TY - JOUR
T1 - Mathematical Games σ-game, σ+-game and two-dimensional additive cellular automata
AU - Barua, Rana
AU - Ramakrishnan, S.
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 1996/2/5
Y1 - 1996/2/5
N2 - 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 pn(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 2k - 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.
AB - 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 pn(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 2k - 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.
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U2 - 10.1016/0304-3975(95)00091-7
DO - 10.1016/0304-3975(95)00091-7
M3 - Article
AN - SCOPUS:0030569853
VL - 154
SP - 349
EP - 366
JO - Theoretical Computer Science
JF - Theoretical Computer Science
SN - 0304-3975
IS - 2
ER -