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 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.
100%. Superimpose the days: Friday starts walking up at 2:00,
Saturday starts walking down at 2:00. Since they are on the same
path, they must meet.