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Miracle Mountain

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.


Puzzle ID:#17877
Fun:*** (2.8)
Difficulty:** (2.09)
Submitted By:jimbo*au******
Corrected By:Winner4600




A hiker climbs all day up a steep mountain path and arrives at the mountain top where he camps overnight. The next day he begins the descent down the same trail to the bottom of the mountain when suddenly he looks at his watch and exclaims, "That is amazing! I was at this very same spot at exactly the same time of day yesterday on my way up."
What is the probability that a hiker will be at exactly the same spot on the mountain at the same time of day on his return trip, as he was on the previous day's hike up the mountain?
Is the probability closest to (A) 99% or (B) 50% or (C) 0.1% ?

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Jun 10, 2004

Exellent teaser. I was convinced this didn't work, till another editor sujested graphing it . Time on one side and distance the other. One line up and one down. Where they cross was the spot at which time and distance meet. It was clear then this was an exellent teaser , really got us thinking. Keep em coming.
Jun 10, 2004

I don't see how this is possible. You go down a mountain faster than you go up. Plus, the time that he wakes up is a very crucial point.
Jun 10, 2004

The idea being, if he goes down faster than he went up, the location that is the crossing point will just be lower down the mountain. Obviously getting up at a time to allow him to start at the same time is required.
Jun 11, 2004

in skool we're learin about probability's so i new the answer rite away!
Jun 12, 2004

Wow, this one really made me think, since initially it was counter intuitive for me. Great teaser!
Jun 14, 2004

very good. any time on his way down the mountain he is passing what he wnet by the day before. like you said, timing is crucial, but it's also irrelivent. good work
Jun 14, 2004

To fishmed. He doesn't need to start at the same time. As long as he starts before the time that he finished the day before. If he starts very late, then he will cross yesterdays path very soon after leaving but it will still be late in the day because he left late. It will be towards the top of the mountain also. If he leaves bright and early and travels very quickly on the way down, he will cross the path early in the day and towards the bottom of the mountain.
Jun 15, 2004

I had not realized that, but that's true. Cool.
Jun 22, 2004

I personaly think this is one of, if not the best, teaser on the site.
Jun 25, 2004

Gee Mog that's high praise. Thank you very much!
Jun 27, 2004

good one
Aug 16, 2004

Very nice teaser ... Got me going .... I really got shocked when seeing the answr but it all added up with the explanation
Nov 20, 2004

The deception happens when time time of measurement is considered as a factor. If you see the problem with two people walking simultaneously (one up and one down) and re-read the problem, then the trick is exposed. FUN! Giving 3 possible answers makes it much easier. If you had to solve for p, then I'm certain I would have spent more time on paper before realizing the trick.
Dec 12, 2004

I understand it. Kind of
Dec 21, 2004

That was easy
Dec 29, 2004

I don't even understand the question....
Jan 27, 2005

Mathematically flawed I'm afraid. The answer is that the probability tends to zero.
Feb 10, 2005

but what if one guy oversleeps and the other dosent?
Feb 26, 2005

Terrific teaser !!!
I first encountered this one with a math class about twenty years ago. Most thought the probability to be quite low. Then, and since, I've used it with other students. Most recently it was with an advanced College-level class in calculus. I was AMAZED that (still) most of the class - on first cut - said the probability would be quite low. Then they did a bit of figgerin' and changed their minds.
Mar 01, 2005

Actually Dave625, the statement would still be true if one of the guys oversleeps. Taking it a bit further... it would be true even if he never left his campsite. I love this teaser.
Apr 18, 2005

In the storytelling, adding the part: "looks at his watch and exclaiming what a coincidence: same CLOCK time AT the SAME location" is the whole trick, done to mislead. Or else, the question boils to :"what is the probability that next day he'll be somewhere on the same slope as today, given that he descends by the same slope?" . No need to draw any graphics...
Apr 19, 2005

Great teaser! I was a little confused at the teaser at first, but then I got it!! U should do another one
Aug 12, 2005

very clever!
it made perfect sense onece I read the answer !!!!!!!!
(user deleted)
Oct 29, 2005

jesus christ that is effin insane
(user deleted)
Oct 29, 2005

i made all these graphs and stuff haha its just so crazy.
Jul 06, 2006

I still don't get it ...but that's okay!
Mar 10, 2007

I can't agree. It is true that at some point in time the hiker will be at the exact same place he was exactly 24 hours earlier, but the problem was written that at that time and place, a third event happened in that he checked his watch, and noted the occurence. Unless he was checking all the way down and was looking for the event to happen, it was pure chance that he checked at that time, and that chance would be approaching zero.
Mar 13, 2007

Sorry Vital but the problem was stated "What is the probability that he will be at the same place at the same time." There was no mention in the question that he would also look at his watch and notice that it was the coincicing place and time.
Aug 03, 2007

I really liked this one! Got it right too!
(user deleted)
Nov 06, 2007

what if the man climbed the mountain entirely in the afternoon the first day and went down the mountain entirely before noon the next day? although he did it on two seperate calendar days, it could have been still within 24 hours and so he wasn't on the mountain at the same time at any point in the two days. Sorry to be picky but this isn't exactly absolute.
(user deleted)
Nov 06, 2007

...wait, nevermind I didn't see the part about him climbing all day. nevermind.
May 22, 2008

The answer is incorrect. You assume his speed never changes. Sure it took him all day going up but maybe not going down. And even if it did, maybe he got a slow start so he had to pick up the pace later on to make up for lost time. He would be at the same place because he is retracing his steps..but NOT at the same time.
May 22, 2008

And before you disagree with my previous statement, consider this: if he was at the halfway point at 30 on his way up, that doesn't mean it's going to be 30 on his way could be anytime. This applies to any point on the mountain not just the middle.

Plus he probably got near the top pretty late, but on his way down he was near the top but it was early. Think about it.
May 22, 2008

In my attempt to prove this teaser wrong I have seen the error in my ways. I've thought about it and the answer is right. Please ignore my two lame commets above. Good teaser :-) But I will say that the chances he happened to look at his watch at that one moment is pretty slim ;-)
Aug 19, 2008

i cant believe what u are saying!!!
it certainly depends on the speed and the time when he starts both the journey and may be the distance.
lets assume, with constant speed up and down, if he starts hiking up at 90 am and hiking down at at 70 am next day . can any one tell me the place and time when he is at the same place & time as the previous day
Aug 19, 2008

sorry! i have to think it over once again
he will meet at 10:30 in my previous Q
may be taking constant speed is a flaw.. but i cant believe that he will be at the same place at the same time
Apr 18, 2009

The multiple choice doesn't come off very well. If the options were 3%, 2%, and 1%, then 3% would have been the best answer!
It does remind me of the old saw that even an unwound watch (watch with a dead battery) is right twice a day.
Jan 26, 2010

Great teaser! Here's a more intuitive way to see why the probability must be 1.

Put 1 guy at the top of the mountain, who will go down. Call this guy "Today." Put another guy at the bottom of the mountain, who will go up. Call this guy "Yesterday." Now lets first suppose they both start at the same time, say 7am. Then they are bound to cross each other at some point (relative to height of the mountain). This is the time at which they are standing in the same spot at the same time.

What if they started at different times, say top starts at 9am, bottom starts at 7am? Then consider only the time at which the guy who starts later starts. At that moment, the other guy is already somewhere between the top and bottom of the mountain. Notice this is now the same situation as 2 guys starting from 9am, who are bound to still cross each other.
Jan 26, 2010

Oops, I just realized my comment is exactly same as the solution -_-
Sep 11, 2010

To improve on my solution, a faster and more intuitive way to see the times they start at doesn't matter is to note they always must pass each other as long as one doesn't finish before the other starts. The problem give us the information that this doesn't happen.
Jul 27, 2013

Very clever, good one! I did get it right after a few minutes of thinking it through...BUT, it's not necessarily true. First of all, do am/pm times count as different times, or is 1:34am the same as 1:34pm? And even if they do count as the same times, how do you know it takes him so long that he's even climnbing and descending the mountain at the same time of day? What if he climbs the mountain from 3pm to 8pm, and descends from 10am to 2pm the next day? Then it's 0% chance.
Jul 27, 2013

Sorry, missed the words "all day" >_
Mar 26, 2014

"What is the probability that a hiker will be at exactly the same spot on the mountain at the same time of day on his return trip, as he was on the previous day's hike up the mountain?"

As phrased I think the probability approaches zero as there is only one possible instance for the statement to be true. It is false at all other times. The term "will be" does not equate to "could be". The question is not "Can it happen?" which is a certainty but rather "Is it true at any given time?" which it most certainly isn't.
Mar 26, 2014

If you pick a particular time say 11 am and ask what is the probability he will at the same spot on both days at 11 am then the probability is quite low. But the question does not ask about a particular time. It asks about whether or not he will be at the same spot at the same time he was the previous day. Of course that is a certainty. On the descent he will cross every point on the previous path and at one of those points it will be also the same time as the previous day. Easiest way to see it is to graph it.

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