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The Maximum Number of Sets for 12 Cards is 14

Justin Stevens, Duncan Wilson

TL;DR

This work determines the maximum number of sets with four properties on a 12-card board to be $14$, by a novel geometric proof that embeds SET cards as points in $\mathbb{F}_3^4$ and sets as lines, augmented by a construction based on a magic square. It combines a constructive geometric argument with computer-assisted proofs to establish exact maxima for $3 \le n \le 12$ and to explore $3$-property variants up to $n=27$, using dedicated search programs and a consecutive maximization heuristic. The key contributions are the explicit $12$-card configuration achieving $14$ sets, a rigorous demonstration that 15 sets are impossible on 12 cards, and the development of computational tools and algorithms (e.g., set_searcher_complete.py, set_searcher_3d_table.py, and the Consecutive Maximization Algorithm) to obtain near-optimal or exact results for related instances. This work advances extremal combinatorics in the SET setting and provides algorithmic strategies that can guide future proofs for larger $n$ and higher-property variants.

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.

The Maximum Number of Sets for 12 Cards is 14

TL;DR

This work determines the maximum number of sets with four properties on a 12-card board to be , by a novel geometric proof that embeds SET cards as points in and sets as lines, augmented by a construction based on a magic square. It combines a constructive geometric argument with computer-assisted proofs to establish exact maxima for and to explore -property variants up to , using dedicated search programs and a consecutive maximization heuristic. The key contributions are the explicit -card configuration achieving sets, a rigorous demonstration that 15 sets are impossible on 12 cards, and the development of computational tools and algorithms (e.g., set_searcher_complete.py, set_searcher_3d_table.py, and the Consecutive Maximization Algorithm) to obtain near-optimal or exact results for related instances. This work advances extremal combinatorics in the SET setting and provides algorithmic strategies that can guide future proofs for larger and higher-property variants.

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.
Paper Structure (14 sections, 10 theorems, 8 equations, 4 figures, 3 tables, 2 algorithms)

This paper contains 14 sections, 10 theorems, 8 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Introduction to Geometric Proof
  4. A Geometric Proof with Magic Squares
  5. Construction of a 14 Set Board for $n=12$
  6. Proof that $15$ sets is impossible for $n=12$
  7. Computer-Assisted Proofs
  8. Large Language Model Disagreement
  9. Computer Proof of the maximum number of sets for $3\le n\le 7$ cards
  10. Computer Proof for $3$ properties
  11. Consecutive Maximization Algorithm
  12. Related Work
  13. Open Questions & Conclusion
  14. Consecutive Maximization Algorithm Pseudocode

Key Result

Theorem 1.1

The maximum number of sets for $3$ and $4$ cards is $1$.

Figures (4)

  • Figure 1: A configuration of $5$ cards (points) that lead to two sets. Notice that each point is represented as a dot in $\mathbb{F}_3^2$ since the first two elements of each card in the configuration above are $0$. Observe that each of the lines is equivalent to a set in the graph above.
  • Figure 2: A configuration of $6$ cards that lead to three sets. Notice among the three points $(1,1), (1,0)$ and $(2,0)$ there are no triangles. Similarly, among $(0,1), (1,1)$, and $(0,2)$ there are no triangles.
  • Figure 3: A configuration of $7$ cards that lead to five sets. Each set is shown with a line or curve.
  • Figure 4: The maximum number of sets for $n=8$ is $8$. Notice that each line or curve forms a set.

Theorems & Definitions (20)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • proof
  • Theorem 1.3
  • proof
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • proof
  • ...and 10 more