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Mutually orthogonal Latin squares

The maximum size of a set of MOLS

If q is a prime power, there exist q-1 MOLS of order q. These are constructed by the "finite field method". For there exists a Galois field F of order q. Now, for each non-zero element a od F, let L_a be the array, with rows and columns indexed by F, such that the symbol in row x and column y of L_a is ax+y. Then the arrays L_a form the required q-1 MOLS.

For example, suppose that q=3. The Galois field GF(3) consists of the integers mod 3, and we can take the elements to be 0,1,2. Now, indexing the rows and columns of the squares by 0,1,2, the first square has x+y in row x and column y, and the second has 2x+y (in other words, y-x):

0 1 2 1 2 0 2 0 1

0 1 2 2 0 1 1 2 0

Magic squares

Given two orthogonal Latin squares A and B with entries 1 . .  n, the square S with (i,j) entry n(a_ij-1)+b_ij has these properties. Here is an example:

1 2 3 2 3 1 3 1 2

1 2 3 3 1 2 2 3 1

------>

1 5 9 6 7 2 8 3 4

For a magic square, it is customary to assume also that the sums of the two diagonals are the same as the row and column sums. This condition is not automatically satisfied by the square constructed from orthogonal Latin squares, but can usually be achieved by some rearrangement.

A translation of Euler's paper can be found here on the arXiv.

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Peter J. Cameron
13 December 2006