Chopsticks

I programmed the game and derived the data to make this visualization, including the layout. Each state is vs. where are the number of fingers on the current player's hands, and are the number of fingers on the opponent player's hands. The rules my friend goes by are as follows:

• Each player starts with 1 finger on each of their 2 hands.
• On each turn, you choose one of your hands to tap one of your opponent's hands. This will change the number of fingers on your hand. If yours has and theirs has , yours now has mod 5. That is, add the two together, but 5 spills over to become 0, 6 spills over to become 1, and so forth. (Actually, my friend uses 10 instead of 5, but that game had too many states to make a pretty graph out of.) (I find this rule the most confusing, since I had always played the game where you cause your opponent "damage" rather than gaining "points" for yourself.)
• You cannot tap or be tapped by a hand with 0 fingers.
• You lose if you have no legal moves; this happens when your opponent has 0 on both hands.

Strategy

We know that the game's end states are the ones in which you have no legal moves; these are marked in red on the graph. A state is defined as "losing" if all legal moves leave the opponent in a "winning" state, and defined as "winning" if there exists a legal move that leaves the opponent in a "losing" state. Everything that isn't "winning" or "losing" is considered a "draw". In a "draw", an optimal player will only make moves that leave their opponent in another "draw" state. In this way, optimal play involves cycling around the "draw" states forever.

There isn't much of a general rule in this variant's strategy, which is why the graph looks so beautiful. What I find most interesting is the fact that the second player can give the opponent a win on the first move (state 1,2 vs 1,3), and otherwise that opponent can give him the exact same win on the following move.