The solution to Berlekamp's switching game (Q1115849)

Berlekamp's game consists of a 10\(\times 10\) array of light-bulbs, with 100 switches at the back, one for each bulb, and 20 switches at the front that can complement any row or column of bulbs. For any initial set S of bulbs turned on using the back switches, let f(S) be the minimal number of lights that can be achieved by throwing any combination of row and column switches. The probem is to find the maximum of f(S) over all choices of S. We show that the answer is 34. We also determine the solution for \(n\times n\) arrays with \(1\leq n\leq 9\).