The n-queens completion problem

Stefan Glock; David Munhá Correia; Benny Sudakov

doi:10.1007/s40687-022-00335-1

The n-queens completion problem

Res Math Sci. 2022;9(3):41. doi: 10.1007/s40687-022-00335-1. Epub 2022 Jul 6.

Authors

Stefan Glock¹, David Munhá Correia², Benny Sudakov²

Affiliations

¹ Institute for Theoretical Studies, ETH, 8092 Zurich, Switzerland.
² Department of Mathematics, ETH, Zurich Switzerland.

Abstract

An n-queens configuration is a placement of n mutually non-attacking queens on an $n \times n$ 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.