The $n$-queens problem

TL;DR

This paper proves the asymptotic number of toroidal n-queens configurations for n ≡ 1,5 mod 6 and establishes bounds for classical n-queens configurations, resolving longstanding conjectures using advanced combinatorial methods.

Contribution

The paper confirms the asymptotic count of toroidal n-queens solutions and provides bounds for classical solutions, settling multiple longstanding conjectures.

Findings

Proves T(n) ~ ((1+o(1))ne^{-3})^n for n ≡ 1,5 mod 6.

Establishes Q(n) ≥ ((1+o(1))ne^{-3})^n for large n.

Completes the resolution of conjectures by Rivin, Vardi, and Zimmerman.

Abstract

The famous $n$-queens problem asks how many ways there are to place $n$ queens on an $n \times n$ chessboard so that no two queens can attack one another. The toroidal $n$-queens problem asks the same question where the board is considered on the surface of the torus and was asked by P\'{o}lya in 1918. Let $Q(n)$ denote the number of $n$-queens configurations on the classical board and $T(n)$ the number of toroidal $n$-queens configurations. P\'{o}lya showed that $T(n)>0$ if and only if $n \equiv 1,5 \mod 6$ and much more recently, in 2017, Luria showed that $T(n)\leq ((1+o(1))ne^{-3})^n$ and conjectured equality when $n \equiv 1,5 \mod 6$. Our main result is a proof of this conjecture, thus answering P\'{o}lya's question asymptotically. Furthermore, we also show that $Q(n)\geq((1+o(1))ne^{-3})^n$ for all $n$ sufficiently large, which was independently proved by Luria and Simkin. Combined with our main result and an upper bound of Luria, this completely settles a conjecture of Rivin, Vardi and Zimmmerman from 1994 regarding both $Q(n)$ and $T(n)$. Our proof combines a random greedy algorithm to count 'almost' configurations with a complex absorbing strategy that uses ideas from the recently developed methods of randomised algebraic construction and iterative absorption.