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.
Keyphrases