Chapter 10: Brainteasers 385
mud on his own forehead, although he can see the
mud on the foreheads of any other muddy children. An
adult comes to collect the children and announces that
at least one of the children has a dirty forehead, and
then asks the group to put up their hand if they know
that they have mud on their forehead. How can each
child determine whether or not their forehead is muddy
without communicating with anyone else?
(Thanks to weaves.)
Solution
If there is only one child with mud on their forehead
they will immediately know it because all of the other
children are clean. He will therefore immediately raise
his hand.
If there are two children with muddy foreheads they
will not immediately raise their hands because they will
each think that perhaps the adult is referring to the
other child. But when neither raises their hand both
will realize that the other is thinking the same as them
and therefore both will raise their hands.
Now if their are three muddy children they will follow
a similar line of thinking but now it will take longer for
them all to realize they are muddy. And so on for an
arbitrary number of muddy children.
To make this work we really need something to divide
time up into intervals, a bell perhaps, because no doubt
not all children will be thinking at quite the same speed!
Pirate puzzle
There are 10 pirates in a rowing boat. Their ship has
just sunk but they managed to save 1000 gold dou-
bloons. Being greedy bastards they each want all the