超主教、超后和超后宫：更高维度棋盘上安全方格的百分比
Hyper-bishops, Hyper-rooks, and Hyper-queens: Percentage of Safe Squares on Higher Dimensional Chess Boards
AI总结：
研究通过组合与概率方法，探讨了更高维度棋盘上随机放置超后、超车和超主教时安全方格比例的收敛性，揭示了不同棋子攻击模式对安全方格分布的影响。
摘要：
n后问题考虑的是在n×n的棋盘上放置n个后时，安全方格的最大数量；答案仅已知于小n。Miller、Sheng和Turek考虑的是随机放置n个超后，证明安全方格的比例收敛到1/e²。我们推广并解决当在k维棋盘上随机放置n个超后和n^{k-1}条线后时的情况，使用组合和概率方法，安全方格的比例收敛到1/e^k。我们证明在n×n棋盘上，二维中放置主教的安全方格比例收敛到2/e²。这个问题显著更有趣和困难；虽然车攻击的方格数量在放置位置不变，但主教则不同。我们扩展到k维棋盘，定义线主教沿二维对角线攻击，超主教在k-1维子空间中攻击。然后结合车和主教的移动，考虑二维中的后移动，以及k维中的线后和超后移动。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
