[en] In this talk, we consider Wythoff’s game and many variations studied by
Fraenkel and others. In this game, two players are taking turns removing tokens
from two piles with the following rules: either you take some tokens from only
one of the two piles, or you withdraw the same amount from both. The first
player unable to play loses the game.
More precisely, we are interested here in characterising the P-positions (i.e.
the losing positions) of the game. We present Walnut, a software commonly used
in combinatorics on words, and show how to use it to obtain short automatic
proofs of several results from the literature, as well as a long-standing conjecture
stated by Duchêne et al. regarding additional moves not changing the set of
P-positions. Moreover, Walnut allows us to state new results and conjectures
about generalisations of Wythoff’s game. This work is linked with non-standard
numeration systems for which addition is recognisable by a finite automaton.
In particular, we make use of the works of Frougny and Sakarovitch to build an
adder for these specific numeration systems.
The talk is based on a joint work with Bastien Mignoty, Michel Rigo and
Markus Whiteland.
