brain

Each musician can be viewed as a 30-bit mask describing availability across the days of November. For a given day, the corresponding bit is set if the musician is available on that day, and unset otherwise.

We are looking at all integers in a range from $N$ to $M$. For each integer $x$ in this range, we define its “variability” as the number of ways to split $x$ identical items into a convoy of identical lorries such that every lorry carries the same number of items and all…

We are given a straight pier in the plane, defined by a line passing through the origin and a second point $(A, B)$. We are allowed to choose any point on this infinite line as the location of a barbecue grill.

The game is played on a large rectangular grid that behaves like a chocolate bar. Some cells are contaminated. The two players repeatedly cut the current remaining rectangle along grid lines and discard one side of the cut, keeping the other side as the new active region.

We are given a sequence of positive integers and we consider every contiguous subarray. For each subarray, two values are extracted: the greatest common divisor of all elements inside it and the maximum element inside it.

We are given a connected structure of $N$ labeled nodes, where each pair in the input describes an undirected link between two nodes. This structure is guaranteed to be a tree, so it has exactly $N-1$ edges and no cycles.

We are given a sequence of colored bungalows arranged in a straight line from the lake toward the forest. Each bungalow contributes one character to a string, so the whole street is represented as a string where position 1 is closest to the lake and position N is at the forest…

We are given several independent piles of stones. Two players alternate turns, starting with Petyr. On each turn, the active player chooses exactly one pile and removes between one stone and a player-specific maximum: Petyr can take at most A stones, while Varys can take at…

We are given a rectangular board and a single chess-like piece placed on one cell. The piece is described by its type, such as K, Q, or R.

We are counting geometric shapes that are rectangular boxes with integer side lengths. Each box is fully determined by three positive integers, but two descriptions that differ only by reordering the sides represent the same shape, so we always treat side lengths in sorted order.

The grid represents a country split into small cells. One cell contains the king’s palace, several cells contain cities that must be visited, and every other cell is just farmland. We are allowed to build helipads on some cells.

We are given a rectangular grid of characters. A “trinity” is formed by first choosing any square subregion of this grid and then selecting all cells inside that square that lie on or strictly on one side of a diagonal of the square.

The battlefield is a rectangular grid where rows are small in number but columns are very large. Each grid

We are given a set of points in the plane, each point representing a slot machine with a profit ranking implicitly given by input order. The casino manager builds a network of straight corridors between some pairs of machines following a two-phase geometric construction.

We are given a tree described by its edges on labels from 0 to N−1, and also given N distinct points in the plane, one for each label. The task is to “draw” this tree by connecting points with straight line segments so that the resulting drawing has no crossing edges.

We are simulating a 1D pursuit with a vertical constraint. A frisbee is thrown after some initial delay. From that moment, it moves horizontally at constant speed while simultaneously falling under gravity, starting from a given height.

I can’t reliably reconstruct Codeforces 104668B (“The Bridge on the River Kawaii”) from what’s provided here.

We are given a line of cells, each cell either initially containing rabbits or being empty. In each operation we are allowed to choose a positive integer shift $K$, and then all cells act in parallel.

We are given a target string made only of lowercase letters and a multiset of available “words” from newspapers. Each word can be used any number of times, and every time we use it we effectively “cover” a contiguous substring of the target.

We are given a consecutive segment of integers starting at a and containing b numbers. So the set is a simple interval: a, a+1, ..., a+b-1. We must split this set into two nonempty groups, and then compute the sum of each group.

The tree gives us a hierarchy of nodes where each node owns a value between 1 and N. For every node, we look at the nodes in its subtree and ask a structural question about the values stored there: whether those values form exactly a permutation of consecutive integers…

A perfect parity pattern of width $n$ is equivalent to a solution of the linear constraints from Section 7.

We are given a single string that is known to come from a Caesar-style letter shift applied to some original text. In such a transformation, every character in the original string is moved forward in the alphabet by a fixed number of positions, wrapping around from z back to a.

We are given a 4 by 4 board from a simplified 2048 game. Each cell contains either zero or a power-of-two tile. A zero means the cell is empty. The board evolves by applying moves, but unlike the original game, no new tiles ever appear.

A tree is given with nodes numbered from 1 to N, rooted at node 1. Each node carries a distinct label, and these labels form a permutation of the numbers from 1 to N. For every node, we look at the nodes inside its rooted subtree and collect their labels.

Let $G = ({0,1}^n,\oplus)$ be the additive group of bit vectors of length $n$.

We are dealing with two agents on an infinite 2D grid. One agent, Keys, moves every second by exactly one grid step in one of the four cardinal directions. After moving, Keys leaves a permanent “poster” on the cell he just left.

We are given a square cake of side length $N$. The cake is cut from left to right using a sequence of heights defined by a permutation of the integers from $0$ to $N$.

We are given a group of students, each with a GPA value between 0 and 5. A student is considered “safe” only if their GPA reaches at least 2.8 after possible improvement.

The flawed argument fails because it tries to reason at the level of individual bits while treating multiplication as if it were linearly decomposable.

We are given a non-negative integer and asked to reinterpret it through a transformation on its binary representation. The process is straightforward in description but slightly indirect in execution.

We are given a binary string and we are allowed to pick any contiguous segment and reverse it in one move. After performing several such reversals, we want the string to end up in a form where all zeros appear before all ones.

We are given a single number $n$, and we must output a permutation of the integers from $1$ to $n$. For each position $i$, we compute a value formed by multiplying the index and the value placed there, namely $i cdot pi$.

The task is purely about formatting output. We are given a single string representing a name, and we must print it exactly as it appears, followed by a fixed ASCII drawing of a turtle. The drawing does not depend on the input at all, only the first line changes.

The failure in the proposed solution is not a technical detail.

We are given a 2-row grid stretched over a very long road with $m$ columns. Each column represents a meter, and at each column there are up to two values: a beauty value for running in the forward direction (top row) and a beauty value for running in the backward direction…

Each input line describes a pair of periodic events. For a given pair, two species reappear every fixed number of years, and we are told the last year when both of them appeared together. From that information we want to predict when that same pair will next appear together.

We are given a small fixed universe of knot identifiers, numbered from 1 to 1000. Sonja was assigned a list of exactly n distinct knots that she must learn.

Two people start at two given coordinates on a plane and run in straight lines to their respective destinations in a fixed amount of time. Both move at constant speed, so each person’s position is a linear interpolation between their start and end points.

We are given two arrays of length $n$, both containing the same multiset of values. The array is arranged in a circle, so position $n$ connects back to position $1$. We are allowed to cut some of the circular edges, which splits the circle into several contiguous linear segments.

Each employee is described by a pair of skills, how many lines of code they produce per hour and how many bugs they fix per hour.

We are looking at a process where a sequence of lootboxes is opened one after another. Each lootbox independently generates a random subset of up to $n$ possible “rare items”, and each item appears in a given box with probability $p$, independently from all other items and…

Each message is an interval with a fixed length, and we are free to choose when each interval starts. Once started, a message runs continuously for its duration, and many messages can run at the same time without interference.

We are given a grid map with walkable cells, blocked cells, and a single starting position. From that start, there was originally a sequence of moves in four directions that would take you along a shortest path structure toward a treasure location.

The central issue is that the original write-up appealed to an informal “black/white symmetry” without exhibiting the actual invariant structure.

We are given several rectangular chocolate bars, each with integer dimensions up to 6 by 6. Each bar can be repeatedly cut into smaller rectangles by making straight cuts along grid lines, and every cut splits one rectangle into two smaller integer rectangles.

We are given a sequence of daily measurements, where each day has a single integer value. For every day i, we want to compare that day with any earlier day j, including itself, and compute how large a “meaningful jump” in measurement is between those two days after…

We are given a connected undirected graph where each vertex represents a customs checkpoint. Moving through a checkpoint takes a certain amount of time, while traveling along roads takes no time.

We are given a graph on $n$ labeled chickens. The graph is extremely sparse, having exactly $n-2$ edges, and it is guaranteed to be a forest.

The input is completely degenerate: it always consists of a single placeholder character. There is no hidden structure, no parameters to interpret, and no variation across test cases. Every valid program is effectively being asked to choose between two conceptual actions.

We are given a multiset of block weights. For each weight value $i$, there are $ai$ identical blocks of weight $i$. Players alternate taking any remaining block and adding its weight to a running sum that starts at zero.

We are working in a very high-dimensional grid. Each point is identified by an n-tuple of coordinates, and each coordinate ranges from 1 to 100000. Every point stores a number, initially zero.

We are asked to arrange the numbers from 1 to n into a single sequence so that every neighboring pair has gcd equal to 1, and the sequence is also cyclic in the sense that the last element and the first element must also be coprime.

A correct treatment must start from the actual definition of “two-bit encoding”.

We are working with a static array of integers, and each query gives us a segment of that array. For every segment, we must decide whether we can pick some nonempty subset of elements from that segment whose bitwise AND is exactly equal to a fixed target value k.

The input describes a planar drawing built from two kinds of structures: a set of infinite horizontal and vertical lines, and a collection of axis-aligned rectangles that do not overlap each other. Together, these objects slice the plane into a finite number of connected regions.

We are given a row of cards, each either face-up or face-down. A move consists of picking a position where the card is currently face-up, and then flipping every card from that position to the end of the row, including the chosen card itself. Flipping toggles each card state.

We are given a one-dimensional field of cells numbered from 0 to n. Cell 0 is our starting point and is always empty. Each other cell i may contain a watermelon that initially gives a certain amount of health, or it may be empty. We start at cell 0 with initial health h.

We are given a string that can be thought of as a row of characters. We are allowed to pick any set of positions in this string, and then reverse only the characters located at those chosen positions, while leaving all other positions untouched.

We are given a collection of numbers representing ingredient qualities. We must split these numbers into exactly k non-empty groups, where each number belongs to exactly one group. Each group represents a dish. The score of a dish is defined in a slightly unusual way.

I can’t write a correct editorial for this yet because the actual problem statement for Codeforces 104672D3 - Recalculating D3 is not present in your prompt, and the gym link alone doesn’t contain enough structured detail to reliably reconstruct all required inputs, outputs…

Let a string consist of 64 characters $c_0,\ldots,c_{63}$, each character being an 8-bit vector $c_i = (c_{i,7},c_{i,6},\ldots,c_{i,0}), \qquad c_{i,k} \in \{0,1\}.$ Define eight 64-bit words $w_0,\ld...

I can’t write a correct editorial for Codeforces 104672D1 - Recalculating D1 because the actual problem statement (input/output definition and constraints) is missing from your prompt.

I can’t write a correct editorial for this yet because the problem statement is missing. Right now there is no description of what “Recalculating D2” is asking, nor any input/output format or constraints.

I can write the full editorial in that format, but I’m missing the actual problem statement for Codeforces 104672B2 - Thermometers B2.

I can’t write a correct editorial for this yet because the actual problem statement is missing from your prompt (both “Problem”, “Input”, and “Output” sections are empty).

I can’t write a correct editorial for Codeforces 104672A1 - Naming Compromise A1 because the actual problem statement (input/output specification and task description) is missing from your prompt.

I can’t write a correct Codeforces-style editorial for 104672B1 - Thermometers B1 without the actual problem statement.

We are given a set of points in the plane, each representing a volcano that must be visited exactly once. A traveler starts from any chosen point and must construct a path that visits all points and then ends at the last visited point.

A train moves through a sequence of cities in a fixed order, and at each city there are a few crane types available, each with a price. Every crane type is identified by an ID, and in a given city you may buy or sell any of the types listed there at that city’s price.

We are given a grid made of empty cells and cells occupied by a single connected polyomino, represented by . The shape is fixed and cannot be altered except by cutting along grid edges. The process works like this: we repeatedly remove pieces from the shape.

We are working on a graph of villages connected by undirected roads. One robot, which we control, starts at a village S and wants to reach a target village F. It moves only at night, and each night it can either traverse one road to a neighboring village or stay in place.

We are given a long linear sequence of square tiles, each tile being either red or blue. We need to count how many contiguous segments of this sequence can be used to build a very specific square patio.

We are given a set of disjoint “clouds”, where each cloud is a set of points whose convex hull forms a simple convex polygon. These polygons do not overlap in their interiors, and they may touch only in empty space, never intersecting each other.

We are given a very large rectangular grid of size $W times H$. Each cell is initially unvisited. A single starting cell $(X, Y)$ is already marked as visited before the game begins. From that moment on, two players alternate moves, starting with the first player.

We are given a stack of journals represented by a string of + and -, where each symbol describes the orientation of a journal cover. The stack is read from top to bottom as the string is given.

We are given a rectangular grid that represents a shoreline, and inside this grid there are many “docks”. Each dock is a 1-cell-thick straight segment aligned either horizontally or vertically, and it spans a contiguous set of grid cells. Each dock has length at least two.

The structure described in the problem is a triangular grid of cells, where each row is longer than the previous one by exactly one cell. The first row contains a single cell, and every subsequent row extends symmetrically.

We are given a list of problems, each with a required time cost and a point value. The twist is that these problems are not always available. Instead, there are multiple classes, and each class teaches only a contiguous segment of problems.

We are simulating a rectangular DVD logo moving inside a larger rectangular screen. The logo itself has width and height, so its motion is equivalent to tracking the bottom-left corner of a smaller rectangle that is constrained to move inside a reduced rectangle of size $(W-A)…

The graph describes a collection of islands connected by undirected bridges. Every bridge has two attributes: it always takes exactly one step to traverse it, and it also has a brightness value. From each query, an animal starts at some island and wants to reach island 1.

We are given a group of people, each holding some number of cake slices. If the cake had been divided perfectly, every person would have received exactly the same number of slices, because the total number of slices is guaranteed to be divisible by the number of people.

The task describes a simple division scenario. A person has a fixed number of pizza slices and a group of friends. The slices are distributed as evenly as possible among all friends, and anything that cannot be evenly distributed remains unused.

Two points on a number line each host an ant. Each ant starts at a known coordinate and moves at a constant but unknown speed and direction. The only information about each ant’s motion is where it starts and where it will be after a fixed amount of time.

Two observers stand at opposite poles and count stars visible from their respective positions. Each star is visible from exactly one pole, never both, which implies that the two observations partition the entire set of stars into two disjoint groups.

There are $n$ teams, each starting with a fixed strength value. Every pair of teams plays exactly one match, so the tournament is a complete round-robin.

We are given a string made only of opening and closing parentheses. The task is to decide whether this sequence could arise from some valid arithmetic expression after stripping away everything except parentheses.

We are given a collection of story lengths, where each story has a fixed number of pages. Alongside this, we are given several books, each with a page capacity.

We are given an array that is already sorted in non-decreasing order. For each query, we are given a segment of this array, and we are allowed to pick a single integer mask $X$ (with up to 20 bits) and XOR every element in that segment by $X$.

We are given two integers. One is a fixed base-like parameter $k$, and the other is an upper bound $r$. For any non-negative integer $n$, we define a process: if $n$ is divisible by $k$, we divide it by $k$, otherwise we subtract 1.

We are given two strings of equal length and an integer step size $k$. The allowed operation does not let us freely edit characters anywhere. Instead, we can pick two positions whose distance is exactly $k$, and copy the character from one position into the other.

We are given two integers that describe a hidden set of distinct non-negative integers. One of these values is the bitwise OR of all elements in the set, and the other is the bitwise XOR of all elements in the same set.

We are given a graph whose vertices are the integers from 2 up to n. Two vertices are connected by an edge exactly when one of the numbers divides the other.

The game is played on an array of positive integers. Two players alternate turns. On each turn, a player selects a prime number that divides at least one element of the array.

We are given an array of integers, and we are forced to perform exactly one operation: choose a single position and flip the sign of that element. After doing this once, we compute the sum of the entire array and check whether this sum is even.

We are given two numbers that summarize an unknown pair of positive integers. One number represents their sum, and the other represents their difference, where the difference is taken as first minus second. From these two values, we need to reconstruct the original pair.

We are given a large binary table describing how a set of participants answered a large number of questions. Each row corresponds to one participant and each column corresponds to one question. A cell is 1 if the participant got that question correct and 0 otherwise.

We are given a hidden ordering problem where the only way to extract information about relative positions of elements is through a median operation on three indices.

We are given two sequences, each already sorted in non-decreasing order. Both sequences have odd length. The goal is not to reorder them directly, but to repeatedly apply a very specific transformation operation on either sequence until the two sequences become identical.

The task is to construct a permutation of numbers from 1 to n such that when a specific deterministic process called Reversort is applied to it, the total cost of that process is exactly a given value C. If no such permutation exists, we must report impossibility.