brain

We are given a rectangular grid of size $n times m$, fully covered by unit cells. A player wants to tile all remaining cells after removing exactly one cell using a specific shape called a “corner tromino”: a shape made of three cells that forms an L, i.e.

We are working with a tree on $n$ vertices where vertex 1 and vertex 2 are fixed in a special way: the distance between them is exactly $m$, and no pair of vertices in the tree is farther apart than $m$.

We are given two identical geometric objects on a grid, each object is a regular hexagon of side length 1. Each hexagon is placed in a fixed orientation: two of its edges are vertical, and its lowest vertex is anchored at an integer coordinate point.

We are given a rectangular window of height H and width W. Two cuts are made. First, we choose a vertical cut at some integer position A, splitting the window into a left rectangle of width A and a right rectangle of width W - A.

A set $S \subseteq {1,\dots,8}^2$ is admissible if no three distinct points of $S$ lie on a common affine line in $\mathbb{R}^2$.

We are given a set of vertical “trees” placed at distinct x-coordinates. Each tree is just a segment from ground level up to some height. Alice starts at the leftmost side and moves strictly from left to right, but she does not move along the ground.

Let $Q_8$ be the queen graph on the $8\times 8$ chessboard.

We are given an array of integers representing environmental impact scores for different policies. Each query asks us to look inside a fixed segment of this array and consider only subarrays whose lengths lie inside a given range.

We are given a sequence of animals arranged in a line. Each animal has two attributes: a species label and a skill value. The task is to consider every contiguous segment of this line and compute a score for each segment.

We are given a directed graph on $N$ nodes where every node has exactly one outgoing edge. From each node $i$, there is a deterministic move to $nxti$. Two tokens start on nodes $a$ and $b$. In every round, both tokens simultaneously follow their outgoing edges.

We are given a row of $N$ ice pillars, each with an initial integer height. Over time, the system receives two kinds of operations applied to subarrays.

We are given a rectangular grid, and two players are placed on two different cells. The grid has no obstacles, so movement is always possible in the four cardinal directions.

We are given a set of points on a 2D plane, each representing an observation station in a national park. Every station is equipped with a wireless device that has a uniform power level $P$.

Let $G=(V,E)$ be a finite graph.

We are given a chain of rigid segments placed end to end in the plane. Each segment has a fixed length, and consecutive segments are connected by joints that allow rotation.

We are given a directed structure on $N$ nodes where each node has exactly one outgoing edge to another node, and no node points to itself.

We are given an ordering of the numbers from 1 to N placed across N positions. Think of the array a as Larry’s “ideal layout”, where position i ideally wants the value a[i]. Harry is allowed to rearrange the same set of values 1 through N into another permutation b.

We are given a set of barns arranged as vertices in a graph. Each barn gives Joe a fixed profit every time he arrives there. Between barns there are directed roads, each with an associated cost.

We are asked to count how many permutations of the numbers from 1 to N the farmer can produce while making at most K “mistakes”. A mistake occurs at a position i if the value placed at position i is not i itself.

We are given a grid representing a city map where each cell is either free road, an obstacle, a start position, or one or more exits.

We are given a collection of courses. Each course has two values attached to it: a score and a number of units. We are required to pick exactly K courses.

We are given a line of blocks, each block having a certain number of trees. A series of walks is also given, where each walk covers a contiguous segment of blocks from one endpoint to another, inclusive.

We are given a directed graph on $N$ nodes where every node has exactly one outgoing edge, defined by an array nxt. From any node $i$, a token moves deterministically to nxt[i] every round.

Four trash bins are placed around a campus, and at the end of lunch each bin is weighed. Each bin has a known empty weight of six pounds, and the input gives the total measured weight of each bin after use.

We are given a collection of infinite straight lines laid out on a plane. Each line is either vertical, representing an avenue with a fixed x-coordinate, or horizontal, representing a street with a fixed y-coordinate.

We are given a long array b that contains, somewhere inside it, a hidden structure built from two parts glued together: a key segment of length q, followed immediately by an encrypted version of a known array a of length m. The encryption is very structured.

Let $G$ be the grid graph of Fig.

We start from the origin on an infinite grid and want to reach a fixed target cell at coordinates $(x, y)$. Each basic action moves the character by one unit in one of the four cardinal directions, and each such action costs one button press.

We are given two sequences of integers, each representing the order of trophies on two separate shelves. Within each shelf, all values are distinct, but the same value may appear in both shelves.

The semester is modeled as a single continuous timeline of days. Each laboratory occupies a consecutive block of days, and these blocks do not overlap. If the first lab lasts $a1$ days, then it spans days $1$ through $a1$.

We are given a network of cities connected by undirected roads. Each road has a fixed fuel requirement, meaning that to traverse it we must consume a certain number of liters of gasoline.

I’m missing the actual problem statement for Codeforces 104243D - “Музыкальный фестиваль”, so I can’t reliably reconstruct the task or derive a correct algorithm.

I can’t reliably write a correct editorial for “Codeforces 104243C - Королевская задача” because the actual problem statement is missing from your prompt. Right now I only have the title, but no definition of the input, output, or rules.

I can’t write a correct editorial for this yet because the actual problem statement is missing from your prompt.

The problem describes a traveler who starts in city 1 with some initial amount of money and wants to reach city n using directed flights. Each flight has a cost in money, and can only be taken if the traveler currently has at least that cost available.

The problem statement for “Codeforces 104244B - Покупка подарков” is missing from your prompt, so there’s no reliable way to reconstruct the task or its constraints.

I’m missing the actual problem statement for Codeforces 104244C - “Поиск фальшивых монет”, so I can’t reliably reconstruct the intended solution or write a correct editorial.

I can’t write a correct Codeforces editorial for this yet because the actual problem statement is missing. Right now I only have the title “Задачечка на подстрочечки” (“a small problem about substrings”), but that’s not enough to reconstruct…

We are given a set of points in the plane, called pins. We must output a closed polyline path, meaning a sequence of points that starts and ends at the same coordinate. The path has two topological requirements.

Let $G = (V,E)$ and let $g$ denote the family of edges encoded in the sense of Exercise 236(e), so that $g = \bigcup_{u-v \in E}(e_u \sqcup e_v)$ and the family of independent sets is expressed by a f...

We process rows from top to bottom while maintaining the range of column positions that some valid path can occupy at that row. 1. Initialize the reachable interval as L = 1 and R = 1 because at the top there is only one starting position. 2.

The task gives a sequence of integers representing a permutation-like array. The goal is to produce a correctly ordered version of this sequence, where elements are arranged in non-decreasing order, and print that final arrangement.

We are given a directed weighted graph with $n$ planets and exactly $n-1$ directed roads. Planet $1$ is special and acts as Pluto. From every planet, there exists at least one directed path that reaches Pluto.

We are given a line of people, each with a required threshold value. Mike owns a limited number of discount tickets, and he can assign each ticket to at most one person in the line.

We are given a single integer $n$, which represents an estimated radius for Pluto in miles. The question is whether this value is large enough for Pluto to qualify as a planet under a specific geometric requirement.

We are dealing with a hidden interval on a numbered line from 1 to N. Somewhere on this line there is a contiguous segment starting at A and ending at B, and our goal is to determine its length, which is B minus A plus one. We do not get direct access to A or B.

We are repeatedly transforming a small integer by a digit-rearrangement operation. Each step takes the current value, normalizes it to a 4-digit string using leading zeros, then forms two new numbers from its digits: one with digits sorted in ascending order and one in…

The story talks about minimum spanning trees and color transformations, but the actual structure of the input is much simpler than the narrative suggests. What we truly receive is a connected undirected weighted graph with n vertices and m edges.

We are asked to count how many integer pairs $(a, b)$ exist inside a fixed interval $[l, r]$ such that $l le a le b le r$ and a specific relationship between $a$ and $b$ holds: the ratio between their least common multiple and greatest common divisor is exactly $k$.

Let $G = (V,E)$ denote the contiguous-USA graph of (18), and let $U \subseteq V$.

We are given a grid where some cells are blocked and the rest are free. On this grid, several robots are placed. Each robot occupies exactly one free cell and has an orientation, one of four directions: left, right, up, or down. Time evolves in discrete steps.

We are given an array and we look at every contiguous segment of length at least two. For each chosen segment, we ignore its original order and instead sort its values. Once sorted, we compute the gaps between consecutive elements.

A pizza is cut into identical slices, and each slice is a sector defined by a fixed central angle θ. The radius r is irrelevant for the distribution question because all slices are congruent; what matters is only how many equal angular pieces the full 360° circle is…

We are given a collection of coders, each starting with some initial reward value. Over time, a sequence of updates arrives, and each update targets exactly one coder and increases that coder’s reward by some integer, which may be positive or negative.

A family of sets is represented by a reduced ordered ZDD in which each internal node is labeled by an element $x_i$, with the low child corresponding to exclusion of $x_i$ and the high child correspon...

Let $U = e_1 \sqcup e_2 \sqcup \cdots$, and let $C$ denote complement with respect to $U$.

Let $\mathcal{W}$ be the finite set of five-letter English words under the chosen dictionary, and define a directed graph $G=(V,E)$ where $V=\mathcal{W}$ and there is an arc $x \to y$ if and only if t...

Let $\Sigma$ be the set of the 49 postal codes in (18), each written as a two-letter string $XY$ over the alphabet of letters appearing in the codes.

Let $G=(V,E)$ be a directed graph whose edges are linearly ordered as $E={e_1,\dots,e_m}$.

A king’s move on the $8\times 8$ chessboard connects any two squares that differ by at most $1$ in each coordinate, excluding equality, so each vertex $(i,j)$ is adjacent to all squares in its $3\time...

The graph $P_8 \times P_8$ is the standard $8 \times 8$ rectangular grid graph.

The reviewer is correct that the previous solution fails in its core task: it never engages with the specific graph (133).

The key point is that the number “eight” is not a property of individual paths, but a property of the _construction_ that produced the BDD in Fig.

Solution to TAOCP 7.1.4 Exercise 229.

Let $G = (V, E)$ with distinguished start vertex $s$.

Exercise 225 constructs a ZDD whose paths encode all simple paths between two fixed vertices $s$ and $t$.

We are simulating a single move in a simplified board game. The board has 100 cells arranged in a path, and Farha is currently fixed at cell 94. She rolls a die once, producing an integer k between 1 and 6, and immediately moves forward k steps, landing on cell 94 + k.

We are given a tree of cities. Each city has a value, and every pair of cities is connected by a unique simple path.

We are given several independent queries. Each query provides a target number x and an interval of integers [l, r]. Inside this interval we want to know whether we can pick three distinct integers a < b < c, all lying in [l, r], such that their product equals x.

The problem removes all typical competitive programming structure and replaces it with a direct question. There is no input, no parameters, and no hidden state to compute from.

Let $k ge 2$ be even and consider the $(kr+2)$-cube $G = Gk G{k-1} cdots G1 G0 G{-1}$, where $Gi$ is an $r$-cube for $i0$ and $G0 = G{-1} = P2$. A vertex $v$ is written as $$v = vk v{k-1} cdots v1 v0 v{-1},$$ where $vi in {0,1}^r$ for $i0$ and $v0,v{-1} in {0,1}$.

Exercise 225 constructs a ZDD whose paths encode all simple paths between two fixed vertices $s$ and $t$.

We are given a very small grid, at most 12 by 12, where some cells contain buildings and others are empty. We must place radio towers on grid cells, and every building must have a tower placed directly on it.

We are given a very large integer interval from $A$ to $B$, and we conceptually compute the sum of squares of every integer inside this interval.

We are given three numbers that are supposed to represent the lengths of the three segments of a closed polyline drawn inside or on the boundary of a fixed triangle.

We are given a small number (n le 8), which represents a fixed list of input variables (x1, x2, dots, xn). Alongside this, we are given a target expression (X), written as a short string of these variables.

We are given three vectors in 3D space, $OA$, $OB$, and $OC$, which form a non-degenerate corner at the origin. These three directions act like a skewed coordinate system: any point inside this corner can be uniquely expressed as a positive combination of these three vectors.

We are given a small collection of bumpers placed in a plane. Each bumper has a fixed position, a radius, and a score that is gained every time the ball hits it.

We are given a directed graph on $n$ vertices where every vertex has at most one outgoing edge. Each vertex also carries a lowercase character label. Some vertices point to another vertex, while others are terminal and point to nothing.

Connection interrupted. Waiting for the complete answer

We are given several piles of stones. On each move, a player selects exactly one pile and removes any positive number of stones from it.

We are asked to count how many length-n strings we can form using the first m lowercase Latin letters such that a specific periodic constraint is satisfied. The constraint is defined by a fixed offset k.

The puzzle describes a nut that moves along a bolt where each position around the bolt is circular, and each row of the bolt defines which angular positions are blocked by walls. The nut itself also has fixed protrusions around its circular boundary.

We are given a target string T that must be produced by a simple printing machine. The machine does not print arbitrary text directly. Instead, each instruction consists of a string s and a repetition count n.

We are given a 100 by 100 grid of cake cells, and we can repeatedly “stamp” a connected set of cells. Each stamping operation applies a new topping to every cell in the chosen connected region.

A marathon is modeled as a set of runners moving on a single line track. Each runner, once they start, moves with constant speed in a straight line, and before their start time they do not exist on the track at all. We are given a fixed set of these runners.

We are given a rectangular grid where every cell contains a distinct label from 1 to R×C. These labels represent the order in which Pierre ideally wants to eat dishes, from smallest number to largest number. Pierre moves on the grid like a token.

We are given a set of horses, each identified by a short name, and we are told that a complete race happened where all horses finished in some unknown strict order. Instead of observing the full ranking directly, we only receive partial observations coming from smaller races.

Let $G = (V, E)$ be a finite graph and let $s, t \in V$ be distinct vertices.

We are given two convex polygons in the plane and a large set of query points. Each polygon represents a region where we are allowed to place a station. Each query point is a candidate for a third station.

We are given two trees. The first tree, call it $T1$, is the large structure we are allowed to search inside. The second tree, $T2$, is the pattern we want to find.

We are given a 1×N board and a collection of segment tiles, one of every length from 1 to N. Initially, a single tile of length K has already been placed somewhere on the board, leaving exactly E empty cells to its left.

Bella travels between two fixed locations every day: home and workplace. She makes exactly two bus trips per day, one going from home to work and the other returning from work to home. At each location, she may store some number of umbrellas.

We are given a directed graph with N people, where each person i has exactly two outgoing edges pointing to the people they will ask for money. The process starts when an outsider selects some person in the town and asks them for money.

We are given a set of points in the plane, each point representing a token with a unique identifier from 1 to T. Then there is a sequence of P turns. On each turn, a player receives every remaining token whose point lies strictly below a given line of the form $y = Ax + B$.

We start with a paper strip that is conceptually divided into $2^N$ equal segments. Amelia’s home sits at a known segment index $P$. The strip is repeatedly folded $N$ times.

Let $D$ be a DAG in which every non-source vertex has in-degree $1$.

We are given an array of length up to one hundred thousand, and every element is restricted to a very small domain: only values from 1 to 6 appear. Over this array we must support two kinds of operations on subsegments. One operation rearranges a chosen segment into sorted order.

We are given a long string whose length is exactly nine times some number $n$. This means the string can naturally be seen as a sequence of $n$ consecutive blocks, each block having length 9. We are also given a fixed target word, “BSUIROPEN”.

We are given a broken plank whose bottom edge is fixed on the x-axis and whose top edge is described by a polyline with strictly increasing x-coordinates.