Berlekamp's Switching Game on Finite Projective and Affine Planes

TL;DR

This paper adapts Berlekamp's switching game to finite geometries, analyzing worst-case configurations on projective and affine planes and extending findings to higher-dimensional spaces.

Contribution

It introduces a novel adaptation of Berlekamp's game to finite geometries and provides analysis of worst arrangements across different plane orders and dimensions.

Findings

Identifies worst-case bulb arrangements for finite projective and affine planes.

Extends analysis to higher-dimensional finite spaces.

Provides insights into the structure of switching configurations in finite geometries.

Abstract

I adapt Berlekamp's light bulb switching game to finite projective plans and finite affine planes, then find the worst arrangement of lit bulbs for planes of even and odd orders. The results are then extended from the planes to spaces of higher dimension.