Extensions and restrictions of Wyhtoff's game preserving Wythoff's P-positions

Eric Duchene (1) , Aviezri S. Fraenkel (2) , Richard J. Nowakowski (3) , Michel Rigo (4)

1 LIESP - Laboratoire d'Informatique pour l'Entreprise et les Systèmes de Production
2 Dept. of Computer Science and Applied Mathematics
3 Department of Mathematics and Statistics [Canada]
4 Université de Liège

We consider extensions and restrictions of Wythoff's game having exactly the same set of $\PP$ positions as the original game. No strict subset of rules give the same set of $\PP$ positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of $\PP$ 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 number system. With these tools, we provide new two-dimensional morphisms generating an infinite picture encoding respectively $\PP$ positions of Wythoff's game and move that can be adjoined.

Domaines: Mathématique discrète [cs.DM]

Eric Duchene, Aviezri S. Fraenkel, Richard J. Nowakowski, Michel Rigo. Extensions and restrictions of Wyhtoff's game preserving Wythoff's P-positions. Journal of Combinatorial Theory, Series A, 2009. ⟨hal-00408012⟩