Rigidity and the chess board theorem for cube packings

Queens in exile: non-attacking queens on infinite chess boards

Number the cells of a (possibly infinite) chessboard in some way with the numbers 0, 1, 2, ... Consider the cells in order, placing a queen in a cell if and only if it would not attack any earlier queen. The problem is to determine the positions of the queens. We study the problem for a doubly-infinite chessboard of size Z x Z numbered along a square spiral, and an infinite single-quadrant chessboard (of size N x N) numbered along antidiagonals. We give a fairly complete solution in the first case, based on the Tribonacci word. There are connections with combinatorial games.

Rook Endgame Problems in $m$ by $n$ Chess

We consider Chess played on an $m \times n$ board (with $m$ and $n$ arbitrary positive integers), with only the two kings and the white rook remaining, but placed at arbitrary positions. Using the symbolic finite state method, developed by Thanatipanonda and Zeilberger, we prove that on a $3 \times n$ board, for almost all initial positions, White can checkmate Black in $\leq n+2$ moves, and that this upper bound is sharp. We also conjecture that for an arbitrary $m \times n$ board, with $m,n \geq 4$ (except for $(m,n)=(4,4)$ when it equals $7$), the number of needed moves is $\leq m+n$, and t…

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

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。

Successful strategies for a queens placing game on an n x n chess board

In his list of open problems, Martin Erickson described a certain game: "Two players alternately put queens on an n x n chess board so that each new queen is not in range of any queen already on the board (the color of the queens is unimportant). The last player who can move wins." Then he asked: "Who should win?" Obviously, for n up to 3, the first player wins, if he does not miss to start at the central position in the case n=3. In this article, we give very simple always winning strategies for the first player if n is 4 or odd. The additionally (in the source package) provided computer prog…

关于由子系统划分的拉丁方的棋盘上的车的分布

Distribution of rooks on a chess-board representing a Latin square partitioned by a subsystem

研究n阶拉丁方d维推广棋盘上车的分布，通过对由子集族诱导的子系统进行分析，证明df(T_k) = (-1)^k df(T_0)，给出多线性证明，此结果有自由度，还针对d = 3得出相关结论及证明棋盘代表一个主要类别。

表示部分拉丁方的棋盘上的车的子系统分析（第2部分）

Analysis of subsystems with rooks on a chess-board representing a partial Latin square (Part 2.)

该论文分析部分拉丁方对应的棋盘车子系统，证明Cruse容量函数的闭式恒等式，得出非退化远砖块对的容量性质，推导部分拉丁方可完成的容量条件，并提出层可完成的平衡条件。