An entry in Fortnightly Topic Challenge #35: Restricted Title 1. Title based on this xkcd.
Your biggest dream has come true: you have an infinite supply of your favorite five-flavor candy button paper. Unfortunately, you are a very picky eater. In fact, you are so picky that it is physically impossible for you to eat the same flavor pattern twice in a row. For example, if we label the flavors a-e, and you eat
aebceb, then it is impossible for you to eat flavor
b (can't repeat
c (can't repeat
ebc) next. But you can eat
Question: Is it always possible to eat another candy, regardless of the pattern you've previously eaten? If so, prove it. If not, what is the minimal number of candies you can eat before it is impossible to eat any more, and prove that it is the minimal number.