An Latin square is a Latin rectangle with . Specifically, a Latin square consists of sets of the numbers 1 to arranged in such a way that no orthogonal (row or column) contains the same number twice. For example, the two Latin squares of order two are given by

(1)

the 12 Latin squares of order three are given by

(2)

and two of the whopping 576 Latin squares of order 4 are given by

(3)

The numbers of Latin squares of order , 2, ... are 1, 2, 12, 576, 161280, ... (Sloane's A002860). The number of isotopically distinct Latin squares of order , 2, ... are 1, 1, 1, 2, 2, 22, 564, 1676267, ... (Sloane's A040082).

A pair of Latin squares is said to be orthogonal if the pairs formed by juxtaposing the two arrays are all distinct. For example, the two Latin squares

(4)

are orthogonal. The number of pairs of orthogonal Latin squares of order , 2, ... are 0, 0, 36, 3456, ... (Sloane's A072377).

The number of Latin squares of order with first row given by is the same as the number of fixed diagonal Latin squares of order (i.e., the number of Latin squares of order having all 1s along their main diagonals). For , 2, ..., the numbers of such matrices are 1, 1, 2, 24, 1344, 1128960, ... (Sloane's A000479) and the total number of Latin squares of order is equal to this number times .

A normalized, or reduced, Latin square is a Latin square with the first row and column given by . General formulas for the number of normalized Latin squares are given by Nechvatal (1981), Gessel (1987), and Shao and Wei (1992), but the asymptotic value of is not known (MacKay and Wanless 2005). The total number of Latin squares of order can then be computed from

(5)

The numbers of normalized Latin squares of order , 2, ..., are summarized in the following table (Sloane's A000315).

		reference
1	1
2	1
3	1
4	4
5	56	Euler (1782), Cayley (1890), MacMahon (1915; incorrect value)
6	9408	Frolov (1890) and Tarry (1900)
7	16942080	Frolov (1890; incorrect), Norton (1939; incomplete), Sade (1948), Saxena (1951)
8	535281401856	Wells (1967)
9	377597570964258816	Bammel and Rothstein (1975)
10	7580721483160132811489280	McKay and Rogoyski (1995)
11	5363937773277371298119673540771840	McKay and Wanless (2005)
12		McKay and Rogoyski (1995)
13		McKay and Rogoyski (1995)
14		McKay and Rogoyski (1995)
15		McKay and Rogoyski (1995)

Sudoku is a special case of a Latin square.

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Cayley, A. "On Latin Squares." Oxford Cambridge Dublin Messenger Math. 19, 135-137, 1890.

Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, 1996.

Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 183, 1974.

Euler, L. "Recherches sur une nouvelle esp ece de quarrès magiques." Verh. Zeeuwsch Gennot. Weten Vliss 9, 85-239, 1782.

Frolov, M. "Sur les permutations carrés." J. de Math. spéc. 4, 8-11 and 25-30, 1890.

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Hunter, J. A. H. and Madachy, J. S. Mathematical Diversions. New York: Dover, pp. 33-34, 1975.

Kraitchik, M. "Latin Squares." §7.11 in Mathematical Recreations. New York: W. W. Norton, p. 178, 1942.

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McKay, B. D. and Rogoyski, E. "Latin Squares of Order 10." Electronic J. Combinatorics 2, No. 1, R3, 1-4, 1995. http://www.combinatorics.org/Volume_2/Abstracts/v2i1r3.html.

McKay, B. D. and Wanless, I. M. "On the Number of Latin Squares." Ann. Combin. 9, 335-344, 2005.

Nechvatal, J. R. "Asymptotic Enumeration of Generalised Latin Rectangles." Util. Math. 20, 273-292, 1981.

Norton, H. W. "The Squares." Ann. Eugenics 9, 269-307, 1939.

Rohl, J. S. Recursion via Pascal. Cambridge, England: Cambridge University Press, pp. 162-165, 1984.

Ryser, H. J. "Latin Rectangles." §3.3 in Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 35-37, 1963.

Sade, A. "Enumération des carrés latins. Application au 7ème ordre. Conjectures pour les ordres supérieurs." 8 pp. Marseille, France: Privately published, 1948.

Saxena, P. N. "A Simplified Method of Enumerating Latin Squares by MacMahon's Differential Operators; II. The Latin Squares." J. Indian Soc. Agricultural Statist. 3, 24-79, 1951.

Shao, J.-Y. and Wei, W.-D. "A Formula for the Number of Latin Squares." Disc. Math. 110, 293-296, 1992.

Sloane, N. J. A. Sequences A000479, A002860/M2051, A000315/M3690, A040082, and A072377 in "The On-Line Encyclopedia of Integer Sequences."

Tarry, G. "Le problème de 36 officiers." Compte Rendu de l'Assoc. Français Avanc. Sci. Naturel 1, 122-123, 1900.

Tarry, G. "Le problème de 36 officiers." Compte Rendu de l'Assoc. Français Avanc. Sci. Naturel 2, 170-203, 1901.

Wells, M. B. "The Number of Latin Squares of Order Eight." J. Combin. Th. 3, 98-99, 1967.

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Weisstein, Eric W. "Latin Square." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LatinSquare.html