The problem of kings (Q1899824)

Summary: Let \(f(n)\) denote the number of configurations of \(n^2\) mutually non- attacking kings on a \(2n\times 2n\) chessboard. We show that \(\log f(n)\) grows like \(2n\log n- 2n\log 2\), with an error term of \(O (n^{4/5} \log n)\). The result depends on an estimate for the sum of the entries of a high power of a matrix with positive entries.