A Purely Geometric Variant of the Gale--Berlekamp Switching Game

We introduce the following variant of the Gale--Berlekamp switching game. Let $P$ be a set of n noncollinear points in the plane, each of them having weight $+1$ or $-1$. At each step, we pick a line $\ell$ passing through at least two points of $P$, and switch the sign of every point $p \in P\cap\ell$ to its opposite. The objective is to maximize the total weight of the elements of $P$. We show that one can always achieve that this quantity is at least $n/3$. Moreover, this can be attained by a polynomial time algorithm.