Two Latin squares are orthogonal when every ordered pair occurs once in their superposition.
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?
1Definitions
Definition 1. A Latin square has every symbol exactly once in each row and each column.
Definition 2. Two squares are orthogonal when their 100 superimposed ordered pairs are all distinct.
2What 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.
1Status
Current status (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.[1][3][4]
1Records
3 records
Record
Kind
Assessment
Result
Supported
claim · Claim 1
A peer-reviewed 2026 SAT investigation identifies the existence of 3 MOLS(10) as an open problem.[1][3][4]
Relevance to this problem
For Three mutually orthogonal Latin squares of order ten, record mols10-claim-currently-open (“Existence of three MOLS of order 10 remains open”) records a bound, answer, status fact, or structural consequence. The record states: A peer-reviewed 2026 SAT investigation identifies the existence of 3 MOLS(10) as an open problem.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Record state
reported
Scope
the reported existence status of three mutually orthogonal Latin squares of order 10, checked on 2026-07-24
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.
Result
Reproduced
claim · Computation 1
All three arrays are Latin; the A-B and A-C overlays have 100 distinct pairs, while the B-C overlay has 91.[2]
Relevance to this problem
For Three mutually orthogonal Latin squares of order ten, record mols10-claim-published-near-triple (“The published near triple contains exactly two orthogonal pairs”) records a bound, answer, status fact, or structural consequence. The record states: All three arrays are Latin; the A-B and A-C overlays have 100 distinct pairs, while the B-C overlay has 91.
Evidence
ReproducedA computation someone reran from the artifact on this page.
Record state
supported
Scope
the first three complete Latin squares in the public neartripleMOLS10.txt file
Egan and Wanless publish three order-10 Latin squares \(A,B,C\) in their Section 8 and on Wanless's companion-data page. The public file uses letters `a` through `j` for symbols 0 through 9 and has SHA-256 digest `04db7b4b17a2321b754b694b6b3f894dd5181808a4580dbd6b3a42587dfb488a`.
The executable replay checks every row and column of all three squares. It then checks all 100 cells in each overlay. The \((A,B)\) and \((A,C)\) overlays each contain all 100 ordered symbol pairs. The \((B,C)\) overlay contains 91 distinct pairs. Its nine repeated pairs are `bi`, `cj`, `dh`, `ei`, `fj`, `gh`, `hi`, `ij`, and `jh`, each occurring twice. Its nine missing pairs are `bj`, `ch`, `di`, `ej`, `fh`, `gi`, `hj`, `ih`, and `ji`.
Thus the file supplies two explicit 2 MOLS(10) witnesses sharing \(A\). The collisions in the third overlay keep it outside the acceptance condition for a triple.
Self-contained Python verifies the source-file digest, every Latin constraint, and every overlay multiplicity.[2]
Relevance to this problem
For Three mutually orthogonal Latin squares of order ten, record mols10-artifact-near-triple-replay (“Exact replay of the order-10 near triple”) supplies evidence or a replay used to check the packet. The record states: Self-contained Python verifies the source-file digest, every Latin constraint, and every overlay multiplicity.
Evidence
ReproducedA computation someone reran from the artifact on this page.
Record state
available
Scope
all row, column, and pair-overlay conditions for the three complete squares in neartripleMOLS10.txt
Join source_lines with LF characters, save as check.py, download neartripleMOLS10.txt from source_url, then run python3 check.py /local/path/neartripleMOLS10.txt
Runtime
CPython 3, standard library only
Details
Download `neartripleMOLS10.txt` lawfully from the cited author-hosted page and pass its local path to the checker. The artifact first checks the 701-byte file's SHA-256 digest. It then parses the first three complete squares, tests all 30 rows and 30 columns against the ten-symbol alphabet, and forms a `Counter` for each of the three 100-cell overlays. Exact assertions pin the 100, 100, and 91 distinct-pair counts, as well as every repeated and missing pair in the deficient overlay. The source bytes are not stored in this packet.
The normalized 30-row payload has SHA-256 digest `7deb5d7d764dc59aec0c8eafa7280b460c4db68545a11204002c5732edd80daf`. The eight-line standard output has digest `b78eb4944a1951d054abb12ecebde2423b4c01424ae5d82475240245ec6b0752`.
Notes and companion materialContext, examples, and computations
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.
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.
How the 3 records connectTyped relations and evidence flowHow the records connect to the problem
ProblemThree mutually orthogonal Latin squares of order ten
TheoremDB contributors, “Three mutually orthogonal Latin squares of order ten,” TheoremDB research memory, snapshot of July 24, 2026. https://theoremdb.org/statements/three-mols-order-10
This problem includes 3 records joined by 3 typed links, sourced from doi.org[1], current as of July 24, 2026.
1Lean verification
Lean formalization needed
An informal proof is recorded. A Lean formalization still needs to be attached. TheoremDB Researcher can start from the exact statement and pinned world.
The prefilled request prepares the target and checks drafts. It submits the accepted proof and polls verification through any packet-review handoff.
1References
Packet source. 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. ↗open copy ↗scholarly publication · reference source · arXiv:2503.10504v2 · checked 2026-08-01Source use: citation only.States that three MOLS of order 10 remain open and proves exhaustive results for the restricted subsquare cases.Also cited at Bright, Keita, and Stevens, Electronic Journal of Combinatorics 33(1) (2026), P1.30, abstract and Introduction.Source named by the research packet.
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. ↗dataset · dataset source · neartripleMOLS10.txt checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supplies the three order-10 arrays whose replay verifies exactly two orthogonal pairs.Also cited at Public companion file for Egan and Wanless, downloaded and independently replayed on 2026-07-24.
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. ↗preprint · reference source · arXiv:2509.09633v2 · checked 2026-07-24Source use: citation only.Settles only the restricted case in which the associated pair of order-10 squares has at least two nontrivial relations.
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. ↗preprint · reference source · arXiv:2204.10996v1 · checked 2026-07-24Source use: citation only.Classifies pairs of order-10 MOLS with nontrivial relations, leaving the general triple problem open.
A direct finite formulation of the long-standing three-MOLS problem at the first unresolved order.
