Mathematical_Recreations-Kraitchik-2e

Permutational Problems 229
go back to their original positions. The first such operationyields
the permutations ACEBGDIFKHJA. The others
are found similarly.
If the number of children is even we may modify the problem
by requiring that every child have every other child but
one as his neighbor. Circular permutations satisfying this
requirement can be found from Figure 107 by marking an
extra point on the other diameter of the figure.
Here is a variation of the problem. n boys and n girls are
A
M
B
K
E
o
FIGURE 109.
H
FIGURE 110.
dancing in a circle, a boy between each two girls. Can one
make successive permutations in their order so that every
boy has every girl as neighbor just once?
In all there are n 2 pairs to be formed. In anyone round
there are 2n such pairs. If there are to be no repetitions or
OffilSSlOns, .. 2n n2 = 2" n arrangements must f orm t h e n d' d
2 eSlre
pairs, so n must be even, say n = 2p. Join the alternate vertices
of an inscribed regular n-gon, so as to form two inscribed
regular p-gons (as in Figure 109). Denote the boys by capital
letters and the girls by corresponding small letters. Letter
the points on the circle with the capital letters in order,
and letter the points of intersection of the two p-gons correspondingly,
as in the figure. The successive arrangements