==> geometry/topology/fixed.point.p <==
A man hikes up a mountain, and starts hiking at 2:00 in the afternoon
on a Friday.  He does not hike at the same speed (a constant rate), and
stops every once in a while to look at the view.  He reaches the top in
4 hours.  After spending the night at the top, he leaves the next day
on the same trail at 2:00 in the afternoon.  Coming down, he doesn't
hike at a constant rate, and stops every once in a while to look at the
view.  It takes him 3 hours to get down the mountain.

Q: What is the probability that there exists a point along the trail
that the hiker was at on the same time Friday as Saturday?

You can assume that the hiker never backtracked.