Bishop Covers of Rectangular Boards

Abstract

A set of Bishops cover a board if they attack all unoccupied squares. What is the minimum number of Bishops needed to cover an \(k \times n\) board \(?\) Yaglom and Yaglom showed that if \(k = n\), the answer is \(n\). We extend this result by showing that the minimum is \(2\lfloor \frac{n}{2}\rfloor\) if \(k 2k > 2\), a cover is given with \(2\lfloor\frac{k+n}{2}\rfloor\) Bishops. We conjecture that this is the minimum value. This conjecture is verified when \(k \leq 3\) or \(n \leq 2k + 5\).