Collapse: A Fibonacci and Sturmian Game

We explore the properties of Collapse, a number game closely related to Fibonacci words. In doing so, we fully classify the set of periods (minimal or not) of finite Fibonacci words via careful examination of the Exceptional (sometimes called singular) finite Fibonacci words. Collapse is not a game in the Game Theory sense, but rather in the recreational sense, like the 15-puzzle (the game where you slide numbered tiles in an attempt to arrange them in order). It was created in an attempt to better understand Sturmian words (to be explained later). Collapse is played by manipulating finite sequences of integers, called words, using three rules. Before we introduce the rules, we need some notation. For an alphabet A, a word w is one of the following: a finite list of symbols w = w1w2w3 · · ·wn (finite word), an infinite list of symbols w = w1w2w3 · · · (infinite word), or a bi-infinite list of symbols w = · · ·w−1w0w1w2w3 · · · (bi-infinite word). The wi in each of these cases are called the letters of w. The number of letters of a finite word w is called the length of w and denoted by |w|. A subword of the word w = w1w2w3 · · · is a finite word u = wkwk+1wk+2 · · ·wk+n composed of a contiguous segment of w and denoted by u ⊆ w. For words u = u1u2 · · ·um and w = w1w2 · · ·wn, the concatenation of u and w is the word uw = u1u2 · · ·umw1w2 · · ·wn. Similarly, w is the concatenation of w with itself k times. ∗I would like to acknowledge the University of Victoria for their support and Professor Robert Burton for introducing me to the structures in the Fibonacci word.