Thresholds of Queen covers

Abstract: We study optimal configurations of Queens on a square chessboard, defined as those covering the maximum number of squares. For a fixed number of Queens, $q$, we prove the existence of two thresholds in board size: a non-attacking threshold beyond which all optimal configurations are pairwise non-attacking, and a stabilizing threshold beyond which the set of optimal configurations becomes constant. Related studies on Queen domination, such as Tarnai and G\'asp\'ar (2007), focus on minimizing the number of Queens needed for full board coverage. Our approach, by contrast, fixes the number of Queens and analyzes optimal cover via a certain loss-function due to {\em internal loss} and {\em decentralization}. We demonstrate how the internal loss can be decomposed in terms of defined concepts, {\em balance} and {\em overlap concentration}. Moreover, by using our results, for sufficiently large board sizes, we find all optimal Queen configurations for all $2\le q\le 9$. And, whenever possible, we relate those solutions in terms of the classical problem of placing $q$ non-attacking Queens on a $q\times q$ board. For example, in case $q=8$, out of the twelve classical fundamental solutions, only three apply here as centralized patterns on large boards. On the other hand, the single classical fundamental solution for $q=6$ is never cover optimal on large boards, even if centralized, but another pattern that fits inside a $q\times (q+1)$ board applies.