Fixed-Point Problem

One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit.

The monk ascended the path at varying rates of speed, stopping many times along the way to rest and to eat the dried fruit he carried with him. He reached the temple shortly before sunset. After several days of fasting and meditation, he began his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way. His average speed descending was, of course, greater than his average climbing speed.

Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.

Originally posted by HandyAndy
One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit.

The monk ascended the path at varying rates of speed, stopping many times along the way to rest and to eat the dried fruit he carried with him. He reached th ...[text shortened]... a spot along the path that the monk will occupy on both trips at precisely the same time of day.

Originally posted by celticcountry
7 /\ 0
6//\\1
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them monk passed the same point on my mountain at precicely 3:30

Originally posted by HandyAndy
Proof? Bear in mind that his average speed descending was greater than his average climbing speed.

Originally posted by PBE6
I think the easiest way to see the answer is to draw a graph of the monk's path starting at (sunrise, 0) and proceeding to (sunset, H). The points define a box, which gets cut into two pieces by the monk's path. It's impossible to draw the return trip starting at (sunrise, H) and ending at (sunset, 0) without crossing the first line unless you travel down the ...[text shortened]... neously.

Not quite a proof, but instructive. I'll work on something more rigorous later.

Imagine we have two monks and when one is travelling down the other is travelling up (both starting at sunrise).

At some point the two monks must pass each other. When they do they have both occupied the same point at the same time. And this is obviously true no matter what speed (constant or varying) the two monks travel at.

Originally posted by HandyAndy
[snip]
Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.

The elegant solution: Let ascent and descent be divided between two persons on the same day. They must meet.

Bingo for Xanthos.. honorable mention to Diapason

I like this problem. It is hard to imagine the answer til you get the solution. Easy to formulate, easy to prove the answer.

I have another problem at Thread 60353 which somewhat resembles this one.

By the way - was this Buddhist monk born in Buddhapest?

Originally posted by XanthosNZ
Imagine we have two monks and when one is travelling down the other is travelling up (both starting at sunrise).

At some point the two monks must pass each other. When they do they have both occupied the same point at the same time. And this is obviously true no matter what speed (constant or varying) the two monks travel at.

let F(x) = distance from base camp with respect to time on day one (ranging from 0 at t=0 to H at t=sunset. clearly this is a continuous function, provided the monk hasn't learned to teleport.

let G(x) = distance from base camp with respect to time on day two (ranging from H at t=0 to 0 at t=sunset). also continuous.

now consider the function F(x) - G(x). this is a composition of continuous functions, along the same domain, and thus is also continuous. Now, it ranges from -H to +H and must achieve every value in between at some point along the way (by the definition of continuity). But the moment when [F(x) - G(x)] achieves a height of 0 is exactly the moment when the monk was equidistant to the base camp on subsequent days.

note that the monk must not teleport, and the time interval must be the same so as to not allow for the possibility that he could roll down the mountain really quick after sunset. 🙂

p.s. i like the idea of two monks doing it on the same day - i've always thought of it as running the tape of the monk on the first day, superimposed over the tape of his trip on the second day, like the "ghost car" in the original super mario kart? anybody know that reference?

Brilliant, thanks a lot.

I dont like the variable speeds part of the question.
Is this a budhist reference to 'emptiness'...not - 'you can't step in the same river [mountin] twice' but that you can't step in the same river once?
1 and 0 are the same....the points of the journey are the same no matter which direction [life] you travel.