An upper bound for the minimum number of queens covering the \(n {\times} n\) chessboard (Q1613386)

A set of queens on an \(n\times n\) chessboard \(Q_n\) dominates the chessboard if every field is either occupied or attacked by a queen. It is an old problem to determine the exact value of the minimum number \(\gamma(Q_n)\) of queens needed to dominate \(Q_n\). Using explicit constructions the authors prove that \(\gamma(Q_n)\leq \frac{8}{15}n+O(1)\) which considerably narrows the gap between the best known lower bound (essentially due to Spencer) and the previously known best upper bound due to the same authors and Cockayne.