## Wednesday, April 21, 2010

### Mountain Top Viewing

A man leaves home for a mountain at 1pm and reaches the top at 3pm. The following day he departs from the top at 1pm and gets home at 3pm, by following the same path as the day before. Was he necessarily ever at the same point on the path at the same time on both days?

1. Well, it probably depends on if he walked a consistent speed both days. Logically, it should take him longer to climb the mountain than to descend because it's more strenuous (this would probably still be the case even if he's driving...you never said he was actually climbing, I guess, but cars go faster downhill too). So if he was in a car, then no, because he would be on the other side of the same road. But if he was walking, and he did walk at a consistent speed both days, he would probably be at the same point at 2pm both days.

2. Of course!

The question is not at *what* time but just *whether*.

Imagine two people doing the up and down trip in opposite direction. They will rest at either end to ensure they are on the path (including beginning and end) for the whole time between 1 and 3. Obviously they have to meet. Thus, obviously, they are at the same point at the same time ...

3. trick question... "same point at the same time both days?" ... different days so it is impossible

4. Nice thinking Garret, but not what I had intended. Certainly true, though. Maybe I should re-write the question somehow?

Anonymous has the reasoning behind the answer. If you imagined two people on the same day, at some point they have to pass each other. It's just harder to imagine it when it's one person on two different days for some reason.

5. Yeah, I do not thing it's possible giving the constraints. I tried graphing it putting a time zone change on the path of the mountain. Same results different location you hit both points at a least one of the same instance of time.

The only way this could be possible is if the times were changed to 1am to 3am and one of those days daylight savings time takes effect.

That would allow for a hole in time of 1hr to squeeze through that point.

6. sorry to clarify my post, of course he will be in at least one place at the same time. Saying it's not possible I mean, it's not possible to miss the event of him being at the same place at the same time.

7. I agree with Mike's answer. This refers to Brouwer's fixed point theorem.

8. If he continued on the same path that means he kept walking. It doesn't say he turned around and walked backwards. Picture how a path may curve, and it may be the same distance back, but he won't be at the same exact place at the same time.

9. What is the answer to the same question if the start time of both trips is the same but the end time is different for the up and down trips?
Does the speed of travel make any difference?

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