[Guest]

On day 1 at exactly 9 AM a person starts to go up a tall mountain along a narrow path. The path spiraled around the mountain to the summit. On his way, he stopped many times to eat some food and to take some rest. His speed at the start of the journey was slightly higher than when he reached the top of the mountain at exactly 5 PM the same day (day 1).

He camped at the mountain top for 2 days and on day three at exactly 9 AM, he started his descent along the same path that he had taken on day 1. Again, stopping many times to take some rest and to eat some food along the way. His speed coming down was of course higher than the speed with which he was climbing up the mountain and he reached the foot of the mountain at exactly 3 PM the same day (day 3).

He wondered if during his descent there ever was a point on the path where he was at the exact same time while climbing up on day 1. Well, he was tired and thought there must have definitely been one such point! Do you think he was right?

[Guest]

No

[Guest]

Yes. Picture the graphs (because I can't be bothered to draw and upload one) of distance from the foot of the mountain against time. The graph of the ascent will have a positive gradient. The graph of descent will have a negative gradient. They must cross at some point.

[Guest]

On day 1 at exactly 9 AM a person starts to go up a tall mountain along a narrow path. The path spiraled around the mountain to the summit. On his way, he stopped many times to eat some food and to take some rest. His speed at the start of the journey was slightly higher than when he reached the top of the mountain at exactly 5 PM the same day (day 1).

He camped at the mountain top for 2 days and on day three at exactly 9 AM, he started his descent along the same path that he had taken on day 1. Again, stopping many times to take some rest and to eat some food along the way. His speed coming down was of course higher than the speed with which he was climbing up the mountain and he reached the foot of the mountain at exactly 3 PM the same day (day 3).

He wondered if during his descent there ever was a point on the path where he was at the exact same time while climbing up on day 1. Well, he was tired and thought there must have definitely been one such point! Do you think he was right?

In my opinion yes there was a point. This can be seen if we make a graph of distance vs time. we plot two curves. the one that starts at 0 distance to the total distance and the other that starts in reverse. since these curves will be continous so they must intersect at some point.

[Guest]

Yes. Picture the graphs (because I can't be bothered to draw and upload one) of distance from the foot of the mountain against time. The graph of the ascent will have a positive gradient. The graph of descent will have a negative gradient. They must cross at some point.

here's a graph. Note that even if he waited until after 3 to start up, there would still be a point (at the base) at which they intersect.

mntn.bmp

Edited July 24, 2009 by tpaxatb

[Guest]

No graphing is necessary. Imagine two people - one leaving the top and one leaving the bottom at the same time (9 AM). They move at the same pace that the person originally did two days apart. The only difference is that they are doing it on the same day. At some point they will pass each other. That point is the answer to the puzzle.

[Guest]

I think this is an important point

EXACT same time

[Guest]

I think this is an important point

EXACT same time

We are, of course, assuming that the spirit of the teaser doesn't imply that the words "exact same time" means the same time on the same day. Of course that would be impossible (unless our intrepid mountain climber was also a time traveller ).

It really doesn't matter where the breaks are taken, when they are taken. Distance from the top over time is bounded at 0 and the peak. The time constraints are overlapping and have the same start point. Across all time points, the climber is somewhere on this curve, so the curve is continuous (i.e. even if he waits at the top or the bottom, he is still at the top or the bottom). Due to these constraints, we know the paths cross at SOME point. EVEN if he waited until 3:01PM to start up the mountain on day 1, the paths would still cross at 3:00PM on day three, at the very bottom of the mountain, since the climber was at the bottom from 9:00AM onward. EVEN if he were to run at an infinite speed up the mountain top on day 1, the point at which the paths cross on day 3 would be at the exact top of the mountain at exactly 9:00AM.

The point is that there is some t at which the paths cross. We don't know what it is, because there is not enough information to determine it (i.e. the curves themselves are unknown). But the point t does exist, which is the answer to the riddle.

[Guest]

What are all of you guys talking about. Read the last paragraph of the riddle: "He wondered if during his descent there ever was a point on the path where he was at the exact same time [today - Day 3] while climbing up on day 1."

He took the SAME PATH UP AS DOWN so THE ENTIRE PATH should intersect with his new route.

So rayates55 is right.The point exists sometime after 12:00 but before 1:00.

Edited July 24, 2009 by Chriswa84

[Guest]

There are several wrong answers

Those of you who drawn a graph with two intercepting lines assumed a constant velocity. But the problem is this guy stopped several times in the way. And the initial speed was higher than the final one. You can't draw a line unless you are certain the speed is the same all the way.

As tpaxatb said. The way for representing position over time for each guy (the one climbing and the one going down) is a curve. It will be continuous function if you draw position over Y and time over X axis, form a mathematical point of view.

Nevertheless, at some point they will meet (same place and same time). It is even possible that they meet in a stop position which will let us with several time spots for the same position (space).

[Guest]

According to my estimates, depending on the breaks the paths will intersect at approx 12:25 pm about 43% of the way up the mountain.

[Guest]

We are, of course, assuming that the spirit of the teaser doesn't imply that the words "exact same time" means the same time on the same day. Of course that would be impossible (unless our intrepid mountain climber was also a time traveller ).

It really doesn't matter where the breaks are taken, when they are taken. Distance from the top over time is bounded at 0 and the peak. The time constraints are overlapping and have the same start point. Across all time points, the climber is somewhere on this curve, so the curve is continuous (i.e. even if he waits at the top or the bottom, he is still at the top or the bottom). Due to these constraints, we know the paths cross at SOME point. EVEN if he waited until 3:01PM to start up the mountain on day 1, the paths would still cross at 3:00PM on day three, at the very bottom of the mountain, since the climber was at the bottom from 9:00AM onward. EVEN if he were to run at an infinite speed up the mountain top on day 1, the point at which the paths cross on day 3 would be at the exact top of the mountain at exactly 9:00AM.

The point is that there is some t at which the paths cross. We don't know what it is, because there is not enough information to determine it (i.e. the curves themselves are unknown). But the point t does exist, which is the answer to the riddle.

I never said on the same day. Only the same time.

[Guest]

There are several wrong answers

Those of you who drawn a graph with two intercepting lines assumed a constant velocity. But the problem is this guy stopped several times in the way. And the initial speed was higher than the final one. You can't draw a line unless you are certain the speed is the same all the way.

As tpaxatb said. The way for representing position over time for each guy (the one climbing and the one going down) is a curve. It will be continuous function if you draw position over Y and time over X axis, form a mathematical point of view.

Nevertheless, at some point they will meet (same place and same time). It is even possible that they meet in a stop position which will let us with several time spots for the same position (space).

It doesn't matter what shape the graph is. Whether or not there is a constant velocity, they will meet.

[Guest]

for a quick proof by contradictionaryism assume he was never at the same point same time that means there are three possibilities.

Day1 guy is always higher the guy 2 at time A

Day 1 guy is always below at time A

or Day1 guy has some time where he is above and some below

in case 1

guy 2 could never make it down as then he would be below guy1 at time A so contradiction

in case 2

guy1 could never make it up so contradiction

in case 3

there must be a time A that is the transitional time. this actually cannot be assumed but for envisioning it, its fine. So take the point B earlier to A on the time and positional line and point c on the other side. These two points have to cross each other without ever touching while traveling on the same line. so contradiction.

[Guest]

well i read this about five times and i still do not know what the freak it is about...

Edited July 24, 2009 by VideoGamer86

[Guest]

Granted there are breaks and such but using simple mathematics and assuming a constant pace which must be assumed even though we no this is not the cas then the rates of going up the hill and comming down the hill would intersect / collide at exactly 3hours 52minutes 30seconds after departing which would make it 30seconds past 12:53pm - this time is can only be assumed as estimated seeing as to how the speeds are not kept constant throughout the journey. But it is important to note the there was an exact time when he was at the same spot going up as he was going down.

Edited July 24, 2009 by netmasterdavid

[plainglazed]

I would say not necessarily. Suppose the climber actually changed time zones on his way up the mountain? On day one he may have ended up at 5:00 p.m. at the top of the mountain but only traveled for seven hours (5:00 at the top of the mountain could be 4:00 at the bottom). On the third day he left at 9:00 a.m. at the top of the mountain which would be 8:00 at the bottom and got to the bottom again seven hours later. His rate of speed coming down the mountain may have been greater than going up the mountain, but that could be offset by more/longer breaks coming down. There are adjacent time zones that dont both observe daylight savings time so I guess the climber could actually go up the mountain upt to an hour faster or come down the mountain up to an hour slower, as well.

Edited July 24, 2009 by plainglazed

[Guest]

No graphing is necessary. Imagine two people - one leaving the top and one leaving the bottom at the same time (9 AM). They move at the same pace that the person originally did two days apart. The only difference is that they are doing it on the same day. At some point they will pass each other. That point is the answer to the puzzle.

What he said. Simplest way to explain it.

[plainglazed]

I would say not necessarily. Suppose the climber actually changed time zones on his way up the mountain? On day one he may have ended up at 5:00 p.m. at the top of the mountain but only traveled for seven hours (5:00 at the top of the mountain could be 4:00 at the bottom). On the third day he left at 9:00 a.m. at the top of the mountain which would be 8:00 at the bottom and got to the bottom again seven hours later. His rate of speed coming down the mountain may have been greater than going up the mountain, but that could be offset by more/longer breaks coming down. There are adjacent time zones that dont both observe daylight savings time so I guess the climber could actually go up the mountain upt to an hour faster or come down the mountain up to an hour slower, as well.

Well you would think this fool would at least read the question to be answered. Oops.

[Guest]

I would have to disagree with everyone who said the climber would cross paths at the same time and same spot. Picture this in you mind. The climber leaves in the morning on the first day going slower than when he climbs down. They don't cross paths at the same time because as the climber goes up his time goes up and the same for when he goes down. If he were going the same speed he would cross at the same, but the problem is he will only meet once at the same time in his paths and the way going down will be at an earlier time in the day than when he climbed up. The answer is no! He will not cross paths at the same spot and time.

Edited July 24, 2009 by superman123

[Guest]

Well going to show that I'm a bit of a simpleton but the Q was do you think he was right? Yes I do! Through probability and through certain assumptions;EG Getting tired and or hungry at same time and looking for a nice spot to rest etc....no not worked out thouroughly or mathematically with graphs or equations BUT considering he followed the same spiral path up and down, and knowing he stopped several times on his journey up AND down I think it is fair to say the answer is yes.I think alot of you failed to consider the rest stops he took and as we don't know how long he stopped for there is a good chance - without using maths - that he was right!!R.Q xx

Clue is in the wording of the teaserAmount of Times that it is mentioned

[Guest]

to be completely honest this isn't at all one of the more complicated logic puzzles on this forum and i'm surprised it has so much debate. The concept is very simple and anyone who has ever learned the mean value theorem in calculus should instantly know the anwser

imagine the times 9-3. ignore 4-5 because it doesnt matter. at time 9 am, the descending climber is above the ascending climber. At time 3 pm, the descending climber is below the ascending climber. at some point in between they must cross paths unles this mountain climber knows how to teleport.

Think about 2 people walking along the same path in opposite directions. they start on one end each and switch to the opposite end. at some point in time they must cross each other regardless of what thier velocities were.

I dont think its possible to give an exact time with the information given. we aren't asked to. And there are way too few details to know.

Edited July 24, 2009 by sparx1

[plainglazed]

Starting with the question this time: Is it possible *if during his descent there ever was a point on the path where he was at the exact same time (of day) while climbing up on day 1*? I think with the parameters given and normal assumptions it is not likely. Again, thinking laterally, it is possible. Suppose the mountain was more of a hill where actual climbing or assending time is an hour or two. If in both cases he climbed (up or down) for a bit rested for five to seven hours (purely estimated times and by no means meant to suggest how long he would need to rest; just that it would be far greater than actual travel time) then resumed, he could have been at that same point (while resting) on the path on both days at the same time of day.

Edited July 24, 2009 by plainglazed

[bonanova]

Let us imagine on the day of his descent his twin brother duplicates his ascent.

The question is now equivalent to the simple question: will he and his brother meet?

The answer is clearly that they will.

"meet" implies same place at same time.

[Guest]

Well, for all the people who said there is a point where the person was at the same time during both the journeys - pat your back! That is the right answer indeed. The explanation is as given by rayates55 and bonanova is the simplest way to figure it out.

PS: Plainglazed's first post on this topic deserves a special mention indeed!! Excellent thinking my friend. Although it didnt quite relate to the question, but I myself would probably not have thought about it had the question been in some way related to this line of thinking.

PSPS: The above special mention is in no way an encouragement to post anything unrelated to the topics!

Edited July 27, 2009 by DeeGee