Mathematical Games σ-game, σ⁺-game and two-dimensional additive cellular automata

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} x {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.