The Maximum Number of Sets for 12 Cards is 14

TL;DR

This paper proves that the maximum number of sets with four properties among 12 cards is 14, using advanced mathematical techniques and computer algorithms, and explores methods for finding and constructing such maximum sets.

Contribution

It introduces a novel proof for the maximum number of sets with four properties among 12 cards and develops algorithms for finding and constructing maximum sets.

Findings

Maximum number of sets with 4 properties for 12 cards is 14

Complete set solver can find maximum sets for small card counts

Algorithms for constructing near-optimal maximum sets are presented

Abstract

We present a novel proof that the maximum number of sets with 4 properties for 12 cards is 14 using the geometry of the finite field F_3^4, number theory, combinatorics, and graph theory. We also present several computer algorithms for finding the maximum number of sets. In particular, we show a complete set solver 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. However, with this method, we compute the maximum number of sets for 3 properties for all possible numbers of cards. We also present an algorithm for constructing near-optimal maximum sets.