Graph all the things
analyzing all the things you forgot to wonder about
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:
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.