Three mutually orthogonal Latin squares of order 10

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.

Problem

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 $n$ .

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: $n = 10, 12, 14, 15, 18, 20, 21, 22, \dots$

Background

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.

What is known

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.
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.
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.

What would count as progress

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.
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.
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.
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).

References

E. T. Parker, "Orthogonal Latin squares", Proceedings of the National Academy of Sciences 45 (1959), 859–862. doi:10.1073/pnas.45.6.859
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
D. T. Todorov, "Four mutually orthogonal Latin squares of order 14", Journal of Combinatorial Designs 20 (2012), 363–367. doi:10.1002/jcd.21298
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
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

Source

Imported from Handbook of Combinatorial Designs, MOLS table: new results page by J. H. Dinitz, 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.