(σ-game, σ+-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.