Latin Square

An Latin square is a Latin rectangle with . Its entries are chosen from symbols, each appearing exactly once in every row and every column. 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 576 Latin squares of order 4 are given by

(3)

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 numbers 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

(4)

are orthogonal. The numbers of unordered pairs of orthogonal Latin squares of order , 2, ... are 1, 0, 36, 3456, ... (OEIS A072377). The count at order 1 includes the unique Latin square paired with itself.

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 (McKay 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 (OEIS A000315). The entries for , 13, 14, and 15 are estimates (McKay and Rogoyski 1995).

		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.

Greenhill et al. (2026) proved that a uniformly random Latin square of order has every row an even permutation with probability as . Their stronger theorem prescribes which row, column, and symbol permutations are even or odd, subject to the necessary parity identity.

The fewest entries needed to specify a Latin square uniquely are studied using Latin square critical sets, minimal collections of entries with a unique completion. Bean (2026) reports that the least size over all Latin squares of order is , where is the floor function. The claimed proof is based on an all-orders parity theorem for completions, not merely on finite checks. Bean (2026) attributes substantial proof and exposition generation to Codex and Claude with human direction and review. The general argument has not been independently verified here.