I guess I don't really understand the point of the question. Of course if you walk a path one and then walk it back you'll be in the same spot. It doesn't seem worded properly or maybe it's just a "duh!". On what read you immediately think 100% but think there must be something more because that is too obvious. What am I missing?
Let me try that again:
The question isn't perfectly clear unless you're thinking about puzzles.
If that's the case, then my answer would something like "infintessimally small probability".
Yet the correct answer is 100%
Think of it this way. Suppose walking up the mountain is the same as moving from (0,0) to (1,1) on a Cartesian graph. Suppose walking down the mountain is the same as moving from (0,1) to (1,0) on the same Cartesian graph. (That is, the vertical axis represents a position on the mountain and the horizontal represents the time.) Then it's pretty clear that at some point, the line from (0,0) to (1,1) will cross the line from (0,1) to (1,0). The point of intersection represents when he is at the same position at the same time but on two different hikes.
Or also: Imagine that you start walking up the mountain at a certain time of the day. Exactly at the same time, your alter-ego is at the top of the mountain and starts walking down. Both of you will follow the same path. There's no way you could avoid meeting your alter-ego somewhere along the way!
The whole trick of the question is that the wording encourages you to consider the same person hiking up one day and down another day.
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