Thresholds of Queen covers
Tirthankar Adhikari, Harman Agrawal, Anjali Bhagat, Ankita Dargad, Sahana Jahagirdar, Prem Kant, Urban Larsson, Sahil Wagh

TL;DR
This paper investigates optimal Queen configurations on large chessboards, identifying thresholds where configurations become non-attacking and stable, and classifies all optimal solutions for 2 to 9 Queens.
Contribution
It introduces thresholds for non-attacking and stable configurations, and characterizes all optimal solutions for small Queen counts on large boards.
Findings
Existence of non-attacking threshold for optimal configurations.
Existence of stabilizing threshold where solutions become constant.
Complete classification of optimal configurations for 2 to 9 Queens.
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, , 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…
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Taxonomy
TopicsGame Theory and Voting Systems · Markov Chains and Monte Carlo Methods · Stochastic processes and statistical mechanics
