Hyper-bishops, Hyper-rooks, and Hyper-queens: Percentage of Safe Squares on Higher Dimensional Chess Boards

Abstract

The $n$ queens problem considers the maximum number of safe squares on an $n \times n$ chess board when placing $n$ queens; the answer is only known for small $n$. Miller, Sheng and Turek considered instead $n$ randomly placed rooks, proving the proportion of safe squares converges to $1/e^2$. We generalize and solve when randomly placing $n$ hyper-rooks and $n^{k-1}$ line-rooks on a $k$-dimensional board, using combinatorial and probabilistic methods, with the proportion of safe squares converging to $1/e^k$. We prove that the proportion of safe squares on an $n \times n$ board with bishops in 2 dimensions converges to $2/e^2$. This problem is significantly more interesting and difficult; while a rook attacks the same number of squares wherever it's placed, this is not so for bishops. We expand to the $k$-dimensional chessboard, defining line-bishops to attack along $2$-dimensional diagonals and hyper-bishops to attack in the $k-1$ dimensional subspace defined by its diagonals in the $k-2$ dimensional subspace. We then combine the movement of rooks and bishops to consider the movement of queens in 2 dimensions, as well as line-queens and hyper-queens in $k$ dimensions.