Geometric Variants of the Gale--Berlekamp Switching Game
Adrian Dumitrescu
TL;DR
This work extends the Gale–Berlekamp switching game to arbitrary boards and switch budgets, deriving tight asymptotics for maximum signed discrepancy under a variety of settings. Using probabilistic methods and projections $u,v$ of boards, it shows $F(H)=\Theta(A^{3/4})$ for dense boards and provides precise constants in several special configurations, including inscribed squares and disks. It also analyzes upgrades and downgrades of the switch set, proving $F(H)=\Theta(n^{3/2} t^{1/2})$ with $t n$ extra switches and characterizing discrepancy under budgeted removals. Finally, the paper extends the framework to a hyperbola-bound planar board and a cubic 3D board, obtaining $\Theta(n)$– and $\Theta(n^{5/2})$ growth rates respectively, thus generalizing Brown–Spencer’s original $\Theta(n^{3/2})$ result and illustrating a unified approach to lattice-board discrepancy problems.
Abstract
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 $F(n)= Θ(n^{3/2})$, and more precisely, that $F(n) \geq \left( 1+ o(1) \right) \sqrt{\frac{2}π} n^{3/2} \approx 0.797 \ldots n^{3/2}$. 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 $Ω(n^{3/2})$. 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 $Θ(A^{3/4})$. 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 $Ω(n)$ and $O(n (\log n)^{1/2})$.