Miracle Mountain
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.
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% ?
HintThis is not a trick. His watch works perfectly well. He does not sit in the same spot all day or any other such device, although it would not change the answer if he did!
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Answer
The answer is (A). Since it must happen, the probability is actually 1 (100%).
Explanation: Firstly, consider 2 men, one starting from the top of the mountain and hiking down while the other starts at the bottom and hikes up. At some time in the day, they will cross over. In other words they will be at the same place at the same time of day.
Now consider our man who has walked up on one day and begins the descent the next day. Imagine there is someone (a second person) shadowing his exact movements from the day before. When he meets his shadower (it must happen) it will be the exact place that he was the day before, and of course they are both at this spot at the same time.
Contrary to our common sense, which seems to say that this is an extremely unlikely event, it is a certainty.
NOTE: There is one unlikely event here, and that is that he will notice the time when he is at the correct location on both days, but that was not what the question asked.
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Comments
od1
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. 
accordionman
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. 
fishmed
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. 
Charmed_fan_3
Jun 11, 2004
 in skool we're learin about probability's so i new the answer rite away! 
Rowsdower
Jun 12, 2004
 Wow, this one really made me think, since initially it was counter intuitive for me. Great teaser! 
robxxx
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 
jimbo
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. 
fishmed
Jun 15, 2004
 I had not realized that, but that's true. Cool. 
Mogmatt16
Jun 22, 2004
 I personaly think this is one of, if not the best, teaser on the site. 
jimbo
Jun 25, 2004
 Gee Mog that's high praise. Thank you very much! 
shelleyz
Jun 27, 2004
 good one 
Rashad
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 
sillywilly
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 reread 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. 
Nicky101
Dec 12, 2004
 I understand it. Kind of 
Fhqwhgads
Dec 21, 2004
 That was easy 
I_am_the_Omega
Dec 29, 2004
 I don't even understand the question.... 
martir01
Jan 27, 2005
 Mathematically flawed I'm afraid. The answer is that the probability tends to zero. 
Dave625
Feb 10, 2005
 but what if one guy oversleeps and the other dosent? 
badbunnee
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 Collegelevel 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.
GREAT ONE !!!

Tree
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. 
epsilon
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... 
xconfusedx
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 
sftball_rocks13
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. 
tonjawithaj
Jul 06, 2006
 I still don't get it ...but that's okay! 
VitalStatistic
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. 
Jimbo
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. 
andrewsmomma01
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. 
zembobo
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. 
zembobo
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 down..it 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. 
zembobo
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 ;) 
donga
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 
donga
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 
stil
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. 
opqpop
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. 
opqpop
Jan 26, 2010
 Oops, I just realized my comment is exactly same as the solution _ 
opqpop
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. 
eighsse
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. 
eighsse
Jul 27, 2013
 Sorry, missed the words "all day" >_ 
Zylantex
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. 
Jimbo
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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