Extensions and restrictions of Wythoff's game preserving its P positions

  • Authors:

  • Eric Duchêne;Aviezri S. Fraenkel;Richard J. Nowakowski;Michel Rigo

  • Affiliations:

  • Laboratoire LIESP, Univ. Claude Bernard Lyon 1, Bât. Nautibus (ex 710), 843, Bd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France;Department of Computer Science & Applied Mathematics, Weizmann Institute of Science, 76100 Rehovot, Israel;Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5;Institute of Mathematics, University of Liège, Grande Traverse 12 (B 37), B-4000 Liège, Belgium

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2010

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Abstract

We consider extensions and restrictions of Wythoff's game having exactly the same set of P positions as the original game. No strict subset of rules gives the same set of P positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of P positions. Testing if a move belongs to such an extended set of rules is shown to be doable in polynomial time. Many arguments rely on the infinite Fibonacci word, automatic sequences and the corresponding numeration system. With these tools, we provide new two-dimensional morphisms generating an infinite picture encoding respectively P positions of Wythoff's game and moves that can be adjoined.