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Nim-induced dynamical systems over Z2

Pages: 453 - 462, Issue Special, August 2005

 Abstract        Full Text (136.3K)              

Michael A. Jones - Montclair State University, Department of Mathematical Sciences, Upper Montclair, NJ 07043, United States (email)
Diana M. Thomas - Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043, United States (email)

Abstract: Winning and losing positions in the well-known two-player game, Nim, are defined recursively as a two symbol sequence depending on a $k$-parameter set known as the subtraction set. In this paper, we write the recursion as a nonlinear dynamical system defined on the phase space $\mathbb Z_2^{s_k}$ with the binary sequence for Nim generated by the appropriate initial conditions. The transient dynamics and Garden of Eden points are completely determined for arbitrary-sized subtraction sets. A characterization of cycle lengths for two parameter subtraction sets is determined. Extensions of the two parameter case to an arbitrary-sized subtraction set are explored.

Keywords:  Dynamical systems, nonlinear, finite field.
Mathematics Subject Classification:  Primary: 91A46, 94A55; Secondary: 65Q05.

Received: September 2004;      Revised: April 2005;      Published: September 2005.