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.
Categories: math.CO