Orthogonal Latin Squares


An nxn Latin square is an nxn array of the numbers 1,2,3,...,n so that each number occurs exactly once in each row and each column. Latin squares are useful in the design of experiments. The existence of a complete set of (n) mutually orthogonal Latin squares is equivalent to the existence of a projective plane of order n.

Here is a numerical representation of the Latin squares pictured below: 

        inner               outer  

     1  2  3  4          1  2  3  4

     3  4  1  2          2  1  4  3

     4  3  2  1          3  4  1  2

     2  1  4  3          4  3  2  1
The pair is called orthogonal because every (i,j) occurs once and only once in the superimposed array. Can you find a third Latin square orthogonal to each of these?