sigma-Game, sigma+-Game, and Two-Dimensional Additive Cellular Automata

Abstract The σ-game, introduced by Sutner, is a combinatorial game played on a graph G and is closely related to the σ-automaton first studied by Lindenmayer. A related game is the σ + -game. In this article, we study the σ-game σ + -game played on the rectangular grid {1, 2, …, m } × {1, 2, …, n }. We analyse the σ + -game by studying the divisibility properties of the polynomials P n ( λ ) which we have introduced here. (Similar polynomials were earlier studied by Sutner). We give a simple algorithm for finding the number of solutions for the σ + -game and also give a necessary and sufficient condition for the existence of a unique solution for the σ + -game, thus partially answering a question posed by Sutner. Further, we compute the number of solutions of the σ + -game when one of n , m is of the form 2 k − 1. Finally, we look at the σ-game and the σ + -game played on cylinders and tori and give necessary and sufficient conditions for the existence of unique solutions for these games.