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A fair coin is flipped n times, and two finite sequences of heads and tails with the same length are given, say A and B. Each time A appears in the sequence of fair coin flips, Alice gets a point, and each time B appears, Bob gets a point. Who is more likely to win? This puzzle is a slight extension of Litt’s game 10. In this note, we show that the game is fair for any value of n and any two words A, B that have the same auto-correlation structure by building up a bijection that exchanges Bob and Alice scores. It is remarkable that the inter-correlations between A and B do not play any role in this case. Additionally, we propose a conjecture for cases where the game is unfair, providing insights into the underlying structure of the game for a fixed n.
Basdevant et al. (Fri,) studied this question.