# P2534: Three mutually orthogonal Latin squares of order ten

- ID: `P2534`
- Reference: `three-mols-order-10`
- Page: https://theoremdb.org/statements/P2534
- Record maturity: Reviewed problem with recorded work

## Problem

Do there exist three arrays \(L_1,L_2,L_3\in\{0,\ldots,9\}^{10\times10}\) such that each \(L_i\) is a Latin square and every pair \((L_i,L_j)\) is orthogonal?

### Definitions

- **Definition.** A Latin square has every symbol exactly once in each row and each column.
- **Definition.** Two squares are orthogonal when their 100 superimposed ordered pairs are all distinct.

### What counts as a solution

- Supply three 10 by 10 arrays and verify the Latin condition for all rows and columns plus all three pairwise orthogonality conditions, or give a complete machine-checkable nonexistence certificate.

## Status

A peer-reviewed 2026 SAT investigation identifies the existence of 3 MOLS(10) as an open problem. [1](#reference-1) [3](#reference-3) [4](#reference-4)

## Work

### Evidence for the current status

**Claim 1 (Existence of three MOLS of order 10 remains open).** A peer-reviewed 2026 SAT investigation identifies the existence of 3 MOLS(10) as an open problem.

Let \(N(10)\) be the largest number of pairwise orthogonal Latin squares of order 10. Parker's 1959 construction gives \(N(10)\geq2\). Bright, Keita, and Stevens state in their 2026 paper that it is unknown whether \(N(10)\geq3\), and call the construction or exclusion of 3 MOLS(10) a longstanding open problem.

Their exhaustive SAT results address a restricted hypothesis: one square in the proposed triple contains a \(4\times4\) Latin subsquare. Myrvold classified 28 possible mate-pattern pairs under that hypothesis. The SAT investigation verifies the absence of an orthogonal pair in 20 pattern cases and constructs an orthogonal pair in each of the other eight. Those eight cases still require analysis of the third square. The general three-square question remains undecided.

### Background and intake notes

Order ten is the first order whose maximum number of mutually orthogonal Latin squares is unknown. A witness or exhaustive refutation would settle a classical design problem.

- Original intake status: As of 2026-07-24, a pair of orthogonal Latin squares of order ten is known, while the existence of a triple remains open.
- Fix row, column, symbol, and square symmetries before a SAT or constraint-programming search. Save the exact normalization because incompatible symmetry conventions can make learned clauses unusable.
- Near triples give objective functions for local search. Record the number and locations of repeated ordered pairs, along with every neighborhood exhausted around the incumbent.

### Other known results

- **Computation 1** (reproduced): All three arrays are Latin; the A-B and A-C overlays have 100 distinct pairs, while the B-C overlay has 91. [2](#reference-2)

### Runnable artifacts

- **Artifact 1** (reproduced): Self-contained Python verifies the source-file digest, every Latin constraint, and every overlay multiplicity. [2](#reference-2)

### Computational notes

- The public neartripleMOLS10.txt file, SHA-256 04db7b4b17a2321b754b694b6b3f894dd5181808a4580dbd6b3a42587dfb488a, was parsed independently. All three arrays satisfy every Latin row and column check. Their three pair overlays contain 100, 100, and 91 distinct ordered pairs, respectively.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `three-mols-order-10`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Curtis Bright, Amadou Keita, and Brett Stevens, Myrvold's Results on Orthogonal Triples of 10 x 10 Latin Squares: A SAT Investigation, Electronic Journal of Combinatorics 33(1) (2026), P1.30. Curtis Bright, Amadou Keita, and Brett Stevens, Myrvold's Results on Orthogonal Triples of 10 x 10 Latin Squares: A SAT Investigation, Electronic Journal of Combinatorics 33(1) (2026), P1.30, DOI 10.37236/13960; abstract and pages 1-3 of arXiv:2503.10504v2 https://doi.org/10.37236/13960
   - Also cited at Bright, Keita, and Stevens, Electronic Journal of Combinatorics 33(1) (2026), P1.30, abstract and Introduction
   - scholarly_publication; reference source; arXiv:2503.10504v2; checked 2026-08-01
   - Open copy: https://arxiv.org/abs/2503.10504v2
   - Source use: citation_only
   - States that three MOLS of order 10 remain open and proves exhaustive results for the restricted subsquare cases.
   - Source named by the research packet.
2. <a id="reference-2"></a>Judith Egan and Ian M. Wanless, Enumeration of MOLS of small order, Mathematics of Computation 85(298) (2016), 799-824, DOI 10.1090/mcom/3010, Section 8 (Order 10), displayed squares A, B, C and the following paragraph; public companion file neartripleMOLS10.txt. Complete companion data file; source-file SHA-256 is recorded in the packet https://users.monash.edu.au/~iwanless/data/MOLS/neartripleMOLS10.txt
   - Also cited at Public companion file for Egan and Wanless, downloaded and independently replayed on 2026-07-24
   - dataset; dataset source; neartripleMOLS10.txt checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Supplies the three order-10 arrays whose replay verifies exactly two orthogonal pairs.
3. <a id="reference-3"></a>Curtis Bright, Amadou Keita, and Brett Stevens, Orthogonal Latin Squares of Order Ten with Two Relations: A SAT Investigation, arXiv:2509.09633v2 (2025). Bright, Keita, and Stevens, Orthogonal Latin Squares of Order Ten with Two Relations: A SAT Investigation, abstract; exhaustive only for pairs whose associated nets have at least two nontrivial relations https://arxiv.org/abs/2509.09633
   - preprint; reference source; arXiv:2509.09633v2; checked 2026-07-24
   - Source use: citation_only
   - Settles only the restricted case in which the associated pair of order-10 squares has at least two nontrivial relations.
4. <a id="reference-4"></a>Michael J. Gill and Ian M. Wanless, Pairs of MOLS of order ten satisfying non-trivial relations, arXiv:2204.10996v1 (2022). Michael J. Gill and Ian M. Wanless, Pairs of MOLS of order ten satisfying non-trivial relations, Designs, Codes and Cryptography 91 (2023), 1293-1313, DOI 10.1007/s10623-022-01149-6; abstract and conclusion https://arxiv.org/abs/2204.10996
   - preprint; reference source; arXiv:2204.10996v1; checked 2026-07-24
   - Source use: citation_only
   - Classifies pairs of order-10 MOLS with nontrivial relations, leaving the general triple problem open.