brain

We are given an array of integers where each value represents a “jumping energy” of a substance. When two substances with energies $a$ and $b$ are mixed, they contribute an energy of $ab$.

We are given a circular arrangement of candy blocks, where each block has a flavor represented by a lowercase letter. The circularity means the first and last positions are adjacent, so any segment we take can wrap around the end of the string.

We are given a collection of words and a long string written on a monster’s body. The task is to determine how many ways we can cut this long string into consecutive pieces so that each piece exactly matches one of the given words.

We are given a tree where every node carries a positive weight. A process runs for exactly $n$ rounds. In each round, one remaining node is chosen uniformly at random.

We are given a pile of chips and two players who alternate turns, starting with Jane. On each turn, the player removes between 1 and a bounded number of chips, where the bound grows with the turn index.

We are given a company hierarchy that forms a rooted tree. Each employee corresponds to a node, and every node has a subtree consisting of all employees they supervise directly or indirectly, including themselves. The company processes a sequence of bonus events.

We maintain a multiset of integers with two operations: inserting or modifying the structure in a specific ordered way, and answering how many elements lie inside a numeric interval. The collection is not just a static bag.

We restart the argument from the ROBDD construction rules and avoid manipulating the coupled recurrences in an unjustified way.

We are given a set of points in the plane, each representing a star. We want to count how many valid “spoke constellations” can be formed. A configuration is defined by choosing one star as a center and then selecting other stars around it in a very structured way.

Each person’s full name is given as a sequence of words. From that name, a compact identifier called a NAME is constructed by taking the first letter of every word and concatenating these letters in order. So a name like “jose osorio jimenez orozco” becomes “jojo”.

We are given a situation where coffee is first packed into several identical small containers. Each small container always holds exactly $K$ units of coffee, and there are $N$ such containers. So the total amount of coffee is simply $N times K$.

We are given a set of points in the plane, each point carrying a positive or negative value. A fisherman stands at the origin and can deploy a net in a very flexible way.

We are given a sequence of 2×2 matrices that play the role of convolution kernels, and another implicit sequence of 2×2 matrices that acts as the inverse under the same convolution rule. More concretely, each input test case gives a list of matrices $A0, A1, dots, A{n-1}$.

We are given a fixed-height ASCII picture consisting of 5 rows and 18 columns. Each column contains either a star character or a dot-like blank representation, and together these characters encode three digits written side by side.

We are given a sequence of integers, and we are asked to count how many of its subsequences satisfy a structural constraint that depends on the MEX of every prefix of that subsequence.

We are given two players, each owning a collection of string “tiles”. The first player has $n$ distinct tile types, and each type can be used an unlimited number of times. The second player has $m$ tile types, also with unlimited supply.

We are given a full matrix describing the outcome probabilities between every pair of 8 teams. For any two teams $i$ and $j$, the entry $a{i,j}$ gives the probability that $i$ defeats $j$ in a single match, with the complementary probability $a{j,i}$ ensuring that exactly one…

We are given a sequence of $n$ pillars, each with a height. From this sequence we are allowed to choose a non-empty subsequence while preserving the original order. After selecting, we only keep the chosen pillars.

We are given a list of integers and asked to select the largest value in that list that satisfies a specific structural property: every prime factor of that value must also appear somewhere in the list.

A Boolean function $f(x_1,\dots,x_n)$ is **read-once** if it can be expressed by a formula in which each variable $x_i$ appears exactly once.

We are standing in a 2D plane and there is a hidden point representing a car. We are given one crucial parameter, a radius r, and we can interactively move a point anywhere in the plane and receive a binary answer: whether our current position is within distance r of the…

We start with an array of length $n$. The array is repeatedly modified by choosing a random interval $[l, r]$ uniformly among all $frac{n(n+1)}{2}$ possible subsegments, and assigning all elements in that interval a new value that has never been used before.

We are given two ordered sequences of integers, both of the same length. Each sequence comes from a sensor reading over time, so the index order is fixed and meaningful.

We are given an array and a random process that repeatedly samples indices uniformly from a chosen segment of this array. After taking k independent samples, we look at the smallest and largest sampled indices and return the sum of the array over that interval.

We are asked to look at all ordered arrays of length $m$ consisting of non-negative integers whose total sum is fixed to $n$. Every such array contributes a value equal to the bitwise XOR of all its elements, and we need the sum of these XOR values over all valid arrays.

We are given an n by n matrix and a sequence of operations. Each operation selects a square submatrix using its top-left and bottom-right coordinates, then applies one geometric transformation to that submatrix.

We are given an undirected graph where each edge carries a weight that behaves like a threshold. You start at node S holding a value k, and want to reach node T.

A read-once Boolean function $f(x_1,\ldots,x_n)$ is represented by a formula tree in which each variable occurs exactly once.

Let $X = (x_{ij})$ be a $6 \times 6$ matrix with entries in ${0,1}$.

We are given a sequence of numbers and we build a deque by processing them in order. For each element, we decide independently whether it is inserted at the front or at the back.

We are repeatedly building a sequence of length n by processing the array from left to right. At each step, the current element is inserted either at the front or at the back of an initially empty deque.

The dungeon is a straight line, and the knight walks from position 0 to position D without ever turning back. Along this path there are two kinds of encounters: monsters and shops, each placed at fixed positions.

We start at the origin in a grid and want to reach a target coordinate $(x, y)$. From any current position, the rabbit has three possible moves: it can move one step right for cost $A$, one step up for cost $B$, or it can scale both coordinates by a factor of two for cost $C$.

We are given a grid representing a maze where some cells are blocked and others are free. A robot starts in the top-left cell and must execute a sequence of moves of fixed length.

Each tower in this problem acts like a light source placed on a number line. A tower at position $ai$ emits a brightness that starts at $pi$ at its own location and then decreases linearly as you move away from it.

We are given a consecutive list of integers starting from 0 up to n. Between every pair of adjacent numbers, we are allowed to insert either a plus or a minus sign, effectively deciding whether each number contributes positively or negatively to a running sum.

We are given a binary grid that represents a pond, where each cell is either water or empty ground. On top of this grid, we consider every possible square region of fixed size $k times k$.

We are given a single positive integer x, and we need to move strictly downward to find the closest smaller integer that avoids a specific digit constraint: none of its decimal digits may be 7.

We are given a binary string that evolves over time. Two kinds of operations happen: flipping a single character, and answering a query on a substring.

We are given a sequence of exam grades for a student, each grade being an integer between 0 and 100. The task is to produce a cleaned version of this sequence where every grade below 60 is removed, while keeping the relative order of the remaining grades exactly the same as in…

We are given a set of people labeled from 1 to $n$, but $n$ can be extremely large, so we cannot afford to explicitly build any structure over all individuals.

We are given a rectangular grid where every cell contains a number describing how many mines are present in a specific neighborhood around that cell.

The earlier solution fails at the point where it replaces the actual definition of $L$ from Exercise 160 with an assumed linear involution structure.

We are given a generalized knight piece that moves on an infinite integer grid. From any cell $(x,y)$, it can jump to eight symmetric positions obtained by permuting and flipping the vectors $(p,q)$ and $(q,p)$ with independent sign changes.

Let $N(i,j)$ denote the Moore neighborhood of $(i,j)$, i.

Let the given BDD represent a Boolean function $f(x_1,\dots,x_n)$ in ordered and reduced form as defined in Section 7.

Let $p=n-m$ and write $k_1=\lfloor p/3\rfloor$, $k_2=\lceil 2p/3\rceil$.

Let $p=n-m$ and write $k_1=\lfloor p/3\rfloor$, $k_2=\lceil 2p/3\rceil$.

The key failure in the previous argument is the unproven monotonicity claim: it is not true in general that swapping an adjacent data–selector inversion preserves or improves ROBDD size.

Algorithm J (sifting) for dynamic variable reordering in reduced ordered BDDs operates by selecting a variable and moving it through the current ordering by adjacent swaps, evaluating the cost functio...

We restate the problem in the language of TAOCP BDD equivalence classes.

The mistake in the previous solution is the assumption that the movement of each state is determined by the induced permutation between (104) and (106).

Let the vertices of the $n$-cube be identified with $n$-bit strings.

Let $h_n$ denote the hidden weighted bit function on variables $x_1,\dots,x_n$, where the value of $h_n(x_1,\dots,x_n)$ is $x_k$ with $k = x_1 + \cdots + x_n$, interpreted in the standard way of Exerc...

Algorithm J performs _sifting_ by repeatedly moving a chosen variable through all possible positions in the variable ordering, exchanging it with adjacent variables to minimize the BDD size.

Let $f_1,\dots,f_m$ be Boolean functions represented by a shared reduced ordered BDD, with node set size $B(f_1,\dots,f_m)$ in the sense of Section 7.

Let $C_n$ have vertices $1,2,\dots,n$ with edges $i\sim i\pm1 \pmod n$.

The central issue is that the original argument tries to maintain a global “ancestry” of nodes through repeated reductions.

The reviewer is correct: the statement is **false**, so the original proof attempt cannot be repaired.

Let level $0$ contain the root nodes of the BDD base, level $1$ the next variable layer, and level $2$ the layer below.

Let level $0$ contain the root nodes of the BDD base, level $1$ the next variable layer, and level $2$ the layer below.

Let the input variables be two binary words $x = x_1x_2x_3x_4,\qquad y = y_1y_2y_3y_4,$ and let $f_1,\dots,f_5$ denote the five output bits of the addition $x+y$ as defined in (36), where $f_1$ is the...

The addition functions $f_1, f_2, f_3, f_4, f_5$ in (36) are the Boolean functions that determine the carry propagation structure of binary addition for increasing word lengths, where $f_k(x_1,\dots,x...

Let $f(x)=\langle x_1^{w_1}\cdots x_{20}^{w_{20}}\rangle$ denote the self-dual threshold function in which the weights are those listed in the statement.

Let $f(x)=\langle x_{w_1}\cdots x_{w_n}\rangle$ denote the threshold function defined in Section 7.

Let $d(n)$ denote the quantity arising in Exercises 45–47, interpreted as the number of Hamiltonian cycles produced by the Gray-cycle constructions in the $(kr+2)$-cube after the reductions and gluing...

Number the vertices of the cycle $C_n$ as in the statement, so edges are ${x_i,x_{i+1}}$ for $1 \le i < n$ and ${x_n,x_1}$.

Number the vertices of the cycle $C_n$ as in the statement, so edges are ${x_i,x_{i+1}}$ for $1 \le i < n$ and ${x_n,x_1}$.

Let $k \ge 2$ be even and consider the $(kr+2)$-cube $G = G_k G_{k-1} \cdots G_1 G_0 G_{-1}$, where $G_i$ is an $r$-cube for $i>0$ and $G_0 = G_{-1} = P_2$.

We interpret a QDD as a shared representation of all cofactors of $f$ with respect to a variable ordering, where each internal node is labeled by a variable and edges correspond to 0/1 restriction.

Let $\Gamma_6 = g(0), g(1), \dots, g(2^6-1)$ be the 6-bit Gray binary code, where g(k) = k \oplus \lfloor k/2 \rfloor.

Let $\Gamma_6 = g(0), g(1), \dots, g(2^6-1)$ be the 6-bit Gray binary code, where g(k) = k \oplus \lfloor k/2 \rfloor.

Let $\Gamma_6 = g(0), g(1), \dots, g(2^6-1)$ be the 6-bit Gray binary code, where g(k) = k \oplus \lfloor k/2 \rfloor.

Let $\Gamma_n = g(0), g(1), \dots, g(2^n-1)$ be the $n$-bit Gray binary code defined in Section 7.

Let $\mathcal{S}(f)$ denote the set of all distinct subfunctions of $f(x_1,\dots,x_n)$ obtained by repeated Shannon decomposition with respect to variables $x_1,\dots,x_n$, as represented in the maste...

Let $f(x_1,x_2,x_3,x_4,x_5)$ 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, includi...

Let C(x_1,\dots,x_p;\,y_{11},\dots,y_{pq}) = \bigwedge_{j=1}^{q}\left(\bigvee_{i=1}^{p}(x_i\wedge y_{ij})\right) be the covering function from the statement.

Let $G=(V,E)$ be an ordinary (undirected) graph, viewed as a digraph by replacing each edge ${u,v}\in E$ with the two arcs $u\to v$ and $v\to u$.

Let $G=(V,E)$ be an ordinary (undirected) graph, viewed as a digraph by replacing each edge ${u,v}\in E$ with the two arcs $u\to v$ and $v\to u$.

Let $S={1,\dots,m}$ denote the selector variables and $T={m+1,\dots,m+2^m}$ the data variables of the multiplexer $M_m$.

Let $S={1,\dots,m}$ denote the selector variables and $T={m+1,\dots,m+2^m}$ the data variables of the multiplexer $M_m$.

Let $h_n(x_1,\ldots,x_n)$ be the hidden weighted bit function and let $h_n^\pi$ denote its permutation under $\pi$, evaluated in the fixed variable order $x_1,\ldots,x_n$.

Let $h_n(x_1,\ldots,x_n)$ denote the hidden weighted bit function and let $h_n^\pi(x_1,\ldots,x_n)=h_n(x_{\pi(1)},\ldots,x_{\pi(n)})$ be its permutation by $\pi$.

Let $h_n(x_1,\ldots,x_n)$ be the hidden weighted bit function, and let $B(h_n)$ denote the number of nodes in its reduced ordered binary decision diagram, including the two sink nodes $\bot$ and $\top...

Let a slate of offset $s$ be defined as in the construction preceding formula (97), where each slate is determined by a choice of $s$ distinguished positions among $n$ ordered positions, and offset me...

Let a slate of offset $s$ be defined as in the construction preceding formula (97), where each slate is determined by a choice of $s$ distinguished positions among $n$ ordered positions, and offset me...

Let $h_n(x_1,\dots,x_n)$ denote the hidden weighted bit function, and let $x^\psi$ be the permutation on ${0,1}^n$ defined in part (c) of Exercise 121 by \epsilon^\psi=\epsilon,\quad (x_1\cdots x_n0)^...

Let $f^{D}(x_1,\dots,x_n)=\overline{f(\overline{x_1},\dots,\overline{x_n})}$ and $f^{R}(x_1,\dots,x_n)=f(x_n,\dots,x_1)$.

The hidden-weighted-bit function $h_n$ assigns a value to a bit vector $(x_1,\dots,x_n)$ by interpreting the input as indexing into a truth table and then extracting a selected bit.

Let $\Gamma_n = {g(0), g(1), \ldots, g(2^n-1)}$ be the $n$-bit Gray binary code defined in Section 7.

Let $N \ge 1$.

Let $g(k)$ be the Gray binary code defined in (7), equivalently $g(k)=k\oplus \lfloor k/2\rfloor$ by (9).

Let $f = M_m(x_1,\ldots,x_m; x_{m+1},\ldots,x_{2m})$, where $M_m$ denotes the equality function on two $m$-bit blocks, so that $f=1$ if and only if $x_i = x_{m+i}$ for all $1 \le i \le m$.

Let the quasi-profile of a BDD for a Boolean function $f(x_1,\dots,x_n)$ be the sequence $Q_k(f)$, where $Q_k(f)$ counts the number of distinct nodes (equivalently distinct subfunctions) at level $k$,...

Let $p_k$ denote the number of beads (BDD nodes) at level $k$, and let $q_k$ denote the number of distinct subtables produced at level $k$ before reduction, in the sense of Section 7.

Let the function depend on six variables $x_1,\dots,x_6$.

Let $F_4$ denote the set of Boolean functions of four variables.

Let $\hat b_k$ denote the quantity defined in (80) of Theorem U, where $\hat b_k$ is obtained from the recurrence counting BDD nodes via subtables of order $n-k$ and their bead structure.

Let $\Gamma_3 = (g(0), g(1), \dots, g(7))$ be the 3-bit Gray binary code in cyclic order, so consecutive terms including $g(7) \to g(0)$ differ in exactly one bit, by the defining property of Gray cod...

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