Brain teaser

This is a discussion on Brain teaser within the A Brief History of Cprogramming.com forums, part of the Community Boards category; This isn't homework, just for fun. I got this from a book years ago. If you have read the book ...

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    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.

  2. #2
    Captain Crash brewbuck's Avatar
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    Quote Originally Posted by medievalelks View Post
    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

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    (?<!re)tired Mario F.'s Avatar
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    Quote Originally Posted by medievalelks View Post
    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.
    Last edited by Mario F.; 04-23-2008 at 04:54 PM.
    The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
    The programmer comes home with 12 loaves of bread.


    Originally Posted by brewbuck:
    Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.

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    Captain Crash brewbuck's Avatar
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    Quote Originally Posted by Mario F. View Post
    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.

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    (?<!re)tired Mario F.'s Avatar
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    Quote Originally Posted by brewbuck View Post
    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.
    The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
    The programmer comes home with 12 loaves of bread.


    Originally Posted by brewbuck:
    Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.

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    Devil's Advocate SlyMaelstrom's Avatar
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    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.
    Sent from my iPad®

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    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.

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    Quote Originally Posted by medievalelks View Post
    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.
    Sent from my iPad®

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    and the Hat of Guessing tabstop's Avatar
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    Quote Originally Posted by SlyMaelstrom View Post

    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!
    Last edited by tabstop; 04-23-2008 at 05:37 PM.

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    (?<!re)tired Mario F.'s Avatar
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    Quote Originally Posted by SlyMaelstrom View Post
    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
    The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
    The programmer comes home with 12 loaves of bread.


    Originally Posted by brewbuck:
    Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.

  11. #11
    Devil's Advocate SlyMaelstrom's Avatar
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    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)
    Sent from my iPad®

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    (?<!re)tired Mario F.'s Avatar
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    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
    The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
    The programmer comes home with 12 loaves of bread.


    Originally Posted by brewbuck:
    Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.

  13. #13
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    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...
    Last edited by SlyMaelstrom; 04-23-2008 at 05:57 PM.
    Sent from my iPad®

  14. #14
    (?<!re)tired Mario F.'s Avatar
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    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
    The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
    The programmer comes home with 12 loaves of bread.


    Originally Posted by brewbuck:
    Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.

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    Devil's Advocate SlyMaelstrom's Avatar
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    Quote Originally Posted by Mario F. View Post
    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...
    Sent from my iPad®

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