From Combinatorial Games to Shape-Symmetric Morphisms

Abstract

Siegel suggests in his book on combinatorial games that quite simple games provide us with challenging problems: “No general formula is known for computing arbitrary Grundy values of Wythoff’s game. In general, they appear chaotic, though they exhibit a striking fractal-like pattern.”. This observation is the first motivation behind this chapter. We present some of the existing connections between combinatorial game theory and combinatorics on words. In particular, multidimensional infinite words can be seen as tilings of \(\mathbb {N}^d\). They naturally arise from subtraction games on d heaps of tokens. We review notions such as k-automatic, k-regular or shape-symmetric multidimensional words. The underlying general idea is to associate a finite automaton with a morphism.