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.