Beth Kupin

Subtraction-Division Games, Patterns, and Self-similarity

Abstract: This talk will investigate a class of two-player combinatorial games with parameters a, b, and n. The game starts at n, and is a race to say the number 1. Each player on his or her turn can either subtract a from the current number, or divide the current number by b and round up. Each game has a Sprague-Grundy value associated to it, that among other things indicates whether or not that game is a first player win. We look at sequences of these values for fixed pairs of a and b. While these sequences are not periodic, for many pairs of a and b there are interesting and beautiful patterns that appear. By showing that many of these sequences fit into the category of k-automatic sequences, we will also be able to answer the following more practical question: Is there a simple formula for which player wins?