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2005, 2005(Special): 453-462. doi: 10.3934/proc.2005.2005.453

Nim-induced dynamical systems over Z2

Michael A. Jones 1, and  Diana M. Thomas 2,

1. 

Montclair State University, Department of Mathematical Sciences, Upper Montclair, NJ 07043, United States
2. 

Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043, United States

Received  September 2004 Revised  April 2005 Published  September 2005

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: 65Q0.
Citation: Michael A. Jones, Diana M. Thomas. Nim-induced dynamical systems over Z2. Conference Publications, 2005, 2005 (Special) : 453-462. doi: 10.3934/proc.2005.2005.453
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