Hyper-Bishops, Hyper-Rooks, and Hyper-Queens: Percentage of Safe Squares on Higher Dimensional Chess Boards

SummaryChess has inspired an abundance of mathematical problems, especially in combinatorics and probability. One such problem, initially studied by Miller, Sheng, and Turek, considers the proportion of safe spaces when randomly placing n rooks on an n×n chess board. They show that as n approaches infinity, the proportion of safe spaces converges to 1/e2. We first generalize their results to bishops and queens. This problem is significantly more interesting and difficult; while a rook attacks the same number of spaces regardless of its position, this is not so for bishops and queens. We prove that the proportion of safe spaces on an n×n board with n randomly placed bishops converges to 2/e2, while it converges to 2/e4 for n randomly placed queens. We then extend this result to the k-dimensional chessboard and consider two natural extensions. We define line-pieces to attack along lines, and define hyper-pieces attack along (k−1)-dimensional hyper-planes. We provide exact results for line-rooks and hyper-rooks for arbitrary k, and give bounds and initial observations for line-bishops, line-queens, hyper-bishops and hyper-queens.