In the study of words, a square is a nonempty sequence of letters followed by itself. For example, "couscous" and "hotshots" are squares on the alphabet {a, b, ..., z}. A word that does not contain any squares is called square-free. For example, "hah" is square-free, but "alfalfa" is not because it contains the square "alfalf" (and also "lfalfa").

In this project we will be interested in long square-free words on the alphabet of non-negative integers. One family of such words can be built by starting with the initial prefix 0 and extending by one letter at a time, always writing down the smallest letter that doesn't introduce any squares. This successively produces the words
0
01
010
0102
01020
010201
and so on. By construction, these words are the lexicographically least (that is, earliest in dictionary order) square-free words of each length. They are all prefixes of the infinite word
0102010301020104...
whose structure is relatively simple. In particular, the nth letter of this word can be computed quickly from the binary representation of n.

What happens when we start with a different initial prefix? Does the trajectory eventually merge with the word generated from the prefix 0? If so, when does that happen? If not, what other behaviors occur, and can we classify them?