Three mutually orthogonal Latin squares of order 10

Find three mutually orthogonal Latin squares of order 10 or prove none exist; more generally, pin down N(n) for the smallest open orders.

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State the problem, from MOLS table, Handbook of Combinatorial Designs (with updates by J. H. Dinitz)

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	1	+# Three mutually orthogonal Latin squares of order 10
	2	+
	3	+Two Latin squares are orthogonal if, when superimposed, every ordered pair of symbols appears exactly once. Pairs of orthogonal $10 \times 10$ squares have been known since 1959, but whether three mutually orthogonal ones exist is one of the most prominent open problems in design theory. A candidate triple is verified instantly.
	4	+
	5	+## Problem
	6	+
	7	+A Latin square of order $n$ is an $n \times n$ array in which each of $n$ symbols appears exactly once in every row and every column. Two Latin squares $A$ and $B$ of order $n$ are orthogonal if the $n^2$ pairs $(A_{ij}, B_{ij})$ are all distinct. Let $N(n)$ be the largest number of pairwise orthogonal Latin squares (MOLS) of order $n$.
	8	+
	9	+Question: is $N(10) \ge 3$? That is, do three mutually orthogonal Latin squares of order 10 exist? More generally, determine $N(n)$ for the smallest orders where it is unknown: $n = 10, 12, 14, 15, 18, 20, 21, 22, \dots$
	10	+
	11	+## Background
	12	+
	13	+Euler conjectured in 1782 that $N(n) = 1$ whenever $n \equiv 2 \pmod 4$ (the case $n = 6$ is the problem of the 36 officers), and Tarry confirmed $N(6) = 1$ in 1900. In 1959 the conjecture collapsed: Parker found two orthogonal squares of order 10, and Bose, Shrikhande and Parker went on to show $N(n) \ge 2$ for every $n > 6$. Since $N(n) = n - 1$ exactly when a projective plane of order $n$ exists, and planes exist for prime powers, 10 is the first order where $N(n)$ is unknown. The Handbook of Combinatorial Designs tabulates the best lower bounds for all $n < 10000$, and Dinitz maintains a page of later improvements.
	14	+
	15	+## What is known
	16	+
	17	+- Order 10: $2 \le N(10) \le 6$. Nine MOLS would give a projective plane of order 10, which Lam, Thiel and Swiercz (1989) proved does not exist, and by a theorem of Shrikhande any seven MOLS of order 10 would extend to nine.
	18	+- Partial results for a triple: a square with a $5 \times 5$ Latin subsquare has no orthogonal mate (Mann, 1944), orthogonal pairs with orthogonal $3 \times 3$ subsquares do not extend to a triple (Parker), and Myrvold (1999) studied squares with a $4 \times 4$ subsquare and excluded 20 of 28 cases. Bright, Keita and Stevens (2026) re-derived Myrvold's exclusions with a SAT solver and found orthogonal pairs in the remaining eight cases, so any proof of nonexistence must involve all three squares.
	19	+- Next orders: lower bounds $N(12) \ge 5$, $N(14) \ge 4$ (Todorov, 2012), $N(15) \ge 4$, $N(18) \ge 5$ (Abel, 2015), $N(20) \ge 4$, $N(21) \ge 5$, $N(22) \ge 3$, from the Handbook table and its updates.
	20	+
	21	+## What would count as progress
	22	+
	23	+- Three Latin squares of order 10 that are pairwise orthogonal. Checking takes microseconds: verify the Latin property and that each of the three superpositions contains all 100 pairs.
	24	+- A proof that no such triple exists, for example a SAT-based search with a DRAT or similar proof certificate that independent checkers can verify, or a further reduction of the cases that a search must cover.
	25	+- An improved lower bound for one of the next open orders, such as $N(14) \ge 5$ or $N(22) \ge 4$, again given by explicit squares and checked in milliseconds.
	26	+- formal-conjectures states the related question of complete sets of MOLS in Lean, whose smallest open case is 11 MOLS of order 12 ([LatinSquare.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Paper/LatinSquare.lean)).
	27	+
	28	+## References
	29	+
	30	+1. E. T. Parker, "Orthogonal Latin squares", Proceedings of the National Academy of Sciences 45 (1959), 859–862. [doi:10.1073/pnas.45.6.859](https://doi.org/10.1073/pnas.45.6.859)
	31	+2. C. W. H. Lam, L. Thiel and S. Swiercz, "The non-existence of finite projective planes of order 10", Canadian Journal of Mathematics 41 (1989), 1117–1123. [doi:10.4153/cjm-1989-049-4](https://doi.org/10.4153/cjm-1989-049-4)
	32	+3. D. T. Todorov, "Four mutually orthogonal Latin squares of order 14", Journal of Combinatorial Designs 20 (2012), 363–367. [doi:10.1002/jcd.21298](https://doi.org/10.1002/jcd.21298)
	33	+4. R. J. R. Abel, "Existence of five MOLS of orders 18 and 60", Journal of Combinatorial Designs 23 (2015), 135–139. [doi:10.1002/jcd.21384](https://doi.org/10.1002/jcd.21384)
	34	+5. C. Bright, A. Keita and B. Stevens, "Myrvold's results on orthogonal triples of 10 × 10 Latin squares: a SAT investigation", Electronic Journal of Combinatorics 33(1) (2026), P1.30. [doi:10.37236/13960](https://doi.org/10.37236/13960)
	35	+
	36	+## Source
	37	+
	38	+Imported from [Handbook of Combinatorial Designs, MOLS table: new results page by J. H. Dinitz](https://site.uvm.edu/jdinitz/?p=389), curated by Charles J. Colbourn and Jeffrey H. Dinitz. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.