Torus Queen Independence

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.