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.