I have a rook confined to a $1$ by $8$ chessboard, with the squares numbered 1 through 8. As my rook moves to different spots on the board, I create a sequence of positions $a_n$.
For example, we could have $(a_n) = (1,2,1,2,3,4,...)$.
Say a sequence $a_n$ violates threefold repetition if it contains $3$ identical consecutive sequences. (Obviously this is not how chess actually works).
For example, $(1,1,2,1,1,1)$ and $(1,2,3,2,3,2,3,8)$ both violate threefold repetition, but $(1,2,1,3,1,4)$ does not.
What is the length of the longest possible sequence that does not violate threefold repetition?
Apparently there are infinite sequences that don't even have twofold repetition:
https://en.m.wikipedia.org/wiki/Square-free_word
Á.