Mathematical_Recreations-Kraitchik-2e

}'1 athematical Recreations
it has not yet delivered out of the way to the left of the siding,
and then pull its own cars to the left of the siding. Then B
can pick up its cars and get out of the way. This procedure
is repeated until train B is all assembled.
3. DISTRIBUTIONS
1. THE DAILY PROMENADE. Years ago, when girls were
called young ladies and were never permitted more violent
exercise than walking, the headmistress of a boarding school
wished to arrange matters so that her pupils would derive the
maximum amount of companionship in their daily walks without
forming boisterous groups. She therefore ordered the
young ladies, of whom there were 2n, to walk in pairs, but to
form new pairs each day in such a way that no young lady
had the same companion a second time before she had
walked with every other young lady. This worked well for
a day or two, but presently the young ladies began to spend
more and more time each day trying to find partners. They
would be nearly ready when it was discovered that the last
two young ladies had already walked together. Can you
help the headmistress?
Here is one solution. Divide the circumference of a circle
into 2n - 1 equal parts by successive points al, a2, ..• , ~n-l.
To simplify, let us say 2n - 1 = 11, and letter the points
from A to K. Mark the center ~n with an L, draw the diameter
through A, and draw the n - 1 chords perpendicular to
this diameter and through the remaining 2n - 2 points,
which are symmetrically placed with respect to this diameter.
The result is Figure 105. The pairs of points located
on the diameter and on the chords give an arrangement of
the promenade for one day. For the following days the diameter
and the chords are rotated so that the diameter passes
successively through B, C, ... , K, and the pairs of points
determined by the new positions of the chords and the diam-