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 2; 2 1],[2 1; 1 2],](/images/equations/LatinSquare/NumberedEquation1.svg) |
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|
the 12 Latin squares of order three are given by
![[1 2 3; 2 3 1; 3 1 2],[1 2 3; 3 1 2; 2 3 1],[1 3 2; 2 1 3; 3 2 1],[1 3 2; 3 2 1; 2 1 3],
[2 1 3; 1 3 2; 3 2 1],[2 1 3; 3 2 1; 1 3 2],[2 3 1; 1 2 3; 3 1 2],[2 3 1; 3 1 2; 1 2 3],
[3 2 1; 1 3 2; 2 1 3],[3 2 1; 2 1 3; 1 3 2],[3 1 2; 1 2 3; 2 3 1],[3 1 2; 2 3 1; 1 2 3],](/images/equations/LatinSquare/NumberedEquation2.svg) |
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and two of the 576 Latin squares of order 4 are given by
![[1 2 3 4; 2 1 4 3; 3 4 1 2; 4 3 2 1] and [1 2 3 4; 3 4 1 2; 4 3 2 1; 2 1 4 3].](/images/equations/LatinSquare/NumberedEquation3.svg) |
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Latin squares correspond to minimum vertex colorings of the 
rook graph, which are illustrated above for 
and 3.
The numbers 
of Latin squares of order 
,
2, ... are 1, 2, 12, 576, 161280, ... (OEIS A002860).
The number 
of isotopically distinct Latin squares of order 
, 2, ... are 1, 1, 1, 2, 2, 22, 564, 1676267, ... (OEIS 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
![[3 2 1; 2 1 3; 1 3 2] [2 3 1; 1 2 3; 3 1 2]](/images/equations/LatinSquare/NumberedEquation4.svg) |
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are orthogonal. The number of pairs of orthogonal Latin squares of order 
, 2, ... are 0, 0, 36, 3456, ... (OEIS 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, ... (OEIS 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
 |
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|
The numbers of normalized Latin squares of order 
, 2, ..., are summarized in the following table (OEIS 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.
Latin Squares." Disc.
Math. 11, 93-95, 1975.Cayley, A. "On Latin Squares."
Oxford Cambridge Dublin Messenger Math. 19, 135-137, 1890.Colbourn,
C. J. and Dinitz, J. H. (Eds.). 
Squares." Ann. Eugenics 9,
269-307, 1939.Rohl, J. S. 
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