
Encyclopaedia of DesignTheory: MOLS 
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If q is a prime power, there exist q1 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 nonzero 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 q1 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, yx):


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:


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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