Latin squares and orthogonal arrays

Three mutually orthogonal Latin squares of order 10

Collaboration beta

Determine whether three Latin squares of order ten can be pairwise orthogonal. The source gives exact tests for fixed residuals, restricted exclusions, and necessary count conditions, while existence of a complete triple remains unresolved.

\exists A, B, C : {0, \dots, 9}^{2} \to {0, \dots, 9} Latin and pairwise orthogonal \Leftrightarrow \exists OA (100,5,10,2)

Known results and sources

Illustration: Three unknown Latin squares must satisfy every pairwise orthogonality condition; no completed triple is shown.

Research problem

Exact mathematical statement

Determine whether there exist functions

A,B,C: \{0,…,9\}^2\longrightarrow\{0,…,9\}

such that every row and column of each function is a permutation, and each map

(r,c)\longmapsto(A(r,c),B(r,c)),\quad (r,c)\longmapsto(A(r,c),C(r,c)),\quad (r,c)\longmapsto(B(r,c),C(r,c))

is a bijection onto $\{0,…,9\}^2$ . Equivalently, determine whether an

\operatorname{OA}(100,5,10,2)

exists: 100 distinct five-tuples whose every two-coordinate projection is bijective. No coordinate is intrinsically privileged. A positive answer requires actual square entries or five-tuples; a negative answer must exclude all such arrays.

Problem infographic

Problem at a glance

Illustration: Three unfilled order-ten Latin-square grids highlight one common cell and map to the five coordinates row, column, A, B and C. — Illustration: Every row and column is a permutation, and every pair of coordinates must contain each ordered symbol pair once. Existence of the triple remains open.

Current mathematical picture

Where work on Three mutually orthogonal Latin squares of order 10 stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureNormalizing the first column of every square

The resulting trivial unsatisfiability comes from invalid symmetry restrictions. Independent symbol relabelings normalize every first row; a common row permutation then normalizes the first column of only one chosen square. Use the exact three-array or five-coordinate model with proved orbit coverage for each normalization and inspect the returned explicit entries.

Route status · Narrowed route

Main reductionAt most 128 residual tests after fixing the outer pair

Fix a valid orthogonal pair and seven disjoint common transversals. The source reports rank_Q H3≥23, hence at most 128 exact binary pivot assignments for an additional transversal; a final 20-cell graph is bipartite exactly when it supplies the last two. At rational rank 28, three kernel-row types are necessary and sufficient for completion, unique up to missing-symbol relabeling. This is not a global 128-case enumeration, does not prove rank≥24, and does not eliminate ranks 23–27. Ternary kernels impose separate nonmonochromatic-line conditions.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve necessary compatibility constraints that identify the same physical point and line across overlapping rooted charts.Task status · Ready to work on

Work mapped so far

Three mutually orthogonal Latin squares of order 10 in numbers

1.1kretained lines of mathematical investigation1,120 in the current working snapshot

Argument development: 702 · 63%
Explored or eliminated routes: 69 · 6%
Computational analysis: 232 · 21%
Open obligations: 51 · 5%
Definitions and setup: 66 · 6%

9selected mapped statements2routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Prove necessary compatibility constraints that identify the same physical point and line across overlapping rooted charts.

Suggested move: Suggested next step: Build a named ten-point line with all 45 root-pair charts, common companion lines, and direction-preserving overlap maps; prove necessity from the array axioms and test whether it adds constraints beyond the seven feasible charts.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

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Research route map

14 selected steps

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Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeNormalizing the first column of every square

Route status · Narrowed route

Narrowed routeRetaining the older degree-two overlap frequency table

The exact indexed-pair identity requires Σ_{p=2}t=27000. The older table is superseded even though it satisfies earlier low moments; the current positive integer certificate respects the corrected identity. Use the corrected indexed-pair and four-point conditions, then add proved shared-identity or higher constraints beyond the feasible current model.

Route status · Narrowed route

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Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove necessary compatibility constraints that identify the same physical point and line across overlapping rooted charts.Suggested move: Suggested next step: Build a named ten-point line with all 45 root-pair charts, common companion lines, and direction-preserving overlap maps; prove necessity from the array axioms and test whether it adds constraints beyond the seven feasible charts.

Ready to work on

Supply an outer generator or structural theorem covering every extending orthogonal pair and its seven disjoint common transversals.Suggested move: Suggested next step: Retain the rank-37 full-pair requirement and all residual rational ranks 23–28, then prove that each generated residual embeds with the complementary seventy cells or produce actual complete five-tuples.

Ready to work on

Find a proved necessary count or incidence condition outside the specified feasible 307-variable system and limited rooted marginals.Suggested move: Suggested next step: Derive a direction-labeled, positive-semidefinite, or cross-root condition with exact normalization, evaluate it on the retained integer and chart witnesses, and provide either a separating calculation or an explicit remaining feasibility boundary.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen problem

Existence of three pairwise orthogonal Latin squares of order ten remains open. Restricted exclusions and near-examples do not settle it.

[3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
Peer reviewedSAT confirms twenty of Myrvold's twenty-eight pair-type exclusions associated with a 4×4 subsquare. Pairs exist in the eight other types, so excluding pairs alone cannot finish the route.[3]
2023
Peer reviewedGill and Wanless studied pairs of order-ten MOLS satisfying nontrivial binary relations. This is a restricted family of pairs.[2]
2016
Peer reviewedEgan and Wanless enumerated MOLS of small orders and supplied a near-triple of order ten. A near-triple is not three pairwise orthogonal Latin squares.[1]

Mathematical neighborhood

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction2 of 92
lemma3 of 93
negative result2 of 92
computational claim1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included

retained route statementCan three order-ten Latin squares realize all ordered pairs in every overlay?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementAt most 128 residual tests after fixing the outer pairintermediate
retained route statementThe Egan–Wanless pair and 63 fixed-entry regionsintermediate
retained route statementAnalytic affine subcases survive; the full exclusion does notintermediate
retained route statementUniversal selected-line limits and bounded hole repairsintermediate
retained route statementCorrected overlap totals and universal four-point boundsintermediate
retained route statementFeasible count system and limited rooted chartsintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureNormalizing the first column of every squarereported failure
Useful failureRetaining the older degree-two overlap frequency tablereported failure
Research targetProve necessary compatibility constraints that identify the same physical point and line across overlapping rooted charts.open
Research targetSupply an outer generator or structural theorem covering every extending orthogonal pair and its seven disjoint common transversals.open
Research targetFind a proved necessary count or incidence condition outside the specified feasible 307-variable system and limited rooted marginals.open
Narrowed routeNormalizing the first column of every squareThe resulting trivial unsatisfiability comes from invalid symmetry restrictions. Independent symbol relabelings normalize every first row; a common row permutation then normalizes the first column of only one chosen square. Use the exact three-array or five-coordinate model with proved orbit coverage for each normalization and inspect the returned explicit entries.
Narrowed routeRetaining the older degree-two overlap frequency tableThe exact indexed-pair identity requires Σ_{p=2}t=27000. The older table is superseded even though it satisfies earlier low moments; the current positive integer certificate respects the corrected identity. Use the corrected indexed-pair and four-point conditions, then add proved shared-identity or higher constraints beyond the feasible current model.

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A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve necessary compatibility constraints that identify the same physical point and line across overlapping rooted charts.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointProve necessary compatibility constraints that identify the same physical point and line across overlapping rooted charts.

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Sources and references3 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Enumeration of MOLS of small orderpeer reviewed result · Judith Egan, Ian M. Wanless · Mathematics of Computation 85, 799–824 · 2016 · ARXIV 1406.3681 · accessed Sep 7, 2026
2
Pairs of MOLS of order ten satisfying non-trivial relationspeer reviewed result · Michael J. Gill, Ian M. Wanless · Designs, Codes and Cryptography 91, 1293–1313 · 2023 · ARXIV 2204.10996 · accessed Sep 7, 2026
3
Myrvold's Results on Orthogonal Triples of 10×10 Latin Squares: A SAT Investigationpeer reviewed result · Curtis Bright, Amadou Keita, Brett Stevens · Electronic Journal of Combinatorics 33(1), P1.30 · 2026 · ARXIV 2503.10504 · accessed Sep 7, 2026

Important qualifications

This is a scoped administrative literature search, not an exhaustive survey or an independent mathematical review.
Submitted packets are not external evidence. Their novel claims, corrections and proof obligations retain their separate source-reported posture.
No external code, proof development, data file or certificate was executed or independently reproduced during collection. Public links confer no acceptance or redistribution authority.
An empty formalization or recognition list means no suitable item was established in this search; it does not establish nonexistence. Unknown origin dates and original attributions are left unresolved.
No exclusion of the full affine family or of all possible triples is inferred from the current work or from restricted pair classifications. The surviving Myrvold pair types are not asserted to extend to triples.

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Three mutually orthogonal Latin squares of order 10

Exact mathematical statement

Problem at a glance

Where work on Three mutually orthogonal Latin squares of order 10 stands

Three mutually orthogonal Latin squares of order 10 in numbers

Prove necessary compatibility constraints that identify the same physical point and line across overlapping rooted charts.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

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Formal and computational footholds

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