Mountain Man

[WorkHorse]

mountain man aha:!

at 6 a.m. a man starts hiking a path up a mountain. he walks at a variable pace, resting occasionally, but never actually reversing his direction. at 6 p.m. he reaches the top. he camps out overnight. the next morning he wakes up at 6 a.m. and starts his descent down the mountain. again he walks down the path at a variable pace, resting occassionally, but always going downhill. at 6 p.m. he reaches the bottom. what is the probability that at some time during the second day, he is in the exact same spot he was in on the first day?

solution: mountain man

answer: the probability is 100%. the easiest way to see it is, consider that on the second day when the man is going down the mountain, a ghost follows his original pace up the mountain. so even if he varies his pace as he goes down the mountain, at some point in time, he will be in the same spot as the ghost, and therefore, the same spot he was in the day before

Is it just me or is this complete hooey? It only has an "aha" rating of one exclamation point meaning it is not a "trick" question.

My answer is infinity! Provided the man follows the exact same path, he will ALWAYS be in a spot that he was yesterday. There is not enough information to determine if he was ever at the same spot at the same time.

The "answer" seems to only be answering the question "what is the probability that at some time during the second day, he is in the exact same spot he was in on the first day--at the exact same time, subtracting the total amount of time from the time he started down." In other words, if two men follow the exact same path from two directions what is the probability that they will meet? Seems to be an entirely different question, or am I missing something?

[jemidiah]

I'd say it's true if time doesn't matter, though if it does, then it really isn't calculable. Without time, this is really a mean value theorem example which states that, given a differentiable and continuous interval [a, b], sometime between a and b there will be a point with the slope of (f(b) - f(a))/(b-a). This makes sence, though there are better examples out there. Now, if it IS including time, a counterexample is very readily reachable making it false.

[Something Else]

he reaches the bottom. what is the probability that at some time during the second day, he is in the exact same spot he was in on the first day?

Doesn't it go like:

he reaches the bottom. what is the probability that at some time during the second day, he is in the exact same spot he was exactly 24 hours earlier?

Ignore the following.

Imagine 2 cars, programmed to travel exactly 20 miles in 12 hours, but at any point in time, the cars speed is completely random, within limits.

Imagine a gully wide enough for a car to drive thru, but no wider. and it just so happens to be 20 miles long.

Car A starts on 1 end, Car B starts on the other.

It is guarenteed they will crash into each other within that 20 miles.

[wossname]

Originally posted by Something Else
Doesn't it go like:



Ignore the following.

Imagine 2 cars, programmed to travel exactly 20 miles in 12 hours, but at any point in time, the cars speed is completely random, within limits.

Imagine a gully wide enough for a car to drive thru, but no wider. and it just so happens to be 20 miles long.

Car A starts on 1 end, Car B starts on the other.

It is guarenteed they will crash into each other within that 20 miles.

Nope that's not true. The council would tow at least one of them away before long before they collided, for obstructing traffic and various other petty violations that the warden can think up on the spot.

[sql_lall]

WorkHorse: yes, the actual problem should be:
Prove there is at least one spot where the Mountain man was at the same time on both days.
(Alternate proof is graph his ascent/descent and show they intersect)

With the cars, i'm guessing that neither car can fly/travel up the side of the valley, or travel into the Future Back-to-the-future style... hehe

[fundu]

I believe that there will be only one spot where they will meet. here I do not mean only one time as then U all will say that may be both are resting on the same point during the same time.