In a village, friendship among girls is mutual. Each girl has either exactly one
friend or exactly two friends among themselves. One morning, all girls with two
friends wear red hats and the other girls all wear blue hats. It turns out that any
two friends wear hats of different colours. In the afternoon, 10 girls change their
red hats into blue hats and 12 girls change their blue hats into red hats. Now it
turns out that any two friends wear hats of the same colour. How many girls are
there in the village?
【Solution】
Consider a girl with only one friend. She will be wearing a blue hat in the morning,
and her only friend will be wearing a red hat. This friend has a second friend who
must be wearing a blue hat, and the friendship network stops there. It follows that the
girls in the village may be divided into groups of three, with one being friends with
the other two, but the other two are not friends of each other. In the afternoon, in
order for any two friends to be wearing hats of the same colour, all three girls in each
group must wear hats of the same colour. This can be accomplished by either the girl
in red changing her hat, or the two girls both changing their hats. It follows that in 10
groups, the girl in red hat changes, and in 12 ÷ 2 = 6 groups, the girls in blue hats
change. The total number of groups is 10+6=16 and the total number of girls is 16 × 3
= 48.
ANS: 48