Torus Queen Independence

TL;DR

This paper extends the classic n-queens problem to higher dimensions on a torus, providing new bounds and constructions for the maximum number of non-attacking queens in any dimension.

Contribution

It introduces the first results for the d-dimensional torus n-queens problem, showing bounds and constructions that surpass trivial upper limits.

Findings

The trivial upper bound n^{d-1} cannot be achieved if n is a multiple of 5 but not 25.

For every dimension d, at least n^{d-1}-O(n^{d-2}) queens can be placed independently.

The paper generalizes the classical 2D problem to higher dimensions with new bounds and constructions.

Abstract

Define a queen on $\mathbb{Z}_n^d$ with admissible moves parallel to $\mathbf{x}\in\{-1,0,1\}^d$ at arbitrary length. How many queens can be placed on $\mathbb{Z}_n^d$ without any two in conflict? In two dimensions, this problem was initiated by P\'{o}lya in 1918 and resolved by Monsky in 1989. We give the first known results in $d$ dimensions, showing that the trivial upper bound $n^{d-1}$ cannot be attained if $n$ is a multiple of $5$, not $25$. We demonstrate, for every $d$, how $n^{d-1}-O(n^{d-2})$ queens can be placed independently.