# P2660: Number of reduced Latin squares of order 12

- ID: `P2660`
- Reference: `reduced-latin-squares-order-12`
- Page: https://theoremdb.org/statements/P2660
- Record maturity: Reviewed problem with recorded work

## Problem

Determine the exact number \(R_{12}\) of \(12\times12\) Latin squares whose first row and first column are both \((0,1,\ldots,11)\).

### Remarks

- **Remark.** A Latin square uses each symbol once in every row and column.
- **Remark.** The stated normalization defines a reduced Latin square.

### What counts as a solution

- Give the exact integer R_12 and an independently checkable weighted enumeration whose class totals sum to it.

## Status

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](#reference-1) [3](#reference-3) [4](#reference-4) [5](#reference-5)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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: The known reduced counts begin R_3=1, R_4=4, R_5=56, and R_6=9408.

### Other known results

- **Computation 1** (reproduced): The published values satisfy L_11=11!10!R_11 exactly, separating reduced squares from all labelled squares. [1](#reference-1)
- **Computation 2** (reproduced): Published lower and divisibility theorems plus a direct row-choice bound give 4410573106297854867267286926659737440 <= R_12 <= (11!)^11. [1](#reference-1)
- **Claim 2** (supported): McKay and Rogoyski estimated R_12 at 1.62e44 using an unbiased sequential-extension estimator over 1.1 million trials. [2](#reference-2)

### Runnable artifacts

- **Artifact 1** (reproduced): Standard-library Python enumerates R_2 through R_6 and checks the order-11 identity, lower bound, divisibility rounding, and elementary upper bound. [1](#reference-1)

### 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `reduced-latin-squares-order-12`, 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>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 https://doi.org/10.1007/s00026-005-0261-7
   - 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
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Open copy: https://users.cecs.anu.edu.au/~bdm/papers/ls11.pdf
   - Source use: citation_only
   - Gives the exact reduced count at order 11 and explains the computational barrier at order 12.
   - Source named by the research packet.
2. <a id="reference-2"></a>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 https://doi.org/10.37236/1222
   - journal_article; primary source; version of record; checked 2026-08-01
   - Source use: original_summary
   - Gives the exact order-10 count used to calibrate the packet's enumeration and normalization.
3. <a id="reference-3"></a>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 https://doi.org/10.37236/487
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Lists exact reduced Latin-square counts through order 11 and states that the order-12 count is unknown.
4. <a id="reference-4"></a>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 https://arxiv.org/abs/0909.3402
   - preprint; reference source; arXiv:0909.3402v2; checked 2026-07-25
   - Source use: citation_only
   - Reports the independent exact enumeration of Latin squares of order 11.
5. <a id="reference-5"></a>Ian Wanless's author-maintained reduced-count table, checked 2026-07-25 https://users.monash.edu.au/~iwanless/
   - website; reference source; web version checked 2026-08-01; checked 2026-07-25
   - Source use: citation_only
   - Provides the author-maintained table of exact reduced counts through order 11.