information:filtering, stylistic:tone, information:extraction
The number of mutually orthogonal Latin squares

The mutual orthogonality property of a set of MOLS is unaffected by

Using these operations, any set of MOLS can be put into standard form, meaning that the first row of every square is identical and normally put in some natural order, and one square has its first column also in this order. The MOLS(4) and MOLS(5) examples at the start of this section have been put in standard form.

By putting a set of MOLS(n) in standard form and examining the entries in the second row and first column of each square, it can be seen that no more than n − 1 squares can exist. A set of n − 1 MOLS(n) is called a complete set of MOLS. Complete sets are known to exist when n is a prime number or power of a prime (see Finite field construction below). However, the number of MOLS that may exist for a given order n is not known for general n, and is an area of research in combinatorics.