Improved asymptotic upper bound on the $n$-queens completion threshold

Abstract

The $n$-queens completion threshold $qc(n)$ is the largest integer $k < n$ such that any placement of $k$ mutually non-attacking queens on an $n \times n$ chessboard can be completed to an $n$-queens configuration by adding $n - k$ queens. For all sufficiently large $n$, we improve the previously best-known upper bound on $qc(n)$ from $qc(n) \leq 0.241n$ to $qc(n) \leq 0.216n$, by constructing a non-completable partial configuration of fewer than $0.216n$ queens.