# Brain teaser

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• 04-23-2008
medievalelks
Brain teaser
This isn't homework, just for fun. I got this from a book years ago. If you have read the book or solved it otherwise in the past, please refrain from blurting out the answer. :-)

A man walks up a path that circles a moutain in a spiral. He starts at 6:00 in the morning and arrives at the top at 3:00 in the afternoon. The next day, he begins his descent at 6:00 in the morning and arrives at the bottom at 3:00 in the afternoon.

The grade of the path is a constant 15 degrees, the altitude at the top is 5000 feet, and the man is 5'10" and weights 160 lbs. He is carrying nothing but a small 32 oz. water bottle that is full at the start of both trips.

Problem: Prove that he was at the exact same spot on the path at the exact same time on both days.
• 04-23-2008
brewbuck
Quote:

Originally Posted by medievalelks
Problem: Prove that he was at the exact same spot on the path at the exact same time on both days.

Simple application of the intermediate value theorem.

Let U(t) be his position on the mountain on the way up, on the first day.
Let D(t) be his position on the way down, on the second day.
Define q(t) = U(t) - D(t).
Let T be the total climb time for both of the two days.

Obviously, U(0) = D(T) where T is the total time to climb, and D(0) = U(T). This is because it took an equal amount of time to climb up vs. go down.

Therefore, q(0) = U(0) - D(T), and q(T) = U(T) - D(0). These are obviously the negations of each other. Assuming U and D are continuous and single-valued, q is also continuous and single-valued, so the IVT applies. It is obvious that q(0) = -q(T), therefore, there must have been some point in time m where q(m) = 0. This would imply that U(m) = D(m), which proves the result. QED
• 04-23-2008
Mario F.
Quote:

Originally Posted by medievalelks
Problem: Prove that he was at the exact same spot on the path at the exact same time on both days.

Hmm... can I prove he wasn't? Because I can't see how he was, since on both days he was traveling on opposite directions.

Edit: Oh. I see the point now after brewbuck reply.
• 04-23-2008
brewbuck
Quote:

Originally Posted by Mario F.
Hmm... can I prove he wasn't? Because I can't see how he was, since on both days he was traveling on opposite directions.

The result is somewhat unintuitive, but true. For the same reason, there are always two points on the surface of the earth which are exactly opposite each other and have precisely the same temperature. Baffling, but true.

EDIT: I think I see your confusion -- the object is to prove that there was at least ONE moment in time where he was in the same spot at the same time on both days, not that he was in the same spot at the same time at all times.
• 04-23-2008
Mario F.
Quote:

Originally Posted by brewbuck
I think I see your confusion -- the object is to prove that there was at least ONE moment in time where he was in the same spot at the same time on both days, not that he was in the same spot at the same time at all times.

Precisely. I shouldn't have hurried into an answer.
• 04-23-2008
SlyMaelstrom
Doesn't seem like a Brain Teaser to me so much... the proof is simple calculus and really the logic you would learn somewhere in intermediate grade school. You have a position over time graph in which two plotted lines starting at the exact same time continue towards another precise time. Both are going from A to B on the time scale and on the position scale, each starting point is the other's ending point. I'm pretty sure if you drew that out you'd see there is no way for these two lines not to cross paths (where position and time are exactly the same). Interestingly enough... as basic as the logic is, I also immediately was thinking what Mario was thinking and was going to say it's not possible.

Here is a brain-teaser for you all:

There is a common English word that is nine letters long. Each time you remove a letter from it, it still remains an English word - from nine letters right down to a single letter. What is the original word, and what are the words that it becomes after removing one letter at a time? You may find that the answer is startling.
• 04-23-2008
medievalelks
Actually, the proof can be done by simple inspection. Imagine a camera filming both trips. Superimpose one video over the other and naturally, at some point, the images of the traveler will meet.

That's the solution the book was looking for, anyway, in an effort to get you to think outside the box.
• 04-23-2008
SlyMaelstrom
Quote:

Originally Posted by medievalelks
Actually, the proof can be done by simple inspection. Imagine a camera filming both trips. Superimpose one video over the other and naturally, at some point, the images of the traveler will meet.

That's the solution the book was looking for, anyway, in an effort to get you to think outside the box.

And outside of my standard budget of things... I don't need to buy any video editing software to figure that one out. :p
• 04-23-2008
tabstop
Quote:

Originally Posted by SlyMaelstrom

Here is a brain-teaser for you all:

There is a common English word that is nine letters long. Each time you remove a letter from it, it still remains an English word - from nine letters right down to a single letter. What is the original word, and what are the words that it becomes after removing one letter at a time? You may find that the answer is startling.

Guess: (although I needed the hint)

startling
starling
staring
string
sting
sing
sin
in
I

Edit: Ooh! Two paths!
• 04-23-2008
Mario F.
Quote:

Originally Posted by SlyMaelstrom
There is a common English word that is nine letters long. Each time you remove a letter from it, it still remains an English word - from nine letters right down to a single letter. What is the original word, and what are the words that it becomes after removing one letter at a time? You may find that the answer is startling.

Startling, I guess :)

Startling
Starting
Stating <-- this one didn't come easy
Sating
Sting
Sing
Sin
Si
I
• 04-23-2008
SlyMaelstrom
Si, Mario? :)

I think your Portuguese came out a little bit on that one.

(Yes, I had to check a translation dictionary to find out that "si" was "if" in Portuguese cause I wasn't sure if it meant "yes" in portuguese)
• 04-23-2008
Mario F.
Nope. This should be fun, hehe

Noun: si. The syllable naming the seventh (subtonic) note of any musical scale in solmization

EDIT: Oh, and that's Spanish. In Portuguese "if" is se
• 04-23-2008
SlyMaelstrom
Uggg... stupid stong. Now I need to watch Kids in the Hall.

...and wait. Isn't it Doe-Ray-Mi-Fa-So-La-Ti-Do? "Tea" a drink with jam and bread...
• 04-23-2008
Mario F.
hehe. Don't worry. To your credit I did think in "in" as tabstop did. But was getting carried away with starting all words with S and had to lookup myself for the meaning of Si ;)
• 04-23-2008
SlyMaelstrom
Quote:

Originally Posted by Mario F.
hehe. Don't worry. To your credit I did think in "in" as tabstop did. But was getting carried away with starting all words with S and had to lookup myself for the meaning of Si ;)

Hmm... I see dictionary.com citing it as the definition, however, any place that you look up the gamut of notes, it will say "Ti"

http://en.wikipedia.org/wiki/Musical_scale

Surely we can't argue with The Sound of Music...
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