The connection between the PQ penny flip game and the dihedral groups
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This paper is inspired by the PQ penny flip game. It employs group-theoretic concepts to study the original game and also its possible extensions. We show that the PQ penny flip game can be associated with the dihedral group D_8. We prove that within D_8 there exist precisely two classes of winning strategies for Q. We establish that there are precisely two different sequences of states that can guaranteed Q's win with probability 1.0. We also show that the game can be played in the all dihedral groups D_8 n, n ≥ 1, with any significant change. We examine what happens when Q can draw his moves from the entire U(2) and we conclude that again, there are exactly two classes of winning strategies for Q, each class containing now an infinite number of equivalent strategies, but all of them send the coin through the same sequence of states as before. Finally, we consider general extensions of the game with the quantum player having U(2) at his disposal. We prove that for Q to surely win against Picard, he must make both the first and the last move.
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