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medievalelks
04-23-2008, 04:03 PM
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.

brewbuck
04-23-2008, 04:44 PM
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

Mario F.
04-23-2008, 04:50 PM
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.

brewbuck
04-23-2008, 04:51 PM
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.

Mario F.
04-23-2008, 04:57 PM
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.

SlyMaelstrom
04-23-2008, 05:04 PM
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.

medievalelks
04-23-2008, 05:10 PM
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.

SlyMaelstrom
04-23-2008, 05:11 PM
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

tabstop
04-23-2008, 05:30 PM
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!

Mario F.
04-23-2008, 05:33 PM
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

SlyMaelstrom
04-23-2008, 05:47 PM
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)

Mario F.
04-23-2008, 05:52 PM
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

SlyMaelstrom
04-23-2008, 05:54 PM
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...

Mario F.
04-23-2008, 05:58 PM
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 ;)

SlyMaelstrom
04-23-2008, 06:01 PM
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...

Mario F.
04-23-2008, 06:28 PM
It's a little bit confusing. But the word does exist and means the same as Ti.

Check http://en.wikipedia.org/wiki/Note#History_of_note_names for a description. And http://en.wikipedia.org/wiki/Solfege for a more elaborate description along with some interesting trivia on how these words came to be.

Si is kept in most romantic speaking countries, and then some more. I'm unsure why English speaking countries felt Si could be confused for So(l). But the noun is still existing notheless. Both because of the chromatic scale, and because my Webster's says so. :)

medievalelks
04-24-2008, 05:14 AM
Si is kept in most romantic speaking countries, and then some more. I'm unsure why English speaking countries felt Si could be confused for So(l). But the noun is still existing notheless. Both because of the chromatic scale, and because my Webster's says so. :)

Oh, I'm not doubting you. The dozens of Scrabble victims left in my wake can attest to that :-)

SlyMaelstrom
04-24-2008, 07:38 AM
Oh, I'm not doubting you. The dozens of Scrabble victims left in my wake can attest to that :-)Yah... if there is anything I learned from Scrabble, it's that almost any two letter combination makes a word.

laserlight
04-24-2008, 07:40 AM
Yah... if there is anything I learned from Scrabble, it's that almost any two letter combination makes a word.
gg :o

Oh wait, is that also a word? I obviously do not play Scrabble.

SlyMaelstrom
04-24-2008, 07:50 AM
gg :o

Oh wait, is that also a word? I obviously do not play Scrabble.Heh, not that I know of. I'm exagerrating when I say "almost any," however, there are certainly more two letter words than you would think.

JMJ_coder
04-24-2008, 06:23 PM
Hello,

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...

Si was the original name for the note (leading tone), but it was changed to Ti. Same with Do (tonic), which was originally Ut.

Currently, Si is the solfege syllable for a raised dominant. An obvious application would be an augmented triad - (do mi si).