Saved packet status (The exact enumeration record ends at order 11). The exact value of R_12 remains open. The last published exact count is R_11, while the published order-12 figure is a randomized estimate.[1][3][4][5]
1Packet records
5 records
Record
Kind
Assessment
The exact value of R_12 remains open. The last published exact count is R_11, while the published order-12 figure is a randomized estimate.
Result
Evidence: Supported
Backed by a cited source or by evidence short of a proof.
claim · Claim 1
The exact value of R_12 remains open. The last published exact count is R_11, while the published order-12 figure is a randomized estimate.[1][3][4][5]
Relevance to this problem
For Number of reduced Latin squares of order 12, record rls12-claim-current-status (“The exact enumeration record ends at order 11”) records a bound, answer, status fact, or structural consequence. The record states: The exact value of R_12 remains open.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Record state
reported
Scope
the published exact enumeration record for reduced Latin squares through order 12, audited on 2026-07-25
McKay and Wanless completed the exact enumeration through order 11. Their paper gives \(R_{11}=5363937773277371298119673540771840\) and says that the same graph-classification method was unlikely to reach \(R_{12}\) soon because there are more than \(10^{11}\) regular bipartite graphs of order 24 and degree 6.
McKay and Rogoyski had earlier reported a randomized estimate for order 12. Stones's 2010 survey explicitly says that \(R_{12}\) is unknown and lists the exact sequence only through order 11. A search on 2026-07-25 of the exact-count papers, the later order-11 class enumeration, the survey literature, and the current author-maintained count table found no subsequent exact value. The candidate therefore remains an open exact-enumeration problem.
The published values satisfy L_11=11!10!R_11 exactly, separating reduced squares from all labelled squares.
Result
Evidence: Reproduced
A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1
The published values satisfy L_11=11!10!R_11 exactly, separating reduced squares from all labelled squares.[1]
Relevance to this problem
For Number of reduced Latin squares of order 12, record rls12-claim-order-11-calibration (“The order-11 exact count fixes the normalization”) records a bound, answer, status fact, or structural consequence. The record states: The published values satisfy L_11=11!10!R_11 exactly, separating reduced squares from all labelled squares.
Evidence
ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope.
Record state
established
Scope
all reduced and labelled Latin squares of order 11
For order \(n\), every reduced Latin square has exactly \(n!(n-1)!\) labelled forms, so
\[
L_n=n!(n-1)!R_n.
\]
McKay and Wanless give
\[
R_{11}=5363937773277371298119673540771840
\]
and
\[
L_{11}=776966836171770144107444346734230682311065600000.
\]
The companion replay checks the multiplication exactly. This calibration prevents the common error of substituting a count of labelled squares, isotopy classes, or main classes for the reduced count requested by the candidate.
Published lower and divisibility theorems plus a direct row-choice bound give 4410573106297854867267286926659737440 <= R_12 <= (11!)^11.
Result
Evidence: Reproduced
A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 2
Published lower and divisibility theorems plus a direct row-choice bound give 4410573106297854867267286926659737440 <= R_12 <= (11!)^11.[1]
Relevance to this problem
For Number of reduced Latin squares of order 12, record rls12-claim-rigorous-bracket (“A rigorous elementary bracket contains the estimate”) records a bound, answer, status fact, or structural consequence. The record states: Published lower and divisibility theorems plus a direct row-choice bound give 4410573106297854867267286926659737440 <= R_12 <= (11!)^11.
Evidence
ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope.
The standard van der Waerden permanent bound quoted by McKay and Wanless gives
\[
L_n\ge\frac{(n!)^{2n}}{n^{n^2}}.
\]
After dividing by \(12!11!\), the order-12 reduced count is at least the ceiling of the resulting rational number:
\[
R_{12}\ge4410573106297854867267286926659737034.
\]
Their Theorem 2 also proves that \(R_{2m}\) is divisible by \(m!\). Thus \(R_{12}\) is divisible by \(6!=720\), and rounding the preceding lower bound to the next multiple gives
\[
R_{12}\ge4410573106297854867267286926659737440.
\]
For an elementary upper bound, fix the first row and column. Each of the remaining 11 rows has at most \(11!\) possible fillings once its first entry is fixed. Ignoring the column restrictions can only increase the count, so
\[
R_{12}\le(11!)^{11}=409933016554924328182440935903164918932547530146724293451448320000000000000000000000.
\]
The published estimate \(1.62\times10^{44}\) lies inside this broad rigorous interval.
McKay and Rogoyski estimated R_12 at 1.62e44 using an unbiased sequential-extension estimator over 1.1 million trials.
Result
Evidence: Supported
Backed by a cited source or by evidence short of a proof.
claim · Claim 2
McKay and Rogoyski estimated R_12 at 1.62e44 using an unbiased sequential-extension estimator over 1.1 million trials.[2]
Relevance to this problem
For Number of reduced Latin squares of order 12, record rls12-claim-published-estimate (“The primary order-12 estimate is 1.62 times 10^44”) records a bound, answer, status fact, or structural consequence. The record states: McKay and Rogoyski estimated R_12 at 1.62e44 using an unbiased sequential-extension estimator over 1.1 million trials.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Record state
reported
Scope
the 1,100,000 randomized trials reported for reduced Latin squares of order 12
McKay and Rogoyski generated a normalized Latin square row by row. At each step they chose uniformly among legal extensions with the required first-column entry. If \(e_i\) is the number of legal choices at step \(i\), then \(e_1e_2\cdots e_{n-1}\) is an unbiased estimator of \(R_n\). Their Table 2 reports 1,100,000 trials for \(n=12\) and the estimate
\[
R_{12}\mathrel{\approx}1.62\times10^{44}.
\]
The authors described the displayed digits as probably accurate to one unit in the last shown digit. This value is statistical evidence. It is neither an exact integer nor a rigorous confidence interval.
Standard-library Python enumerates R_2 through R_6 and checks the order-11 identity, lower bound, divisibility rounding, and elementary upper bound.
Artifact
Evidence: Executable material
Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1
Standard-library Python enumerates R_2 through R_6 and checks the order-11 identity, lower bound, divisibility rounding, and elementary upper bound.[1]
Relevance to this problem
For Number of reduced Latin squares of order 12, record rls12-artifact-small-order-replay (“Exact small-order enumeration and order-12 arithmetic replay”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python enumerates R_2 through R_6 and checks the order-11 identity, lower bound, divisibility rounding, and elementary upper bound.
Evidence
Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay.
Record state
available
Scope
all reduced Latin squares of orders 2 through 6, the published order-11 normalization identity, and exact order-12 bound arithmetic
The row-by-row backtracker fixes the natural first row and first column. Bit masks enforce that every symbol occurs once per row and once per column. A minimum-remaining-values choice selects the next column inside each row. The complete searches reproduce \(1,1,4,56,9408\) for orders 2 through 6.
The same program checks \(L_{11}=11!10!R_{11}\), evaluates the van der Waerden rational lower bound with `Fraction`, rounds it upward, applies the published factor \(6!\), and computes \((11!)^{11}\). Its complete standard output has SHA-256 digest `8c167454c650bc023b4ea8f3c237fa06fb595ede7def12744d31b4e89ebc8696`.
Saved work that is not yet in the reviewed packet, one entry per attempt.
Notes and companion material
Class-level subtotals, canonical graph hashes, and independent modular sums can preserve years of enumeration work.
Original intake status. McKay and Wanless determine the order-11 count in their primary paper. The exact order-12 count was not established by that source; current status remains unverified.
Decompose by the second row or by one-factorizations of regular bipartite graphs, quotienting automorphisms with orbit-stabilizer weights.
Trap: counting main classes, isotopy classes, or all Latin squares gives a different integer. Every symmetry weight must map back to the reduced convention.
Recorded example 1. The known reduced counts begin R_3=1, R_4=4, R_5=56, and R_6=9408.
Computational notes
A fresh row-by-row bitmask enumerator independently reproduced R_3=1, R_4=4, R_5=56, and R_6=9408. The same code verified row, column, and normalization constraints before counting each completion.
How the 5 records connectThe overview places each record once. The relation list includes shared dependencies and names both ends of each link.
ProblemNumber of reduced Latin squares of order 12
Contribute to this problemCite this problem statement
Cite the original sources separately.
Plain text
“Number of reduced Latin squares of order 12.” TheoremDB. P2660. Problem statement; statement text SHA-256 9c34565aea8a37c154f013eecaef3ff8ed7cde0240ccd599128a7f2e669b57ae. https://theoremdb.org/statement/?ref=P2660
BibTeX
@misc{theoremdb-problem-9c34565aea8a37c154f013eecaef3ff8ed7cde0240ccd599128a7f2e669b57ae,
title = {{Number of reduced Latin squares of order 12}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 9c34565aea8a37c154f013eecaef3ff8ed7cde0240ccd599128a7f2e669b57ae},
url = {https://theoremdb.org/statement/?ref=P2660}
}
Once Lean accepts the draft, you can submit it while target review is pending. Your submission request is saved and continues after approval. Private checks stay private until you choose to submit.
1References
Packet source. Brendan D. McKay and Ian M. Wanless, On the Number of Latin Squares, Annals of Combinatorics 9(3) (2005), 335-344. McKay and Wanless, On the Number of Latin Squares, Annals of Combinatorics 9 (2005), 335-344, Section 3 and Table 1; Douglas S. Stones, The Many Formulae for the Number of Latin Rectangles, Electronic Journal of Combinatorics 17 (2010), #A1, Figure 1. ↗open copy ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.Gives the exact reduced count at order 11 and explains the computational barrier at order 12.Also cited at Brendan D. McKay and Ian M. Wanless, On the Number of Latin Squares, exact order-11 enumeration and order-12 computational barrier.Also cited at McKay and Wanless, On the Number of Latin Squares, definition in Section 1 and R_11 in Table 1; independently checked in rls12-artifact-small-order-replay.Also cited at McKay and Wanless, On the Number of Latin Squares, Theorem 2 and the standard lower bound quoted in Section 7; arithmetic replayed in rls12-artifact-small-order-replay.Source named by the research packet.For Number of reduced Latin squares of order 12: The published values satisfy L_11=11!10!R_11 exactly, separating reduced squares from all labelled squares.
Brendan D. McKay and Eric Rogoyski, Latin Squares of Order 10, Electronic Journal of Combinatorics 2(1) (1995). Brendan D. McKay and Eric Rogoyski, Latin Squares of Order 10, Electronic Journal of Combinatorics 2 (1995), #N3, Section 3 and Table 2. ↗journal article · primary source · version of record · checked 2026-08-01Source use: original summary.Gives the exact order-10 count used to calibrate the packet's enumeration and normalization.For Number of reduced Latin squares of order 12: McKay and Rogoyski estimated R_12 at 1.62e44 using an unbiased sequential-extension estimator over 1.1 million trials.
Douglas S. Stones, The Many Formulae for the Number of Latin Rectangles, Electronic Journal of Combinatorics 17(1) (2010). Douglas S. Stones, The Many Formulae for the Number of Latin Rectangles, exact-count table through order 11 and explicit order-12 unknown status. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.Lists exact reduced Latin-square counts through order 11 and states that the order-12 count is unknown.
Alexander Hulpke, Petteri Kaski, and Patric R. J. Östergård, The number of Latin squares of order 11, arXiv:0909.3402v2 (2009). Abstract and enumeration results (i)-(v) in version 2. ↗preprint · reference source · arXiv:0909.3402v2 · checked 2026-07-25Source use: citation only.Reports the independent exact enumeration of Latin squares of order 11.Also cited at McKay, Meynert, and Myrvold, The Number of Latin Squares of Order 11, later exact enumeration of order-11 equivalence classes.
Ian Wanless's author-maintained reduced-count table, checked 2026-07-25. ↗website · reference source · web version checked 2026-08-01 · checked 2026-07-25Source use: citation only.Provides the author-maintained table of exact reduced counts through order 11.
CC0 next-order exact-counting target with an independently replayed small-order calibration.
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