The n-queens completion problem

11/22/2021

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by   Stefan Glock, et al.

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An n-queens configuration is a placement of n mutually non-attacking queens on an n× n chessboard. The n-queens completion problem, introduced by Nauck in 1850, is to decide whether a given partial configuration can be completed to an n-queens configuration. In this paper, we study an extremal aspect of this question, namely: how small must a partial configuration be so that a completion is always possible? We show that any placement of at most n/60 mutually non-attacking queens can be completed. We also provide partial configurations of roughly n/4 queens that cannot be completed, and formulate a number of interesting problems. Our proofs connect the queens problem to rainbow matchings in bipartite graphs and use probabilistic arguments together with linear programming duality.