THE NINE SCHOOLBOYS.
(
Combination and Group Problems)
This is a new and interesting companion puzzle to the "Fifteen
Schoolgirls" (see solution of No. 269), and even in the simplest
possible form in which I present it there are unquestionable
difficulties. Nine schoolboys walk out in triplets on the six week days
so that no boy ever walks _side by side_ with any other boy more than
once. How would you arrange them?
If we represent them by the first nine letters of the alphabet, they
might be grouped on the first day as follows:--
A B C
D E F
G H I
Then A can never walk again side by side with B, or B with C, or D with
E, and so on. But A can, of course, walk side by side with C. It is here
not a question of being together in the same triplet, but of walking
side by side in a triplet. Under these conditions they can walk out on
six days; under the "Schoolgirls" conditions they can only walk on four
days.
Answer:
The boys can walk out as follows:--
1st Day. 2nd Day. 3rd Day.
A B C B F H F A G
D E F E I A I D B
G H I C G D H C E
4th Day. 5th Day. 6th Day.
A D H G B I D C A
B E G C F D E H B
F I C H A E I G F
Every boy will then have walked by the side of every other boy once and
once only.
Dealing with the problem generally, 12n+9 boys may walk out in triplets
under the conditions on 9n+6 days, where n may be nought or any integer.
Every possible pair will occur once. Call the number of boys m. Then
every boy will pair m-1 times, of which (m-1)/4 times he will be in the
middle of a triplet and (m-1)/2 times on the outside. Thus, if we refer
to the solution above, we find that every boy is in the middle twice
(making 4 pairs) and four times on the outside (making the remaining 4
pairs of his 8). The reader may now like to try his hand at solving the
two next cases of 21 boys on 15 days, and 33 boys on 24 days. It is,
perhaps, interesting to note that a school of 489 boys could thus walk
out daily in one leap year, but it would take 731 girls (referred to in
the solution to No. 269) to perform their particular feat by a daily
walk in a year of 365 days.
