The Maximum Number of Sets for 12 Cards is 14
Combinatorics
2025-01-23 v1 Discrete Mathematics
Number Theory
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.
Cite
@article{arxiv.2501.12565,
title = {The Maximum Number of Sets for 12 Cards is 14},
author = {Justin Stevens and Duncan Wilson},
journal= {arXiv preprint arXiv:2501.12565},
year = {2025}
}
Comments
11 pages, 7 figures, 3 tables, 2 algorithms