The n-Queens Problem in Higher Dimensions

The Electronic Journal of Combinatorics

By means of the Ehrhart theory of inside-out polytopes we establish a general counting theory for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen, on a polygonal convex board. The number of ways to place $q$ identical nonattacking pieces on a board of variable size $n$ but fixed shape is (up to a normalization) given by a quasipolynomial function of $n$, of degree $2q$, whose coefficients are polynomials in $q$. The number of combinatorially distinct types of nonattacking configuration is the evaluation of our quasipolynomial at $n=-1$. The quasipolynomial has an exact formula that depends on a matroid of weighted graphs, which is in turn determined by incidence properties of lines in the real affine plane. We study the highest-degree coefficients and also the period of the quasipolynomial, which is needed if the quasipolynomial is to be interpolated from data, and which is bounded by some function, not well understood, of the board and ...

We describe various computing techniques for tackling chessboard domination problems and apply these to the determination of domination and irredundance numbers for queens' and kings' graphs. In particular we show that γ(Q 15 ) = γ(Q 16 ) = 9, confirm that γ(Q 17 ) = γ(Q 18 ) = 9, show that γ(Q 19 ) = 10, show that i(Q18) = 10, improve the bound for i(Q 19 ) to 10 ≤ i(Q 19 ) ≤ 11, show that ir(Q n ) = γ(Q n ) for 1 ≤ n ≤ 13, show that IR(Q 9 ) = Γ(Q 9 ) = 13 and that IR(Q 10 ) = Γ(Q 10 ) = 15, show that γ(Q 4k+1 ) = 2k + 1 for 16 ≤ k ≤ 21, improve the bound for i(Q 22 ) to i(Q 22 ) ≤ 12, and show that IR(K 8 ) = 17, IR(K 9 ) = 25, IR(K 10 ) = 27, and IR(K 11 ) = 36.

2014

We apply to the n× n chessboard the counting theory from Part I for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen. Part I showed that the number of ways to place q identical nonattacking pieces is given by a quasipolynomial function of n of degree 2q, whose coefficients are (essentially) polynomials in q that depend cyclically on n. Here we study the periods of the quasipolynomial and its coefficients, which are bounded by functions, not well understood, of the piece's move directions, and we develop exact formulas for the very highest coefficients. The coefficients of the three highest powers of n do not vary with n. On the other hand, we present simple pieces for which the fourth coefficient varies periodically. We develop detailed properties of counting quasipolynomials that will be applied in sequels to partial queens, whose moves are subsets of those of the queen, and the nightrider, whose moves are extended knight's moves...

Information Processing Letters, 1987

Discrete Mathematics

In Parts I-III we showed that the number of ways to place q nonattacking queens or similar chess pieces on an n × n chessboard is a quasipolynomial function of n whose coefficients are essentially polynomials in q. In this part we focus on the periods of those quasipolynomials. We calculate denominators of vertices of the inside-out polytope, since the period is bounded by, and conjecturally equal to, their least common denominator. We find an exact formula for that denominator of every piece with one move and of two-move pieces having a horizontal move. For pieces with three or more moves, we produce geometrical constructions related to the Fibonacci numbers that show the denominator grows at least exponentially with q. Contents 1. Introduction 1 2. Essentials 3 2.1. From before 3 2.2. Periods and denominators 5 3. One-move riders 6 4. Two-move riders 7 5. Pieces with Three or More Moves 9 5.1. Triangle configurations 10 5.2. The golden parallelogram 11 6. Pieces with Four or More Moves 16 Dictionary of Notation 20 References 21

Parts I and II showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ square chessboard is a quasipolynomial function of $n$ in which the coefficients are essentially polynomials in $q$. We explore this function for partial queens, which are pieces like the rook and bishop whose moves are a subset of those of the queen. We compute the five highest-order coefficients of the counting quasipolynomial, which are constant (independent of $n$), and find the periodicity of the next two coefficients, which depend on the move set. For two and three pieces we derive the complete counting functions and the number of combinatorially distinct nonattacking configurations. The method, as in Parts I and II, is geometrical, using the lattice of subspaces of an inside-out polytope.

Open Journal of Discrete Mathematifcs, 2025

In this paper, we consider chessboard graphs in higher dimensions and the number of edges of their corresponding graphs. First, we solve for the number of edges for some chessboard graphs of 3 dimensions. Then, we obtain results for the number of edges for higher dimensional chessboard graphs with all dimensions sizes being the same. For the higher dimensional queen's graph, a double series solution for the number of edges was found - given both the dimension size and the number of dimensions. This solution didn't reduce readily to a closed form solution. For the higher dimensional bishop's graph, a series solution was found that involves the generalized Harmonic number. This solution series also didn't readily reduce to a closed form solution. Results for the number of edges for two types of higher dimensional knights' graphs were also found. Finally, a series solution for the number of edges for the higher dimensional king's graph was found as well. To reduce this series solution to a closed form solution, the Wolfram Alpha series calculator was utilized.

Discrete Applied Mathematics, 2002

We show that the minimum number of queens required to cover the n × n chessboard is at most 8 15 n + O(1). ?

Journal of Algebraic Combinatorics, 2014

We apply to the n × n chessboard the counting theory from Part I for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen. Part I showed that the number of ways to place q identical nonattacking pieces is given by a quasipolynomial function of n of degree 2q, whose coefficients are (essentially) polynomials in q that depend cyclically on n. Here we study the periods of the quasipolynomial and its coefficients, which are bounded by functions, not well understood, of the piece's move directions, and we develop exact formulas for the very highest coefficients. The coefficients of the three highest powers of n do not vary with n. On the other hand, we present simple pieces for which the fourth coefficient varies periodically. We develop detailed properties of counting quasipolynomials that will be applied in sequels to partial queens, whose moves are subsets of those of the queen, and the nightrider, whose moves are extended knight's moves. We conclude with the first, though strange, formula for the classical n-Queens Problem and with several conjectures and open problems. Contents 1. Introduction 2. Preliminaries 3. Two Pieces 4. Coefficients of Subspace Ehrhart Functions 4.1. Odd and even functions 4.2. The second leading coefficient of a subspace Ehrhart function 5. The Form of Coefficients on the Square Board 6. One-Move Riders 7. A Formula for the n-Queens Problem 8. Questions, Extensions 8.1. Detailed improvements 8.2. Recurrences and their lengths 8.3. Period bounds 8.3.1. Plausible bounds 8.3.2. Observed bounds 8.3.3. The geometry of coefficient periods

2020

The n-queens problem is a generalization of the eight-queens problem of placing eight queens on a standard chessboard so that no queen attacks any other queen. The original eight-queens problem was first posed in 1848 by Bezzel, a German chess player, in the Berliner Schachzeitung (or the Berlin Chess Newspaper). The generalization is due to Linolet, who asked the same question later in 1869, but now for n queens on an n x n board. The problem still retains much fascination, and continues to be studied. Why study this problem if it has already been solved? It was initially studied for “mathematical recreation.” However today, the problem is applied in parallel memory storage schemes, VLSI testing, traffic control and deadlock prevention in concurrent programming. Other applications include neural networks, constraint satisfaction problems, image processing, motion estimation in video coding, and error-correcting codes. Additionally, the problem appears naturally in biology, where it...