Mathematical_Recreations-Kraitchik-2e

228 Mathematical Recreations
child has every other child just once as neighbor, on one side
or on the other?
Solution: Since each child has 2 neighbors at a time, the
total number of children must be odd, say 2n + 1. (In Fig-
K
F
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FIGURE 107.
ure 107 is shown the case where n = 5.) Mark the circumference
of the circle with 2n equally spaced points and letter
them in the order al<l2a4· .. <l2,,<l2n-l<l2,,-3· •• a3. Letter the center
au and join the points aoal<l2a3· .. <l2"ao by a zigzag line.
Each successive rotation of the zigzag line through a (2n)th
part of a revolution yields a new circular permutation of the
FIGURE 108.
2n + 1 letters. The full set satisfies all the requirements.
We can also use the two-rowed rectangular box. For this
problem one of the letters is repeated, and the arrangement
corresponding to the circular permutation ABCDEFGHIJK
is given in Figure 108. To obtain a new permutation the two
cubes A are removed, the two rows are slid in opposite directions,
and the end cubes are moved so that the cubes A can