maximum number of courses before their deadlines.
A clear explanation of finding the minimum-cost path with bounded jumps, blocked cells, and lexicographic tie-breaking.
A recursive guide for converting a binary tree into a preorder parenthesized string while preserving the one-to-one mapping between the tree and the string.
A clear explanation of counting connected components in an undirected graph represented by an adjacency matrix.
A clear SQL guide for computing the overall friend request acceptance rate with duplicate pairs counted once.
A clear explanation of Squirrel Simulation using Manhattan distance and the special first trip.
A clear explanation of checking whether a word uses capital letters correctly by counting uppercase letters.
A clear explanation of Subtree of Another Tree using recursive tree matching and DFS.
A clear explanation of randomly flipping zero cells in a matrix without repetition using hash mapping and virtual swapping.
A clear SQL guide for finding classes that have at least five students.
A clear explanation of Find Median Given Frequency of Numbers using cumulative frequency and SQL window functions.
A clear explanation of counting coin-change combinations using dynamic programming.
A clear explanation of maximizing remove-box scores using interval dynamic programming with memoization.
A clear explanation of Managers with at Least 5 Direct Reports using grouping and a self join.
A clear explanation of collecting the boundary of a binary tree using separate left boundary, leaves, and right boundary traversals.
A clear SQL guide for finding countries with either large area or large population.
A clear explanation of Median Employee Salary using SQL window functions to rank employees inside each company.
A clear explanation of balancing dresses across washing machines using greedy prefix flow.
A clear explanation of building the final tournament bracket by repeatedly pairing strongest and weakest teams.
A clear hash map solution for finding the longest subsequence whose maximum and minimum differ by exactly one.
A clear explanation of finding the length of the longest palindromic subsequence using interval dynamic programming.
A clear explanation of finding the longest path between any two nodes in a binary tree using DFS height computation.
A clear explanation of Maximum Vacation Days using dynamic programming over weeks and cities.
A clear explanation of finding the maximum value at every depth of a binary tree using level-order traversal.
A clear geometry solution for checking whether four unordered points form a valid square.
A clear explanation of Permutation in String using a fixed-size sliding window and character frequency counts.
A clear explanation of computing the distance to the nearest zero in a binary matrix using multi-source BFS.
A clear explanation of finding the minimum steps to spell a key on a circular ring using dynamic programming and memoized DFS.
A clear parsing and math solution for evaluating fraction addition and subtraction expressions.
A clear stack-based parser for validating nested XML-like tags with CDATA sections.
A clear explanation of Reshape the Matrix using index mapping from the original matrix to the reshaped matrix.
A clear explanation of finding the leftmost value in the deepest row of a binary tree using level-order traversal.
A clear explanation of finding the first device used by each player using SQL aggregation and a join.
A clear explanation of finding each player's first login date using SQL aggregation.
A clear DFS solution for returning the postorder traversal of an N-ary tree.
A clear explanation of Array Nesting using cycle detection over a permutation.
A clear explanation of reversing the first k characters in every 2k block of a string.
A clear explanation of finding the inorder successor in a binary search tree when nodes contain parent pointers.
A clear explanation of computing Fibonacci numbers using dynamic programming and iterative state transitions.
A clear explanation of finding the most frequent subtree sum in a binary tree using postorder DFS and a frequency map.
A dynamic programming solution for counting permutations of 1 to n with exactly k inverse pairs.
A clear explanation of formatting a binary tree into a 2D string matrix using tree height and recursive placement.
A greedy guide for determining whether a given number of flowers can be planted without violating the no-adjacent-flowers rule.
A clear DFS solution for returning the preorder traversal of an N-ary tree.
A clear explanation of Find the Closest Palindrome using prefix mirroring and a small candidate set.
A clear explanation of finding the only non-duplicate element in a sorted array using binary search.
A clear explanation of checking whether a number equals the sum of its positive divisors excluding itself.
A clear explanation of counting car fleets by sorting cars by position and tracking arrival times.
A reverse simulation and union-find solution for counting how many bricks fall after each hit.
A clear explanation of Cracking the Safe using a de Bruijn sequence and depth-first search over password states.
Check each number in a range by extracting its digits and testing whether every digit divides the original number.
A clear explanation of searching in a sorted array when the array length is hidden behind an ArrayReader interface.
A clear explanation of assigning athlete ranks from scores using sorting while preserving original indices.
Check whether a string containing parentheses and wildcard stars can be made valid using a greedy range of possible open counts.
A clear explanation of finding the minimum difference between 24-hour clock times using minute conversion and sorting.
A guide to implementing a lazy iterator over a run-length encoded string without fully decompressing it.
A clear explanation of Binary Tree Tilt using postorder DFS to compute subtree sums and accumulate tilt.
A clear explanation of constructing a maximum binary tree recursively using divide and conquer.
A clear design guide for implementing an in-memory file system with directory listing, directory creation, file append, and file read operations.
A clear explanation of finding the largest product of three numbers using sorting or constant-space tracking.
A clear explanation of finding the shortest rolling distance in a maze using Dijkstra’s algorithm.
Design a map that supports key-value insertion and prefix-sum queries using a hash map and trie.
A clear explanation of Longest Line of Consecutive One in Matrix using dynamic programming over four directions.
A SQL guide for finding all cinema seats that are free and adjacent to at least one other free seat.
A clear explanation of finding whether two different nodes in a binary search tree sum to a target value.
A clear convex hull solution for returning all trees that lie on the fence boundary.
A SQL update solution for swapping all m and f values in the Salary table using a single statement.
A clear explanation of converting a BST into a greater tree using reverse inorder traversal and a running sum.
A clear explanation of converting an integer into its base 7 string representation using repeated division.
A clear explanation of finding the next greater element in a circular array using a monotonic stack.
A clear explanation of solving Three Equal Parts by counting ones, locating the three binary patterns, and comparing them in one pass.
A clear explanation of sorting squared values from a sorted array using two pointers.
A clear explanation of solving Largest Component Size by Common Factor using prime factorization and union find.
A clear explanation of the Stone Game problem using game theory and interval dynamic programming.
A clear explanation of counting numbers less than or equal to N using digit-by-digit construction and combinatorics.
A clear explanation of finding the peak index in a mountain array using binary search.
A clear explanation of the Making A Large Island problem using connected component labeling and island size lookup.
A graph traversal solution for finding all nodes that cannot reach a directed cycle.
A clear explanation of validating string transformation using two pointers and movement constraints.
A clear explanation of solving Open the Lock using breadth-first search over lock states.
Find the shortest substring of s1 that contains s2 as a subsequence using dynamic programming.
A clear explanation of finding the largest valid triangle perimeter using sorting and a greedy scan.
A clear explanation of checking whether two binary trees are equivalent after swapping left and right children at any number of nodes.
A clear explanation of solving Flip String to Monotone Increasing with a one-pass dynamic programming approach.
A clear explanation of Online Stock Span using a monotonic decreasing stack with accumulated spans.
A clear explanation of finding the middle node of a singly linked list using slow and fast pointers.
A clear explanation of Loud and Rich using graph traversal, DFS, and memoization.
A clear explanation of the Most Profit Assigning Work problem using sorting, greedy choice, and two pointers.
A dynamic programming solution for finding the minimum number of same-index swaps needed to make two arrays strictly increasing.
A clear explanation of splitting a binary search tree into two BSTs using recursion and pointer rewiring.
A clear explanation of converting a range of IPv4 addresses into the shortest list of CIDR blocks using greedy bit manipulation.
Parse a chemical formula with nested parentheses, atom names, and multipliers using recursive descent.
A clear explanation of inserting a value into a binary search tree using recursive and iterative traversal.
Design a dictionary that can check whether a word can match a stored word after changing exactly one character.
A clear explanation of finding duplicate binary tree subtrees using postorder traversal, serialization, and a hash map.
A SQL guide for counting friendships from both requester and accepter sides, then returning the user with the most friends.
A clear explanation of maximizing capital by selecting at most k projects using sorting and a max heap.
A clear explanation of multiplying complex numbers represented as strings using algebraic expansion.
A clear SQL guide for finding the customer who placed the most orders.
A clear explanation of Array Partition using sorting and adjacent pairing to maximize the sum of pair minimums.
A dynamic programming solution for maximizing the number of A characters printed with a limited number of keyboard operations.
A SQL solution for swapping every pair of adjacent student seats while leaving the final seat unchanged when the row count is odd.
A SQL guide for finding stadium records that belong to runs of at least three consecutive ids where each row has at least 100 people.
A clear explanation of finding the most frequent value or values in a binary search tree using inorder traversal.
A clear explanation of parsing a parenthesized string recursively to construct a binary tree.
A clear SQL guide for summing 2016 investments for policies with repeated 2015 investment values and unique locations.
A clear explanation of Subarray Sum Equals K using prefix sums and a hash map to count matching subarrays in linear time.
A clear explanation of designing a simple URL encoder and decoder using a hash map and generated keys.
A clear explanation of computing cumulative games played per player and date using SQL window functions.
A clear SQL guide for selecting customers who were not referred by customer 2, including customers with no referee.
A clear explanation of Maximum Depth of N-ary Tree using recursive depth-first search.
A clear dynamic programming solution for finding the minimum deletions needed to make two strings equal.
A clear explanation of merging two quad-trees using recursive logical OR operations.
A clear explanation of counting black lonely pixels using row counts, column counts, and duplicate row patterns.
A clear explanation of Reverse Words in a String III using two-pointer scanning and string reversal.
A clear explanation of counting unique pairs whose absolute difference is k using frequency counting.
A clear graph traversal solution for finding all processes terminated when killing a target process.
A clear linear-time solution for finding the shortest subarray that must be sorted to make the whole array sorted.
A clear explanation of Next Greater Element III using the next permutation algorithm on the digits of an integer.
A clear explanation of counting black pixels that are alone in both their row and column.
A clear explanation of Split Concatenated Strings using string reversal choices and enumeration of every possible cut point.
A clear SQL guide for counting students in every department, including departments with zero students.
A clear explanation of finding the minimum difference between two BST node values using inorder traversal.
A clear SQL guide for computing each employee's 3-month cumulative salary while excluding their most recent month.
A clear explanation of Brick Wall using prefix sums and a hash map to find the best vertical cut position.
A clear explanation of updating a Minesweeper board using DFS flood fill and adjacent mine counting.
A clear explanation of Optimal Division using the structure of division expressions to build the maximum-value expression.
A clear SQL guide for finding the question with the highest answer rate from survey logs.
A clear explanation of weighted random sampling using prefix sums and binary search.
A clear SQL guide for finding employees whose bonus is less than 1000 or missing.
A clear explanation of Student Attendance Record II using dynamic programming over absence count and late streak.
A clear explanation of generating minimal unique word abbreviations using grouping and trie prefixes.
A clear dynamic programming solution for counting paths that move a ball out of a grid boundary.
A clear explanation of Student Attendance Record I using simple string checks and a one-pass counter solution.
A clear explanation of counting beautiful arrangements using backtracking and divisibility pruning.
Arithmetic geometry studies solutions of polynomial equations by combining algebra, geometry, and number theory. Its basic objects are spaces defined by polynomial equations....
A field is a number system in which addition, subtraction, multiplication, and division by nonzero elements are always possible. The rational numbers $\mathbb{Q}$, the real...
The harmonic series is the infinite series
A Diophantine equation is an equation whose solutions are required to be integers. The unknowns are not allowed to range over the real numbers or complex numbers unless...
Division of integers does not always produce an integer. For example,
The idea of number arose long before formal mathematics. Early civilizations used numbers for counting objects, measuring land, recording trade, and tracking time.
The integers extend infinitely in both directions:
Many mathematical objects are defined recursively. A recursive definition specifies:
Ordinary induction proves a statement $Pn$ by showing that truth passes from one case to the next:
Many statements in number theory concern all natural numbers. For example, one may wish to prove that
The order relation distinguishes positive and negative integers, but in many situations the sign of a number is less important than its magnitude. For example, the integers
The integers are not merely a collection of numbers equipped with arithmetic operations. They also possess an order structure. Given two integers $a$ and $b$, one can...
An arithmetic operation is a rule that combines numbers to produce another number. The most basic operations on integers are addition, subtraction, multiplication, and division.
The natural numbers are sufficient for counting and addition, but they are not sufficient for subtraction. For example,
The natural numbers arise from the basic act of counting. When we count objects in a collection, we assign successive numbers:
Automatic differentiation works naturally on pure mathematical functions:
Automatic differentiation works naturally on pure mathematical functions:
Automatic differentiation works naturally on pure mathematical functions:
A minimal automatic differentiation engine can compute correct gradients on small programs. A production system must survive long-running workloads, large tensors, distributed...
Automatic differentiation began as a numerical technique for computing gradients of scalar functions.
Sparse and structured differentiation studies how to compute derivatives without materializing dense derivative objects. Many real systems have enormous Jacobians and...
Physics-informed models combine data fitting with equations from physics or applied mathematics. The model is trained not only to match observed samples, but also to satisfy...
Automatic differentiation is easiest to define for pure functions. A pure function behaves like a mathematical mapping: it consumes inputs, produces outputs, and has no...
Automatic differentiation can be performed before a program runs, while it runs, or in a staged phase between the two.
This section studies reverse mode automatic differentiation through concrete examples. Each case has the same structure:
Forward mode automatic differentiation computes derivatives by propagating tangent values alongside ordinary values. The ordinary value is called the primal. The derivative...
Differentiation describes how a function changes locally. A Taylor expansion extends this idea by approximating a function with a polynomial around a point.
The systems in this chapter show that automatic differentiation is not one implementation technique. It is a family of program transformations. Each system chooses a different...
The systems in this chapter show that automatic differentiation is not one implementation technique. It is a family of program transformations. Each system chooses a different...
An automatic differentiation engine is only useful if its derivatives are correct. A small mistake in a backward rule can silently corrupt optimization, training, or...
The preceding sections described automatic differentiation through algebraic, categorical, logical, and denotational models. These viewpoints converge on one central idea:
Automatic differentiation systems are usually trusted because they implement mathematically established rules such as the chain rule, product rule, and linearization of...
Distributed gradient computation appears when a differentiable program no longer fits comfortably on one device or one machine. The reason may be model size, data volume,...
Differentiable systems architecture extends automatic differentiation beyond isolated functions and neural network layers. The central idea is to treat larger systems as...
Probabilistic programming represents uncertainty using executable probabilistic models. A probabilistic program defines a distribution rather than only a deterministic computation.
Reinforcement learning studies learning systems that act in an environment. Unlike supervised learning, the training signal is not a target label for each input. The model...
Automatic differentiation interacts deeply with type systems because differentiation changes the structure of computation. A derivative operator maps one function into another...
Kernel fusion combines several small operations into one larger executable unit.
Modern automatic differentiation systems are fundamentally tensor compiler systems. Their performance depends less on mathematical differentiation rules than on how...
Higher-order automatic differentiation faces a fundamental problem: derivative structure grows combinatorially with order.
Automatic differentiation is deeply connected to functional programming and lambda calculus. Programs can be viewed as mathematical functions, and differentiation can be...
Reverse mode automatic differentiation is the mathematical and systems basis of backpropagation. In deep learning, the objective is usually a scalar loss depending on many...
Forward mode automatic differentiation appears in many numerical systems where directional derivatives, local sensitivities, or small parameter sets are important. This...
Automatic differentiation is a method for computing derivatives by transforming programs into derivative-propagating computations. It does not approximate derivatives...
Tinygrad is a small deep learning framework centered around a minimal reverse-mode automatic differentiation engine. It was created by entity"people","George...
Tinygrad is a small deep learning framework centered around a minimal reverse-mode automatic differentiation engine. It was created by entity"people","George...
Performance benchmarking measures whether an automatic differentiation engine is fast, memory-efficient, and scalable under realistic workloads. It also protects the engine...
Operational semantics explains how automatic differentiation executes. Denotational semantics explains what differentiable programs mean.
Automatic differentiation began as a transformation applied to numerical programs. A differentiable programming language instead treats differentiation as a native semantic...
Modern automatic differentiation systems are built around accelerator hardware. GPUs and TPUs provide enormous throughput for tensor operations, making large-scale...
A hybrid symbolic-numeric system combines discrete symbolic reasoning with continuous numerical computation. In the context of automatic differentiation, it means a pipeline...
Robotics and control systems interact with the physical world through sensing, estimation, planning, and actuation. Automatic differentiation is important because modern...
Meta-learning studies systems that improve how they learn. Instead of only optimizing model parameters for one task, a meta-learning method optimizes some part of the learning...
Swift became an important experiment in language-integrated automatic differentiation because it attempted to make differentiation a core compiler feature rather than a...
Memory planning determines where values are stored, how long they remain alive, and when storage can be reused.
Most real computational problems are sparse. Large matrices and tensors often contain mostly zeros, structured blocks, or local interactions. Sparse representations reduce...
Programs do not only branch between valid computations. They also fail, stop early, raise exceptions, return sentinel values, or enter undefined numerical regions. These...
Perturbation confusion is a correctness bug that appears in nested automatic differentiation, especially nested forward mode. It happens when two derivative computations...
Automatic differentiation is deeply connected to functional programming and lambda calculus. Programs can be viewed as mathematical functions, and differentiation can be...
Checkpointing is a technique for reducing the memory cost of reverse mode automatic differentiation by selectively storing intermediate states and recomputing missing values...
Many real-world Jacobians are sparse. Most derivative entries are zero because outputs depend only on small subsets of inputs.
Automatic differentiation can be understood as a transformation from one program into another program.
A differentiable subprogram is a program fragment that can participate in derivative propagation as a coherent unit. Instead of differentiating an entire application...
Enzyme is a compiler-based automatic differentiation system for LLVM and MLIR. Instead of differentiating source code directly, or recording tensor operations at runtime,...
Enzyme is a compiler-based automatic differentiation system for LLVM and MLIR. Instead of differentiating source code directly, or recording tensor operations at runtime,...
A custom gradient gives the user direct control over the backward rule of an operation. The forward computation still produces an ordinary value, but the derivative no longer...
Automatic differentiation systems are trusted infrastructure. Scientific computing, machine learning, optimization, simulation, and control systems depend on gradients being...
Quantum computation introduces a computational model fundamentally different from classical programs.
Automatic differentiation is usually described as a transformation of programs or computational graphs. In real systems, it is also a parallel execution problem. Large...
A differentiable operating system is an execution environment whose resource-management decisions can be optimized using gradients or gradient-like feedback. Instead of...
Signal processing studies how information is represented, transformed, filtered, compressed, reconstructed, and estimated from signals. A signal may be a time series, an...
An implicit layer defines its output as the solution of an equation, not as a fixed sequence of explicit operations. Instead of computing
Julia was designed for high-performance technical computing. It combines interactive syntax with a compiler capable of specializing code aggressively based on types. This...
Staging is the separation of a program into phases.
The singular value decomposition SVD is one of the most important matrix factorizations in numerical linear algebra. It appears in dimensionality reduction, least squares,...
A stateful system is a program whose output depends not only on its explicit inputs, but also on stored state. The state may live in variables, objects, arrays, files, random...
Higher-order derivatives contain rich geometric information, but naïve computation quickly becomes impractical.
Automatic differentiation can be described operationally through dual numbers and computational graphs. It can also be described abstractly using category theory.
Reverse mode automatic differentiation is computationally efficient for scalar-output functions, but it has a major systems cost: it needs information from the forward pass...
Forward mode automatic differentiation computes Jacobian-vector products:
Automatic differentiation computes derivatives exactly with respect to the executed floating point program. This distinguishes AD from numerical differentiation, which...
A pure computation is easier to differentiate because every output is determined only by its explicit inputs. There is no hidden state, no external mutation, and no dependence...
Forward mode and reverse mode propagate different kinds of objects.
Automatic differentiation became important because derivatives are required everywhere numerical models are optimized, controlled, calibrated, or analyzed. Once a system can...
Zygote is a source-to-source reverse-mode automatic differentiation system for the Julia programming language. It was designed to differentiate high-level Julia code directly,...
Zygote is a source-to-source reverse-mode automatic differentiation system for the Julia programming language. It was designed to differentiate high-level Julia code directly,...
An automatic differentiation engine becomes useful only after it supports a sufficiently rich set of primitive operations. The collection of these primitives is the operator...
Automatic differentiation transforms programs. A fundamental semantic question therefore arises:
Classical automatic differentiation computes derivatives of deterministic programs.
Automatic differentiation systems are often assumed to be deterministic. Given identical inputs, identical parameters, and identical code, many users expect identical...
A differentiable compiler is a compilation system that supports gradient propagation through compilation decisions, generated programs, or execution behavior. Instead of...
Computational finance uses numerical models to price contracts, measure risk, and optimize portfolios. Automatic differentiation is useful because most financial computations...
Attention is a sequence operation that lets each position read information from other positions. Instead of compressing the whole past into one recurrent hidden state,...
Rust is an attractive language for automatic differentiation because it combines low-level performance with strong static guarantees. It gives the programmer control over...
Tracing is an implementation strategy where an AD system observes a program while it runs and records the operations that occur.
Eigenvalue problems are fundamental in numerical analysis, optimization, physics, graph methods, control theory, and machine learning. They are also among the most subtle...
A non-smooth program contains operations where the derivative is undefined, discontinuous, set-valued, or unstable under small perturbations. These programs arise naturally in...
Taylor mode automatic differentiation computes derivatives by propagating truncated Taylor series through a program.
Dual numbers and hyper-dual numbers are special cases of a broader algebraic structure called a differential algebra. This framework abstracts differentiation away from...
A Wengert list is a linear representation of a computation in which every intermediate result is assigned to a unique variable. It is one of the earliest and most influential...
So far, forward mode has propagated a single tangent direction:
Automatic differentiation is fundamentally a computational technique. Its practical importance comes from the fact that derivatives can often be computed with asymptotic cost...
Automatic differentiation operates on computations, but computations execute inside a memory model. Variables occupy storage locations, arrays are mutated, buffers are reused,...
Linearization is the operation of replacing a nonlinear function by its best local linear approximation at a chosen point. Automatic differentiation can be understood as a...
Automatic differentiation developed from a simple observation: a numerical computation already contains the structure needed to compute its derivative. The program evaluates...
JAX is an automatic differentiation and array programming system for Python. It combines NumPy-like syntax with composable program transformations. Its core transformations...
JAX is an automatic differentiation and array programming system for Python. It combines NumPy-like syntax with composable program transformations. Its core transformations...
Memory management is the main systems problem in reverse mode automatic differentiation. The derivative rules are usually small. The hard part is deciding which primal values,...
Automatic differentiation becomes substantially more difficult once programs contain higher-order functions.
Classical neural networks apply a finite sequence of transformations:
Gradient-based optimization relies on propagating derivative information through many layers, time steps, or computational transformations. In deep systems, these gradients...
Differentiable search and retrieval systems integrate information access into gradient-based learning. Instead of treating retrieval as an external symbolic operation, the...
Molecular simulation models the behavior of atoms and molecules using physical interaction laws. Automatic differentiation is important because many molecular methods require...
Sequence models process ordered data. The input is not one independent vector, but a series:
Python became the dominant language for modern machine learning and differentiable computing because it combines a simple programming model with access to high-performance...
A graph intermediate representation models a program as nodes and edges.
Matrix factorizations rewrite a matrix into structured factors. They are used because the factors make later computations cheaper, more stable, or easier to interpret. In...
A piecewise differentiable function is built from several differentiable pieces joined by boundaries. Each piece has an ordinary derivative inside its region. At the...
Nested automatic differentiation means applying automatic differentiation inside another automatic differentiation computation.
Dual numbers compute first derivatives exactly. Truncated polynomial algebras extend this to higher-order derivatives, but practical higher-order differentiation introduces an...
Most reverse mode automatic differentiation systems require a mechanism for recording the forward computation so that the reverse pass can later traverse it backward. This...
Forward mode automatic differentiation has a simple cost model. It evaluates the original program and, at the same time, evaluates the tangent program. Each primitive...
Mixed-mode differentiation combines forward accumulation and reverse accumulation in the same derivative computation. It is used when neither pure forward mode nor pure...
Loops express repeated computation. Recurrence relations express the same idea mathematically: each state is computed from one or more earlier states.
A computational graph represents a calculation as nodes and edges. Nodes represent operations or values. Edges represent data dependencies. Automatic differentiation uses this...
Derivative computation is not only a mathematical problem. It is also a numerical and systems problem. A derivative method must answer three questions simultaneously:
PyTorch Autograd is a dynamic reverse-mode automatic differentiation system. It records tensor operations as they execute, builds a computation graph at runtime, and then...
PyTorch Autograd is a dynamic reverse-mode automatic differentiation system. It records tensor operations as they execute, builds a computation graph at runtime, and then...
A tape is an append-only record of the operations executed during the forward pass. Reverse mode uses the tape to replay derivative rules backward.
Cartesian differential categories model differentiation in categories with products. Differential categories generalize this idea further by shifting attention from cartesian...
Many systems evolve continuously over time rather than through discrete layers. A state variable changes according to a differential equation:
Reverse-mode automatic differentiation trades computation for memory. To compute gradients efficiently, the backward pass requires access to intermediate values produced...
A differentiable physics engine computes gradients of physical simulation outputs with respect to inputs, parameters, or control signals. Instead of treating simulation as a...
Computational fluid dynamics studies fluid motion by solving discretized forms of the governing equations. Automatic differentiation enters CFD when we want gradients of...
Neural network training is the repeated application of three operations: evaluate a model, differentiate a scalar loss, and update parameters. Automatic differentiation...
C and C++ are important targets for automatic differentiation because much scientific, engineering, graphics, finance, and machine learning infrastructure is written in these...
Static single assignment form, or SSA, is an intermediate representation where each variable is assigned exactly once.
Linear algebra primitives are tensor operations with algebraic structure: matrix multiplication, triangular solves, factorizations, inverses, determinants, norms, and spectral...
A dynamic graph is a computation graph built while the program runs. Its structure depends on ordinary runtime values: branches, loop counts, recursive calls, tensor shapes,...
Reverse mode is efficient for scalar-output functions because it propagates one adjoint backward through the computation and produces a full gradient. For
Dual numbers capture first-order derivatives because the infinitesimal element satisfies
Reverse accumulation is the operational core of reverse mode automatic differentiation. The forward pass evaluates a program and records dependency information. The reverse...
The natural output of forward mode automatic differentiation is a Jacobian-vector product. Instead of constructing the full Jacobian matrix explicitly, forward mode computes...
Reverse accumulation is the reverse-mode form of automatic differentiation. It propagates derivative information backward from outputs to inputs.
Control flow determines which operations a program executes. Straight-line programs have a fixed sequence of operations, but ordinary programs contain branches, loops,...
The chain rule is the central theorem behind automatic differentiation. Every useful AD algorithm is a disciplined way of applying the chain rule to a program.
Automatic differentiation computes derivatives by applying the chain rule to the operations of a program. The input is ordinary code that computes a value. The output is code,...
TensorFlow Autograd refers to TensorFlow’s automatic differentiation system, mainly exposed through tf.GradientTape. It is a reverse-mode AD system designed for tensor...
TensorFlow Autograd refers to TensorFlow’s automatic differentiation system, mainly exposed through tf.GradientTape. It is a reverse-mode AD system designed for tensor...
A graph representation makes the structure of a differentiated computation explicit. In reverse mode, this structure is required because the backward pass must know which...
Algebraic semantics describes differentiation through derivations, tangent maps, and linear structure. Categorical semantics goes further. It studies differentiation as a...
An optimization layer is a program component whose output is the solution of an optimization problem. Instead of computing
Floating point systems represent numbers within a finite range. When a computed value exceeds the largest representable magnitude, overflow occurs. When a value becomes too...
Differentiable rendering is the process of computing derivatives of rendered images with respect to scene parameters. A renderer becomes part of the computational graph rather...
An inverse problem asks for causes from effects. A forward model predicts observations from parameters. An inverse model tries to recover parameters from observations.
Backpropagation is reverse mode automatic differentiation applied to neural networks. In most machine learning writing, the term refers to the whole training procedure: run a...
Differentiable programming treats differentiation as a general programming-language feature. A program can contain numerical kernels, control flow, data structures, solvers,...
An intermediate representation, or IR, is the internal program form used by a compiler or AD system after parsing and before final code generation.
Broadcasting is the rule system that allows tensor operations between arrays of different shapes without explicitly materializing expanded copies. It is one of the most...
Recursion is control flow where a function calls itself. In automatic differentiation, recursion behaves like a loop with a call stack. Each recursive call contributes one...
A Hessian-vector product computes
Dual numbers provide an algebraic mechanism for differentiation, but they also have a precise geometric meaning. A dual number represents a point together with an...
Reverse mode automatic differentiation fundamentally computes vector-Jacobian products. The gradient of a scalar function is a special case of this more general operation.
Forward mode automatic differentiation works by replacing each primitive operation with an extended operation on pairs:
Forward accumulation is the forward-mode form of automatic differentiation. It propagates derivative information in the same order as ordinary program evaluation. Each...
A dependency graph describes how values in a computation depend on earlier values. Automatic differentiation operates on these dependencies.
The gradient is enough when a function has many inputs and one scalar output. More general programs need more general derivative objects. Two of the most important are the...
Symbolic differentiation computes derivatives by manipulating expressions. The input is a formula. The output is another formula.
Tapenade is a source-transformation automatic differentiation system developed at INRIA. Like ADIFOR, it takes an existing program and produces a new differentiated program....
Tapenade is a source-transformation automatic differentiation system developed at INRIA. Like ADIFOR, it takes an existing program and produces a new differentiated program....
Reverse mode automatic differentiation computes derivatives by traversing the program backward after evaluation. Unlike forward mode, which propagates tangents alongside...
Automatic differentiation is often introduced operationally. A program executes elementary operations, and derivative information propagates alongside the computation. This...
A solver is a program that computes a value by search, iteration, or factorization. Instead of evaluating a closed-form expression, it finds a value that satisfies a condition.
Reverse mode automatic differentiation computes gradients by propagating adjoint values backward through a computational graph. In exact arithmetic, the reverse accumulation...
A differentiable database is a data system whose operations participate in gradient-based optimization. Instead of treating storage and querying as external infrastructure,...
Sensitivity analysis studies how changes in inputs affect the outputs of a system. In differential equations, optimization, simulation, and machine learning, the main object...
Stochastic optimization studies optimization when the objective is accessed through samples, noisy estimates, or partial observations. In machine learning, this is the normal...
Functional programming languages provide a natural semantic foundation for automatic differentiation. Programs are expressed as compositions of functions, immutable values,...
Operator overloading implements automatic differentiation by changing the meaning of ordinary arithmetic operations for special numeric objects.
Tensor operations generalize scalar, vector, and matrix operations to arrays with arbitrary rank. In automatic differentiation, a tensor is usually treated as a typed array...
A loop repeats a computation until a condition fails or a fixed iteration count is reached. In automatic differentiation, loops are important because many numerical algorithms...
For a scalar function
The defining feature of dual numbers is the existence of a nonzero element whose square vanishes:
Reverse mode automatic differentiation operates on a computational graph. The forward pass evaluates the graph from inputs to outputs. The reverse pass traverses the same...
Dual numbers give forward mode automatic differentiation a compact algebraic form. Instead of storing a value and a tangent as two unrelated fields, we package them into one...
Automatic differentiation reduces differentiation to a finite collection of elementary operations. Every program, regardless of complexity, is decomposed into primitive...
Intermediate variables are the named values created between program inputs and program outputs. They make automatic differentiation mechanical.
Automatic differentiation is usually applied to functions with many inputs and many outputs. The calculus needed for this setting is multivariate calculus: the study of how a...
Numerical differentiation estimates derivatives by evaluating a function at nearby input values. It treats the function as a black box. The method does not need access to the...
ADIFOR, short for Automatic Differentiation of Fortran, is one of the classical source-transformation systems for automatic differentiation. It was designed for numerical...
ADIFOR, short for Automatic Differentiation of Fortran, is one of the classical source-transformation systems for automatic differentiation. It was designed for numerical...
A minimal forward mode automatic differentiation engine has one job: evaluate a program while carrying both a value and its derivative. The engine does not build a graph. It...
Automatic differentiation is often described by a simple rule:
Many programs do not compute their output by applying a fixed sequence of explicit operations. Instead, they define the output as the solution of another problem.
Automatic differentiation computes derivatives by executing arithmetic. On a real machine, arithmetic uses finite precision. This means AD gives the derivative of the...
An end-to-end differentiable pipeline is a system whose final objective can send derivative information backward through every trainable or tunable stage of computation....
Differential equations are one of the main reasons automatic differentiation matters in scientific computing. Many scientific models are not written as closed-form functions....
Gradient descent is the basic optimization procedure behind much of modern machine learning. It is simple enough to state in one line, but rich enough to expose many of the...
Lisp is one of the natural homes of automatic differentiation. It treats programs as data, has a simple expression syntax, and supports macro systems that can transform code...
Source transformation is an implementation strategy for automatic differentiation in which a program that computes a function is rewritten into another program that computes...
Matrix calculus is the notation and rule system used to differentiate functions whose inputs, outputs, or intermediate values are vectors, matrices, or tensors. Automatic...
A conditional is a program construct that chooses one computation among several possible computations. In ordinary code, this is written as if, else, switch, case, pattern...
First derivatives describe local rate of change. Second derivatives describe how that rate of change itself changes. In optimization, this is curvature. In dynamics, it is...
Dual numbers give the cleanest algebraic model of forward mode automatic differentiation. They extend ordinary real numbers with a formal infinitesimal part. Instead of...
Reverse mode automatic differentiation computes derivatives by propagating sensitivities backward through a computation. In forward mode, each intermediate value carries a...
Forward mode automatic differentiation computes derivatives by carrying two values through a program at the same time: the ordinary value and its tangent. The ordinary value...
Automatic differentiation is built on a simple observation: a complicated derivative can be computed by composing many small local derivatives. Instead of manipulating a full...
A straight-line program is the simplest model of computation used in automatic differentiation. It is a program with a fixed sequence of assignments, no branches, no loops,...
Automatic differentiation begins with a simple object: a function.
A derivative measures how an output changes when an input changes. That sentence is simple, but it is one of the main ideas behind numerical computing, optimization, machine...
A clear explanation of building all word squares using backtracking with prefix pruning.
A clear explanation of finding the minimum heater radius by sorting positions and matching each house to its nearest heater.
A clear explanation of finding the longest substring that can become all one letter using a sliding window.
A clear explanation of solving the largest subset problem as a two-dimensional 0/1 knapsack over zero and one counts.
A clear explanation of reconstructing digits from shuffled English words using character frequency counts and unique identifying letters.
A clear explanation of deciding whether matchsticks can form a square using backtracking, sorting, and pruning.
A clear explanation of filtering words that can be typed using only one row of an American keyboard.
A clear explanation of finding all words that can be formed by concatenating at least two shorter words from the same list.
Delete a node from a binary search tree while preserving the BST property using recursive search and inorder successor replacement.
A clear explanation of finding the nth digit in the infinite integer sequence using digit groups and arithmetic.
A clear explanation of solving division equations using graph traversal and weighted edges.
A clear explanation of finding the minimum guaranteed cost using interval dynamic programming.
A clear explanation of finding the shortest rolling-ball path to the hole using Dijkstra with lexicographic tie-breaking.
A clear explanation of interval dynamic programming for encoding a string into the shortest k[encoded_string] form.
Serialize a binary search tree compactly with preorder traversal and rebuild it using BST value bounds.
A clear explanation of checking whether rows and columns read the same using direct index comparison.
A clear explanation of picking a uniformly random index for a target value using reservoir sampling, with an alternative hash map approach.
A clear explanation of finding the picked number using binary search and the guess API.
A clear explanation of returning matrix elements in diagonal zigzag order by grouping cells with the same row plus column index.
A clear explanation of finding the k smallest pair sums from two sorted arrays using a min heap and best-first search.
A clear explanation of generating a uniform random integer from 1 to 10 using only rand7 and rejection sampling.
Find all missing numbers from 1 to n in O(n) time using in-place index marking.
A clear explanation of finding the maximum XOR of two numbers using greedy bit prefixes.
A clear explanation of reducing an integer to 1 with the fewest operations using greedy bit decisions.
A clear explanation of computing large modular exponentiation using fast power, modular arithmetic, and digit decomposition.
A clear explanation of Maximum Size Subarray Sum Equals k using prefix sums and earliest-index hashing.
A clear explanation of Wiggle Sort II using sorting, median splitting, and virtual indexing.
A clear explanation of maximizing the rotation function using a recurrence instead of simulating every rotation.
A clear explanation of counting connected components using Union-Find and graph traversal.
A clear explanation of checking whether ordered points form a convex polygon using cross products.
Count ordered boomerang tuples by fixing each point as the center and grouping other points by squared distance.
A clear explanation of uniformly picking an integer point from non-overlapping rectangles using prefix sums and binary search.
A clear explanation of checking the minimum edits needed to make a password strong using greedy handling of length, missing character types, and repeated runs.
A clear explanation of adding two integers without using plus or minus by using XOR, AND, carry, and a 32-bit mask.
A clear explanation of Intersection of Two Arrays II using frequency counting.
A clear explanation of Coin Change using dynamic programming for minimum coin count.
A clear explanation of finding the next greater element using a monotonic decreasing stack and hash map.
A clear explanation of counting battleships in a board using one-pass observation without modifying the grid.
A clear explanation of Create Maximum Number using monotonic stacks for subsequences and greedy merging.
A clear explanation of calculating total poisoned duration by merging overlapping attack intervals.
A clear explanation of validating IPv4 and IPv6 addresses by checking segment count, length, characters, range, and leading-zero rules.
Count arithmetic subsequences of length at least three using dynamic programming with one hash map per ending index.
A clear explanation of fitting a sentence onto a screen using cyclic string simulation and greedy row transitions.
A clear explanation of finding the longest substring where every character appears at least k times using divide and conquer.
A clear explanation of applying many range updates efficiently using a difference array and prefix sums.
A clear explanation of Intersection of Two Arrays using hash sets for uniqueness and fast lookup.
A clear explanation of Generalized Abbreviation using backtracking to choose whether each character is kept or abbreviated.
A dynamic programming and patience sorting solution for finding the longest strictly increasing subsequence in an array.
A clear explanation of counting sign assignments that reach a target using recursion first, then subset-sum dynamic programming.
A clear explanation of finding cells that can flow to both oceans using reverse graph traversal from the borders.
A clear explanation of counting unique substrings that appear in the infinite alphabet wraparound string using dynamic programming by ending character.
A clear explanation of Bulb Switcher using divisor parity and perfect squares.
A counting solution for producing the Bulls and Cows hint while handling duplicate digits correctly.
Add two numbers stored in forward-order linked lists using stacks and carry propagation.
A clear explanation of Design Tic-Tac-Toe using row, column, and diagonal counters for constant-time winner checks.
A clear explanation of decoding nested repeat expressions using a stack.
A clear explanation of adding one to a number stored as a linked list using the rightmost non-nine digit.
A clear explanation of counting pairs where nums[i] is greater than twice nums[j] using merge sort.
A clear explanation of Maximum Product of Word Lengths using bit masks to test disjoint character sets efficiently.
A clear explanation of deciding whether an array can be split into two equal-sum subsets using 0/1 knapsack dynamic programming.
A clear explanation of validating a byte sequence as UTF-8 using bit masks and a continuation-byte counter.
Check whether nums is the unique shortest supersequence of given subsequences using topological sorting.
A clear explanation of counting how many repeated copies of one string can be obtained as a subsequence of another repeated string.
A clear explanation of finding the largest subset where every pair is divisible using sorting, dynamic programming, and parent reconstruction.
A clear explanation of finding rectangle dimensions with a fixed area and the smallest length-width difference.
A clear explanation of Top K Frequent Elements using frequency counting and bucket sort.
A DFS solution for finding the longest parent-to-child path where each node value increases by exactly one.
A preorder DFS codec for converting a binary tree to a string and reconstructing the same tree from that string.
A clear explanation of implementing a LIFO stack using only FIFO queue operations.
Compress a character array in-place using two pointers and grouped character counting.
A clear explanation of checking whether one string is a subsequence of another using two pointers.
A clear explanation of checking whether an integer is a perfect square using binary search without sqrt.
A clear explanation of adding two non-negative integer strings using manual digit-by-digit simulation.
A clear explanation of Shortest Distance from All Buildings using BFS from each building with distance and reach accumulation.
A clear explanation of generating all distinct non-decreasing subsequences using DFS, backtracking, and per-level duplicate control.
A clear explanation of Moving Average from Data Stream using a queue and rolling sum.
A clear explanation of minimizing debt-settlement transactions using net balances, backtracking, and memoization-style pruning.
A clear explanation of evaluating an expression with plus, minus, spaces, and parentheses using a stack.
A median-based solution for minimizing total Manhattan distance in a grid.
A clear explanation of computing the total covered area of two axis-aligned rectangles by subtracting their overlap.
Find all duplicated numbers in an array in O(n) time and O(1) extra space using index marking.
Find the maximum number of complete staircase rows that can be formed using binary search and triangular numbers.
A clear explanation of finding the third distinct maximum number using one pass and constant space.
A clear explanation of Reverse Vowels of a String using two pointers and selective swaps.
A clear explanation of grouping binary tree nodes by the round in which they become leaves using postorder DFS.
A clear explanation of Remove Duplicate Letters using a greedy monotonic stack.
A clear explanation of checking whether many small axis-aligned rectangles form one exact rectangular cover using area and corner parity.
A clear explanation of the H-Index II problem using binary search on a sorted citations array.
A two-heap data structure for adding numbers from a stream and returning the current median in constant time.
A clear explanation of counting nodes in a complete binary tree faster than visiting every node.
A clear explanation of deciding whether a rolling ball can stop at the destination using BFS or DFS over stopping cells.
A clear explanation of Reverse String using two pointers and in-place swaps.
A clear explanation of counting arithmetic subarrays using dynamic programming and consecutive differences.
A clear explanation of solving the Can I Win game using minimax recursion, bitmask state compression, and memoization.
A clear explanation of Count of Smaller Numbers After Self using coordinate compression and a Fenwick Tree.
Find the k-th integer in lexicographical order without generating all numbers, using prefix counting over a conceptual trie.
A clear explanation of solving the Water and Jug Problem using Bézout's identity and greatest common divisor.
A clear explanation of finding the last remaining number after alternating left-to-right and right-to-left eliminations.
A clear explanation of the H-Index problem using sorting, then an optimized counting approach.
A recursive game theory solution with memoization for deciding whether the starting player can force a win.
A clear explanation of finding the largest square of 1s in a binary matrix using dynamic programming.
A simple string scanning solution for generating every possible next state after flipping one consecutive ++ pair into --.
A clear explanation of the Integer to English Words problem using three-digit chunks and scale words.
A game theory solution for deciding whether the first player can win by using the losing-position pattern of multiples of four.
A clear explanation of checking nearby indices with nearby values using a sliding window and bucket hashing.
A clear explanation of the Closest Binary Search Tree Value II problem using inorder traversal and a fixed-size sliding window.
A backtracking solution for matching a pattern string to a target string using a bijective character-to-substring mapping.
A clear explanation of detecting whether equal values appear within distance k using a hash map or sliding window set.
A hash map solution for checking whether a pattern string and a space-separated word string form a bijection.
A clear explanation of the Encode and Decode Strings problem using length-prefix encoding.
An in-place matrix simulation for computing the next state of Conway's Game of Life using temporary encoded states.
A clear explanation of checking whether two strings are anagrams using character frequency counting.
A clear explanation of the Closest Binary Search Tree Value problem using the BST property to walk toward the target.
A clear explanation of computing the skyline formed by buildings using sweep line and a max-heap.
A clear explanation of the Alien Dictionary problem using graph construction and topological sorting.
A clear explanation of detecting duplicates in an array using a hash set and sorting.
A clear explanation of generating all possible results from different parenthesizations using divide and conquer recursion.
A clear explanation of the Missing Number problem using sum formula and XOR.
A clear explanation of finding k distinct numbers from 1 to 9 that sum to n using backtracking.
A hash map design for checking whether a word's abbreviation is unique in a dictionary.
A clear explanation of searching a row-sorted and column-sorted matrix using the top-right corner elimination method.
A clear explanation of finding the kth largest element using sorting, a min-heap, and Quickselect.
A clear explanation of the Palindrome Permutation II problem using character counts and backtracking over half of the palindrome.
A clear explanation of finding the maximum value in every sliding window using a monotonic deque.
A clear explanation of building the shortest palindrome by finding the longest palindromic prefix using KMP.
A Floyd cycle detection solution for finding the repeated number without modifying the array and using constant extra space.
A clear explanation of the Palindrome Permutation problem using character parity counting.
A multi-source BFS solution for filling each empty room with its shortest distance to the nearest gate.
A clear explanation of computing each product except self using prefix and suffix products without division.
A clear explanation of maximizing robbed money from circularly arranged houses using dynamic programming.
A detailed guide to solving Same Tree with recursive DFS and structural comparison.
A detailed guide to solving Recover Binary Search Tree with inorder traversal and two misplaced nodes.
A clear explanation of checking whether a string is a palindrome after ignoring non-alphanumeric characters and case.
A detailed guide to solving Validate Binary Search Tree with recursive lower and upper bounds.
Evaluate an arithmetic expression written in Reverse Polish Notation using a stack.
Find the maximum number of points lying on the same straight line using slope counting and normalization.
A clear explanation of finding the maximum path sum in a binary tree using bottom-up depth-first search.
A clear explanation of maximizing stock profit with at most two transactions using dynamic programming.
A detailed guide to solving Interleaving String with two-dimensional dynamic programming.
A clear explanation of maximizing stock profit with unlimited transactions using a greedy single-pass method.
A detailed guide to solving Unique Binary Search Trees with dynamic programming and the Catalan recurrence.
A clear guide to sorting an array of 0s, 1s, and 2s in place using the Dutch National Flag algorithm.
Sort a singly linked list in ascending order using merge sort with fast and slow pointers.
A clear explanation of finding the maximum profit from one stock transaction using a single pass.
A clear explanation of computing the minimum initial health needed to survive a dungeon using reverse dynamic programming.
A clear guide to searching a sorted 2D matrix using binary search over a virtual one-dimensional array.
A clear explanation of finding the minimum path sum in a triangle using bottom-up dynamic programming.
Sort a singly linked list using insertion sort by splicing each node into a growing sorted list.
A clear explanation of designing an iterator over a BST using controlled inorder traversal with a stack.
A detailed guide to solving Unique Binary Search Trees II with recursive tree generation over value ranges.
Design an LRU cache with O(1) get and put operations using a hash map and doubly linked list.
A clear explanation of counting trailing zeroes in n! by counting factors of 5 instead of computing the factorial directly.
A clear guide to setting matrix rows and columns to zero in place using the first row and first column as markers.
A detailed guide to solving Binary Tree Inorder Traversal with recursion and an iterative stack.
Return the postorder traversal of a binary tree using recursion or an iterative stack-based approach.
A clear explanation of converting an Excel column title into its numeric index using base 26 accumulation.
A clear guide to computing the minimum number of insert, delete, and replace operations needed to convert one string into another.
Return the preorder traversal of a binary tree using recursion or an explicit stack.
A clear explanation of designing a data structure that supports add and find operations for pair sums.
A detailed guide to solving Restore IP Addresses with backtracking over four valid IP segments.
A clear guide to simplifying Unix-style file paths using a stack.
A clear explanation of generating a single row of Pascal's Triangle using in-place dynamic programming.
A clear explanation of counting connected groups of land cells in a grid using DFS or BFS.
A clear explanation of finding the element that appears more than half the time using Boyer-Moore voting.
Reorder a singly linked list in-place by finding the middle, reversing the second half, and merging the two halves alternately.
A clear explanation of generating Pascal's Triangle row by row using dynamic programming.
A detailed guide to solving Reverse Linked List II with a dummy node and in-place sublist reversal.
A clear guide to counting distinct ways to climb stairs using dynamic programming.
A clear explanation of returning the visible nodes from the right side of a binary tree using level-order traversal.
A clear explanation of maximizing robbery profit without robbing adjacent houses using dynamic programming.
Find the node where a linked list cycle begins using Floyd’s tortoise and hare algorithm with cycle entry mathematics.
A clear explanation of connecting next pointers in any binary tree using constant extra space.
A detailed guide to solving Decode Ways with dynamic programming and careful handling of zeroes.
A clear guide to computing the integer square root using binary search without built-in exponent functions.
A clear explanation of converting a positive integer into an Excel column title using bijective base 26.
A clear explanation of connecting next pointers in a perfect binary tree using constant extra space.
A clear explanation of finding two numbers in a sorted array using two pointers and constant extra space.
A clear explanation of counting distinct subsequences using dynamic programming.
Detect whether a linked list contains a cycle using Floyd’s tortoise and hare two-pointer algorithm.
A clear explanation of flattening a binary tree into a linked list in preorder traversal order using recursive depth-first search.
A detailed guide to solving Subsets II with sorting, backtracking, and duplicate skipping.
A clear guide to formatting text with greedy line packing and even space distribution.
A clear explanation of counting set bits in an integer using bit manipulation and Brian Kernighan's algorithm.
A clear explanation of converting a fraction into decimal form and detecting repeating fractional parts with a hash map.
A clear explanation of finding all root-to-leaf paths whose values add up to a target sum using depth-first search and backtracking.
A clear guide to adding two binary strings using two pointers and a carry.
A detailed guide to solving Gray Code using the binary-to-Gray-code formula.
A clear explanation of checking whether a binary tree has a root-to-leaf path whose values add up to a target sum.
Return all valid sentences formed by inserting spaces into a string so every word belongs to the dictionary, using DFS with memoization.
A clear guide to adding one to a large integer represented as an array of digits.
Decide whether a string can be segmented into dictionary words using dynamic programming over prefixes.
A clear explanation of finding the minimum depth of a binary tree using breadth-first search.
A detailed guide to solving Merge Sorted Array in-place by merging from the back with three pointers.
A clear guide to validating whether a string is a valid number using grammar rules and one left-to-right scan.
A clear explanation of the Paint House II problem using optimized dynamic programming with minimum and second minimum tracking.
A clear explanation of checking whether a binary tree is height-balanced using bottom-up depth-first search.
Create a deep copy of a linked list with next and random pointers using hash maps or interleaved node cloning.
A clear explanation of comparing version strings revision by revision while ignoring leading zeros.
A clear explanation of finding unique combinations that sum to a target when each array element may be used at most once.
A clear explanation of finding multiple words in a character board using a Trie and DFS backtracking.
A clear explanation of the Fizz Buzz problem using direct simulation and divisibility checks.
A clear explanation of reversing the bits of a 32-bit integer using bit manipulation.
A clear explanation of cleaning an unknown grid using DFS, relative coordinates, and physical backtracking.
A clear explanation of Binary Tree Vertical Order Traversal using BFS with column indices.
A clear explanation of counting the perimeter of an island in a grid by adding land-cell edges and subtracting shared edges.
A detailed guide to solving Scramble String with recursive dynamic programming and memoization.
A clear explanation of Integer Break using dynamic programming, with a note on the greedy math solution.
A clear explanation of computing inverse depth weighted sum using level-order traversal.
A clear explanation of finding the extra character added to a shuffled string using counting and XOR.
A clear guide to finding the minimum path sum in a grid using dynamic programming.
A binary-search-style solution for finding the smallest node greater than p in a binary search tree.
A clear explanation of deleting a node from a singly linked list when only that node is given.
A clear explanation of converting a sorted linked list into a height-balanced binary search tree using slow and fast pointers.
Evaluate a nested ternary expression using a right-to-left stack parser.
A clear explanation of the Ugly Number II problem using dynamic programming with three pointers.
Find the number that appears once when every other number appears three times using bit counting or finite-state bit manipulation.
A clear explanation of finding all unique combinations that sum to a target using backtracking.
A clear explanation of designing a word dictionary with addWord and wildcard search using a Trie and DFS.
A clear explanation of finding the shortest abbreviation that does not conflict with any dictionary word using bit masks.
A clear explanation of rotating an array to the right by k steps using in-place reversal.
A clear explanation of finding the maximum adjacent gap in sorted order using buckets and the pigeonhole principle.
A clear explanation of Super Ugly Number using dynamic programming with one pointer per prime.
A clear explanation of solving Zuma Game with DFS, memoization, and chain-removal simulation.
A clear explanation of Power of Four using bit manipulation and binary properties.
A clear explanation of why the median minimizes the number of moves needed to make all array elements equal.
A detailed guide to solving Partition List with two dummy lists while preserving relative order.
A clear explanation of reducing a 2D rectangle problem to a 1D prefix-sum problem with binary search.
A clear guide to counting unique paths in a grid with obstacles using dynamic programming.
A wrapper iterator design that supports peeking at the next element without advancing the iterator.
A clear explanation of computing the longest absolute path to a file from a serialized file system string using path lengths by depth.
A clear explanation of building a height-balanced binary search tree from a sorted array using divide and conquer.
Find all starting indices where an anagram of p appears in s using a fixed-size sliding window.
Find the only number that appears once using the XOR operator, while every other number appears exactly twice.
A clear explanation of the Ugly Number problem using repeated division by the only allowed prime factors.
A clear explanation of finding a valid course ordering using topological sorting and cycle detection.
A clear explanation of finding the lowest common ancestor in a normal binary tree using recursive depth-first search.
A clear explanation of generating the count-and-say sequence using run-length encoding.
A clear explanation of solving a Sudoku board using backtracking and constraint checking.
A clear explanation of minimizing the largest subarray sum using binary search on the answer and greedy validation.
A clear explanation of maximizing stock trading profit with at most k transactions using dynamic programming.
A clear explanation of finding all missing ranges inside an inclusive interval by scanning sorted unique numbers.
A clear explanation of Burst Balloons using interval dynamic programming and the last-burst idea.
A clear explanation of finding the longest run of 1s after flipping at most one 0 using a sliding window.
A clear explanation of Flatten Nested List Iterator using lazy stack-based flattening.
A clear explanation of computing the Hamming distance between two integers using XOR and bit counting.
A two-pointer in-place solution for moving all zeroes to the end while preserving the relative order of non-zero elements.
A clear guide to counting unique paths in a grid using dynamic programming.
A clear explanation of designing a hit counter for the last 5 minutes using a queue with compressed timestamps.
A clear guide to rotating a linked list to the right by k places using a circular list.
A clear explanation of finding the first non-repeating character in a string using character frequency counting.
A clear explanation of returning binary tree levels from bottom to top using breadth-first search.
Count downward paths in a binary tree whose values sum to targetSum using DFS and prefix sums.
A clear explanation of finding the lowest common ancestor in a binary search tree using BST ordering properties.
A detailed guide to solving Maximal Rectangle by converting each matrix row into a histogram and applying a monotonic stack.
A clear explanation of the Trips and Users SQL problem using joins, filtering, grouping, and conditional aggregation.
Compute the minimum candies needed using two greedy passes, one from the left and one from the right.
A clear explanation of finding the shortest contiguous subarray whose sum is at least target using a sliding window.
A clear explanation of finding repeated 10-letter DNA substrings using a fixed-size sliding window and hash sets.
A clear explanation of checking whether a partially filled Sudoku board is valid using hash sets.
A clear explanation of finding the longest palindrome length that can be built from given letters using character counts.
A clear explanation of finding any peak element using binary search on the slope of the array.
A clear explanation of Sparse Matrix Multiplication using non-zero entries to avoid wasted work.
A clear explanation of predicting whether Player 1 can win using minimax dynamic programming over score difference.
A clear explanation of Longest Substring with At Most K Distinct Characters using a sliding window and character counts.
A clear explanation of Nested List Weight Sum using depth-first search over a nested structure.
A clear explanation of Counting Bits using dynamic programming and bit manipulation.
A backtracking solution for inserting operators into a numeric string so the expression evaluates to a target value.
A clear explanation of designing an LFU cache with O(1) average get and put operations.
A clear explanation of checking whether a string can be built by repeating one of its proper substrings.
A clear guide to finding the kth permutation sequence using factorial blocks instead of generating all permutations.
A clear explanation of finding the best bomb placement in a grid using cached row and column segment counts.
A clear explanation of sorting values after applying a quadratic function using two pointers.
A detailed guide to solving Largest Rectangle in Histogram with a monotonic increasing stack.
Find, for each interval, the interval with the smallest start point greater than or equal to its end point using sorting and binary search.
A clear explanation of the Graph Valid Tree problem using Union Find to detect cycles and verify connectivity.
A clear explanation of checking whether a singly linked list is a palindrome using fast and slow pointers plus in-place reversal.
Find the unique starting gas station index using a greedy scan with total fuel balance and current tank balance.
A clear explanation of implementing a Trie with insert, search, and startsWith operations.
A clear explanation of reversing the order of words in a character array in-place using two reversals.
A clear explanation of finding the index of a target, or where it should be inserted, using binary search.
A clear explanation of generating numbers from 1 to n in lexicographical order using an iterative DFS-style traversal.
A clear explanation of validating a word abbreviation using two pointers and number parsing.
A clear explanation of checking whether two strings are exactly one edit apart using a linear scan.
A clear explanation of finding the longest streak of 1s in a binary array with a single pass.
A clear explanation of Minimum Height Trees using leaf trimming to find the center of a tree.
A clear explanation of House Robber III using tree dynamic programming with rob and skip states.
A clear guide to generating an n x n matrix filled from 1 to n squared in spiral order.
A clear explanation of the Single Number III problem using XOR partitioning to isolate the two unique numbers.
A clear explanation of designing a logger that prints each message at most once every 10 seconds using a hash map.
A detailed explanation of counting how many times digit one appears from 0 to n using positional digit analysis.
A detailed guide to solving Remove Duplicates from Sorted List with one pointer and in-place linked list rewiring.
A queue-based iterator design for returning elements from two vectors in alternating order, with a clean extension to k vectors.
Remove the minimum number of intervals so the remaining intervals do not overlap, using greedy sorting by end time.
Count the number of word segments in a string by detecting transitions from spaces to non-space characters.
Create a deep copy of a connected undirected graph using DFS and a hash map from original nodes to cloned nodes.
A clear explanation of detecting cycles in a prerequisite graph using topological sorting and DFS.
A clear explanation of the combinatorics behind finding the minimum number of pigs needed to identify the poisonous bucket.
A clear explanation of finding the first and last index of a target in a sorted array using two binary searches.
A clear explanation of rebuilding a binary tree from inorder and postorder traversals using recursion and an index map.
A clear explanation of finding the node where two singly linked lists intersect using two pointers.
A clear explanation of constructing the lexicographically smallest permutation that matches an I and D pattern.
A clear explanation of Best Time to Buy and Sell Stock with Cooldown using dynamic programming states.
A clear explanation of the 3Sum Smaller problem using sorting and the two-pointer technique.
A clear guide to solving Length of Last Word by scanning the string from right to left.
A clear explanation of trapping rain water in a 2D elevation map using a min heap and boundary expansion.
A clear explanation of parsing a serialized nested integer string using a stack.
A detailed explanation of implementing a FIFO queue using two LIFO stacks with amortized constant time operations.
A clear SQL solution for finding employees whose salaries are in the top three unique salary levels within their department.
A clear explanation of rearranging a string so equal characters are at least k positions apart using a heap and cooldown queue.
A clear explanation of Palindrome Pairs using reversed-word lookup and palindrome split checks.
A clear explanation of reversing a singly linked list using iterative and recursive approaches.
A detailed guide to solving Remove Duplicates from Sorted List II with a dummy node and pointer rewiring.
Find the minimum number of cuts needed to split a string into palindromic substrings using palindrome precomputation and dynamic programming.
A greedy in-place solution for rearranging an array into a non-strict wiggle pattern.
Find the minimum number of valid one-character gene mutations using breadth-first search.
A detailed explanation of reversing linked-list nodes in groups of k using pointer manipulation and constant extra space.
A detailed explanation of swapping every two adjacent nodes in a linked list using pointer manipulation.
A detailed explanation of merging k sorted linked lists using a min heap.
A clear explanation of searching a rotated sorted array in logarithmic time using modified binary search.
A clear explanation of Range Sum Query 2D - Mutable using a 2D Fenwick Tree for efficient updates and rectangle sum queries.
A clear explanation of reconstructing a queue using greedy sorting and indexed insertion.
A clear explanation of the Add Digits problem using repeated digit sums first, then the digital root formula.
A clear explanation of finding the smallest base where n is written as all ones using geometric series and binary search.
A clear explanation of determining whether an integer is a power of two using binary properties and bit manipulation.
A clear explanation of rebuilding a binary tree from preorder and inorder traversals using recursion and an index map.
A clear explanation of detecting a valid cycle in a circular array using fast and slow pointers.
A clear explanation of finding the longest substring with at most two distinct characters using a sliding window.
A clear explanation of shuffling an array uniformly using the Fisher-Yates algorithm while supporting reset.
A clear explanation of checking whether two strings follow the same character mapping pattern.
A clear guide to solving Insert Interval with one linear scan over sorted, non-overlapping intervals.
A clear explanation of Self Crossing using constant-space checks for the only possible crossing patterns.
A clear explanation of finding the longest well-formed parentheses substring using a stack of indices.
A clear explanation of counting numbers with unique digits using combinatorics.
A clear SQL solution for finding every employee who earns the highest salary in their department.
A clear explanation of Range Sum Query - Mutable using a Fenwick Tree for efficient updates and range sums.
A detailed guide to solving Search in Rotated Sorted Array II with modified binary search and duplicate handling.
A clear explanation of converting integers to hexadecimal using bit manipulation and two's complement representation.
A detailed explanation of generating all well-formed parentheses strings using backtracking.
A detailed explanation of merging two sorted linked lists using a dummy node and pointer splicing.
A clear explanation of reformatting a license key by removing dashes, uppercasing characters, and grouping from the right.
A clear explanation of the Binary Tree Paths problem using DFS backtracking to collect every root-to-leaf path.
A clear explanation of finding the maximum depth of a binary tree using recursive depth-first search.
A clear explanation of finding the kth smallest value in a binary search tree using inorder traversal.
A clear explanation of checking whether one string can be constructed from another using character frequency counting.
A clear explanation of counting prime numbers less than n using the Sieve of Eratosthenes.
A clear guide to solving Merge Intervals by sorting intervals and merging them in one pass.
A dynamic programming solution for finding the least number of perfect square numbers that sum to n.
A clear explanation of implementing read with read4 when read may be called multiple times.
A clear explanation of Increasing Triplet Subsequence using greedy tracking of two minimum values.
Design a data structure that supports increment, decrement, get minimum key, and get maximum key in average O(1) time.
Generate all ways to split a string so that every piece is a palindrome, using backtracking with palindrome precomputation.
A clear SQL solution for finding customers who have no matching rows in the Orders table.
A clear explanation of detecting a 132 pattern using reverse traversal and a monotonic stack.
A clear explanation of checking whether 2D points are symmetric around a vertical line using min and max x-coordinates.
A clear explanation of finding the next lexicographically greater permutation in place using a right-to-left scan.
A detailed explanation of checking whether a bracket string is valid using a stack.
A clear explanation of Additive Number using split enumeration and deterministic checking.
A clear explanation of the Paint House problem using dynamic programming with constant space.
A clear explanation of constructing the magical string by using the string itself as run-length instructions.
A clear explanation of the Sum of Left Leaves problem using depth-first traversal of a binary tree.
A clear explanation of zigzag level order traversal using breadth-first search and alternating level direction.
A detailed guide to solving Remove Duplicates from Sorted Array II with an in-place two-pointer method.
A clear explanation of finding all elements that appear more than n/3 times using the extended Boyer-Moore voting algorithm.
A clear explanation of finding all starting indices where a substring is formed by concatenating every word exactly once.
A clear guide to solving Jump Game with greedy reachability.
A clear explanation of removing all linked list nodes with a target value using iteration and a dummy node.
A binary search solution for finding the first bad version while minimizing calls to the isBadVersion API.
A clear explanation of implementing read using the given read4 API and copying only the needed characters.
A clear explanation of Largest BST Subtree using postorder traversal and subtree state propagation.
A clear explanation of selecting a random linked list node with equal probability using reservoir sampling.
Convert an N-ary tree into a binary tree and reconstruct it using the left-child right-sibling representation.
A detailed explanation of removing the nth node from the end of a singly linked list using two pointers and a dummy node.
A clear explanation of Reconstruct Itinerary using a directed graph and Hierholzer's algorithm.
A clear explanation of flipping a binary tree upside down by rewiring pointers from the left spine.
Capture surrounded O regions by marking border-connected O cells first, then flipping the remaining O cells.
A clear explanation of designing a randomized multiset with average O(1) insert, remove, and getRandom operations.
A clear SQL solution for reporting email values that appear more than once in the Person table.
A detailed explanation of finding all unique quadruplets that sum to a target using sorting and two pointers.
A clear explanation of verifying preorder serialization using slot counting without reconstructing the tree.
A clear explanation of the greedy two-pointer solution for maximizing the number of content children.
A clear SQL solution for finding employees whose salary is greater than their manager's salary using a self join.
Flatten a multilevel doubly linked list in-place using depth-first traversal and pointer splicing.
A detailed explanation of generating all possible phone keypad letter combinations using backtracking.
A clear explanation of Number of Islands II using Union-Find to dynamically merge connected land cells.
A clear explanation of implementing a simplified Twitter using hash maps, sets, timestamps, and a heap.
A clear explanation of Patching Array using a greedy smallest-missing-sum invariant.
Compute the sum of all numbers formed by root-to-leaf paths using depth-first search and decimal accumulation.
Traverse an N-ary tree level by level using breadth-first search.
A clear explanation of designing a randomized set with average O(1) insert, remove, and getRandom operations.
A clear explanation of maintaining the median of each fixed-size window using two heaps and lazy deletion.
A clear explanation of designing a stack that can return the current minimum element in constant time.
A clear SQL solution for finding numbers that appear at least three times consecutively in the Logs table.
A clear explanation of Range Sum Query 2D - Immutable using a 2D prefix sum matrix for constant-time rectangle queries.
A clear explanation of the Verify Preorder Sequence in Binary Search Tree problem using a monotonic stack and lower bound tracking.
A clear explanation of solving Russian Doll Envelopes using sorting and longest increasing subsequence.
A clear explanation of counting zero-sum tuples across four arrays using pair sums and a hash map.
A detailed explanation of finding the sum of three integers closest to a target using sorting and two pointers.
A clear explanation of Longest Increasing Path in a Matrix using DFS with memoization.
A clear explanation of designing a phone directory that can allocate, check, and release numbers efficiently.
Find the longest run of consecutive integers in an unsorted array using a hash set and sequence-start detection.
A clear explanation of arranging non-negative integers to form the largest possible concatenated number using a custom sort order.
A clear explanation of finding the largest palindrome made from the product of two n-digit numbers by generating palindrome candidates directly.
A clear explanation of finding the minimum element in a rotated sorted array that may contain duplicates.
A clear explanation of the Factor Combinations problem using DFS backtracking with non-decreasing factors.
A clear explanation of the math behind making all array elements equal by incrementing n - 1 elements at a time.
Serialize an N-ary tree into a string and reconstruct the same tree using preorder traversal with child counts.
A clear explanation of Range Sum Query - Immutable using prefix sums for constant-time range queries.
A clear explanation of implementing Snake Game with a deque for body order and a set for constant-time collision checks.
A detailed explanation of finding all unique triplets that sum to zero using sorting and two pointers.
A clear explanation of binary tree level order traversal using breadth-first search and a queue.
Use breadth-first search to find the shortest transformation sequence length between two words.
A clear explanation of the Frog Jump problem using dynamic programming with reachable jump sizes.
A clear explanation of Odd Even Linked List using in-place pointer rewiring.
A clear explanation of finding the kth smallest value in a row-sorted and column-sorted matrix using binary search on values.
A clear SQL solution for ranking scores with dense ranking, where ties share the same rank and no rank numbers are skipped.
A clear explanation of generating uniformly random points inside a circle using polar coordinates.
A clear explanation of the Meeting Rooms II problem using a min heap to track active meeting end times.
A clear explanation of summarizing a sorted unique integer array into compact consecutive ranges.
A detailed guide to solving Word Search with depth-first search and backtracking on a grid.
A clear explanation of finding the minimum element in a rotated sorted array using binary search.
A clear explanation of integer division without using multiplication, division, or modulo, using repeated doubling with bit shifts.
A detailed explanation of finding the longest common prefix among an array of strings by comparing characters column by column.
A clear guide to reading a matrix in spiral order using shrinking boundaries.
A clear explanation of the greedy interval solution for finding the minimum number of arrows needed to burst all balloons.
Build a quad tree from a binary square grid using recursive divide and conquer.
A clear explanation of the Remove K Digits problem using a greedy monotonic stack.
A clear explanation of Combination Sum IV using dynamic programming to count ordered combinations that sum to a target.
A clear explanation of maintaining disjoint sorted intervals from a stream using insertion and merging.
A clear explanation of Count of Range Sum using prefix sums and merge sort counting.
A clear explanation of Smallest Rectangle Enclosing Black Pixels using binary search on rows and columns.
A two-pass solution for finding a celebrity using the knows API with O(n) calls and O(1) extra space.
A clear explanation of detecting whether repeated digit-square sums eventually reach 1.
A clear SQL solution for finding the nth highest distinct salary from the Employee table.
A clear explanation of the Meeting Rooms problem using interval sorting to detect overlaps.
A detailed explanation of evaluating arithmetic expressions with stack-based parsing and operator precedence.
A detailed explanation of tracking both maximum and minimum products while scanning the array.
A detailed guide to solving Subsets with backtracking and the include-or-skip recursion idea.
A clear explanation of finding the first occurrence of one string inside another using direct string matching.
A clear guide to solving Maximum Subarray with brute force first, then Kadane's dynamic programming algorithm.
A detailed explanation of converting a Roman numeral string into an integer using symbol values and the subtraction rule.
A clear explanation of computing the total Hamming distance across all pairs by counting different bits column by column.
A clear explanation of finding the bitwise complement of a positive integer using a binary mask.
A clear explanation of sorting characters by decreasing frequency using a hash map and sorting.
Convert a BST into a sorted circular doubly linked list in-place using inorder traversal.
A clear explanation of the Binary Watch problem using bit counting over all valid times.
A clear explanation of the Wiggle Subsequence problem using dynamic programming intuition and an optimized greedy solution.
A clear explanation of Android Unlock Patterns using backtracking, a jump table, and symmetry optimization.
A clear explanation of the Power of Three problem using repeated division and integer arithmetic.
A clear explanation of Remove Invalid Parentheses using BFS to guarantee the minimum number of removals.
A dynamic programming solution for counting ways to paint fence posts with no more than two adjacent posts sharing the same color.
A clear explanation of the Flatten 2D Vector problem using row and column pointers to implement an iterator.
A clear explanation of inverting a binary tree using recursive depth-first traversal.
A clear explanation of finding the bitwise AND of every number in an inclusive range using the common binary prefix.
A clear SQL solution for finding the second highest distinct salary from the Employee table.
A detailed guide to solving Combinations with backtracking and pruning.
A clear guide to solving N-Queens II by counting valid queen placements with backtracking.
A clear explanation of removing all occurrences of a value from an array in place using a write pointer.
A detailed explanation of converting an integer into a Roman numeral using a fixed value-symbol table and greedy subtraction.
A detailed explanation of finding the maximum water container area using two pointers.
A detailed explanation of matching a full string against a simplified regular expression with dot and star using dynamic programming.
A detailed explanation of checking whether an integer is a palindrome using digit operations without converting it to a string.
A detailed explanation of parsing a string into a 32-bit signed integer with whitespace, sign, digit reading, and clamping rules.
A detailed explanation of reversing a signed 32-bit integer while handling overflow correctly.
A detailed explanation of converting a string into a zigzag pattern using row simulation.
A detailed explanation of finding the longest palindromic substring using expand-around-center.
A clear explanation of reversing word order while removing extra spaces.
Find all shortest word transformation sequences using BFS to build shortest-path parents, then backtracking to reconstruct every answer.
A clear explanation of checking whether a binary tree is symmetric using mirror recursion.
A detailed guide to solving Minimum Window Substring with a sliding window and frequency counters.
A clear guide to solving N-Queens with backtracking, row-by-row placement, and constant-time conflict checks.
A detailed explanation of finding the median of two sorted arrays using binary search over partitions.
A clear explanation of removing duplicates from a sorted array in place using two pointers.
A clear explanation of the longest substring problem using sliding window and a hash set.
A detailed explanation of the Add Two Numbers linked list problem, including digit-by-digit addition, carry handling, and linked list construction.
A clear explanation of the Two Sum problem using brute force first, then an optimized hash map solution.
Browse notes by date
Verify both the ordering and permutation properties of sorting implementations using randomized, edge-case, and adversarial test strategies.
Select the appropriate sorting algorithm based on input size, data characteristics, memory constraints, and required guarantees such as stability or worst-case bounds.
Prove that any comparison-based sorting algorithm requires Ω(n log n) comparisons in the worst case using a decision tree argument.
Distribute sorting work across multiple processors to reduce wall-clock time, with analysis of total work, span, communication, and synchronization.
Identify boundary errors, broken invariants, and comparator mistakes that cause sorting implementations to fail on edge cases or duplicate-heavy inputs.
Replace large or sparse keys with small dense ranks that preserve order, making range-based and indexed structures practical on wide-valued data.
Count pairs of elements in the wrong relative order to measure how far an array is from sorted, using a modified merge sort in O(n log n) time.
Find the element at a given rank using quicksort's partition step but recursing into only one side, achieving expected linear time.
Produce only the smallest k elements in sorted order rather than sorting the entire array, reducing unnecessary work when the full order is not needed.
Define correct comparison relations for user-defined types and non-trivial orderings — consistency requirements that sorting correctness depends on.
Find the middle value of a collection in linear time using selection algorithms, without the overhead of a full sort.
Sort structured values by one or more fields while moving the full record, with attention to key extraction, stability, and multi-key ordering.
Find the largest or smallest k elements without sorting the full input, using a heap or partition-based approach.
Sort datasets that exceed main memory by organizing the algorithm around sequential disk access, merge passes, and minimizing I/O operations.
Exploit near-sorted structure in inputs like append-only logs or incremental updates to sort in linear or near-linear time.
Divide the input into halves, recursively sort each half, then merge them — combining local order into global order in O(n log n) time.
Sort keys digit by digit using a stable subroutine, achieving linear time for fixed-width integers without any key comparisons.
Distribute elements into buckets by value range, sort each bucket, then concatenate — achieving linear expected time on uniformly distributed input.
A stable sort preserves the original relative order of equal keys — an extra guarantee required when sorting by secondary fields or compound criteria.
Partition around a pivot so smaller elements go left and larger go right, then recursively sort each partition in expected O(n log n) time.
Build a max-heap in place, then repeatedly extract the maximum to produce a sorted array in O(n log n) worst-case time.
Sort integer keys from a small range in linear time by counting occurrences and reconstructing the output from those counts.
Build a sorted prefix one element at a time by inserting each new element into its correct position within the already-sorted portion.
A sorting algorithm is correct only when its output is both ordered and a permutation of the input — two properties every implementation must preserve.
Sort by repeatedly selecting the minimum element from the unsorted suffix and placing it into the next output position.
Verify correctness, stability, and performance of hash-based structures through randomized and adversarial test strategies.
Distribute keys across a dynamic set of nodes so that adding or removing nodes moves only a minimal fraction of keys.
Remove duplicate entries from a dataset or stream efficiently using hash sets for membership tracking.
Join two collections by key using a hash table to reduce the cost from quadratic to linear expected time.
Understand how memory hierarchy effects cause hash table performance to deviate from asymptotic expectations.
Choose and implement hash tables that perform reliably under mixed key types, uneven access patterns, and adversarial input.
Produce stable hash values that remain consistent across program runs, machines, builds, and language runtimes.
Defend hash tables against adversarial inputs that force worst-case collision behavior using randomized hashing.
Combine hash tables with other data structures to handle skewed distributions, heavy deletions, and mixed workloads.
Examine hash-based structures in complete systems: streaming pipelines, graph algorithms, caches, and distributed workflows.
Achieve predictable worst-case bounds for hash-based structures rather than relying solely on average-case expectations.
Approximate frequency counting for large key streams using a compact probabilistic data structure with bounded error.
Design hash table layouts that minimize cache misses and align memory access patterns with hardware behavior.
Track frequency counts for keys using a hash map that increments a counter on each insertion of an existing key.
Hash and compare multi-field keys correctly by combining all fields that participate in equality into the hash function.
Test set membership approximately using a compact bit array and multiple hash functions, with no false negatives and bounded false positives.
Compute hash values for sliding windows over a sequence in constant time by incrementally updating rather than recomputing.
Implement a hash-based set for fast membership testing, insertion, and deletion without associated values.
Build a hash-based map that associates keys with values and supports insert, lookup, delete, and update in expected constant time.
Partition a collection into groups by key using a hash map that accumulates values into per-key lists or sets.
Control hash table performance by monitoring the load factor and growing the bucket array before collisions accumulate.
Resolve hash collisions using separate chaining or open addressing, with trade-offs in memory, locality, and load tolerance.
Rebuild a hash table's bucket array after resizing so that every stored key satisfies the placement invariant for the new capacity.
A data structure that supports fast insertion, lookup, and deletion by mapping keys to bucket positions via a hash function.
Design and evaluate hash functions that distribute keys uniformly across buckets while remaining fast to compute.
Case analysis splits a goal according to the structure of a value.
The tactic `simp` performs normalization by repeated rewriting using a curated set of lemmas.
Propositional equality is the explicit notion of equality in Lean.
Structures package multiple fields into a single value.
Propositional equality in Lean is generated from a small set of core operations.
Rewriting systems can diverge if rules are poorly oriented or interact cyclically.
Lemmas provide reusable equalities that drive most rewriting.
When types depend on values, rewriting affects both terms and their types.
Rewriting becomes more subtle when the target term appears inside a binder.
Equality between functions requires a different treatment than equality between values.
Complex equalities are rarely achieved in a single step.
Equalities are often too strict.
The behavior of `simp` depends entirely on the set of rewrite rules it uses.
Equality becomes useful when it propagates through larger expressions.
In most proofs, equalities come from the local context.
Substitution is the direct elimination of equalities from the context.
Rewriting is the primary way to apply propositional equalities in Lean.
Definitional equality is the built-in notion of equality used by the kernel of Lean.
Rewriting can target either the goal or the local context.
Pattern matching performs case analysis by selecting a branch based on the shape of a value.
Reasoning often alternates between propositional equality `a = b` and boolean equality `a == b`.
Rewriting is most effective when guided by a small set of consistent strategies.
In Lean, propositions live in `Prop`, a universe where proof irrelevance holds.
Rewriting failures in Lean usually come from a small set of recurring issues.
The `conv` tactic provides fine-grained control over rewriting.
Even when the algorithmic idea is correct, implementations fail in predictable ways.
Data representation is the choice of concrete form used to store the objects in a problem.
Stability and determinism describe how predictably an algorithm behaves when there are ties, repeated values, or multiple valid answers.
Algorithms are usually described with mathematical integers and real numbers.
Benchmarking measures how an implementation behaves on real inputs and real hardware.
Implementation discipline means translating an algorithm into code without changing its meaning accidentally.
A reduction transforms one problem into another problem.
Dynamic programming solves problems by storing answers to subproblems and reusing them.
Pseudocode is a bridge between the problem statement and an implementation.
Randomized algorithms use random choices during execution.
Amortized analysis studies the average cost of operations over a sequence, even when individual operations are sometimes expensive.
A greedy algorithm builds a solution by making one locally best choice at a time.
A brute force baseline is the simplest correct algorithm you can write from the problem statement.
Testing does not prove an algorithm correct, but it exposes mistakes in specifications, invariants, edge cases, and implementation details.
Divide and conquer solves a problem by splitting it into smaller subproblems, solving those subproblems, and combining their answers.
Edge cases are valid inputs that sit at the boundary of the specification.
A lower bound states that every algorithm for a problem must perform at least a certain amount of work in some model of computation.
Big O notation provides a formal way to describe how a function grows.
Space complexity measures how much memory an algorithm uses as a function of input size.
Recursive algorithms replace loop structure with self-reference.
Time complexity describes how the running time of an algorithm grows as the input size grows.
An algorithm does not operate on an abstract idea of data.
Loop invariants are the primary tool for reasoning about iterative algorithms.
An algorithm begins with a precise statement of the problem.
A correctness argument explains why an algorithm returns an acceptable output for every valid input.
This section combines multiple patterns from the chapter into complete, end-to-end linked list algorithms.
An intrusive list stores the linkage fields inside the objects being linked.
Pointer aliasing occurs when two or more references point to the same node.
Edge cases are inputs that sit near the boundary of an algorithm's assumptions.
An LRU cache stores a fixed number of key-value entries and removes the least recently used entry when capacity is exceeded.
A dummy head is a fixed node placed before the real head of a singly linked list.
Pointer code should be tested by checking structure, not only values.
Insertion adds nodes into a linked list by creating new links while preserving reachability of all existing nodes.
A persistent list is a list that preserves older versions after an update.
Memory ownership describes which part of a program is responsible for creating, linking, unlinking, and destroying a node.
A stack is a last-in, first-out (LIFO) structure.
A skip list augments a sorted linked list with multiple levels of forward pointers.
A queue is a first-in, first-out structure.
An iterator is an object or procedure that visits the nodes of a linked list one at a time.
Linked list algorithms are dominated by pointer traversal and constant-time link updates.
Splitting a linked list means cutting one list into two or more lists while preserving the original nodes.
Deletion removes one or more nodes from a linked list by changing links around them.
Merging combines two sorted singly linked lists into one sorted list by relinking nodes.
A sentinel node is an artificial node placed at the boundary of a linked list.
A cycle exists in a linked list when some node’s `next` pointer eventually leads back to a previously visited node.
Reversal transforms a linked list so that the direction of all edges is flipped.
A doubly linked list is a sequence of nodes where each node stores a value, a reference to the next node, and a reference to the previous node.
A singly linked list is a sequence of nodes where each node stores a value and a reference to the next node.
Merging is a family of constructions that combine multiple linked lists into one or more output lists while preserving structural invariants.
Fast and slow pointers are two references that traverse the same linked structure at different speeds.
Array and string problems often look different on the surface, but many reduce to a small number of reusable patterns.
Boundary conditions define the valid domain of indices, ranges, and states in an algorithm.
Spiral traversal visits a matrix layer by layer, moving right across the top row, down the right column, left across the bottom row, and up the left column,...
Flood fill explores a connected region in a grid starting from a seed cell and marks or transforms all cells that belong to the same region.
Matrix traversal processes a two-dimensional array in a defined order.
Rolling hashes assign numeric fingerprints to substrings so that many substring comparisons can be done quickly.
String comparison determines the ordering or equality of two strings.
Parsing expressions converts a sequence of tokens into a structured form that reflects operator precedence and associativity.
A trie is a tree structure for storing a set of strings so that common prefixes are shared.
Anagrams are strings or sequences that contain the same elements with the same multiplicities, possibly in different order.
Substring search locates occurrences of a pattern `p` inside a text `s`.
A palindrome is a sequence that reads the same forward and backward.
Run-Length Encoding (RLE) compresses sequences by replacing consecutive equal values with a pair `(value, count)`.
A frequency table records how many times each value appears in an array, string, or stream.
Partitioning rearranges an array so that elements are grouped by a predicate.
Tokenization converts a string into a sequence of meaningful units called tokens.
In-place modification changes an array without allocating another array of the same size.
Array rotation moves elements by a fixed offset while preserving their relative circular order.
String scanning is the basic operation behind parsing, tokenization, validation, search, and text normalization.
Deduplication removes repeated values while preserving a chosen notion of identity and, optionally, order.
The two pointers technique uses two indices that move through an array or string in a controlled way.
Sliding windows maintain a contiguous subarray `[l, r)` while both endpoints move forward.
A difference array is the inverse pattern of a prefix sum.
Prefix sums are a preprocessing technique for answering repeated range sum queries on an array.
Array traversal is the base operation for all algorithms over linear data.
Lean provides automation to reduce routine proof steps.
Lean proofs fail in predictable ways.
Rewriting with equality is one of the most frequent operations in Lean.
Backward reasoning starts from the goal and reduces it to simpler subgoals.
Forward reasoning starts from the assumptions in the local context and derives new facts until the goal becomes immediate.
An assumption is a local term that Lean may use to solve the current goal or produce another proof.
The local context is the list of variables, hypotheses, instances, and intermediate facts available at a point in a proof.
Every logical connective in Lean has two sides.
Names in Lean carry meaning.
A Lean proof should expose the shape of the argument.
A universal statement asserts that a property holds for every element of a type.
Proof by contradiction is a classical proof pattern.
Case analysis is the proof pattern for using data that has more than one possible constructor.
Lean is constructive by default.
Equality supports two structural operations: reversing direction (symmetry) and chaining steps (transitivity).
Rewriting is the main way to use equality in Lean.
An existential statement says that some object exists with a given property.
`True` is the proposition that always has a proof.
Negation represents "not".
`False` is the proposition with no constructors.
Equality expresses that two terms are identical.
Disjunction represents "or".
Implication is the first logical connective to understand in Lean because it is also the ordinary function type.
Lean identifies propositions with types and proofs with terms.
Conjunction represents "and".
Lean is normally developed inside an editor with live feedback.
Lean development is interactive.
A minimal working example is the smallest complete Lean fragment that demonstrates a definition, theorem, error, or technique.
Lean development is driven by goals.
Imports control what a Lean file can see.
Lean’s error messages report failed constraints during elaboration.
Lean proofs are terms.
A structure is a type whose values are built from named fields.
Tactic mode is an interactive way to build proofs.
Inductive types define data by listing its constructors.
Lean files serve two readers at once: the compiler and the human maintainer.
Type checking is the primary feedback mechanism in Lean.
Pattern matching defines functions by cases on the shape of their inputs.
Rewriting replaces one expression with another using an equality.
Lean can execute many expressions during development.
Lean organizes code as a hierarchy of modules.
Lean notation is ordinary syntax attached to ordinary declarations.
Functions are the main form of computation in Lean.
Lean code is built from expressions.
Lean is a dependently typed system.
Lean treats a theorem as a named proof.
Lean development is organized around projects.
A definition introduces a named term together with its type.
Lean functions often contain arguments that the user does not write.
This book is a working manual for the Lean proof assistant.
Lean is distributed as a small toolchain rather than as a single editor plugin.
Logic programming, type systems, verification, model checking, and program synthesis.
Incompleteness, undecidability, independence, practical implications, and future directions in logic and foundations.
Simple type theory, dependent types, Curry-Howard correspondence, proof assistants, and formalized mathematics.
Grammars, syntax, automata theory, regular and context free languages, parsing, recognition, and applications in compilers.
Logic programming, type systems, verification, model checking, and program synthesis.
Logicism, formalism, intuitionism, structuralism, and modern perspectives on the foundations of mathematics.
Logicism, formalism, intuitionism, structuralism, and modern perspectives on the foundations of mathematics.
Simple type theory, dependent types, Curry-Howard correspondence, proof assistants, and formalized mathematics.
Constructive semantics, proof interpretation, differences from classical logic, Kripke models, and applications in computation.
Proof theoretic ordinals, transfinite induction, strength of theories, applications to arithmetic, and limits of formal strength.
Consequences of incompleteness for truth, provability, independence, and the structure of formal systems.
Stronger forms of incompleteness, Rosser’s improvement, Löb’s theorem, and connections to computability.
Construction of a true but unprovable statement using diagonalization and self-reference.
Proof that no sufficiently strong consistent system can prove its own consistency.
Arithmetization of syntax, the first and second incompleteness theorems, implications for formal systems, and refinements.
Encoding symbols, formulas, and proofs as natural numbers to allow arithmetic to reason about its own syntax.
Normalization of proofs, elimination of detours, and structural simplification of derivations.
Formal notion of deriving formulas from assumptions, including structural properties and inference behavior.
Formal structure of proofs, including derivations, inference rules, axioms, and proof representations.
Syntax of proofs, derivability, normal forms, consistency proofs, proof length, and proof complexity.
Methods for proving consistency of formal systems, including syntactic and semantic approaches.
Quantitative study of proofs, including proof size, efficiency, and connections to computational complexity.
Sets that can be enumerated by algorithms and their role in semi-decidability and computability theory.
Existence of intermediate degrees between computable sets and the halting problem.
Formal methods for comparing decision problems using many-one and Turing reducibility.
Global properties of the Turing degrees including incomparability, density, and jump structure.
Equivalence classes of sets under Turing reducibility and the ordering of computational power.
Reducibility, Turing degrees, recursively enumerable sets, Post's problem, and the structure of degrees.
Extensions of undecidability using reductions and general results such as Rice’s theorem.
The undecidable problem of determining whether a Turing machine halts on a given input.
Step-by-step evolution of Turing machine configurations and how computations are represented as traces.
Machines that simulate any other Turing machine, establishing the concept of programmable computation.
Formal definition of Turing machines, including states, tape, alphabets, and transition functions.
Turing machine definitions, computation traces, universal machines, the halting problem, and undecidability results.
Detailed equivalence proofs between Turing machines, recursive functions, and lambda calculus, with explicit constructions and simulations.
Turing machines, register machines, lambda calculus, recursive functions, and the precise mathematical models used to define computation.
The Church Turing thesis, formal models of computation, equivalence of models, and the distinction between mathematical theorem and foundational principle.
Recursive functions, partial and total functions, the Church-Turing thesis, formal models of computation, and equivalence of models.
Primitive recursive functions, general recursive functions, minimization, and the formal construction of computable numerical functions.
Distinction between partial and total computable functions, undefined values, domains of definition, and the role of nontermination.
Applications of advanced set theory to analysis and topology, including regularity properties, Banach spaces, measure theory, and topological classification.
An introduction to Polish spaces, Borel sets, analytic sets, projective sets, regularity properties, and the role of definability in set theory.
An introduction to infinite games, determined games, the axiom of determinacy, projective determinacy, and consequences for sets of reals.
An introduction to forcing, generic filters, forcing names, the forcing relation, and the basic extension theorem.
An introduction to large cardinal axioms, inaccessible cardinals, measurable cardinals, elementary embeddings, and consistency strength.
Forcing, large cardinals, descriptive set theory, determinacy principles, and applications in analysis and topology.
Independence results in set theory, including the axiom of choice and the continuum hypothesis, and the methods used to establish independence.
The constructible universe, definable subsets, the hierarchy L_alpha, and the axiom of constructibility.
The axiom of choice, choice functions, indexed families, and first consequences in axiomatic set theory.
Equivalent forms of the axiom of choice, including Zorn's lemma, the well ordering theorem, maximal principles, and right inverses of surjections.
Relative consistency, inner models, constructibility, and the role of consistency results in axiomatic set theory.
Axiom of Choice, equivalent formulations, constructible universe, consistency results, and independence phenomena.
The axioms of Zermelo Fraenkel set theory, the role of choice, and the use of axioms as a foundation for mathematics.
Cardinal addition, multiplication, exponentiation, finite and infinite cardinal arithmetic, and basic comparison laws.
Cardinality, finite and infinite sets, countable sets, uncountable sets, and Cantor diagonal arguments.
Basic set theoretic language, including sets, membership, subsets, operations, relations, equivalence relations, order relations, and functions.
Well ordered sets, order isomorphisms, ordinals, successor ordinals, limit ordinals, and transfinite induction.
Basic set theoretic notions including sets, relations, functions, cardinality, ordinals, well ordering, cardinal arithmetic, and the ZF and ZFC axioms.
Detailed introduction to stability theory, counting types, order property, definability of types, and structural consequences.
Definition of definable sets and functions in first order structures, with parameters, examples, closure properties, and proofs.
Definition of complete and partial types, realization of types in structures, examples, consistency, and basic properties.
Saturated models, realization of types, and their role in controlling definability and extensions.
Detailed overview of classification theory, dividing lines such as stability, simplicity, and NIP, and the structural analysis of first order theories.
Definable sets, definable functions, types, realizations, saturated models, stability theory, and classification programs.
Applications of compactness and Lowenheim Skolem to algebraic structures and existence results.
Expressive limitations of first order logic, including inexpressibility of finiteness and categoricity issues.
Detailed development of the compactness theorem, its proof via completeness, and fundamental applications in model theory.
Downward and upward Lowenheim Skolem theorems and their consequences for model sizes in first order logic.
Construction and properties of nonstandard models using compactness and Lowenheim Skolem.
Compactness, completeness, Lowenheim-Skolem theorems, nonstandard models, and limitations of first order logic.
Isomorphisms of first order structures, structural invariants, and properties preserved by isomorphism.
Examples of first order structures from algebra, order theory, graph theory, and geometry.
Formal languages, signatures, and symbols used to describe structures in first order logic.
Elementary equivalence, theories of structures, and preservation of first order sentences.
Basic model theoretic notions including languages, signatures, substructures, embeddings, elementary equivalence, isomorphism, and examples.
Substructures, generated substructures, homomorphisms, embeddings, and preservation of atomic formulas.
The cut rule, its elimination, and consequences for consistency and normalization.
Transformations of proofs, normalization, and structural properties of derivations.
Introduction to natural deduction, inference rules, and structured proofs for propositional logic.
Sequents, structural rules, and introduction rules for logical connectives in the sequent calculus.
Formal systems for deriving logical conclusions including natural deduction, sequent calculus, Hilbert systems, and proof transformations.
Hilbert style proof systems, axioms, and derivations using a minimal set of inference rules.
Satisfaction, truth in a structure, models of sentences, and theories in first order logic.
Validity, semantic entailment, satisfiability, countermodels, and logical consequence in first order logic.
Universal and existential quantifiers, scope, free variables, bound variables, and variable capture.
Structures, domains, and interpretations of symbols in first order logic.
Syntax of first order logic including terms, predicate symbols, and the formation of formulas.
Soundness, completeness, and the relationship between semantic validity and formal provability.
Conjunctive normal form, disjunctive normal form, and systematic conversion of propositional formulas.
Extension of propositional logic with terms, predicates, quantifiers, structures, satisfaction, models, validity, and entailment.
Truth values, valuations, and evaluation of propositional formulas using truth tables.
Definition of propositional variables, logical connectives, and formation rules for well formed formulas.
Foundations of propositional logic including syntax, semantics, equivalence, normal forms, and proof systems.
Logical equivalence, truth preserving transformations, and basic laws for rewriting propositional formulas.
Overview of mathematical logic, its scope, and the structure of the book.
How numbers move from concrete counting to abstract ideas.
Practical rules for writing mathematics in a clear, consistent, and readable way.
Early counting through marks, objects, and physical recording systems.
Common mistakes in mathematical writing and how to avoid them.
Purpose, scope, and approach of this volume on the history and biography of mathematics.
How early humans developed counting, measurement, and basic mathematical thinking before formal notation.
How a mathematical paper or article is organized so that readers can follow the main ideas.
How to write mathematics with enough detail, few distractions, and clear logical structure.
How definitions, theorems, and proofs work together in mathematical writing.
Understanding the difference between manipulating exact mathematical expressions and computing with numerical values.
Overview of how to write mathematical ideas clearly, precisely, and in a useful structure.
How to make computational results repeatable, checkable, and trustworthy.
When to compute exact results and when to use approximations.
How to think step by step and turn mathematical ideas into clear procedures.
Understanding how the cost of an algorithm grows with input size.
Overview of algorithmic thinking, computational methods, complexity, approximation, and verification in mathematics.
Using failures, boundary conditions, and extreme cases to test, refine, and understand mathematical statements.
Using structural similarity between problems to move ideas, methods, and proofs across domains.
Using examples, informal rules, and exploratory computation to guide mathematical problem solving.
Using counting, random choice, and finite structure to prove identities and existence statements.
Overview of general methods used to approach, transform, and solve mathematical problems.
Expanding or restricting a problem to reveal structure and guide solution.
Solving a problem by converting it into a simpler, known, or more structured form.
Proving a statement by assuming its negation and deriving an impossibility.
Proving a statement by starting from its assumptions and deriving its conclusion step by step.
Proving existence by giving explicit witnesses, algorithms, or methods of construction.
Overview of the main methods used to prove mathematical statements.
Using base cases and step rules to prove statements about objects built recursively.
Defining objects step by step and proving properties by following the same construction.
How mathematics studies small pieces first and then assembles them into statements about the whole.
How transformations preserve structure and how invariants record what remains unchanged.
Breaking complex objects into simpler parts and building larger structures from controlled combinations.
Overview of recurring patterns such as duality, symmetry, local-to-global reasoning, decomposition, recursion, and induction.
Understanding how reversing structure reveals parallel theories and results.
Understanding the benefits and costs of abstraction, and choosing the right level for mathematical work.
Raising abstraction from objects and operations to maps, composition, and universal properties.
Studying mathematical systems themselves through languages, axioms, models, proofs, and interpretations.
Replacing concrete values with symbols and rules to express general patterns.
Working with explicit examples, calculations, and finite procedures as the base level of mathematical reasoning.
Overview of how mathematics moves from concrete computation to structural and higher-level reasoning.
Concrete examples showing structural thinking across algebra, topology, and graph theory.
How preserved quantities and properties support comparison, classification, and structural reasoning.
Structure-preserving maps, their role in comparison, composition, and transport of mathematical information.
Distinguishing abstract structures from their concrete instances, and using that distinction to reason across examples.
How isomorphism formalizes structural sameness and separates equality from equivalence.
How mathematical writing balances exact statements with readable exposition.
How formal precision and informal readability work together in mathematical writing.
How mathematical symbols and notation are chosen, scoped, reused, and designed for precision and readability.
Viewing notation as a designed interface that exposes structure, supports composition, and enables efficient reasoning.
How definitions introduce mathematical objects, fix meaning, and support reusable reasoning.
Overview of structures, mappings, invariants, and classification in mathematics.
Overview of abstract objects, structures, equality, finiteness, and viewpoints in mathematics.
How truth, provability, consistency, completeness, and independence appear across major branches of mathematics.
Overview of symbols, notation, definitions, and the balance between precision and readability.
Distinction between semantic truth and syntactic provability, with examples and limits.
Syntax, axioms, inference rules, and the semantic interpretation of mathematical languages.
Core meta-properties of formal systems: avoiding contradiction and deciding statements.
Statements that cannot be proved or refuted from a chosen axiom system, and what independence means in mathematical practice.
Comparison of constructive and classical mathematics, including existence, proof, logic, and computation.
Distinction between finite and infinite objects, methods of reasoning, and consequences across mathematics.
Different notions of sameness in mathematics: strict equality, structural identity, and equivalence relations.
Overview of truth, provability, formal systems, and independence in mathematics.
How mathematics treats objects through the rules they satisfy, the relations they support, and the transformations that preserve them.
Three ways to organize a domain of discourse for mathematics: sets, types, and universes — and how they relate.
How this volume defines the ground layer of mathematics: language, structure, and method before specialization.
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