Gale-Berlekamp转换博弈的变体及其解决方案：平衡矩形和立方体
Variants of the Gale-Berlekamp Switching Game and their Solutions: Balancing the Rectangle and the Cube
摘要：
Gale-Berlekamp灯切换游戏是在一个正方形的灯板上进行的。每个灯都有两种状态，开或关。每一行和每一列都有一个开关。转动此开关将改变该行或列上所有灯的状态。游戏的目的是最大限度地减少棋盘上的不平衡，定义为打开的灯和关闭的灯之间的绝对差异。我们研究了一个$m \times n$矩阵与n偶和$m \le n$的游戏的变体。给出了一个构造性的证明，证明了对于任意的$m \times n$矩形矩阵$A$，存在$x \in (\pm 1)^n$和$y \in (\pm 1)^m$，使得$|yAx| \le 2$ .即列和行开关，以将不平衡减少到最多$2$，构建一个完整的Python例程来找到这些开关，并针对随机生成的初始板配置测试运行算法。然后，我们将游戏扩展为三维$n \times n \times n$立方体，具有相应的行、列和层开关。我们定义了一个最小阈值$P_n$，使得对于立方体的所有初始状态，不平衡总是可以减少到最多$P_n$。然后，我们提供了一个存在性证明，$P_2 = 2$和$P_4 = 4$。
英文摘要：
The Gale-Berlekamp Light Switching Game is played on a square board of lights. Each light has two states, either on or off. There is a switch to every row and column. Turning this switch would change the state of all the lights on that row or column. The aim of the game is to minimise the imbalance in the board, defined to be the absolute difference between the number of lights switched on and that of lights switched off. We investigate variants of the game for an $m \times n$ matrix with n even and $m \le n$. We provide a constructive proof that for any $m \times n$ rectangle matrix $A$, there exists $x \in (\pm 1)^n$ and $y \in (\pm 1)^m$ such that $|yAx| \le 2$. i.e. column and row switches to reduce the imbalance to at most $2$, construct a complete Python routine to find these switches, and test run the algorithm against randomly generated initial board configurations. We then expand the game to a three-dimensional $n \times n \times n$ cube, with corresponding row, column, and layer switches. We define a minimum threshold $P_n$, such that the imbalance can always be reduced to at most $P_n$, for all initial states of the cube. We then provide an existential proof that $P_2 = 2$ and $P_4 = 4$.
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。
