You spend an entire day hiking up a mountain, and camp overnight at the top. The next day, you hike down the mountain along the same trail, starting around the same time you started the day before.

How likely is it that there is a point on the mountain that you passed at the exact same time on both days?



The easiest way to demonstrate this is true:

Assume your friend hikes up the mountain on the second day at the exact same pace as you hiked up the first day. On your way down the mountain, you will inevitably run into your friend at some point – when you run into each other, you are obviously at the same point on the mountain at the same time.

Therefore you must have passed that point on the mountain at the same time on both days.

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