The Maximum Number of Sets for 12 Cards is 14

We analyze a mathematical property of the popular card game SET®. We present a novel proof that the maximum number of sets for 12 cards is 14 using finite field geometry, combinatorics, and graph theory. We present several computer algorithms for finding or approximating the maximum number of sets. We show a complete computer proof that iterates over all possible board configurations. We use this method to compute the maximum number of sets with 4 properties for a small number of cards, but it is generally too inefficient. With this method, we compute the maximum number of sets for 3 properties for all possible numbers of cards. Finally, we compare this to an existing approximation algorithm for finding maximal sets.