Variants of the Gale-Berlekamp Switching Game and their Solutions: Balancing the Rectangle and the Cube

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 $P_4 = 4$.