brain

We are given a binary string and allowed to repeatedly perform a very flexible operation: pick any contiguous segment and rotate it cyclically.

We are given two types of items. There are n items of type A, each contributing value a, and m items of type B, each contributing value b. We want to repeatedly assemble identical “products”.

We are given a hidden $n times n$ grid filled with positive integers in the range $[1, n^2]$. The grid is not arbitrary: values are monotone in both directions, meaning they never decrease as we move right or down.

We are given a tree where a “fake message” starts at a fixed node $r$ and spreads outward one edge per unit time. At time $t$, every node within distance at most $t$ from $r$ has received it, so the infected set is exactly a metric ball centered at $r$.

We are standing on a line of seats, each seat holding a non-negative value. From a chosen starting seat, we may move left, right, or stay in place once per second. Whenever we land on a seat for the first time, we collect its value. Re-visiting a seat later gives nothing new.

We are given a rooted tree with node 1 as the root. Each move lets us pick a node $u$, detach it from its parent, and then perform a “leaf pruning” process inside the component rooted at $u$.

We are given an array of non-negative integers and a very flexible operation that allows us to move any amount of value from one position to another, as long as no element becomes negative.

We are asked to construct a string of decimal digits whose length does not exceed one hundred thousand, but the string is not arbitrary. The requirement is tied to all longest strictly increasing subsequences of that string.

We are given a tree with weighted nodes. The tree is undirected and can be rooted at any node we choose. Once a root is fixed, every other node has exactly one parent defined by the rooted tree structure.

We are asked to construct an $n times n$ binary grid, where each cell is either a mine or an empty cell. The grid must contain exactly $m$ mines.

We are given a rooted tree with root at vertex 1. Every non-root node has a parent, and each edge from a node to its parent has a non-negative weight. We must assign to every vertex a distinct integer from 1 to n, forming a permutation of the vertices.

We are given a continuous range of integers from L to R, where each integer represents a material. From this range, we must choose two materials (they may be the same) and compute a score based on how different their decimal representations are.

We are given a permutation of size n, representing numbers placed in n cells in a line. The goal is to understand how many times we must “reset” a special typing machine in order to restore the permutation into the identity arrangement where number i sits in cell i, but we…

Let the perfect shuffle of Exercise 204 be the MMIX program obtained in (175)–(178), using constants $p, q, r, m$, and let it map an input register state $z$ to an output state $w$.

The failure in the previous construction comes from a false invariant: masking with x \,\&\, 0xFFFFFFFF00000000 does not produce a normalized 32-bit quantity.

We restart from the actual MMIX semantics used in TAOCP.

Let $x = (x_{15}\ldots x_1 x_0)_{16}$, where each $x_i \in {0,\ldots,15}$ is a hexadecimal digit.

In MMIX, register $0$ is the constant zero register, so its contents are $0$.

Let $x>0$ and define $u = x \,\&\, (-x), \qquad v = x + u.$ Let $k$ be the unique index such that $u = 2^k$.

We restart from the actual existence condition.

Let $l$ be the number of bytes in the UTF-8 encoding of $x$.

Let $x$ be a codepoint in $0 \le x < 2^{20}+2^{16}$.

The solution must be rebuilt from the actual definitions, not from byte-range heuristics.

Let $A$ be the binary matrix with rows $\alpha_1,\ldots,\alpha_m \in {0,1}^n$.

Let $A=(a_{i,j})$ be a perfect $m\times n$ parity pattern, so for every $i,j$, a_{i,j}\equiv \sum_{j'\ne j} a_{i,j'}+\sum_{i'\ne i} a_{i',j}\pmod 2, and no row or column of $A$ is identically zero.

We are given a rectangular cake modeled as an $n times m$ grid. Inside this grid there are $k$ distinct cells, each containing exactly one candle.

We restart from the actual combinatorial structure of parity patterns and only use identities for Fibonacci polynomials that can be derived directly from their defining recurrence.

Work in the ring R=\mathbb{F}_2[x,x^{-1}]/(x^N+1), \qquad N=2n+2, so that $x^{-1}=x^{N-1}$.

We correct the solution by rebuilding the argument from the linear structure of the parity condition and avoiding any invalid submatrix or periodicity assumptions.

Each student in the classroom is associated with a range of topics they understand. If the teacher asks about a topic, every student either reacts positively if the topic lies inside their learned interval or negatively if it lies outside.

Let the bitmap be stored as $8$ consecutive rows of bytes per block column.

The failure in the previous solution is fundamental: the bitmap is 1-bit packed, so each pixel must be extracted by bit operations, not by byte-wise `LDB` interpretation.

We are given two strings of equal length. One player can freely change any character of either string at any time, while the other player can flip a whole string end to end in a single move.

The failure in the previous solution is the assumption that the right subsegment must be explicitly stored.

Let B(t) = (1-t)^2 z_0 + 2(1-t)t z_1 + t^2 z_2, \qquad 0 \le t \le 1.

Let the endpoints be rational numbers (\xi,\eta)=\left(\frac{a}{c},\frac{b}{c}\right), \qquad (\xi',\eta')=\left(\frac{a'}{c'},\frac{b'}{c'}\right), where $a,b,a',b' \in \mathbb{Z}$ and $c,c' \in \mat...

Let Algorithm T be applied to the endpoints $(x,y)$ and $(x',y')$ with quadratic form $Q$, producing a sequence of edges determined by the sign changes of $Q$ along the digitized path from $(x,y)$ to...

Let $F(x,y)$ be the integer-valued function defining the conic, as in Algorithm T.

Let F(x,y)=ax^2+bxy+cy^2+dx+ey+g define the conic, and let the algorithm operate on a segment of the curve on which, say, $x$ is strictly increasing (the other case is symmetric).

Let the conic be given by F(x,y)=ax^2+bxy+cy^2+dx+ey+f=0,\qquad a,b,c,d,e,f\in\mathbb{Q}.

Let F(x,y)=y^{2}-x^{2}-13.

We are given a set of points in the plane, and for every pair of points we can measure two different distances: the Manhattan distance, which adds absolute horizontal and vertical displacement, and the Euclidean distance, which is the straight-line distance.

We are given a single integer $n$, and we want to split it into an ordered pair of positive integers $(a, b)$ such that $a + b = n$.

We are given a circular sequence of typed brackets, where each element is an integer. A positive value represents an opening bracket of a certain type, and the corresponding negative value represents its matching closing bracket.

We are given a graph with stations as vertices and tunnels as undirected edges. Each station has a cost, and we also have a modulus value $k$. For any chosen starting station $s$, we consider all stations that are reachable from $s$ using at most $d$ edges.

Let the columns of the original bitmap $X$ be indexed by $0,1,\ldots,N-1$.

We are given two collections of stones, one stored in an inventory and the other in a chest. Each stone has a size, and for every size we know how many stones of that size exist in each location.

We are simulating a race where the cost of each lap depends on how worn the current tire set is. Each tire set starts with some initial wear value, and every time a lap is driven on that set, the lap takes exactly the current wear value in seconds, and then the wear increases…

We are given a square office, but only its left and bottom walls exist. The top and right sides are open and act like a continuous source of incoming light.

We are asked to imagine an infinite increasing sequence built from numbers that can be written in the form $$x = 2^k + 60m$$ where $k$ and $m$ are positive integers (or at least positive for $k$, and non-negative for $m$, depending on interpretation; the important part is that…

We are given a line of segments, each segment indexed from 1 to n. The interesting part is that each segment i has a constraint value a[i] which controls how restrictive the next move becomes after visiting i.

We are given a traffic light that alternates which of two one-way streets is allowed to pass. The pattern of the light is periodic and fully known in advance. Every minute belongs to either street 1 or street 2 depending on this repeating pattern. A set of cars arrives over time.

Let $G$ be a graph on ${1,\ldots,n}$ and let $S={{u_j,v_j}\mid 1\le j\le r}$ be an $r$-family.

The reviewer is correct that the original argument is invalid because it replaces pixel-level adjacency with an invented semantic decomposition.

The previous argument correctly identifies a real obstruction: in three dimensions, simplicity of individual voxels is not preserved under simultaneous deletion.

We restate the definitions precisely and then rebuild the argument from first principles.

Let the three black pixels be $a,b,c$ and assume they are pairwise king-neighbors.

The previous solution failed because it did not use the actual definition of $g$ from (159).

The previous argument fails because it models Guo–Hall thinning as uniform geometric erosion.

An expression $E(x_1,\ldots,x_m)$ is built from integer variables and integer constants using only $+$ and $\oplus$ (and possibly also $&$ in the second part).

The state of a Life automaton on a finite torus is completely determined by the initial bitmap and the update rule given in Exercise 167.

We start by separating three independent issues: the word packing geometry, the toroidal indexing, and the correctness of the bit-parallel update.

Let the eight neighbors be $a_1,\dots,a_8 \in \{0,1\}$ and the center be $b\in\{0,1\}$.

Let $X = \operatorname{custer}(X)$, where \operatorname{custer}(X)(i,j)=\overline{X(i,j)} \;\&\; S(i,j), \quad S(i,j)=X(i-1,j)\lor X(i+1,j)\lor X(i,j-1)\lor X(i,j+1).

We are given two independent progressive tax systems and a fixed total income $X$. Dmitry and Anna must split this income into two parts: Dmitry declares $t$, and Anna declares $X - t$.

Two players simulate a deterministic game on a single integer. The state is just one number, initially $n$. Players alternate turns, starting with the first player. On each turn, the active player tries to apply a division move using their own fixed divisor.

Let the $3\times 3$ configuration at time $t$ be represented by a bit matrix $X(t) = (x_{ij}(t))_{1 \le i,j \le 3}$, where each $x_{ij}(t) \in {0,1}$.

Let the eight neighbors of a cell $X$ be $X_{NW}, X_N, X_{NE}, X_W, X_E, X_{SW}, X_S, X_{SE}$.

The previous solution fails because it _assumes_ finiteness of triangle types without deriving it from the actual construction of Fig.

The board is extremely tall but only two columns wide, so every row is just a left or right cell. A white pawn starts at the bottom-left cell and moves upward row by row until it either gets stuck or reaches the top row at height $10^{18}$.

We are given a set of points on a plane, each representing a shop that yields exactly one collectible item. The key restriction is geometric: we are only allowed to pick items from shops that lie on a single straight line.

We are given a set of rain droplets that each fall onto a point on a horizontal line. Each droplet appears at a specific coordinate and only starts expanding after its own falling time.

We are given five integers that describe how many steps exist in different segments of a staircase structure. The picture (which we do not need explicitly) encodes a set of possible routes from the bottom to the top, where each route corresponds to choosing a sequence of…

The previous solution fails because it replaces the actual object in Fig.

We are given multiple independent test cases. In each test case, there are two integer arrays of the same length. For every index, we are allowed to “adjust” the value at that position, but the adjustment is not arbitrary.

We are given a program consisting of $n$ lines, and a subset of $m$ of these lines contain bugs. The positions of all buggy lines are known in advance and are strictly increasing. Toxel repeatedly performs a debugging operation. In one operation, he chooses a prefix length $i$.

We are given a string consisting of lowercase letters, and we are allowed to change at most $k$ characters. The goal is to determine whether we can turn the string into a very rigid periodic structure.

We are given a five-digit integer where all digits are different. From these five digits we are allowed to rearrange their order arbitrarily, but the resulting number must still be a valid five-digit integer, meaning it cannot start with zero.

We are given a process that builds a multiset dynamically. There are exactly n insert operations and n removal operations, interleaved in a fixed order. Each insertion adds a known value, while each removal deletes a uniformly random element from the current multiset.

We are given a collection of strings and we need to count how many of them satisfy a very specific structural pattern.

We are given a tree with $n$ vertices representing cities connected by $n-1$ roads. After each query, one existing road is removed and a new road is added, and the structure remains a tree. In each resulting tree, we must place troops on vertices.

We are given an array of length $n$, where each element is an integer in the range $[1, n]$. We are allowed to apply a transformation defined by a function $f$, which maps every value in $[1, n]$ to another value in the same range.

We are given a sequence of $n$ game rounds. At the start of each round, exactly one coin is added to T0xel’s wallet, and coins are never lost except when they are spent.

We are given a positive integer n and a digit d. We are allowed to choose another positive integer k, and we look at the product x = n · k. The goal is to make this resulting number satisfy a very specific digit pattern constraint.

We are given an array of exactly four positive integers, each between 1 and 9. The task is to decide whether this array matches a hidden pattern defined by a string “USST”, where identical characters in the string enforce equality constraints between corresponding positions…

We are given a rectangular grid of size $n times m$ whose cells are filled with the integers from $1$ to $n cdot m$. The filling order is not row-wise or column-wise.

We are given a tree with nodes labeled from 1 to n, plus an extra node 0. Node 0 is connected to node 1, so effectively node 0 acts like a root attached above the original tree. Every other edge connects the n student locations into a tree. Each student lives at a unique node i.

The task describes a deterministic way to assign numbers to an n by m grid. Imagine starting with an empty matrix and writing integers beginning from 1, increasing one by one, while always walking along the outer boundary of the remaining unfilled region in a clockwise spiral.

We are given two circles in the plane. Each circle is defined by its center coordinates and radius. For every test case, we need to count how many distinct straight lines exist such that the line is tangent to both circles at the same time.

We are given a collection of hexadecimal numbers written as strings. Each number is supposed to represent a valid non-negative integer in base 16, but the data set has a twist: some entries are correct results of hexadecimal subtraction problems, while others are wrong results…

We start with a single pile of stones. Two players alternate turns, Alice moving first. On a turn, if the pile currently has $x$ stones, the player may add between $1$ and $x$ stones inclusive. After the move, the pile size must not exceed a fixed upper bound $k$.

We are given a collection of distinct numbers written in hexadecimal, and a sequence of queries. For each query, we receive a decimal number $x$.

We restate the problem in graph-theoretic form.

The grid describes a map where each cell is either blocked or available for placing a unit. Over time, we receive a sequence of placement attempts. Each attempt tries to place a directional unit, a snake, on a specific cell facing up, down, left, or right.

We are given an $n times m$ grid that represents a tiled game board. Each cell is either empty or colored with one of three colors labeled 1, 2, and 3.

We are given an $n times n$ grid and a multiset of rectangular tiles that can be placed either horizontally or vertically. Every tile is a $1 times k$ strip for some length $k$, and we are allowed to place each strip anywhere inside the grid as long as it stays inside bounds.

We are given a list of problem difficulties, where each problem also has an implicit identifier given by its position in the input. The task is to reorder the problem indices according to difficulty from smallest to largest.

We are given a large square $ABCD$ with side length $n$. Inside it sits a smaller square $AEFG$ whose side length is a variable integer $m$, restricted to an interval $[l, r]$.

We are given a multiset or array of integers. Two players alternate turns in a game. On each turn, a player is allowed to remove one occurrence of the current maximum value present in the structure.

We restart from the definitions implicit in formulas (150) and (151) and prove directly that they generate identical labels, without introducing unproved intermediate tables.