Answers to 4 and 5:

4.

5. I will give you 3 correct answers. There are lots of variations on these, so they're not the only right answers.

A. If you know John's timeline up the mountain, and his timeline down the mountain, then you can give those timelines to two different people. Matt can go up the mountain following John's up timeline, and Sara can go down the mountain, following John's down timeline. It's hard to predict the exact time, but somewhere, Matt and Sara have to meet on the trail. The time when Matt and Sara meet is the same time when John would be at the same place at the same time of day on his way up as on his way down.

B. Suppose you took a movie of John going up the mountain, and a movie of him coming down, and then you showed the two movies at the same time (one superimposed over the other). They would show John going down at one place and coming up in another place. Somewhere in the middle they would have to meet.

C. Let's say John kept careful notes of his timeline coming up. Then, as he goes down the mountain, he either points down (if he was lower down the mountain at the same time the previous day), or John points up (if he was higher up the mountain at the same time on the previous day). When John first starts down the mountain on Tuesday, it is 1:00. At 1:00 the previous day, John was at the bottom of the mountain, so at 1:00, John is pointing down. When John gets to the bottom of the mountain on Tuesday it is 8:00. At 8:00 on Monday, John was at the top of the mountain, so at 8:00 John is pointing up. Somewhere in between, as John comes down the mountain, he has to switch from pointing down to pointing up. At that point, he is at the same height on the mountain on both days.