New bounds on the number of n-queens configurations

TL;DR

This paper establishes new lower bounds for the number of n-queens solutions and configurations on toroidal boards for specific n, confirming a conjecture and providing new upper bounds using entropy methods.

Contribution

It proves that for n of the form 4^k+1, both Q(n) and T(n) are at least exponential in n, confirming a conjecture and introducing new bounds via entropy techniques.

Findings

Both Q(n) and T(n) are at least n^{Ω(n)} for n=4^k+1.

New upper bounds on Q(n) and T(n) are derived using the entropy method.

An upper bound on perfect matchings in regular hypergraphs is established.

Abstract

In how many ways can $n$ queens be placed on an $n \times n$ chessboard so that no two queens attack each other? This is the famous $n$-queens problem. Let $Q(n)$ denote the number of such configurations, and let $T(n)$ be the number of configurations on a toroidal chessboard. We show that for every $n$ of the form $4^k+1$, $T(n)$ and $Q(n)$ are both at least $n^{\Omega(n)}$. This result confirms a conjecture of Rivin, Vardi and Zimmerman for these values of $n$. We also present new upper bounds on $T(n)$ and $Q(n)$ using the entropy method, and conjecture that in the case of $T(n)$ the bound is asymptotically tight. Along the way, we prove an upper bound on the number of perfect matchings in regular hypergraphs, which may be of independent interest.