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Let $m, k \in \mathbb{Z}^+$, and define the sequence $(X_n)$ by X_1 = X_2 = \cdots = X_k = 0, and, for $n \ge 1$,

Let $X_n$ be the binary sequence generated by method (10) with $k=35$ and CONTENTS$(A)=(a_1a_2\ldots a_{35})_2$, where $a_{35}=1,\quad a_{31}=a_{33}=a_{35}=1,\quad a_i=0 \text{ otherwise in the final...

Let the binary representation of $\mathrm{CONTENTS}(A_n)$ be \mathrm{CONTENTS}(A_n) = (c_{n,1} c_{n,2} \ldots c_{n,k})_2, where $c_{n,i} \in {0,1}$ for $1 \le i \le k$, and $c_{n,1}$ is the most signi...

Let $(X_n)$ and $(Y_n)$ be integer sequences modulo $m$, with periods $\lambda_1$ and $\lambda_2$.

The previous solution fails because it never constructs a valid global structure linking the return-time function $q_n$ with the indexing of the base period of $X_n$, and it incorrectly treats periodi...

The sequence is defined modulo $2^e$ by $X_{n+1} = aX_n + bX_{n-1} + c \pmod{2^e}, \qquad n \ge 1.$ The goal is to choose integers $a,b,c,X_0,X_1$ so that the resulting sequence has maximal possible p...

Let $(X_n)$ and $(Y_n)$ be sequences of integers modulo $m$ with periods $\lambda_1$ and $\lambda_2$, respectively.

Let X_{n+1}=X_n+X_{n-1}\pmod{2^e} and write the state vector

Let $m = 2^e$ and consider the modified middle-square sequence defined by Coveyou: $X_0 \text{ given}, \qquad X_{n+1} = \operatorname{middle}(X_n^2 + 2^{e-1} X_n), \eqno(4)$ where the function $\opera...

Let $R_{p^r} = (\mathbb{Z}/p^r\mathbb{Z})[z]/(f(z))$ with $f(0)=1$, and denote by $\overline{z}$ the residue class of $z$ in $R_{p^r}$.

Let the binary method (10) be the scheme in which a word $X$ is updated by shifting and inserting a random bit, so that each step effectively appends a new random least significant bit while discardin...

Let the MIX machine have accumulator $A$, index register $X$, and overflow toggle $O$.

Work modulo $8$ throughout.

**Corrected Solution to Exercise 3.

We consider the linear congruential generator (LCG) in its standard integer form: X_{n+1} = (a X_n + c) \bmod m, \quad X_0 \in \{0,1,\dots,m-1\}.

From the Fibonacci generator, X_{n+1} = (X_n + X_{n-1}) \bmod m, there exists an integer $t \in {0,1}$ such that

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

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

By Exercise 5, if m=p_1^{e_1}\cdots p_r^{e_r}, \qquad a=1+k\,p_1^{f_1}\cdots p_r^{f_r},

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

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

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

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

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

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

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

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

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

**Corrected Solution for Exercise 3.

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

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

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 $(X_0, a, c, m)$:

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.

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

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

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}.

**Exercise 3.

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

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

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

**Exercise 3.

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

Let $m$ be a positive integer modulus.

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...

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

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

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 $k < 0$.

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 $d \mid m$.

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

**Exercise 3.

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

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

Let $N=m^k$.

**Solution.

**Exercise 3.

**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.

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...

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.

**Exercise 3.

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

**Exercise 3.

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

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

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.

Step K11 is \text{K11.

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.