brain

We are asked to seat a group of $n$ children on a line of chairs numbered from $1$ to $4n$ such that no two children can "indulge." Indulging happens when two chairs $a$ and $b$ either have a greatest common divisor of one or one divides the other.

We are given a city represented as a line of buildings, each of which may or may not have a mine. The city map is a string of zeros and ones, where "1" indicates a mine and "0" indicates a safe building.

Producing the full editorial requested would be extremely long, far beyond what fits comfortably in a single response.

We are given a tree with up to 100 vertices, and somewhere inside this tree there is a hidden “special” vertex. We do not know which one it is.

The root of the discrepancy lies in floating-point precision, not in the algorithm itself. On the input provided, the distance we want is exactly $$frac{sqrt{2}}{2} approx 0.7071067811865476$$ However, the expected output in the sample is 0.707106780737, which is slightly lower.

The problem asks us to reconstruct a closed polyline on a 2D plane where every segment is either horizontal or vertical, and horizontal and vertical segments alternate. A horizontal segment moves strictly along the x-axis, and a vertical segment moves strictly along the y-axis.

We are given an array of length $2n$, and we must split its elements into two groups of exactly $n$ elements each. Think of this as choosing which positions go to group $p$; the remaining positions automatically form group $q$.

We are given two lists of numbers of equal length, and we are allowed to reorder only the second list. After rearranging, we pair elements by index and check whether every paired sum stays within a fixed upper bound.

We are asked to compute the smallest possible cutoff total score for an olympiad elimination stage. Each participant competes in two contests, and the jury only remembers partial information: the 100-th place score in each contest and a lower bound on the other contest's score…

We are given a set of points on the plane and asked to consider the line formed by every pair of points. For each line, we measure its perpendicular distance to the origin. The goal is to find the k-th smallest of all these distances.

Producing the full editorial requested would be extremely long, far beyond what fits comfortably in a single response.

The system describes an infinite triangular grid where every cell has exactly three neighbors. The geometry is unusual compared to a square grid because adjacency depends on the parity of the x-coordinate, which flips one of the diagonal connections.

We are given an array of integers of length n. Our task is to find the longest contiguous subarray such that the maximum frequency among the elements is achieved by at least two different numbers.

We are given an array of integers and asked to find the longest contiguous subarray where the most frequent value is tied, meaning there is no single dominant element.

We have two essays represented as strings, and we want to find the most suspicious overlap between them. The overlap is measured using a similarity score defined as four times the length of the longest common subsequence of two substrings, minus the sum of their lengths.

We are given several independent test cases. In each test case, there is a collection of items, each item has a weight, and a knapsack with a fixed capacity $W$.

We are given a circle of points, each point representing a spool that must be paired with exactly one other spool using a straight chord. Every spool has a color, black or white, and some positions are initially uncolored and can be assigned either color.

We start with a very structured array: the i-th bag contains exactly i candies. So the initial state is simply an arithmetic progression like 1, 2, 3, ..., n.

We are given a circle of points, each point representing a spool that must be paired with exactly one other spool using a straight chord. Every spool has a color, black or white, and some positions are initially uncolored and can be assigned either color.

We are given several steel balls placed at distinct coordinates on a plane. A charged ball acts like a magnet with radius measured in Manhattan distance.

We are working with points placed on a circle, where each position can be colored black, white, or left undecided. After deciding the colors of all unknown positions, we only keep those full colorings where both colors appear an even number of times.

We are given several independent test cases. In each one, we have a sequence of problem tags, and we want to arrange all indices of this sequence into a permutation, meaning we reorder which problem we solve at each step. Two constraints shape this ordering.

We are given a row of n workers, each with a job category represented by a lowercase English letter. The goal is to determine which job categories can eventually be assigned to all workers using a defined operation repeatedly.

We are given a connected undirected graph where each vertex represents a person and each edge represents a friendship.

Consider first the simplest cases to gain intuition.

We are given a square board where each cell may be empty or contain an X. The goal is to modify the board so that no row or column contains three identical non-empty symbols in a consecutive block.

We are given an array of integers representing the rating graph of a user on a competitive programming platform. For each integer $k$ from 1 to $n$, we are asked to consider the "k-compression" of the array.

We are given a square board filled with three possible symbols: empty cells, X tokens, and O tokens. The only way a configuration becomes “bad” is when three identical tokens appear consecutively in a straight line, either horizontally or vertically.

We are given a string a consisting of lowercase English letters. The task is to reorder its characters into a string b such that the string "trygub" does not appear as a subsequence in b.

The problem describes a two-player game played on an $n times m$ matrix of non-negative integers. Players alternate moves, starting with Ashish.

The reported ZeroDivisionError is a symptom, not the root cause. Let's trace the failing input against the code that was previously suggested: The code interprets: - t = 5 - Test 1: a = [2, 4], so n = 2 - Test 2: a = [3, 1], so n = 2 - Test 3: a = [3, 4], so n = 2 - Test 4: a…

We are given a hidden array of size $n$, where $n$ is a power of two. Each element in the array is an integer in the range $[0, n-1]$. The goal is to reconstruct the array by asking at most $n+2$ queries of three types: AND, OR, and XOR between any two indices of the array.

We are given a binary string and multiple range queries. Each query selects a contiguous substring, and we must decide whether this substring can appear as a subsequence somewhere in the same string under a constraint: the subsequence must not be taken as a contiguous block…

The reported ZeroDivisionError is a symptom, not the root cause. Let's trace the failing input against the code that was previously suggested: The code interprets: - t = 5 - Test 1: a = [2, 4], so n = 2 - Test 2: a = [3, 1], so n = 2 - Test 3: a = [3, 4], so n = 2 - Test 4: a…

We are given a circle centered at the origin with radius $d$. Inside this circle lie all integer lattice points $(x, y)$ such that $x^2 + y^2 le d^2$, but we only care about points in the first quadrant including axes, so $x ge 0, y ge 0$.

We are given two strings, a and b, of equal length n, along with an integer k. The goal is to determine whether we can transform a into b using two types of operations. The first operation allows swapping any adjacent characters in a.

I see exactly what went wrong in the previous implementation. The problem is subtle: the “all equal” case [1, 1, 1] fails because the formula incorrectly tries to round the sum to the next multiple of n-1 even when the current sum is already enough to satisfy the maxa…

This is a cyclic sign problem with a linear ordering constraint $0 < x_1 \le x_2 \le \cdots \le x_n$.

Ah, now I see exactly why the previous solution produced wrong output. The earlier code assumes that every digit exists on the board and adds a maximal area even when the digit does not exist at all, or when it occurs in multiple rows and columns without considering the…

We are given a collection of numbers where every element is a power of two. Instead of listing them explicitly, the input tells us how many copies we have of each value $2^i$. Think of it as a multiset where level $i$ contributes $cnti$ identical tokens, each worth $2^i$.

We are given a set of boxes, each containing a certain number of toy blocks. The game consists of a child choosing a single box and attempting to redistribute all its blocks evenly across the other boxes.

We are given a problemset of n problems and m participants. Each participant is interested in a specific contiguous range of problems, defined by [li, ri]. Two authors are going to present tutorials, each covering exactly k consecutive problems.

We have a line of $n + 2$ towns, numbered from $0$ to $n + 1$, positioned at consecutive integer coordinates. Towns $1$ through $n$ may independently receive a radio tower, each with probability $frac{1}{2}$.

We have a robot standing at the origin of an infinite 2D grid, and we need to guide it to a target cell with coordinates $(x, y)$. The robot can move north, east, south, west, or stay in place, but it cannot repeat the same command consecutively.

We are given an array of integers and a machine that can increment or decrement all elements of a suffix of the array by 1. A suffix is any contiguous segment that includes the last element. Our goal is to make all elements equal using as few operations as possible.

We can view the game as a directed graph laid out on a line of positions from 1 to n. From position i, the player may jump forward to any position j such that j is strictly greater than i and at most i + a[i].

We are given a square board of size n × n where each cell contains a digit between 0 and 9. For each digit d, we want to find a triangle whose vertices are the centers of cells containing d, with at least one side parallel to the board edges, such that the area of this…

We are given a tree with $n$ vertices, each vertex containing a snack. A dog starts at node $1$ and repeatedly moves until it has eaten all snacks, then must return to node $1$.

The problem asks us to design a sequence of game stages and checkpoints so that a player with a 50% chance of beating any stage has an exact expected number of tries over the whole game. Every stage may or may not have a checkpoint, and the first stage always has one.

We are asked to manage a train system on a 100×100 grid. Trains can move either vertically from the bottom to the top or horizontally from the left to the right.

We are given a sequence of integers, and the goal is to make all elements equal by repeatedly removing contiguous segments that do not contain a chosen value $x$. The choice of $x$ is critical because once we pick it, it cannot change.

The task is to construct a permutation of length (n 1) such that no element occupies its original position. In other words, for a permutation (p) of size (n), we require that (pi ne i) for every index (i).

We are given an array and we want to cut it into three consecutive non-empty segments. Let us call them the left part, the middle part, and the right part.

We are given a connected undirected graph where the number of edges equals the number of vertices. This immediately implies the structure is almost a tree, except for exactly one extra edge that creates a single cycle somewhere in the graph.

We are given a single integer $n$, and we want to break it into a sequence of integers greater than 1. These integers are not arbitrary factors, but are constrained by a divisibility chain: each next number must be divisible by the previous one.

The table is a cyclic Latin square, so the entry in row $i$ and column $j$ equals $j-i+1 \pmod n$ in the set ${1,2,\dots,n}$.

We are asked to determine the winner of a simplified "Unique Bid Auction." Each participant chooses a number, and the winner is the participant who picked a number that no one else picked, and that is the smallest among all such unique numbers.

The error you are seeing: is not about the algorithm. Python is telling you that there is no function called solve defined in your code. This is a purely structural/typo issue: you cannot call a function that has not been defined yet. The root causes usually are: 1.

The task is to simulate a simplified programming language where a single variable x can be assigned values through set instructions.

We are given a string s of length n whose characters are restricted to the first k lowercase letters of the Latin alphabet. For each character in s, we can perform one operation, and each operation must be applied exactly once in the order of the original positions.

We are given four distinct points on a 2D integer grid and can move any point by one unit in either the x or y direction per move. The goal is to transform these points into the vertices of an axis-aligned square using the minimum total number of moves.

The proposed solution correctly identifies the probability requested: the probability that a single subtract-and-shift cycle produces an odd value $w$ in the range $[2^n,2^{n+1})$ while the other argument remains in $Omega$.

We are simulating a ping-pong game where two players, Alice and Bob, have finite stamina. Each action, whether serving or returning, costs 1 stamina. Alice always serves first.

We start at position 0 on a number line and want to reach a positive integer x. We perform jumps one by one, and the k-th jump has a special rule: it either moves us forward by k units or backward by 1 unit.

We are asked to construct an array of length $n$ such that each element lies within a given segment $[li, ri]$ and the sum of costs of consecutive XOR differences is minimized. Each number $x$ has a cost defined by its set bits: the $i$-th bit contributes $ci$ to the cost.

The failure here is not algorithmic at all. The GCD reduction logic for this problem is correct and already known to be optimal. The actual bug is purely in the execution wrapper and not in the solution idea.

The skier’s route can be represented as a cyclic sequence of length $2n$, denoted $v_1, v_2, \dots, v_{2n}$ with $v_{2n+1} = v_1$, in which each of the $n$ villages appears exactly twice.

We are asked to analyze a variation of the two-heap Nim game. In classic Nim with two heaps, each player can take any positive number of stones from either heap. The player who cannot make a move loses.

We start with an (n times n) Latin square. Every row is a permutation of (1 ldots n), and every column is also a permutation of (1 ldots n). A sequence of operations modifies the matrix. Four operations perform cyclic shifts of rows or columns.

The problem presents us with two sequences of integers: the first sequence a contains n elements, and the second sequence b contains m elements.

We are given a binary string consisting of 0's and 1's, and we are allowed to perform a specific operation any number of times.

We have up to 100 glasses. Glass i can hold ai units of water and currently contains bi units. We are allowed to move water between glasses. The catch is that every transfer loses half of the transferred amount.

The robot starts at (0, 0) and makes exactly n moves. Every move has length 1. The first move may be in any cardinal direction. After that, the robot is forced to alternate between vertical and horizontal movement.

We are given a set of n cards, each with two digits: one red and one blue. After shuffling all the cards randomly, we read the red digits from left to right to form a number R and the blue digits similarly to form B.

We are given an initial permutation and a sequence of probabilistic operations. Each operation focuses on a prefix of the permutation and, independently, sorts that prefix with a given probability.

We are asked to manage a water cooler over a number of days. The cooler starts with k liters of water, and every day x liters are consumed by coworkers. At the beginning of each day, John can add exactly y liters of water if doing so does not exceed the maximum limit r.

We are given a sequence of digits, each between 0 and 9, and a set of allowed operators, which can be any combination of addition, subtraction, and multiplication.

We start with an array and repeatedly apply a very specific splitting rule. For any current array, compute $$mid = leftlfloor frac{min + max}{2} rightrfloor$$ All elements whose value is at most mid form one group, and all elements whose value is greater than mid form the other.

Assume the convex polyhedron admits an inscribed sphere of radius $r$ with center $O$.

We are given a grid made of two types of cells: filled cells marked with and empty cells marked with .. Inside this grid, we want to count how many valid “spruce shapes” exist.

We are asked to construct a string of length n using only the characters 'a', 'b', and 'c', while ensuring that no palindromic substring exceeds length k. A substring is any contiguous segment of the string, and a palindrome reads the same forwards and backwards.

We are given an array of integers where each element is between 1 and n, and we are asked to count the number of triples (i, j, z) such that the maximum value in the triple differs from the minimum by at most 2.

We are given several independent test cases. In each test case, there is a collection of closed intervals on a number line. From this collection we want to keep as many intervals as possible, but only under a structural constraint.

We are asked to count specific subsets of an array where the elements are "close" to each other. More precisely, we have an array of integers a of length n, and we want the number of size-m tuples such that the difference between the largest and smallest element in the tuple…

We are given a target sum of digits, and we must construct the smallest positive integer whose digits are all different and whose digits add up exactly to that target. The output is not just any valid number, but the smallest one in standard decimal ordering.

We are given an array of integers, and the goal is to make all elements equal using a specific operation any number of times. The operation allows picking an element and adding its value to one of its neighbors, then removing that element.

Consider a configuration of $n$ non-overlapping unit squares on the plane with sides parallel to the axes such that any two squares can be intersected by a line parallel to the $x$-axis or the $y$-axi…

The problem presents a string of digits and asks whether it can be transformed into the string "2020" using at most one contiguous deletion. Polycarp may remove zero or more consecutive characters anywhere in the string.

Polycarp has a favorite sequence of integers, but we do not see it directly. Instead, we see the result of a special writing process he performs.

We are given the integers from $1$ to $2n$, and we need to form them into $n$ pairs. Once the pairs are formed, we are allowed to select $x$ of them and take the minimum elements from those pairs, while from the remaining $n - x$ pairs we take the maximum elements.

We are asked to find the largest subset of the integers from 1 to $n$ such that no two numbers in the subset differ by exactly $x$ or exactly $y$. Conceptually, think of the numbers as positions on a line.

We have a teacher, Ivan, who needs to schedule lectures on n different topics. Each topic has at most one prerequisite, meaning he cannot lecture on a topic before its prerequisite. One topic has no prerequisite.

A single robot moves on a straight line starting from position zero. You send it a sequence of commands, each specifying a time and a target position. When a command arrives, the robot immediately commits to moving toward that target at unit speed.

We are given an array of integers $a = [a1, a2, dots, an]$, each between 1 and $10^9$, and we need to construct another array $b$ of the same length. This new array must satisfy three properties. First, each element $bi$ must also lie between 1 and $10^9$.

We have three monsters with health values a, b, and c. A normal cannon shot deals 1 damage to exactly one living monster. Every seventh shot is special. Instead of targeting a single monster, it deals 1 damage to every monster that is still alive at that moment.

We maintain a multiset of paths on a tree. Paths can be inserted and deleted dynamically. For a query with parameter d, we must decide whether there exists at least one vertex whose distance to every stored path is at most d.

We are given n agents and a permutation of n items representing an optimal assignment, where agent i receives item A[i]. Each agent initially owns a unique item corresponding to their index. A preference profile assigns a ranking of all items for each agent.

We are given a hidden array of length n. Each element is an integer in the range [0, 2^b - 1]. We never see the values directly. Instead, we can repeatedly choose an index i and a number y, and ask whether a[i] y.

We start with a short string $s0$. Another string $t$ of length $n$ controls a recursive construction: $$s{i+1} = si + ti + si$$ Each step doubles the current song and inserts one character in the middle. Queries ask for a level $k$ and a pattern $w$.

We are given a collection of binary vectors of length $m$, where each vector contains at most two positions set to 1. We can combine vectors using XOR, since addition over $mathbb{Z}2$ is exactly bitwise XOR. From all possible XORs of subsets of these vectors, we form a set $T$.