Solving Impartial SET Using Knowledge and Combinatorial Game Theory

Standard SET is a card game played between any number of players moving simultaneously, where a move means taking a number of cards obeying some predetermined conditions (a Set). It is mainly a game of pattern recognition and speed. The winner is the player obtaining the most Sets. To enable analyzing SET in a more mathematical and game-theoretic sense we transformed the game into an impartial combinatorial game. SET versions may differ depending on the number of characteristics c on the cards and the number of values v of each characteristic. We indicate a particular version of SET for some v and c as SET-v-c. We analyze different versions of Impartial SET using aß search with several enhancements from mathematical SET theory and from Combinatorial Game Theory. We first show using SET theory that all SET-2-c versions for c> 1 are second-player wins and that SET-v-2 versions are first-player wins for odd v and second-player wins for even v. We next show how to compute solutions using search. We give some results and discuss how solving efficiencies were dependent on the enhancements used. Especially, the use of a pruning method based on symmetry of positions was pivotal for a large efficiency enhancement. Also the use of two methods based on endgame databases filled with Combinatorial Game Theory values proved very useful in further enhancing the solving efficiency. Using all enhancements together the complex SET-4-3 game was solved, needing the investigation of some 2 billion nodes. This game is (maybe counterintuitive) a first-player win.