latin-II-4

Latin Squares in Practice and in Theory II

Feature Column Archive

4. Orthogonal Latin Squares and Magic Squares

Euler was interested in magic squares, square arraysof size n, containing all the numbers from 1 ton², and such that each row and each columnadd up to the same number (which must be n(n²+1)/2).In his article he shows how to derive an ``ordinary'' magicsquare from a pair of orthogonal latin squares. We can see theprocess with our 7 x 7 pair:

We make a square of numbers from the ``ranks'' square bysetting King=0, Queen=7, Rook=14, Bishop=21, Knight=28, Pawn=35, Joker=42and another one from the ``regiments'' square by settingblack=1, red=2, blue=3, green=4, purple=5, brown=6, yellow=7.Then we add the two tables together:

 0  7 14 21 28 35 42    1  2  3  4  5  6  7      1  9 17 25 33 41 4914 21 28 35 42  0  7    2  3  4  5  6  7  1     16 24 32 40 48  7  828 35 42  0  7 14 21    3  4  5  6  7  1  2     31 39 47  6 14 15 23 42  0  7 14 21 28 35 +  4  5  6  7  1  2  3  =  46  5 13 21 22 30 38 7 14 21 28 35 42  0    5  6  7  1  2  3  4     12 20 28 29 37 45  421 28 35 42  0  7 14    6  7  1  2  3  4  5     27 35 36 44  3 11 1935 42  0  7 14 21 28    7  1  2  3  4  5  6     42 43  2 10 18 26 34

Since all 49 combinations must appear, all the numbers from 1 to 49will be in final square. Since the same numbers appear in everyrow of each of the latin squares, the row sums will always be thesame, and so will the column sums. Finally, since the assignmentof numbers to ranks and regiments is completely arbitrary, a greatnumber of different magic squares can be constructed by this method.

• 1. Leonhard Euler's Puzzle of the 36 Officers

   

• 2. Orthogonal latin squares

   

• 3. Mutually orthogonal latin squares (MOLS)

   

• 4. Orthogonal latin squares and magic squares

   

• 5. References in print. For further thought

   

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