The $n$-queens completion problem

TL;DR

This paper investigates the minimal size of partial $n$-queens configurations that guarantee completion to a full solution, establishing bounds and connecting the problem to graph matchings using probabilistic and linear programming methods.

Contribution

It proves that partial configurations of up to $n/60$ queens can always be completed and provides examples of larger partial configurations that cannot be completed.

Findings

Any partial configuration of at most $n/60$ queens can be completed.

Partial configurations of roughly $n/4$ queens may not be completable.

The problem is connected to rainbow matchings in bipartite graphs using probabilistic and LP duality techniques.

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.