The number of $n$-queens configurations

Abstract

The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n×n board. We show that there exists a constant α = 1.942±3×10 −3 such that Q(n) = ((1 ± o(1)ne −α ) n . The constant α is characterized as the solution to a convex optimization problem in P([−1/2, 1/2] 2 ), the space of Borel probability measures on the square.The chief innovation is the introduction of limit objects for n-queens configurations, which we call queenons. These are a convex set in P([−1/2, 1/2] 2 ). We define an entropy function that counts the number of n-queens configurations that approximate a given queenon. The upper bound uses the entropy method. For the lower bound we describe a randomized algorithm that constructs a configuration near a prespecified queenon and whose entropy matches that found in the upper bound. The enumeration of n-queens configurations is then obtained by maximizing the (concave) entropy function in the space of queenons.Along the way we prove a large deviations principle for n-queens configurations that can be used to study their typical structure.

Computing Tighter Bounds on the $n$-Queens Constant via Newton's Method

“…The i, jth entry in N, E, S, W is interpreted as a value associated with the North, East, South, or West triangle, respectively, formed by dividing each square of an n × n chessboard into 4 right triangles. We index these matrices starting at 0, diverging from the notation used in [11,5] which use indexing that begins at 1. Figure 1 shows the n = 8 case.…”

“…This problem formulation is taken from [11,Claim 6.3], except that Simkin maximizes a concave function and we minimize its negative, a convex function. Slack variables.…”

“…Let Q(n) denote the number of ways that n queens can be arranged on an n × n chessboard in such a way that none is threatening another, i.e., no two queens are in the same row, column, or diagonal. Recent work by Simkin [11] has shown that…”

“…In [11], Simkin establishes that α ∈ [1.94, 1.9449], a strong tightening of the best previously known bounds α ∈ [1.58, 3] [6, 7]. His method finds lower and upper bounds by solving two convex optimization problems.…”

The $n$-queens completion problem

“…We refer to the surveys [5,31] for a more detailed account of the history and many related problems. Recently, there has been some exciting progress [6,19,20,32] towards the n-queens problem and its toroidal version which was introduced by Pólya in 1918.…”