The monk on the mountain path

Question

A monk starts at the bottom of a mountain at 6:00 a.m. and walks up a single narrow path, reaching the temple at the summit by 6:00 p.m. He rests overnight. The next morning he leaves the temple at 6:00 a.m. and walks down the same path, reaching the bottom by 6:00 p.m. His speed varies freely on both days, he may stop, backtrack, or sprint.

Prove that there is some spot on the path that the monk occupies at exactly the same time of day on both days.

Accepted Answer

The cleanest argument is a superposition trick. Imagine both walks take place on a single day. One version of the monk starts at the bottom at 6:00 a.m. and heads up; a second version starts at the top at 6:00 a.m. and heads down. Both use the same path and both finish by 6:00 p.m. Two travelers moving toward each other along the same one-dimensional path must meet, the up-monk starts below the down-monk and ends above him, so at some instant they are at the same place. Whatever clock time t that meeting happens, the up-monk (day 1) and the down-monk (day 2) are at the identical spot at the i…

The monk on the mountain path

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