Mathematical_Recreations-Kraitchik-2e

The Problem of the Queens ~43
no solution is left invariant by any of the subgroups (3) ,( 4),
(5), (8), or (10). With one exception the group of a given
solution is also the group of every solution equivalent to it.
In the exceptional case, noted below, half of the solutions of
a set of equivalent solutions are invariant under one group
and the other half under another . We shall put the full set
of these equivalent solutions in a single class. We denote
these classes as 0, C, D, Q, and R. Here is the classification:
(0) Ordinary solutions, invariant only under the identity .
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FIGURE 113. Rook Problem.
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Each such solution is therefore one of 8 distinct equivalent
solutions, and each of them is ordinary. We shall denote by
On the number of complete sets of equivalent ordinary solutions
of order n. There are none for n < 4. For n = 4 there
is just one set: 1342, 4132, 3124, 3241, 4213, 1423, 2431,
2314. They are illustrated in Figure 113.
(C) Centrosymmetric solutions, invariant under subgroup
(2), but under no larger subgroup. Each belongs to a complete
set of 4 equivalent solutions, all centrosymmetric.
Each is symmetric with respect to the center, but has no
other symmetry. The number of complete sets of equivalent
centrosymmetric solutions of order n will be denoted by en.