One morning, at 7:00, a Buddhist monk began hiking up a narrow trail that leads to the top of a tall mountain. The monk ascended the trail at varying rates of speed, and stopped several times along the way to eat and rest. The monk reached the top at 7:00 that evening. After a night of meditation, the monk started going down the trail at 7:00 in the morning. Again, the monk walked at various rates of speed and stopped several times along the route. The monk reached the bottom of the trail at 7:00 in the evening.
Will there always be some spot along the path that the monk passed at precisely the same time on both days?
Surprisingly, the answer is yes, there must be some such spot. The easiest way to show this is to imagine that the monk begins his downhill journey on the same day that he was walking up the hill instead of the next day. So, as the monk starts at the bottom of the hill and walks towards the top of the hill, another copy of the monk starts at the top of the hill at the same time and walks towards the bottom of the hill. Looking at it this way, it's clear that the two must pass each other at some point or another. This is the point that the monk passed at the same time on both days.
This story is a simple example of what is called a fixed-point theorem. It establishes the existence of a point, but does not show where that point is.
Sources used (see bibliography page for titles corresponding to numbers): 55, 56.