On bounds for a board covering problem

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...

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...

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.

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.

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 ...

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

Rivin, I. and R. Zabih, A dynamic programming solution to the n-queens problem, Information Processing Letters 41 (1992) 253-256.

arXiv: Combinatorics, 2016

Parts I-IV showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ chessboard is a quasipolynomial function of $n$ whose coefficients are essentially polynomials in $q$. For partial queens, which have a subset of the queen's moves, we proved complete formulas for these counting quasipolynomials for small numbers of pieces and other formulas for high-order coefficients of the general counting quasipolynomials. We found some upper and lower bounds for the periods of those quasipolynomials by calculating explicit denominators of vertices of the inside-out polytope. Here we discover more about the counting quasipolynomials for partial queens, both familiar and strange, and the nightrider and its subpieces, and we compare our results to the empirical formulas found by Kot\v{e}\v{s}ovec. We prove some of Kot\v{e}\v{s}ovec's formulas and conjectures about the quasipolynomials and their high-order coefficients, and in some instances ...

2013

Q: Can you place 8 nonattacking queens on an 8 × 8 chessboard? Q: In ho w ma ny wa ys A: Yes! 92 The n-Queens Problem: Find a formula for the number of nonattacking configurations of n queens on an n × n chessboard.

arXiv: Combinatorics, 2016

Parts I-III showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ chessboard is a quasipolynomial function of $n$ whose coefficients are essentially polynomials in $q$ and, for pieces with some of the queen's moves, proved formulas for these counting quasipolynomials for small numbers of pieces and high-order coefficients of the general counting quasipolynomials. In this part, we focus on the periods of those quasipolynomials by calculating explicit denominators of vertices of the inside-out polytope. We find an exact formula for the denominator when a piece has one move, give intuition for the denominator when a piece has two moves, and show that when a piece has three or more moves, geometrical constructions related to the Fibonacci numbers show that the denominators grow at least exponentially with the number of pieces. Furthermore, we provide the current state of knowledge about the counting quasipolynomials for queens, bis...