This solution doesn't hold - just because a game is symmetrical, doesn't guarantee that the optimal move in some limiting case works in all cases. For example,

This solution doesn't hold - just because a game is symmetrical, doesn't guarantee that the optimal move in some limiting case works in all cases. For example, consider if the game was played with N>2 players, or if the win condition was to make the first move modulo N, for some N>2. In those cases everything would still be symmetrical, and playing in the center would still be a winning move for huge coins, but not (necessarily) for smaller coins.