# P2614: A projective plane of order 12

- ID: `P2614`
- Reference: `projective-plane-order-12`
- Page: https://theoremdb.org/statements/P2614
- Record maturity: Reviewed problem with recorded work

## Problem

Does a finite projective plane of order \(12\) exist?

### Remarks

- **Remark.** A projective plane of order 12 has 157 points and 157 lines.
- **Remark.** Every line contains 13 points, every point lies on 13 lines, and each pair of points lies on exactly one line.

### What counts as a solution

- Supply a 157 by 157 incidence matrix satisfying the plane axioms, or give a complete nonexistence proof with independently replayable case certificates.

## Status

The literature and catalogue audit found no accepted construction or complete exclusion for a projective plane of order 12. [1](#reference-1) [2](#reference-2) [4](#reference-4)

## Work

### Evidence for the current status

**Proposition 1 (Existence at order 12 remains open).** The literature and catalogue audit found no accepted construction or complete exclusion for a projective plane of order 12.

A projective plane of order 12 would have 157 points and 157 lines. Every line would contain 13 points, every point would lie on 13 lines, and each pair of points would determine one line. Equivalently, it would be a symmetric \(2\text{-}(157,13,1)\) design.

Akiyama, Suetake, and Tanaka state in their 2019 primary paper that the order-12 existence question is still unknown. Their exhaustive computation excludes collineation groups of order 9. Their 2023 sequel sharpens the symmetry restriction to collineation-group orders 1, 2, or 3 while leaving the plane itself unresolved. Kharaghani and Suda give a 2023 equivalence with a balancedly multi-splittable quaternary Hadamard matrix of order 144, which supplies another exact target rather than a resolution.

The research check through 2026-07-25 found no later accepted construction or nonexistence theorem. The answer recorded here is therefore open.

### Background and intake notes

The parameters make this a sharply bounded design search. A full solution is formidable, while smaller symmetry classes and failed canonical augmentations are reusable units of evidence.

- Original intake status: Status remains unverified. The order-12 existence question has a long literature, so a current finite-geometry bibliography check belongs before a large search.
- Encode one line and its pencil in canonical form, then use exact cover for the remaining point pairs. Record canonical augmentation rules and automorphism-group assumptions with every exclusion.
- Trap: the Bruck-Ryser-Chowla obstruction has no force at order 12 because 12 is divisible by 4. Reusing that test as an exclusion silently proves nothing here.

- Recorded example: The incidence matrix B would satisfy BB^T=12I+J and every row and column would have weight 13.

### Other known results

- **Computation 1** (reproduced): The theorem's sum-of-two-squares condition applies to orders congruent to 1 or 2 modulo 4, while 12 is congruent to 0. [5](#reference-5)
- **Proposition 2** (supported): The strongest published symmetry restriction leaves only trivial, involutory, or order-three collineation groups. [2](#reference-2) [1](#reference-1)
- **Proposition 3** (supported): A 1960 construction proves N(12) at least 5; the universal upper bound is 11, and equality is equivalent to an order-12 plane. [3](#reference-3)

### Prior approaches

- **Route 1** (supported): The audit located strong arithmetic and symmetry restrictions, a five-MOLS construction, and exact reformulations, while the existence question stayed open. [6](#reference-6) [5](#reference-5) [3](#reference-3) [1](#reference-1) [2](#reference-2) [4](#reference-4)

### Runnable artifacts

- **Artifact 1** (reproduced): A 1,584-variable finite-domain model is equivalent to the required orthogonal array and has safe first-row and first-column symmetry breaking. [4](#reference-4)

### Computational notes

- Exact parameter checks give b=v=12^2+12+1=157, block size 13, replication 13, and 157 choose 2 = 12246 point pairs, matching 157 times 13 choose 2. The determinant identity det(B)^2=13^2 times 12^156 passes the elementary square test, so this route supplies no contradiction.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `projective-plane-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>Kenzi Akiyama, Chihiro Suetake, and Masaki Tanaka, The nonexistence of projective planes of order 12 with a collineation group of order 9, Australasian Journal of Combinatorics 74(1) (2019), 112-160. Main theorem and corollary on pp. 112-114 https://ajc.maths.uq.edu.au/pdf/74/ajc_v74_p112.pdf
   - Also cited at Akiyama, Suetake, and Tanaka, Australasian Journal of Combinatorics 74(1) (2019), 112-160, pages 112-114
   - Also cited at Akiyama, Suetake, and Tanaka, Australasian Journal of Combinatorics 74(1) (2019), main theorem
   - journal_article; primary source; version of record; checked 2026-07-25
   - Source use: citation_only
   - For A projective plane of order 12: The literature and catalogue audit found no accepted construction or complete exclusion for a projective plane of order 12.
   - order-9 exclusion and 2019 open-status statement
2. <a id="reference-2"></a>Kenzi Akiyama, Chihiro Suetake, and Masaki Tanaka, Projective planes of order 12 do not have a collineation group of order 4, Journal of Combinatorial Designs 31(2) (2023), 87-123. Abstract and main theorem https://doi.org/10.1002/jcd.21869
   - Also cited at Akiyama, Suetake, and Tanaka, Journal of Combinatorial Designs 31(2) (2023), abstract and main theorem
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - For A projective plane of order 12: The strongest published symmetry restriction leaves only trivial, involutory, or order-three collineation groups.
   - all collineation-group orders reduced to 1, 2, or 3
   - Source named by the research packet.
3. <a id="reference-3"></a>R. C. Bose, I. M. Chakravarti, and D. E. Knuth, On Methods of Constructing Sets of Mutually Orthogonal Latin Squares Using a Computer. I, Technometrics 2(4) (1960), 507-516. R. C. Bose, I. M. Chakravarti, and D. E. Knuth, On Methods of Constructing Sets of Mutually Orthogonal Latin Squares Using a Computer. I, Technometrics 2(4) (1960), 507-516, abstract, Figure 1, and Table 2 https://doi.org/10.1080/00401706.1960.10489916
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - For A projective plane of order 12: A 1960 construction proves N(12) at least 5; the universal upper bound is 11, and equality is equivalent to an order-12 plane.
   - two explicit sets of five MOLS of order 12
4. <a id="reference-4"></a>Hadi Kharaghani and Sho Suda, Hadamard Matrices related to Projective Planes, Electronic Journal of Combinatorics 30(2) (2023). The projective-plane, orthogonal-array, and complete-MOLS equivalences are standard and are used in Kharaghani and Suda, Electronic Journal of Combinatorics 30(2) (2023), P2.49; exact parameter audit executed 2026-07-25 https://doi.org/10.37236/11990
   - Also cited at Hadi Kharaghani and Sho Suda, Hadamard Matrices related to Projective Planes, Electronic Journal of Combinatorics 30(2) (2023), P2.49, main equivalence
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - For A projective plane of order 12: A 1,584-variable finite-domain model is equivalent to the required orthogonal array and has safe first-row and first-column symmetry breaking.
   - equivalence with a balancedly multi-splittable quaternary Hadamard matrix of order 144
5. <a id="reference-5"></a>R. H. Bruck and H. J. Ryser, The Nonexistence of Certain Finite Projective Planes, Canadian Journal of Mathematics 1(1) (1949), 88-93. R. H. Bruck and H. J. Ryser, The Nonexistence of Certain Finite Projective Planes, Canadian Journal of Mathematics 1 (1949), 88-93, Theorem 1; order-12 substitution and determinant calculation replayed in pp12-artifact-model-audit https://doi.org/10.4153/CJM-1949-009-2
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - For A projective plane of order 12: The theorem's sum-of-two-squares condition applies to orders congruent to 1 or 2 modulo 4, while 12 is congruent to 0.
   - Bruck-Ryser necessary condition
6. <a id="reference-6"></a>Eric Moorhouse, Projective Planes of Small Order, author-maintained catalogue (revised November 2017). Catalogue description and the order-12 table entry https://ericmoorhouse.org/pub/planes/
   - website; reference source; web version checked 2026-08-01; checked 2026-07-25
   - Source use: citation_only
   - Records the known small projective planes and leaves order 12 unresolved.