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

TL;DR

The paper generalizes the random-placement paradigm for safe squares from MST21 to bishops, queens, and their higher-dimensional analogs, introducing S_{n,k} and mu_{n,k} to quantify safe-space proportions. It proves exact asymptotics for hyper-rooks ($1/e^{k}$) and provides 2D results for bishops and queens with limits $2/e^{2}$ and $2/e^{4}$, respectively, while establishing variance-vanishing results to ensure concentration. For higher dimensions, it yields precise results for hyper-rooks and tight bounds for hyper-bishops/queens; line-pieces are analyzed via explicit integrals, with verified 3D cases. The work highlights a Poisson-like limit structure and opens multiple avenues for sharper bounds and broader generalizations across dimensions and piece types.

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.

Figures (6)

• Figure 1: Outline of the Generalizations of MST21 we present in this paper.
• Figure 2: Depiction of a $5 \times 5 \times 5$ chessboard.
• Figure 3: Depiction of the rings on a 7 by 7 chessboard, as well as the attacking path of a bishop at (3,1).
• Figure 4: Spaces attacked by a hyper-bishop placed at $(3,3,3)$ on a $5 \times 5 \times 5$ board.
• Figure 6: Movement of a line-rook placed at $(3,3,3)$ on a $5 \times 5 \times 5$ chessboard.

Theorems & Definitions (16)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.5: Higher dimensional boards
• Theorem 1.6
• Lemma 2.1
• Remark 2.2
• proof
• Theorem 3.1