Consider the following impartial combinatorial game. Two boxes each contain a nonzero number of chips. Two player alternate turns. A turn consists of a player choosing a box and emptying it, removing 0 or 1 chip from pile in the other box, then dividing the remaining chips so that both boxes again have a nonzero number of chips. Thus, (1,1) is the only terminal position. Required:
Show that (x, y) is a P-position if and only if x mod 3 = 1 and y mod 3 = 1.