Mathematical_Recreations-Kraitchik-2e

Magic Squares 179
12. EULER (GRAECO-LATIN) SQUARES
From n objects (a, (3, "', v) and n objects Ca, b, "', n)
fonn the n2 pairs aa, ab, "', an, {3a, "', vn. An arrangement
of these n2 pairs in the cells of a square of order n in
such a way that the array formed of the first elements of the
pairs is a Latin square and the array formed of the second
elements of the pairs is also a Latin square is called a Graeco­
Latin or Euler square of order n. If the two Latin squares
are both diagonal, the Euler square is called diagonal. Figure
55 shows two diagonal Euler squares of order 4.
11 22 33 44
43 34 21 12
24 13 42 31
32 41 14 23
11 22 33 44
34 43 12 21
42 31 24 13
23 14 41 32
FIGURE 55.
These squares originated in a problem that Euler states
as follows: "A very curious question is the following: A
meeting of 36 officers of 6 different ranks and from 6 different
regiments must be arranged in a square in such a manner
that each orthogonal contains 6 officers from different regiments
and of different ranks. "
This problem is impossible of solution, as Tarry has shown
that no Euler squares of order 6 exist.
If we take as the fundamental position of the n 2 pairs of
numbers the arrangement (Figure 56, p. 180), in which the
first number of the pair gives the row and the second number
the column in which the pair lies, we may apply to this fundamental
square the method of lattices used for forming
magic squares, provided the order of the square is a composite
odd number. Since we are not concerned with the diag-