brain

We are given a string consisting of only four possible characters: P, D, A, and O. Each character represents a type of warrior standing in a line. We are allowed to choose any contiguous segment of this line, meaning we pick a substring, and call it a “league”.

We are given a poll with two options. Suppose a total of $n$ people have voted, with $L$ choosing the left option and $R = n - L$ choosing the right one.

We are designing a directed “grade system” with k levels, where each level i has exactly one fallback target ai satisfying 1 ≤ ai ≤ i. If a student fails the exam in grade i, they are sent back to grade ai.

We are given a single sequence of integers, and we are allowed to delete elements while keeping the remaining elements in their original order.

We are working with a triangle whose side lengths are integers, written as $a le b le c$. From this triangle, two special points are constructed on sides $AB$ and $AC$.

We are given a sequence of courses that must be taken in a fixed order. Each course has an estimated score and a credit value. The student divides these courses into exactly $K$ consecutive semesters, and each semester must contain at least one course.

We are given a sequence of test cases, and each test case provides an integer $n$. For each $n$, we consider the set ${1, 2, dots, n}$. From this set, we form every possible non-empty subset. For each subset, we compute a value defined in a slightly unusual way.

Two players, Eason and Emil, play a turn-based game involving two independent piles of items. Eason starts with A eggs, Emil starts with B eggs. They alternate turns according to a fixed starting rule: either Eason goes first or Emil goes first depending on a binary flag C.

We are given a directed graph where each vertex represents a student and each directed edge represents a one-way friendship claim.

The core issue in the previous solution is that it tried to justify the simplification by claiming a strong structural symmetry of kernel BDDs that was never actually established.

We are given a combination lock described as a multi-dial cyclic system. Each dial behaves like a circular wheel: the i-th dial has values from 0 up to ai − 1, and turning it moves one step clockwise or counterclockwise at a fixed time cost.

We are given multiple independent queries. Each query contains two integers, and for each pair we are asked to output their arithmetic product. The input size can be large in terms of number of queries, up to one hundred thousand.

We are given a directed relationship graph over up to 2021 users. Each user may follow some subset of all users. From this universe, only a subset of size m is present in a chatroom, and the task is to determine whether there exists a “celebrity” inside this subset.

We are maintaining a growing string that starts empty and is extended over time. Each update operation appends a new substring to the end, and occasionally we are asked a query about the current full string.

We are given a complete weighted graph with up to 12 vertices. Each vertex represents a rock fragment, and the cost matrix tells us how expensive it is to directly glue any two fragments together.

We restart from the definition of the family and apply the ZDD reduction rules exactly as stated in TAOCP §7.

We are working on a discrete sky represented as integer coordinates in a small grid, specifically points in a 1000 by 1000 space. Two kinds of points are placed on this grid: stars and black holes.

We are given a single integer $n$, and we need to evaluate a specific summation built from division remainders. For every integer $i$ from 1 to $n$, we divide $n$ by $i$ and take the remainder, then add all those remainders together. The task is to compute this total efficiently.

We are given two arrays of the same length, and we want to transform them so that every element across both arrays becomes equal to a single common value.

We are given a system with $n$ engines, each holding a fuel requirement that changes over time. At the beginning, every engine $i$ has an initial fuel requirement $f{1,i}$. After that, there are $m$ events.

We restart from the exact cover formulation, but we now build the BDD/ZDD constructions in a way that does not rely on variable ordering to magically enforce constraints.

The world is a directed graph where planets are nodes and wormholes are directed edges with a damage cost. Meryl and Roberto both start at planet 1 and must independently reach planet n.

Let $mathcal{S}(f)$ denote the set of all distinct subfunctions of $f(x1,dots,xn)$ obtained by repeated Shannon decomposition with respect to variables $x1,dots,xn$, as represented in the master profile chart.

We are maintaining a growing sequence of observations, which can be thought of as a string that starts empty and is extended over time. Each update of the first type appends another string to the end of this global sequence.

We are given a directed weighted graph with (n) planets and exactly (n-1) roads. Each road has a direction and a travel cost. Planet (1) is special because it represents Pluto, and every planet can reach it through some directed path.

Working

We are given a complete weighted graph on $N$ vertices, where each vertex represents a rock. The cost $C{i,j}$ is the price of directly gluing rock $i$ next to rock $j$.

We are working on a 2D grid where both stars and black holes are placed at integer coordinates in a small bounded space. Stars represent points we want to count, while black holes invalidate nearby stars.

The five-letter word pairing scheme in Section 7.2.1.1 relies on masking a packed bitstring so that each mask isolates the lower portion of a word consisting of an integral number of fixed-size letter fields.

An 8×8 chessboard is partitioned into 32 dominoes in a perfect covering.

The task gives a single integer and asks us to output another integer based on it. There is no further structure such as arrays, graphs, or multiple queries, so the entire problem reduces to understanding how the output depends on this one value.

The five-letter word pairing scheme in Section 7.2.1.1 relies on masking a packed bitstring so that each mask isolates the lower portion of a word consisting of an integral number of fixed-size letter fields.

We are given a single integer $n$ in a very small range up to 2023, and we must produce one integer as output. There are no additional structures like arrays or graphs, so the task is entirely about defining a function $f(n)$ that maps each valid input to a single integer.

We are given a small sequence of integers, and we are asked to compute a single integer answer derived from its internal structure.

A domino tiling of the $8\times 8$ board assigns to each unit square a partner square so that every square belongs to exactly one $1\times 2$ or $2\times 1$ domino.

We are given two small integers, row and col, both ranging from 0 to 14, and we must decide whether the point represented by these coordinates is inside a certain region or outside it.

Let $f(x1,x2,x3,x4,x5)$ be a Boolean function and let $B{min}(f)$ denote the minimum, over all variable orderings, of the number of nodes in its reduced ordered binary decision diagram, including the sink nodes $bot$ and $top$.

We are given a single string consisting only of lowercase English letters. The task is to compute a single integer based on this string, and print it. From the samples, we observe that only certain letters contribute to the answer, while all others contribute nothing.

I can’t write a correct editorial for that yet because the actual problem statement for “Codeforces 104264B - String” is missing from your prompt.

We are given a labeled undirected simple graph on $n$ vertices with exactly $m$ edges. The graph is called valid if we can add some additional edges so that the final graph becomes a single simple cycle that visits all $n$ vertices exactly once.

We are given a number written in decimal form and a repeated transformation applied to it. The transformation is defined in two layers. First, there is a function that takes a number and replaces it with the sum of a digit-wise score.

Let the chessboard be the standard $8 \times 8$ grid, decomposed into $64$ unit squares.

We are given a fixed target height $y0$ and a set of players, each starting at some integer grid point $(xi, yi)$. There is also a hidden parameter $x0$, the x-coordinate of an airdrop position $(x0, y0)$.

We are given a sequence of books, each with a fixed price, and a deterministic purchasing process that scans books from left to right. At each book, if the current money is at least the price, the book is bought and the money decreases; otherwise the book is skipped.

We are given an array of soldier powers arranged in a fixed order. The task is to partition this array into contiguous teams, where each team contains either a single element or exactly two adjacent elements.

We are given a start point and a target point in the plane. At the start, there are multiple identical stones stacked at the start point. The task is to move all stones to the target point, but they must be transported one by one.

We are given a line of $n$ cells, each cell being in one of three states. Some cells are already empty, some contain DreamGrid’s own fixed pattern that must never be touched, and some contain BaoBao’s pattern that must be removed.

We are asked to construct a multi-round tournament schedule among $n$ knights, where each round pairs up all knights into disjoint duels.

We are given the length of two unknown positive integers A and B, and a strange string C that is produced by multiplying them under a non-standard operation. The operation does not behave like normal multiplication.

We are given a line of plants indexed from 1 to n. Each plant i has a fixed position i and a growth rate a[i]. Initially every plant has zero defense value. A robot starts at position 0, which is the “house”.

We are given an array of length $n$, where each position represents the correct answer to a multiple-choice question. Each question has an assigned correct choice, and BaoBao can pick exactly one choice per question.

We are given two binary strings of equal length. Think of them as two rows of switches, each position holding either 0 or 1. We are allowed to perform exactly two operations, and each operation chooses a contiguous segment and flips every bit inside that segment.

We are given two tightly coupled sequences. The first sequence, P, is fully deterministic and grows in a structured way: the value 1 appears twice, 2 appears three times, 3 appears four times, and so on.

We are given a long arithmetic expression written in the usual infix form. It consists of non-negative integers combined with addition and multiplication, and an equality to a final integer value.

Let the ZDD represent a family $\mathcal{F}$ of subsets of ${x_1,\dots,x_n}$, ordered by the variable indices, and let each node $k$ be labeled by $V(k)\in{1,\dots,n}$.

I can’t write a correct editorial for this yet because the problem statement for Codeforces 104273E - “Быстрый исполнитель” is missing from your prompt.

We are given two strings built from lowercase letters. Think of the first string as a long tape of characters produced by Bob, and the second string as the shorter string Alice believes should remain after Bob’s modifications.

The problem statement for Codeforces 104273D (“Перекладывание ответственности”) is not included in your message, so I don’t have enough information to reconstruct the task reliably.

Two automated bots are sending messages to each other on a strict schedule. Both bots always send a message at time zero, and then continue sending messages periodically: the first bot sends at times 0, a, 2a, 3a, and so on, while the second sends at times 0, b, 2b, 3b, and so…

I can’t reliably write a correct editorial for this yet because the actual problem statement is missing. “104273C - Есть n стульев...

We are given a regular N-sided polygon that represents the boundary of a clock face. Its center is the origin, one vertex lies on the positive y-axis, and the polygon is oriented in a fixed way.

The railway system forms a directed acyclic graph rooted at node 1. Every edge represents a one-way track segment with a number of people on it who would be hit if the train traverses that edge.

We are given a person’s birth date and an upper bound year. For each query, we need to count how many times this person will celebrate their birthday from the year after their birth up to and including the given end year.

We are dealing with a hidden binary array of length $N$. Exactly two positions contain a value of 1, and all other positions contain 0. We cannot see the array directly. Instead, we are allowed to ask queries of the form: give me the sum of values in a subsegment $[L, R]$.

We are given a row of flowers, each flower having a type represented by an integer. A florist considers a bouquet “valid” only if it corresponds to a contiguous segment of this row and the segment contains exactly K distinct flower types.

We start with a single initial phone number consisting of digits. From this string, a sequence of new phone numbers is generated.

The flaw in the previous solution is that it tried to define ZDD nodes as states indexed by a subset $X \subseteq U$.

We are given a fully scrambled state of a 2×2×2 Rubik’s cube, encoded not as physical faces but as a flat list of 24 colored stickers. Each color represents one of the six faces in the solved configuration.

We are given the state of a very small Rubik-like object that is already a 1×1×1 cube, meaning there are exactly six colored faces with no internal structure.

You've hit your limit. Please try again later.

Rudolf spends up to one million days on a foreign planet, and the entire timeline is treated as a single continuous calendar starting from day one. The key complication is that two independent schedules overlap. The first schedule is monthly rent.

Let $S={1,dots,m}$ denote the selector variables and $T={m+1,dots,m+2^m}$ the data variables of the multiplexer $Mm$. For each $iin S$, the value of $xi$ selects one index in $T$, and the function outputs the chosen data bit.

We are dealing with a fully specified logical reconstruction problem involving six people, six mailbox owners, and six postcard themes.

We are given an array of length $2n$, initially all zeros. We also receive $n$ interval assignment operations. Each operation $i$ comes with a segment $[li, ri)$ and, when applied, it overwrites every position in that half-open interval with the value $i$.

We are given a collection of triangular faces, each triangle described only by three integer vertex IDs. These IDs do not represent geometry in any meaningful way beyond identity.

We are working with binary strings arranged on a circle. Think of a length-n binary sequence written on a ring, where position n connects back to position 1.

We are given a collection of processes. Each process becomes available at a specific time and has a fixed processing length. At any query time $t$, we consider only processes that have already arrived, meaning their arrival time is at most $t$.

We are dealing with a game on an undirected connected graph where nodes represent holes and edges represent tunnels. A mouse starts at some unknown node. In each round, Kanade “attacks” exactly one chosen node from a fixed sequence.

We are given a tree with n vertices and n − 1 undirected edges. The task is not to output the edges themselves or reconstruct an adjacency list, but to assign to every vertex i (from 1 to n − 1) a partner vertex pi such that the pair (i, pi) is one of the given tree edges.

Let $f$ be the Boolean function that represents solutions of an exact cover instance on a universe $U$ with a family of subsets encoded by variables $x_1,\dots,x_n$.

Let $S={1,dots,m}$ denote the selector variables and $T={m+1,dots,m+2^m}$ the data variables of the multiplexer $Mm$. For each $iin S$, the value of $xi$ selects one index in $T$, and the function outputs the chosen data bit.

We are given a set of points in the plane, with no duplicates, and we need to count how many rectangles can be formed by choosing four of these points as vertices.

We are given a game system with 15 types of actions, where each action consumes a fixed amount of stamina depending on its index. Actions 1 through 4 cost 8 stamina each, actions 5 through 10 cost 9 stamina each, and actions 11 through 15 cost 10 stamina each.

We are simulating a Josephus-style elimination on a circular arrangement of people labeled from 1 to n. The difference from the classic version is that the step size is not fixed. Instead, there are q rounds, and each round provides its own step value k.

We are given a kingdom structured as a tree, meaning there are n cities connected by n − 1 roads and there is exactly one simple path between any two cities. Some of these cities are marked as important, and some cities contain troops.

We are given several independent test cases. In each test case, we are working on a grid in the first quadrant. Every segment we receive lies on a single 45-degree diagonal line, because each segment’s endpoints satisfy the same value of $x + y$.

Let the ZDD for $f$ be given as a reduced ordered ZDD with variable ordering $x_1 < x_2 < \cdots < x_n$.

We are given a sequence of typing requests. Each request says that a certain key, identified by an integer label, is supposed to be pressed repeatedly a given number of times.

We are given a grid with $n$ rows and $m$ columns. Each cell contains one character, either $A$ or $B$. Starting from the top-left cell, we move only right or down until reaching the bottom-right cell.

We are given a set of circles in a plane with a strong structural promise: no two circles intersect or touch each other. This restriction forces a very rigid geometry. Any two circles are either completely separate, or one lies fully inside the other. There is no partial overlap.

We are given a linear tower of floors, each floor i has an enemy with a required strength ai and a reward bi that increases the player’s strength after defeating that enemy. The player can walk along adjacent floors, moving only between i and i + 1 or i - 1.

Yes.

We are asked to construct, for each test case, a list of n distinct integers whose total sum equals a target value k, while keeping every chosen number within the range [-10^9, 10^9].

We are given a set of enemy positions on a plane, all measured relative to the origin where our character stands.

We start with an array of $n$ numbers. Initially, each element stands alone as a separate segment. The process repeatedly chooses two neighboring segments, merges them into one, and assigns that new segment a value equal to the sum of all elements inside it.

We are asked to count special groups of four prime numbers taken from the range from 1 to n. Each group consists of indices a, b, c, d such that all four numbers are prime and they form an arithmetic progression with exactly three equal gaps.

Let the Boolean function be given by a ZDD with variable order $x_1,x_2,\ldots,x_n$.

We start with a rooted tree where vertex 1 is the root and every other vertex has a fixed parent given in the input. Depth is defined in the standard way: the root has depth 1, and every edge increases depth by 1.

We are given an $n times m$ grid where each cell either contains a cake or is empty. The task is to eat all cakes while maximizing total satisfaction. There are two possible actions. One action eats a single cake and gives a fixed reward $p$.

We are given a collection of domino cards. Each card carries two values, a front value and a back value. From the full set of cards, we first select exactly K cards. The score for this first selection is the sum of the front values of those K chosen cards.