This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
A boy, a girl and a dog are standing together on a long, straight road.
Simulataneously, they all start walking in the same direction:
The boy at 4 mph, the girl at 3 mph, and the dog trots back and forth
between them at 10 mph. Assume all reversals of direction instantaneous.
In one hour, where is the dog and in which direction is he facing?
The dog's position and direction are indeterminate, other than that the
dog must be between the boy and girl (endpoints included). To see this,
simply time reverse the problem. No matter where the dog starts out,
the three of them wind up together in one hour.
This argument is not quite adequate. It is possible to construct problems
where the orientation changes an infinite number of times initially, but for
which there can be a definite result. This would be the case if the positions
at time t are uniformly continuous in the positions at time s, s small.
But suppose that at time a the dog is with the girl. Then the boy is at
4a, and the time it takes the dog to reach the boy is a/6, because
the relative speed is 6 mph. So the time b at which the dog reaches the
boy is proportional to a. A similar argument shows that the time the
dog next reaches the girl is b + b/13, and is hence proportional to b.
This makes the position of the dog at time (t > a) a periodic function of
the logarithm of a, and thus does not approach a limit as a -> 0.