The number of n-queens configurations

The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n x n board. We show that there exists a constant 1.94 < a < 1.9449 such that Q(n) = ((1 + o(1))ne^(-a))^n. The constant a 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.

Based on arXiv:2107.13460