Related papers: Asymptotic Results for the Queen Packing Problem
We introduce a two player game on an n x n chessboard where queens are placed by alternating turns on a chessboard square whose availability is determined by the number of queens already on the board which can attack that square modulo two.…
Number the cells of a (possibly infinite) chessboard in some way with the numbers 0, 1, 2, ... Consider the cells in order, placing a queen in a cell if and only if it would not attack any earlier queen. The problem is to determine the…
The number of ways to place $q$ nonattacking queens, bishops, or similar chess pieces on an $n\times n$ square chessboard is essentially a quasipolynomial function of $n$ (by Part I of this series). The period of the quasipolynomial is…
We apply our geometrical theory for counting placements of $q$ nonattacking on an $n\times n$ chessboard, from Parts~I and II, to partial queens: that is, chess pieces with any combination of horizontal, vertical, and $45^\circ$-diagonal…
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…
In this work, we have introduced two innovative quantum algorithms: the Direct Column Algorithm and the Quantum Backtracking Algorithm to solve N-Queens problem, which involves the arrangement of $N$ queens on an $N \times N$ chessboard…
We study k-dependence and half domination problems for king's graphs in dimension n (n>1). Various sharp bounds are provided and a few conjectures are formulated in the cases the estimates are not the best possible.
In this paper, we derive simple closed-form expressions for the $n$-queens problem and three related problems in terms of permanents of $(0,1)$ matrices. These formulas are the first of their kind. Moreover, they provide the first method…
In recent work Simkin shows that bounds on an exponent occurring in the famous $n$-queens problem can be evaluated by solving convex optimization problems, allowing him to find bounds far tighter than previously known. In this note we use…
Quantum computers can potentially solve problems that are computationally intractable on a classical computer in polynomial time using quantum-mechanical effects such as superposition and entanglement. The N-Queens Problem is a notable…
Some preliminary results are reported on the equivalence of any n-queens problem with the roots of a Boolean valued quadratic form via a generic dimensional reduction scheme. It is then proven that the solutions set is encoded in the…
We consider the minimal number of points on a regular grid on the plane that generates $n$ line segments of points of exactly length $k$. We illustrate how this is related to the $n$-queens problem on the toroidal chessboard and show that…
The peaceable queens problem asks to determine the maximum number $a(n)$ such that there is a placement of $a(n)$ white queens and $a(n)$ black queens on an $n \times n$ chessboard so that no queen can capture any queen of the opposite…
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…
Consider a set $P$ of $n$ points on the boundary of an axis-aligned square $Q$. We study the boundary-anchored packing problem on $P$ in which the goal is to find a set of interior-disjoint axis-aligned rectangles in $Q$ such that each…
We generalize the recent results of Chaiken et al. to a rectangular $m\times n$ chessboard. An explicit formula for the number of nonattacking configurations of one-move riders on such a chessboard is calculated in two different ways, one…
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 $\lim_{m,n\to\infty}(M_{m,n})^{1/mn}$…
We study the domination number $\gamma(Q_n^3)$ of the three-dimensional $n \times n \times n$ queen graph. The main result is a stratified theorem computing, for each position type -- corner, edge, face, or interior -- the number 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…
Consider N equally-spaced points on a circle of circumference N. Choose at random n points out of $N$ on this circle and append clockwise an arc of integral length k to each such point. The resulting random set is made of a random number of…