Solution to the monk problem

The monk problem (Duncker, 1945)

One morning a Buddhist monk sets out at sunrise to climb a path up the mountain to reach the temple at the summit. He arrives at the temple just before sunset. A few days later, he leaves the temple at sunrise to descend the mountain, traveling somewhat faster since it is downhill. Show that there is a spot along the path that the monk will occupy at precisely the same time of day on both trips.

How not to represent the problem:

Thinking about the problem in mathematical terms, or in terms of rates and distances will be unproductive.

A more productive way to represent the problem:

Representing the problem visually leads to a fairly obvious solution. Visualize the path of the monk ascending the mountain, starting at dawn, and his path descending the mountain, also starting at dawn. If the paths start at opposite ends of the trail at the same time, they must meet somewhere. Another approach is to think of two monks to represent his two jouneys, one walking up the trail and one walking down at the same time. The two monks must meet somewhere on the trail, therefore occupying the same spot on the trail at the same time of day.