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The monk and the mountain

Perhaps the cleanest miniature of the principle of continuity turned into history: a path, two days and a conclusion that seems inevitable once seen.

A monk climbs a mountain along a narrow path. It leaves at dawn and reaches the summit at dusk.

Spend the night there. The next day go down the same path.

It also leaves at dawn and arrives at the foot at dusk. You may have walked at different speeds in different sections, stopped in different places and rested at different times each day.

Can it be assured that there is at least one point on the path that the monk passes at exactly the same time on both days?

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Hints

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Don't follow just one monk.
Imagine that both tours occur on the same day.
If one monk went up while another went down the same path, they would have to cross paths.

Solution

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Imagine two different monks:

one does the uphill route on the first day;
the other makes the descent of the second; and suppose that they both walk the path on the same day, leaving at dawn from opposite ends. One starts from the foot of the mountain and the other from the top. Since both travel the same path and advance during the same time interval—from dawn to dusk—there will necessarily have to be a moment when they cross paths. This crossing point has an immediate interpretation: it is a place through which the monk on the way up passed at a certain time and through which the monk on the way down passed at that same time. Therefore, yes: there is at least one point on the path that the monk passes at exactly the same time on both days.

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