The Gale-Berlekamp switching game is played on the following device: $G_n=\{1,2,\ldots,n\} \times \{1,2,\ldots,n\}$ is an $n \times n$ array of lights is controlled by $2n$ switches, one for each row or column. Given an (arbitrary) initial configuration of the board, the objective is to have as many lights on as possible. Denoting the maximum difference (discrepancy) between the number of lights that are on minus the number of lights that are off by $F(n)$, it is known (Brown and Spencer, 1971) that $F(n)= \Theta(n^{3/2})$, and more precisely, that $F(n) \geq \left( 1+ o(1) \right) \sqrt{\frac{2}{\pi}} n^{3/2} \approx 0.797 \ldots n^{3/2}$. Here we extend the game to other playing boards. For example: (i)~For any constant $c>1$, if $c n$ switches are conveniently chosen, then the maximum discrepancy for the square board is $\Omega(n^{3/2})$. From the other direction, suppose we fix any set of $a$ column switches, $b$ row switches, where $a \geq b$ and $a+b=n$. Then the maximum discrepancy is at most $-b (n-b)$. (ii) A board $H \subset \{1,\ldots,n\}^2$, with area $A=|H|$, is \emph{dense} if $A \geq c (u+v)^2$, for some constant $c>0$, where $u= |\{x \colon (x,y) \in H\}|$ and $v=|\{y \colon (x,y) \in H\}|$. For a dense board of area $A$, we show that the maximum discrepancy is $\Theta(A^{3/4})$. This result is a generalization of the Brown and Spencer result for the original game. (iii) If $H$ consists of the elements of $G_n$ below the hyperbola $xy=n$, then its maximum discrepancy is $\Omega(n)$ and $O(n (\log n)^{1/2})$.

翻译：Gale-Berlekamp 开关游戏在如下装置上进行：设 $G_n=\{1,2,\ldots,n\} \times \{1,2,\ldots,n\}$ 为一个 $n \times n$ 的灯阵列，由 $2n$ 个开关控制，每行或每列各有一个开关。给定棋盘的（任意）初始配置，目标是使尽可能多的灯点亮。记点亮灯的数量减去熄灭灯数量的最大差值（差异）为 $F(n)$，已知（Brown 和 Spencer，1971）$F(n)= \Theta(n^{3/2})$，更精确地说，$F(n) \geq \left( 1+ o(1) \right) \sqrt{\frac{2}{\pi}} n^{3/2} \approx 0.797 \ldots n^{3/2}$。本文将该游戏推广到其他游戏棋盘。例如：（i）对于任意常数 $c>1$，若巧妙地选择 $c n$ 个开关，则正方形棋盘的最大差异为 $\Omega(n^{3/2})$。从另一个方向看，假设我们固定任意一组 $a$ 个列开关和 $b$ 个行开关，其中 $a \geq b$ 且 $a+b=n$，则最大差异至多为 $-b (n-b)$。（ii）棋盘 $H \subset \{1,\ldots,n\}^2$ 的面积为 $A=|H|$，若对于某个常数 $c>0$ 有 $A \geq c (u+v)^2$，则称其为\emph{稠密的}，其中 $u= |\{x \colon (x,y) \in H\}|$，$v=|\{y \colon (x,y) \in H\}|$。对于一个面积为 $A$ 的稠密棋盘，我们证明最大差异为 $\Theta(A^{3/4})$。该结果是原始游戏的 Brown 和 Spencer 结果的推广。（iii）若 $H$ 由 $G_n$ 中位于双曲线 $xy=n$ 下方的元素组成，则其最大差异为 $\Omega(n)$ 且 $O(n (\log n)^{1/2})$。