Mathematical_Recreations-Kraitchik-2e

9146 Mathematical Recreations
2Qn = 0
for n = 4k + 2 or n = 4k + 3.
The whole set of solutions may be separated into (n - I)!
sets of n solutions each in such a way that each set of n solutions
just fills the board. If each solution in the set is assigned
a different color, each such set is a solution of the
problem of coloring the cells of the board with n colors in
such a way that no color appears twice in the same orthogonal.
If the numbers from 1 to n are used instead of colors
FIGURE 118. FIGURE 119.
we have a Latin square. Since, however, these may not all
be independent, this does not give the total number of independent
Latin squares.
In analogy with magic squares we may call a solution regular
if it is generated by a motion Cu, v). In order to fulfill
the requirements of the problem such a motion cannot have
a factor of n as a factor of either u or v, since then more than
one element would be found in certain orthogonals. All solutions
for n = 2 and n = 3 are regular, and are generated
by (1, 1) or by (1, -1). For n = 4 the solutions 1234,4321,
2143, 3412, 2341, 1432, 4123, 3214 are regular.
A solution is called composite if the board can be broken