Solving Color-Switching Puzzle on an Infinite Grid

Abstract

In this paper, we embark in the study of the puzzle Parity Lights, studying its
properties. The puzzle is a simple one. It is played on a grid with variable dimensions.
Traditionally, each cell is either on or off. The puzzle starts with each cell in a random
state. The player is allowed one move, called a ”tap” at a target cell: by swapping
the state in each cell defined by the von Neumann neighbourhood at r = 1 around the
target cell. The goal of the puzzle is to put the state of each cell in the grid to off. A
modified version of this puzzle will be applied to this such that an on cell is one and
an off cell is zero. We try to generalize the puzzle such that many modulos can be
applied allowing for n-state Parity Lights puzzles. The transition function mentioned
is a representation of ”tapping” each lit cell on an empty grid.