230 14. Finite Geometries, Codes,Latin Squares,and Other Pretty Creatures
second:
0, 0 1, 1 2, 2 3, 3
1, 3 0, 2 3, 1 2, 0
2, 1 3, 0 0, 3 1, 2
3, 2 2, 3 1, 0 0, 1
(14.10)
Do you notice something about this composite square? Every field contains
a different pair of numbers! From this it follows that each of the possible
42 = 16 pairs occurs exactly once (Pigeonhole Principle). If two Latin
squares have this property, we call themorthogonal. One may check the
orthogonality of two Latin squares in the following way: We take all the
fields in the first Latin square that contain 0, and we check the same fields
on the second square, to see whether they contain different integers. We do
the same with 1, 2, etc. If the squares pass all these checks, then the first
square is orthogonal to the second one, and vice versa.
14.5.3Find two orthogonal 3×3 Latin squares.Magic squares.If we have two orthogonal Latin squares, we may very
easily construct from them amagic square. (In a magic square the sums of
the numbers in every row and every column are equal.) Consider the pairs
in the fields in (14.10). Replace each pair (a, b)byab=4a+b(in other
words, considerabas a two-digit number in base 4). Writing our numbers
in decimal notation, we get the magic square shown in (14.11).
0 5 10 15
7 2 13 8
9 12 3 6
14 11 4 1(14.11)
(This is a magic square indeed: Every row and column sum is 30.) From
any two orthogonal Latin squares we can get a magic square using the
same method. In every row (and also in every column) the numbers 0, 1,
2, 3 occur exactly once in the first position and exactly once in the second
position, so in every row (and column) the sum of the elements is exactly
(0+1+2+3)·4+(0+1+2+3)=30,as required in a magic square.
14.5.4In our magic square we have the numbers 0 through 15, instead of 1
through 16. Try to make a magic square from (14.10) formed by the numbers 1
through 16.
14.5.5The magic square constructed from our two orthogonal Latin squares is
not “perfect”, because in a perfect magic square the sums on the diagonals are