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A001438 Maximal number of mutually orthogonal Latin squares (or MOLS) of order n. 4
1, 2, 3, 4, 1, 6, 7, 8 (list; graph; refs; listen; history; text; internal format)



By convention, a(0) = a(1) = infinity.

Parker and others conjecture that a(10) = 2.

It is also known that a(11) = 10, a(12) >= 5.

It is known that a(n) >= 2 for all n > 6, disproving a conjecture by Euler that a(4k+2) = 1 for all k. - Jeppe Stig Nielsen, May 13 2020


CRC Handbook of Combinatorial Designs, 1996, pp. 113ff.

S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer-Verlag, NY, 1999, Chapter 8.

E. T. Parker, Attempts for orthogonal latin 10-squares, Abstracts Amer. Math. Soc., Vol. 12 1991 #91T-05-27.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 58 Penguin Books 1997.


Table of n, a(n) for n=2..9.

Anonymous, Order-10 Greco-Latin square.

R. C. Bose & S. S. Shrikhande, On The Falsity Of Euler's Conjecture About The Non-Existence Of Two Orthogonal Latin Squares Of Order 4t+2, Proc. Nat. Acad. Sci., 1959 45 (5) 734-737.

R. Bose, S. Shrikhande, & E. Parker, Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture, Canadian Journal of Mathematics, 12 (1960), 189-203.

C. J. Colbourn & J. H. Dinitz, Mutually Orthogonal Latin Squares: A Brief Survey of Constructions, preprint, Journal of Statistical Planning and Inference, Volume 95, Issues 1-2, 1 May 2001, Pages 9-48.

M. Dettinger, Euler's Square

David Joyner and Jon-Lark Kim, Kittens, Mathematical Blackjack, and Combinatorial Codes, Chapter 3 in Selected Unsolved Problems in Coding Theory, Applied and Numerical Harmonic Analysis, Springer, 2011, pp. 47-70, DOI: 10.1007/978-0-8176-8256-9_3.

Numberphile, Euler squares, YouTube video, 2020.

E. T. Parker, Orthogonal Latin Squares, Proc. Nat. Acad. Sci., 1959 45 (6) 859-862.

E. Parker-Woodruff, Greco-Latin Squares Problem

N. Rao, Shrikhande, “Euler’s Spoiler”, Turns 100, Bhāvanā, The mathematics magazine, Volume 1, Issue 4, 2017.

Eric Weisstein's World of Mathematics, Euler's Graeco-Roman Squares Conjecture

Wikipedia, Graeco-Latin square.

Index entries for sequences related to Latin squares and rectangles


a(n) <= n-1 for all n>1. - Tom Edgar, Apr 27 2015

a(p^k) = p^k-1 for all primes p and k>0. - Tom Edgar, Apr 27 2015

a(n) = A107431(n,n) - 2. - Floris P. van Doorn, Sep 10 2019


Cf. A287695, A328873.

Sequence in context: A065338 A316272 A294649 * A105587 A319676 A049073

Adjacent sequences:  A001435 A001436 A001437 * A001439 A001440 A001441




N. J. A. Sloane



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Last modified February 27 13:43 EST 2021. Contains 341657 sequences. (Running on oeis4.)