# A SAT-based Resolution of Lam's Problem

arXiv: https://arxiv.org/abs/2012.04715 · HTML: https://mlsanity.com/paper/2012.04715/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Dec 2020.

## Built on by (most cited first, 3 of 3)

- [AlphaMapleSAT: An MCTS-based Cube-and-Conquer SAT Solver for Hard Combinatorial Problems](https://mlsanity.com/paper/2401.13770/) (2024)
- [Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares](https://mlsanity.com/paper/2103.11018/) (2021)
- [Inverting Cryptographic Hash Functions via Cube-and-Conquer](https://mlsanity.com/paper/2212.02405/) (2022)

## Abstract (arXiv)

In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam's problem from projective geometryunicodex2014the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written code and did not produce nonexistence certificates. In this paper, we resolve Lam's problem by translating the problem into Boolean logic and use satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party. Our work uncovered consistency issues in both previous searchesunicodex2014highlighting the difficulty of relying on special-purpose search code for nonexistence results.