Sturmian Words and k-Wythoff Nim

Nim is an ancient game of strategy and variations of Nim are the source of a great deal of research in combinatorial game theory. Variations, such as k-Wythoff Nim have winning strategies that can be described via Beatty sequences and also via Sturmian words. A majority of this talk focuses on the connections among mechanical words, balanced words and Sturmian words presented by Berstel and Seebold (2002). Properties of morphisms on Sturmian words played a key role in answering an inverse question posed by Aviezri Fraenkel (2011): ‘Given a pair of complementary Beatty sequences, find a set of Nim-like rules for which the Beatty sequences form the winning positions’. (Joint work with Urban Larsson)