How many 3 by 3 Latin squares?

Question

A Latin square of order 3 is a 3 by 3 grid filled with the symbols 1, 2, 3 so that each symbol appears exactly once in every row and exactly once in every column. How many 3 by 3 Latin squares are there? Then explain why the count for order 4 is much larger (576) and why exact counts are known only up to order 11.

Accepted Answer

Twelve. First row 1 2 3. The second row must be a rearrangement of 1, 2, 3 with no symbol in the same column as in row 1, that is, a derangement of three items: there are exactly 2 (2 3 1 and 3 1 2). Once the first two rows are fixed, each column has exactly one symbol missing, and those missing symbols form the third row, which automatically has each symbol once (each symbol appears in exactly two columns among the first two rows, so it is missing from exactly one column). So there are 2 squares with first row 1 2 3. Any first row. The first row can be any of 3! = 6 orderings, and permuting …

How many 3 by 3 Latin squares?

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