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In Chess, there is the Threefold Repetition rule where if a sequence of moves is repeated 3 times in a row, either player can claim a draw.

Say two players wanted to play a legal, infinite game of chess. How would they decide on a series of moves that never violates the threefold repetition criteria? Alternately, can it be proved that all infinite sequences (with finite alphabets) must eventually satisfy the threefold repetition criteria? If so, is it possible to know what the maximum number of moves possible in a game would be without either playing being able to claim a draw?

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    $\begingroup$ Aside from other issues, the Threefold Repetition Rule described in this post bears very little resemblance to the Threefold Repetition Rule from the actual game of chess. $\endgroup$
    – Steven Landsburg
    Commented Jan 23, 2015 at 1:23
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    $\begingroup$ projecteuclid.org/euclid.dmj/1077472364 $\endgroup$
    – Benjamin Steinberg
    Commented Jan 23, 2015 at 2:08
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    $\begingroup$ I don't see why this was closed unless people feel the op already new the answer. This question was asked and answered by Morse-Hedlund in the 40s $\endgroup$
    – Benjamin Steinberg
    Commented Jan 23, 2015 at 2:10
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    $\begingroup$ en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence#History $\endgroup$
    – Ian Morris
    Commented Jan 23, 2015 at 2:29
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    $\begingroup$ This is not a mathematical comment but a chess one (and Steven Landsburg has already hinted at this): the opening post misunderstands and misquotes the threefold repetition rule in chess. A player can claim a draw if the same POSITION occurs three times, the sequence of moves if irrelevant (it says just as much on the linked page). As there are only finitely many possible positions, there are only finitely many games that avoid a triple repetition. The mathematical question asked thus has nothing to do with the chess one. $\endgroup$
    – verret
    Commented Jan 23, 2015 at 3:21

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For the question in the title, you're looking for cube-free infinite words. Of special interest are cube-free infinite words that are binary (use an alphabet of two letters). A lot of work has been done on these. (For example, search for questions here with the tag.)

For the question specific to chess, see https://chess.stackexchange.com/q/4113/3489.

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