brain

The generator is X_{n+1}\equiv aX_n \pmod{2^{35}}, \qquad a=2^{17}+3, \qquad

Since \(a\) satisfies the conditions of Theorem 3.

Let $m = 2^e$ with $e \ge 3$.

Let m=p_1^{e_1}p_2^{e_2}\cdots p_r^{e_r}, and let

In (3) the multiplier is $a=B^2+1$, hence $b=a-1=B^2$.

We consider the multiplicative linear congruential sequence modulo $m = 2^{35}$, so we study the multiplicative order of $a$ in the unit group $(\mathbb{Z}/2^{35}\mathbb{Z})^\times$.

Let $B$ be the byte size of MIX, so that $m = B^e$ is the word size.

Assume $e>1$ and that $a$ is a primitive element modulo $p$.

Let $x$ be an odd integer with $x>1$.

Let $p$ be an odd prime and let $e>1$.

**Corrected Solution for Exercise 3.

Write m=2^{e}p_1^{e_1}\cdots p_t^{e_t}, where $p_1,\dots,p_t$ are distinct odd primes.

We are asked to show that if a \equiv 3 \pmod 4, then, for every integer $e>1$,

By Theorem A, the multipliers that yield the maximum period are characterized by the conditions a-1 \equiv 0 \pmod p for every prime divisor $p$ of $m$, together with the additional condition

Let the modulus be m = p_1^{e_1} p_2^{e_2} \cdots p_t^{e_t}, and let $(X_n)$ denote the linear congruential sequence defined by $(X_0, a, c, m)$:

We are asked to find all multipliers $a$ satisfying the conditions of Theorem A when $m = 2^{35} + 1$.

Let $m = 2^e$, and let $(X_n)$ be the linear congruential sequence defined by $X_{n+1} \equiv a X_n + c \pmod{2^e}, \qquad X_0 = 0,$ where $a$ and $c$ satisfy the conditions of Theorem A.

Let \(x_{n+1} \equiv a x_n + c \pmod{m}\) with \(m = 2^k\), and consider the conditions \[ c \text{ is odd}, \qquad a \equiv 1 \pmod{4}.

Let $m = 10^e$ with $e \ge 2$, and let $c$ be odd and not a multiple of 5.

We are asked to perform computations modulo $m = 9999999001$, with multipliers $a = 10$ and $a = 9999999101$.

We first verify the conditions of Theorem A for the given parameters.

**Exercise 3.

**Exercise 3.

Let m=9999999999=10^{10}-1.

Let aX=qw+r,\qquad 0\le r<w.

Let $m$ be a positive integer modulus.

Let $m$ be a positive integer modulus and let $a, c, X_0$ be integers with $0 \le X_0 < m$.

The flawed solution attempts to describe specific factorizations, but the actual question is to identify structural patterns visible in the table of factorizations of numbers of the form $w \pm 1$, wh...

We are asked to discuss the calculation of linear congruential sequences with modulus $m = 2^{32}$ on two's-complement machines such as the IBM System/370 series.

Let $m$ be a positive integer less than the computer word size $w$, and let $x$ and $y$ be nonnegative integers satisfying $0 \le x, y < m$.

Let $w$ be the word size and let $X$ be stored in location $\texttt{XRAND}$.

We are asked to compute (aX + c) \bmod w in MIX using **three instructions** when $m = w$ and $\gcd(a,w)=1$, with the result ending in register X.

Let $w$ be the word size, $0 \le a,x < m < w$, and $\gcd(m,w)=1$.

Equation (6) asserts that, for $k \ge 0$, $X_{n+k} = \bigl(a^k X_n + (a^k - 1)c/b\bigr) \bmod m, \qquad b = a-1. \eqno(6)$ We seek an expression for $X_{n+k}$ when $k < 0$.

Equation (2) defines the linear congruential sequence by X_{n+1}\equiv aX_n+c \pmod m.

Assume that $(a,m)=1$.

If $a$ and $m$ are not relatively prime, there exists a nontrivial common factor $d > 1$ such that $d \mid a$ and $d \mid m$.

Let $f$ be an arbitrary function from ${0,1,\ldots,m-1}$ into itself.

**Exercise 3.

A linear congruential sequence has the form X_{n+1} \equiv aX_n + c \pmod m.

Let K(X) denote one application of Algorithm K.

Let $N=m^k$.

**Exercise 3.

**Solution.

**Exercise 3.

**Exercise 3.

**Solution to Exercise 3.

Let the trajectory be X_0,\;X_1=f(X_0),\;X_2=f(X_1),\ldots and let the eventual cycle have length $\lambda$.

Let $L_m$ denote the length of the longest cycle in the functional digraph of a random mapping $f$ on an $m$-element set.

**Exercise 3.

Let a number in the middle-square method have $2n$ digits in base $b$, and let $X_k$ denote the $k$th number in the sequence.

Let X_{n+1}=f(X_n),\qquad X_n\in\{1,\ldots,m\}, where $f$ is chosen uniformly from the $m^m$ mappings of $\{1,\ldots,m\}$ into itself, and $X_0$ is chosen uniformly from the $m$ possible starting valu...

**Exercise 3.

Let X_0,X_1,X_2,\ldots be a sequence generated by

The sequence takes its values from the finite set \{0,1,\ldots,m-1\}, which contains exactly $m$ elements.

Algorithm K generates each new value of $X$ by a fixed deterministic rule applied to the preceding value.

Step K11 is \text{K11.

In the middle-square method for $10$-digit numbers, we square the current value and take the middle $10$ digits of the resulting $20$-digit number.

Let $X_i$ denote the number of occurrences of digit $i$ in a random sequence of $1{,}000{,}000$ decimal digits.

The desired outcome is a digit distributed as uniformly as possible on the set ${0,1,\ldots,9}$.

Each input item is a permutation of a finite length, and you are allowed to cyclically rotate it. A rotation means taking the last element and moving it to the front, repeated any number of times.

We are given a tree with $N$ nodes, and each node carries a single uppercase letter. The structure of the tree is fixed, but we are allowed to choose any node $u$ as a root. Once rooted, every node defines a rooted subtree consisting of itself and all nodes below it.

We are given a process that starts at position 0 and evolves for $T$ steps. At every second, we either increase the position by 1 or decrease it by 1. The sequence of positions over time forms a walk on the integers, starting at 0.

Two players are playing a turn-based game that changes a single integer, the current number of spaghetti strands in a shared pile. The game always starts from zero. Lario moves first, then Muigi, and they alternate for up to 100 moves each.

We are given a square chess board of size $N times N$. Each cell is identified by integer coordinates, and a single knight piece starts on one cell while a target cell is fixed elsewhere on the board.

We are given an even number of permutations, all of the same length. Each permutation represents a cyclic object: we are allowed to rotate it any number of times, meaning we can choose any cyclic shift of its elements.

We are given a collection of noodle strands, each carrying a numeric flavor value. We need to divide these strands into several dishes. Every dish must contain at least $K$ strands, and the value of a dish is defined as the maximum flavor among the strands placed into it.

We are given a building footprint in the plane, described as an axis-aligned simple polygon. Above every point inside this footprint there is a piecewise linear roof surface.

We are given an undirected graph representing islands and direct influence paths between some pairs of islands. Influence is transitive, meaning if island A can influence B and B can influence C, then A and C are in the same connected environment even without a direct edge.

Codeforces 104666L: The Bugs

We are given two strings of equal length over the alphabet {A, C, G, T}. The second string is a permutation of the first, meaning both contain exactly the same multiset of characters.

We are asked to count how many sequences of length $N$ can be formed from an alphabet of 26 symbols, while avoiding a set of forbidden substrings.

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 $N$.