brain

We are asked to unlock a circular lock with n wheels. Each wheel has a number from 0 to m-1. The lock has a hidden target combination, and we can query the lock with a proposed sequence of wheel positions.

We are asked to help a sheep maximize the "fortune" of a collection of numbers it writes. The sheep must write exactly $k$ non-negative integers that sum to a target $n$. Each digit of each number has an associated fortune, given in a table for digits $0$ through $5$.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

We are given a sequence of fruits represented as a binary string, where 1 stands for an apple and 0 for an orange. The goal is to compute, over all possible contiguous substrings, the length of the longest consecutive apples in that substring and sum these values.

A box sits on a grid point and needs to be moved to another grid point using a very specific interaction with a mover. The mover can walk freely in four directions, paying one second per unit step, but he is not allowed to occupy the box’s cell.

We are given a string consisting only of the characters A and B. We may repeatedly remove any adjacent substring equal to AB or BB. After removing such a pair, the remaining parts of the string join together, potentially creating new removable pairs.

The rooms form a cycle. Between every pair of consecutive rooms there is a conveyor belt, and each belt can be directed clockwise (), directed counterclockwise (<), or usable in both directions (-).

We start with the integers from 1 through n written on a whiteboard. In one operation we choose any two numbers currently on the board, erase them, and write back the value ceil((a+b)/2). After exactly n-1 operations only one number remains.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

We are given a sequence of monster waves that occur over time in a fixed order. Each wave arrives at a specific time interval and spawns a known number of monsters instantly at its start time.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

We are given a binary string of length $n$, and the task is to perform a sequence of operations until the string becomes empty.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

The building is described as being composed of three fixed apartment types, and each type contributes a fixed number of windows. A three-room apartment contributes 3 windows, a five-room contributes 5, and a seven-room contributes 7.

We are tasked with scheduling a set of lectures in an auditorium, each of which will use a shared marker. Each lecturer has a personal tolerance: they will refuse to use a marker that has been used a certain number of times, asking for a new one instead.

Codeforces 1431I: Cyclic Shifts

We are given a sorted array a of length n. Between every consecutive pair of elements in a, we must insert exactly one additional integer, producing a longer array b of length 2n - 1.

We are given a grid of rewards where each run in the game is defined by choosing one species and one class. Executing a run with a pair $(i, j)$ gives a score $c{i,j}$, and this score accumulates over time. At the beginning, only some species and classes are already available.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

Let the two circles be $\Gamma_1$ and $\Gamma_2$, with centers $O_1$ and $O_2$.

We are given a collection of distinct integers. The game is played in a sequence of rounds, and each round removes two numbers and converts their difference into score.

We are given two teams of equal size, each player having a fixed skill value. A match is formed by pairing every player from the first team with exactly one distinct player from the second team, so the pairing is a permutation of indices of the second team.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

We are asked to process Polycarp’s notes written in the VwV language, which only contains letters 'v' and 'w'. The problem arises because a single 'w' can be visually identical to two consecutive 'v's, and consecutive 'v's can themselves form a 'w'.

We are asked to find the optimal price for selling hamburgers to a set of customers, each with a specific amount of money. Each customer will buy a hamburger if and only if its price does not exceed the money they have.

We are asked to identify a dominant piranha in a linear aquarium. Each piranha has a size, and a piranha is dominant if it can eventually eat all other piranhas by repeatedly consuming an adjacent piranha smaller than itself. Every time it eats, its size increases by one.

We are asked to minimize the total delivery cost for couriers traveling along cheapest paths in a city network. The city is modeled as an undirected weighted graph with n districts as nodes and m two-way roads as edges, each with a positive cost.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

We are given a grid of numbers with $n$ rows and $m$ columns. From each row, we are allowed to pick some elements, but with a strict cap: in any single row we cannot pick more than half of its elements, rounded down.

We are given several independent scenarios. In each scenario there are $n$ districts, and each district belongs to some gang identified by an integer label.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

We are given a bookshelf represented as a binary array. A value of 1 means a book is present at that position, while 0 means the position is empty.

We are given several independent games. Each game consists of a strictly increasing array of integers. Two players alternate moves, and on each move a player selects one element from the array.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

We are given a multiset of integers, where each distinct value appears with a given frequency. From this multiset, we consider all pairs of subsets $A$ and $B$ such that $B$ is formed by removing exactly one element from $A$.

The problem presents an array of integers and asks whether it can be reordered so that a specific weighted sum matches a given target value.

We are given a rooted tree with node 1 acting as the starting point of a bandit. Every edge is directed away from the root in such a way that from node 1, every other node is reachable, so the structure is effectively a rooted tree.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

We are asked to count permutations of size n where a particular number x is placed at a fixed index pos, and the standard binary search procedure, as described in the problem, successfully finds x at that position. A permutation here is any ordering of the numbers 1 through n.

We are given an array of integers and a set of positions that are “locked,” meaning we cannot change the values at those indices. The task is to transform the array into a strictly increasing sequence while changing the minimum number of elements outside the locked positions.

We are given a permutation of the vertices of a tree as they were visited by a breadth-first search starting from the root. The root is always vertex 1, and all children of a vertex are visited in increasing order.

We are given a collection of dishes, each associated with an “ideal” time at which it should be removed from the oven. Time in this problem is discrete and increases one minute at a time.

Ah, now I see exactly what is happening. The ValueError: not enough values to unpack occurs because the previous parsing assumed that every test case’s first line has exactly three numbers (n, k, m).

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

We are given a perfect binary tree of height $h$, which contains $n = 2^h - 1$ nodes. Each node has a unique label from 1 to $n$, but the labeling is completely unknown. Our task is to determine the label of the root.

We are given a sequence of positive integers, and we are allowed to perform a very specific transformation: pick three distinct positions and replace all three values by their bitwise XOR.

The key issue is that this is a constructive problem. The output shown in the statement is only one valid sequence of operations, not the unique correct output. Your program's output is different from the sample output, but that does not mean it is wrong.

The problem asks us to construct arrays with a very strict divisibility property. For an array to be called perfect, every subarray, regardless of length or position, must have a sum divisible by its length.

We are given a grid of zeros and ones with dimensions n by m. Each cell can either be off (0) or on (1). Our goal is to turn all the ones into zeros using a specific operation.

The game is played on a very large grid, but only a very special subset of cells is relevant. A cell $(x, y)$ is considered valid only when the bitwise condition $x & y = 0$ holds.

Ah, now the issue is clear. The previous solution is not implementing the correct logic for the minimum time. The code I provided only considered taking a fixed number k of personal pickups in descending order of b[i], which is not what the problem asks.

We have a line of $n$ computers and $m$ participants, each with a preferred computer and an entry direction. Each participant wants to sit at a specific computer, but if it is occupied they continue moving in the direction they entered until they find a free computer.

We are given a sequence of shops arranged in a line, each with a fixed meal price. The prices form a non-increasing array, so each shop to the right is never more expensive than its left neighbor. There are two types of operations we need to support.

Consider triangle $ABC$ in the plane.

The grid contains only two possible states per cell, and the goal is to eliminate all ones by applying a very specific local operation. Each operation looks at a single 2 by 2 block and flips exactly three of its four cells.

We are given a sorted list of $nk$ numbers and asked to split it into $k$ groups, each containing exactly $n$ elements.

We are given a binary string and two ways to pay for it. The first is direct purchase: every character 0 costs c0 coins and every character 1 costs c1 coins.

Thank you. Now I see exactly why the previous code is producing the wrong output. Let’s go carefully. The code currently reads each line and assigns: It prints the first number of each line. That explains why it prints 10 and 12 for the first two lines.

We are dealing with an interactive combinatorial game on a directed graph. Each vertex may hold a token, and two players alternate moves. On a turn, a player chooses any token and moves it along an outgoing edge. If a player cannot move any token, they lose.

We are given a tree in which each node is coloured white, black, or grey. The goal is to remove all nodes in the minimum number of operations, where in each operation we select a connected component of nodes to remove.

We are given several sequences, each already sorted in non-decreasing order. We repeatedly perform an operation where we choose one sequence, take its current first element, add it to our total, and remove that element from the sequence.

We start with a permutation stored in a line. At each step, we remove one element from the current line and, depending on where we removed it, we are forced to append one of its immediate neighbors (left or right, whichever exists at that moment) into a second sequence.

We are given an array of positive integers and a very specific way to reduce it. Each operation picks a prefix or a suffix, and decreases every element in that chosen segment by exactly one.

In this problem, Petya wants to get all his birthday dishes in the minimum amount of time. For each dish, he can either pick it up himself from a restaurant, taking bi minutes, or order a delivery, which will arrive in ai minutes.

We are given the initial permutation of integers from 1 to $n$ in order. Queries ask either for the sum of elements in a subarray of the current permutation or to advance the permutation by a given number of next-permutation steps.

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 $x ge 0, y ge 0$.