相关论文: A $q$-Queens Problem. VI. The Bishops' Period
We consider the problem of placing k queens on an nxn board so that the total number of attacked squares is as small as possible. In particular, we consider the domain where k is small relative to n and derive nearly tight bounds in this…
The famous $n$-queens problem asks how many ways there are to place $n$ queens on an $n \times n$ chessboard so that no two queens can attack one another. The toroidal $n$-queens problem asks the same question where the board is considered…
How many mutually non-attacking queens can be placed on a d-dimensional chessboard of size n? The n-queens problem in higher dimensions is a generalization of the well-known n-queens problem. We provide a comprehensive overview of…
The $n$-queens problem is to determine $\mathcal{Q}(n)$, the number of ways to place $n$ mutually non-threatening queens on an $n \times n$ board. We show that there exists a constant $\alpha = 1.942 \pm 3 \times 10^{-3}$ such that…
In this paper we study the number $M_{m,n}$ of ways to place nonattacking pawns on an $m\times n$ chessboard. We find an upper bound for $M_{m,n}$ and analyse its asymptotic behavior. It turns out that $\lim_{m,n\to\infty}(M_{m,n})^{1/mn}$…
An $n$-queens configuration is a placement of $n$ mutually non-attacking queens on an $n\times n$ chessboard. The $n$-queens completion problem, introduced by Nauck in 1850, is to decide whether a given partial configuration can be…
How many mutually non-attacking queens can be placed on a d-dimensional chessboard of size n? The n-queens problem in higher dimensions is a generalization of the well-known n-queens problem. We present an integer programming formulation of…
On the n x n chessboard, the move totals of distinct pieces satisfy a small number of striking arithmetic identities. The total diagonal mobility of the bishop and the total 8-neighbor mobility of the king are exactly proportional, with…
To count the number of maximum independent arrangements of $n^2$ kings on a $2n\times 2n$ chessboard, we build a $2^n \times (n+1)$ matrix whose entries are independent arrangements of $n$ kings on $2\times 2n$ rectangles. Utilizing upper…
A magic labeling of a graph is a labeling of the edges by nonnegative integers such that the label sum over the edges incident to every vertex is the same. This common label sum is known as the index. We count magic labelings by maximum…
The $m \times n$ king graph consists of all locations on an $m \times n$ chessboard, where edges are legal moves of a chess king. %where each vertex represents a square on a chessboard and each edge is a legal move. Let $P_{m \times n}(z)$…
We consider the problem of placing $n$ nonattacking queens on a symmetric $n \times n$ Toeplitz matrix. As in the $N$-queens Problem on a chessboard, two queens may attack each other if they share a row or a column in the matrix. However,…
In his list of open problems, Martin Erickson described a certain game: "Two players alternately put queens on an n x n chess board so that each new queen is not in range of any queen already on the board (the color of the queens is…
The queen's graph $Q_{m \times n}$ has the squares of the $m \times n$ chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal of the board. A set $D$ of squares of $Q_{m \times n}$ is a…
In 2015, Steinerberger showed that the number of legal chess diagrams without promotion is bounded from above by $2\times 10^{40}$. This number was obtained by restricting both bishops and pawns position and by a precise bound when no…
A linear algorithm is described for solving the n-Queens Completion problem for an arbitrary composition of k queens, consistently distributed on a chessboard of size n x n. Two important rules are used in the algorithm: a) the rule of…
Chess graphs encode the moves that a particular chess piece can make on an $m\times n$ chessboard. We study through these graphs through the lens of chip-firing games and graph gonality. We provide upper and lower bounds for the gonality of…
How many chess rooks or queens does it take to guard all the squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can "see" whichever squares…
We introduce QUEENS, a derivative chess problem based on the classical n-queens problem. We prove that QUEENS is NP-complete, with respect to polynomial-time reductions.
A famous (and hard) chess problem asks what is the maximum number of safe squares possible in placing $n$ queens on an $n\times n$ board. We examine related problems from placing $n$ rooks. We prove that as $n\to\infty$, the probability…