A dynamic programming solution to the< i> n</i>-queens problem

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.

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...

The N-queens problem is a popular classic puzzle where numbers of queen were to be placed on an n x n matrix such that no queen can attack any other queen. The Branching Factor grows in a roughly linear way, which is an important consideration for the researchers. However, many researchers have cited the issues with help of artificial intelligence search patterns say DFS, BFS and backtracking algorithms. The proposed algorithm is able to compute one unique solution in polynomial time when chess board size is greater than 7. This algorithm is based on 8 different series. For each series a different approach is taken to place the queen on a given chess board.

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 ...

In this paper we survey known results for the n-queens problem of placing n nonattacking queens on an n × n chessboard and consider extensions of the problem, e.g. other board topologies and dimensions. For all solution constructions, we either give the construction, an outline of it, or a reference. In our analysis of the modular board, we give a simple result for finding the intersections of diagonals. We then investigate a number of open research areas for the problem, stating several existing and new conjectures. Along with the known results for n-queens that we discuss, we also give a history of the problem. In particular, we note that the first proof that n nonattacking queens can always be placed on an n × n board for n > 3 is by E. Pauls, rather than by W. Ahrens who is typically cited. We have attempted in this paper to discuss all the mathematical literature in all languages on the n-queens problem. However, we look only briefly at computational approaches.

Proceedings of the 1992 ACM annual conference on Communications - CSC '92, 1992

The N-Queens problem is a commonly used example in computer science. There are numerous approaches proposed to solve the problem. We introduce several definitions of the problem, and review some of the algorithms. We classify the algorithms for the N-Queens problem into 3 categories. The fmt category cnmpriaca the algorithms generating all the solutions for a given N. The algorithms in the second category are designed to generate otdy the fundamental solutions [34]. The algorithms in the last category generate only one or several solutions but not necessarily all of them.

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

Journal of the Indonesian Mathematical Society, 2021

The famous eight queens problem with non-attacking queens placement on an 8 x 8 chessboard was first posed in the year 1848. The queens separation problem is the legal placement of the fewest number of pawns with the maximum number of independent queens placed on an N x N board which results in a separated board. Here a legal placement is defined as the separation of attacking queens by pawns. Using this concept, the current study extends the queens separation problem onto the rectangular board M x N, (M&lt;N), to result in a separated board with the maximum number of independent queens. The research work here first describes the M+k queens separation with k=1 pawn and continue to find for any k. Then it focuses on finding the symmetric solutions of the M+k queens separation with k pawns.

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.

2011

The research article examines the three distinguished heuristics approaches for solving the N-Queens problem. The problem is widely recognized as constraint satisfaction problems (CSP) in the domain of Artificial Intelligence. The N-Queens problem demands the non-attacking placements of finite number of queens over chessboard. So that, two or more queens cannot share the horizontal, vertical and diagonal positions in a straight line. In this research work, improved version of Backtracking Recursive Algorithm, modified Min-Conflicts Algorithm and classic Genetic Algorithm are applied to address the problem. The comparative results validate the efficiency of research direction.