brain

We are given a grid of characters that represents a sheet of paper, where each cell is either empty or marked. The task is to determine whether the pattern of marked cells could have been produced by a single rectangular stamp that was pressed one or more times onto an…

Let $\pi$ be a permutation of $\{1,2,\dots,n\}$.

I can write the full editorial in the exact style you requested, but I need the actual problem statement (or at least a link or summary of the task).

Each test case describes a single hidden permutation of the digits 0 through 9 into uppercase letters. In other words, every digit is represented by exactly one letter, and every letter represents exactly one digit. The server encodes numbers using this unknown substitution.

We are working on a grid where both you and a moving target start at known integer coordinates and move in unit time steps along the grid edges.

We are given several circular pancakes that have already been cut into wedge-shaped pieces. Each piece has some angular size, and we are allowed to further split any piece by making radial cuts.

We are given a string of balanced parentheses of length K. Every index is a position in a one-dimensional editor.

We are given a set of points in the plane, each representing a hole. A ball moves along an infinite straight line once we choose its starting position and direction.

We are given a network of computers connected by undirected links. Computer 1 is the source, and every other computer is reachable from it. Each link has an unknown positive integer latency, and these latencies determine how quickly a security update spreads through the network.

We are given two piles of pancakes. The process is a deterministic sequence of customers arriving one by one. The i-th customer always requests exactly i pancakes, and we must satisfy them by taking all pancakes from exactly one of the two stacks.

The flaw in the previous response is that it refused the task instead of engaging with the standard definition of Algorithm $J$ in TAOCP §1.

We are given two sets of points on a sphere, each point described by latitude and longitude. The latitude is fixed by the planet’s geometry, so it stays identical across both maps.

We are given several points in a d-dimensional Euclidean space. Each point is a known vector, and for each of them we are also given the exact Euclidean distance to an unknown hidden vector.

We are given a rectangular grid where each cell independently contains a mine with a probability that depends only on its row and column indices. Specifically, a cell at position $(i, j)$ is mined with probability $pi + qj$.

We are given a base price for a single item, and we are allowed to sell items only in bundles. If each item costs b cents and we bundle k items, then the bundle price becomes k · b. The goal is not to maximize revenue in the usual sense.

The surface of the space station is built from axis-aligned unit cubes that are glued together in 3D. Only the outer skin matters, so every robot moves on exposed square faces of this union.

We are given a weighted tree representing a campus. Buildings are nodes and sidewalks are edges with lengths. Every sidewalk must be cleared at least once using two snow blowers that can be pushed along the tree.

We are given a set of quests. Each quest has a reward value and a required level. Your character gains experience points as you complete quests, and the current level is determined only by total experience divided by a fixed constant.

The error occurs precisely at the point where a congruence modulo $mn-1$ is turned into an equality without first identifying the correct residue system and without using the correct domain restrictio...

We are given a collection of phones, each described by three numbers: price, performance, and user-friendliness. For any phone we decide to buy, we compare it against every other phone and measure how much worse it is along each dimension, but only in the direction where the…

We are given a batch of $n$ machines. Each machine is either working correctly or malfunctioning, but we are guaranteed that strictly more than half of them are correct.

We are given a set of points in the plane and a parameter $t$, which represents how thick a “drawn line” is allowed to be.

We are given a single genetic string made only of the four DNA characters A, C, G, and T. The process allowed on this string repeatedly picks a “cut position” between two adjacent characters, but only if the string is symmetric around that cut: if you look outward from the…

We are given an initially flat landscape of n integer positions, all starting at height zero. Then a sequence of k operations modifies contiguous segments of this array. After applying all operations in order, we must output the final height at every position.

We are given a road system that forms a tree. Every road connects two intersections and already has a speed limit. We are allowed to increase any road’s speed limit, but never decrease it.

We are given a rectangular region, which we can think of as the viewing area where a tourist can stand. Inside this rectangle are several fixed points, called domes.

We are given a rectangular grid of cards arranged in r rows and c columns, filled initially in row-major order with numbers from 1 to r·c.

Let $(a,b,c,d,e,f)$ be given and suppose we want to transform it into $(c,d,f,b,e,a)$ using only exchanges $(x \leftrightarrow y)$.

We are given a starting string and a collection of operations, where each operation globally replaces every occurrence of one character with another fixed character.

We are given a circular harp with $N$ attachment points placed on its boundary. Each attachment point has a fixed angular position around the circle and a personal cost $Li$, which represents extra cord needed to attach a string to that point.

We are simulating a two-player game where tiles labeled from 1 to N are gradually placed into N empty positions arranged in a line. Each move consists of choosing one unused number and placing it into an empty cell.

We are given a list of N fixed hex numbers, each written with exactly D hexadecimal digits. We are also given a target interval $[S, E]$, also expressed as D-digit hexadecimal numbers.

We are given a rooted structure that is effectively a directed tree rooted at node 1. Every node is reachable from the root by exactly one directed path, so although edges may be listed in any orientation in the input, the underlying structure behaves like a tree with a unique…

We are given a hidden set of positions labeled from 1 to N. Some of these positions are “broken”, and the rest are “working”. The number of broken positions is not directly given, but we are allowed to interactively query the system.

Consider the mapping $f(x)=2x \bmod 7$ on ${0,1,2,3,4,5,6}$.

We are dealing with a hidden binary array of length $N$. Each position corresponds to a device that is either working or broken, but we do not know which ones are which.

We are given an encrypted message that was originally formed from a sequence of prime numbers. Each letter was first converted into a prime, and then the encryption replaced the sequence of primes with the product of every two adjacent primes.

We are given a sequence of integers that were produced from an underlying hidden sequence of primes. Each integer represents the product of two neighboring primes in that hidden sequence.

We are given a square grid of size n by n. A person named Lydia has already chosen a path from the top-left corner to the bottom-right corner, moving only right or down.

We are given a square grid of size $N times N$. Someone else, Lydia, has already chosen a valid path from the top-left corner to the bottom-right corner, moving only right or down at each step.

We are given a single grid path that starts at the top-left corner of a square grid and reaches the bottom-right corner. The path is described as a sequence of unit moves, where each move goes either right or down. This sequence represents one valid route through the grid.

Let $N=m+n$.

We are given a number written as a string of decimal digits. The task is to split this number into two other numbers such that adding them together reconstructs the original number exactly, digit by digit with normal base 10 addition, and neither of the two resulting numbers…

We are given a single large integer written in decimal form, and we need to split it into two non-negative integers whose sum equals the original number.

We are given a very large non-negative integer written as a string. The task is to split this number into two non-negative integers, call them A and B, such that when we add them digit-wise we recover the original number, and neither A nor B contains the digit 4 in their…

We are simulating a very small hotel with exactly ten rooms indexed from 0 to 9. Each room can either be empty or occupied by exactly one guest. Over time, guests either arrive from one of two entrances or leave from a specific room.

We are given a string of Latin letters where both uppercase and lowercase characters may appear. The task is to determine whether the string is a pangram, meaning that every letter from 'a' to 'z' appears at least once somewhere in the string, ignoring case differences.

We are given a string of length n that represents a partially decoded genome. Each position is either one of the four nucleotides A, C, G, T, or an unknown character ? that must be replaced.

We are given multiple independent test cases. Each test case consists of a very short string of exactly three characters. The task is to decide whether this string represents the word “YES” when we ignore letter case.

We are given a pipe system with $n$ outlets, each outlet having a size $si$. Arkady pours a fixed amount of water $A$ into the system, but he is allowed to block any subset of outlets before doing so.

We are given a sequence of cafe visits, where each number represents the index of a cafe Vlad visited at that moment in time. Cafes can repeat, meaning Vlad may visit the same cafe multiple times, and we only care about the order of visits.

We are given a list of students, each identified by an integer id from 1 to n. Student 1 is Thomas. Every student has four exam scores, and their overall performance is measured by the sum of these four scores.

We are given a list of steward strengths, and we need to decide which stewards Jon Snow will support. A steward is supported only if there exists at least one steward with strictly smaller strength and at least one steward with strictly larger strength.

Codeforces 104637I: Definite Game

We are given a single sequence of integers that represents what Tanya says while climbing stairs. Each time she enters a new stairway, she starts counting from 1 and continues upward step by step until that stairway ends.

We are given three positive integers, and we want to check whether they could have come from a very specific construction involving three hidden positive integers $a$, $b$, and $c$.

A frog starts at position 0 on a number line and repeatedly jumps left and right in a fixed pattern. The first jump moves it to the right by a, the second jump moves it to the left by b, and this alternation continues for k total jumps.

We are given several independent pairs of positive integers. For each pair, we repeatedly apply a deterministic transformation: identify the larger of the two numbers and subtract the smaller from it.

We are given a sequence describing the order in which doors are opened. Each door belongs to one of two exits of a house, either the left exit or the right exit.

We are trying to produce a target number of torches. Each torch consumes exactly one stick and one coal, so if we want $k$ torches, we ultimately need $k$ sticks and $k$ coal. We start with no useful inventory except that we can manipulate sticks through two trade operations.

We are asked to count how many integers in the range from 1 to n can be written in at least one of two special forms: a perfect square or a perfect cube. If a number can be written both ways, it should still be counted only once.

For each vertex $V_j$, let $\mathcal{T}_j$ be the set of oriented spanning trees of $G$ rooted at $V_j$ in the sense of Section 2.

We are simulating a very particular sleep cycle with a repeating alarm. Polycarp falls asleep and initially waits a fixed number of minutes before the first alarm rings. After that, every time he wakes up, he checks whether he has accumulated enough total sleep.

We are given two piles of indistinguishable items, red beans and blue beans. In each test case we must decide whether it is possible to split all beans into several groups, where each group contains both colors, and within every group the difference between the number of red…

We are given a reference program whose running time is fixed at $T$, and a list of $n$ other programs with known running times.

We are given a convex polygon that represents a safe region in the plane. For every airdrop query, we also get a circle defined by its diameter endpoints. Each airdrop lands uniformly at random anywhere inside that circle.

We are given two geometric objects in the plane. Each object is a circle, but instead of being defined by a center and radius, each circle is specified by the endpoints of a diameter.

We are building a sequence of strings S1 through Sn by processing a string of operations. Starting from an empty string, each step adds exactly one character either to the left or to the right. If the current operation is a lowercase letter, we place it at the front.

We are given a partially known password string of length n. Each position already restricts what the final password character can be.

We are given a process that builds a spanning tree in a somewhat indirect way. Initially, every node is isolated. Then we perform $n-1$ operations. Each operation provides two nodes $ai$ and $bi$, and at that moment both belong to different connected components.

We are given an array of integers and a two-player game defined on it. Alice moves first, and in each move a player picks any two positions in the array and replaces those two values with a new pair of integers that preserves their sum while strictly decreasing their absolute…

We are given an undirected simple graph where some pairs of vertices are already connected by edges. The operation we are allowed to perform is to add new edges, but we cannot delete existing ones and we cannot introduce parallel edges or self loops.

We are given a prime number $n$. We consider all ordered pairs of positive integers $(x, y)$ where both values lie in the range $1 le x, y le n^2 - n$.

We are given the ordered results of two separate programming contests. Each contest ranks individual teams, but what ultimately matters is the performance of universities rather than individual teams.

We are maintaining a dynamic collection of pairs of integers. Each pair behaves like a linear function in a single variable: for a pair $(ai, bi)$, we can evaluate a value $fi(x) = ai cdot x + bi$. The system supports two operations over time.

We are given two strings of equal length, call them S1 and S2. Think of them as two aligned rows of characters, both indexed from 1 to n.

Construct a directed multigraph $G'$ from $G$ by replacing each arc $e_j$ with $E_j$ parallel arcs, for $1\le j\le m$, and omitting $e_0$.

We are given a hidden array of length $n$. The array is strictly increasing, meaning every next value is larger than the previous one. However, we are not allowed to see the array directly. Instead, there is an interactive function $f(x)$.

We are given a tree of $n$ nodes rooted at node $1$. Each edge represents a direct supervision relation in a hierarchy, but the direction is not fixed in the input, only the structure of the tree is known.

Let $A=\{0,1,\dots,m-1\}$.

We are given a circular boundary centered at the origin, and from the origin we imagine emitting rays in every possible direction. Each ray represents a “web line” that travels outward until it either reaches the boundary circle or gets blocked earlier.

We are given a directed graph with $n$ vertices, where each vertex has exactly two outgoing edges and exactly two incoming edges. So the whole structure is a 2-in-2-out directed multigraph, potentially with parallel edges. Each edge has an interval $[ai, bi]$.

We are given a binary string length $n$ that we must construct. The cost of the string is defined through all its contiguous substrings of length $k$.

We are given a grid of size $n times m$ where each cell is either black or white. We are allowed to apply operations that flip an entire row or flip an entire column, toggling all colors in that line.

We are given a collection of axis-aligned rectangles, each defined by its height and width, and each rectangle is also allowed to be rotated by 90 degrees.

We start with a rectangle of size $n times m$. Two players take turns, starting with Gwen, and there are exactly $k$ moves per player, so $2k$ moves in total. On each move, a player chooses one side of the rectangle and increases it by 1.

We are asked to construct a string of length $n$ over lowercase Latin letters. After we output the string, a machine evaluates it in a way that depends only on which distinct letters appear and how many times each appears.

We are given a graph where vertices represent universes and edges represent portals. Each portal has a minimum energy requirement, meaning you can only traverse that portal if your current energy level is at least a given threshold. Energy starts at zero.

We are given two separate directed-unweighted weighted graphs that share the same set of vertex labels from 1 to n.

We repair the argument by giving a complete BEST-theorem style proof with fully justified bijections and a clean treatment of the root and cyclic factors.

We are asked to choose a number of prepared food portions, call it $m$, under a global budget $m le n$. The process has a fixed structure: one portion is always consumed by analysis, leaving $m - 1$ portions.

We are missing the actual problem statement for Codeforces 104642D2 - Cubic UFO D2, so I cannot safely write a correct editorial yet.

The statement for Codeforces 104642C2 - Go, Gopher! C2 is not included in your prompt, and this specific task is not a standard standalone Codeforces problem with a widely indexed description.

The problem statement is missing from your prompt, so there is no way to reconstruct the required model, operations, or solution logic for “Codeforces 104642D1 - Cubic UFO D1”.

I can’t write a correct editorial for this yet because the actual problem statement (and I/O format) is missing. “Codeforces 104642C1 - Go, Gopher!

I can't write a correct editorial because the actual problem statement is missing. The prompt only includes: - Problem: Codeforces 104642B2 - Trouble Sort B2 - Empty Problem Statement - Empty Input - Empty Output Without the statement, it is impossible to determine: - what…

We are given a sequence of commands that controls a simple robot-like system. The program is a string made of two kinds of characters. One character increases the firing power of the system, and the other fires a shot that deals damage equal to the current power at that moment.

We are given a sequence of numbers arranged in a line, and we are allowed to perform a very restricted kind of rearrangement operation. The operation does not let us swap arbitrary elements, only elements that are two positions apart in the array.

I can't write a correct editorial because the essential problem statement is missing. The prompt only identifies the problem as Codeforces 104642A2 - Saving The Universe Again A2, but the sections that define the task are empty: - Problem Statement - Input - Output Without…

I can't write the requested editorial because the essential problem statement is missing. The prompt only identifies Codeforces 104644C2 - Ant Stack C2, but the sections containing the actual statement, input format, output format, constraints, and samples are empty: - Problem…