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Dot Product Riddle

  1. Feb 10, 2013 #1
    I was recently posed a riddle that went like the following:

    There are two people. Person A picks three numbers from 0-99. Person B guesses which three numbers that person A has picked. Then, person A gives the dot product of his picked numbers with person B's guessed numbers. The question is how could person B figure out person A's selected numbers in three guesses. Even more challenging is to provide a solution that allows person B to guess the numbers in one guess.

    I have a solution to the first part:

    Think of person A and person B as having their guess put into vectors [itex]\vec{a}[/itex]=(a[itex]_{1}[/itex], a[itex]_{2}[/itex], a[itex]_{3}[/itex]) and [itex]\vec{b}[/itex]=(b[itex]_{1}[/itex], b[itex]_{2}[/itex], b[itex]_{3}[/itex]) respectively. To get the corresponding component a[itex]_{1}[/itex], person B should select the components (1,0,0) so the dot product will yield a[itex]_{1}[/itex]. Same for a[itex]_{2}[/itex] and a[itex]_{3}[/itex]. Simple enough.

    The next part I am stumped. The only clue I was given is that person B's three guesses are not restricted between 0-99. Anyone have any insight?
  2. jcsd
  3. Feb 10, 2013 #2
    The key is that the numbers have a finite length, and can be separated far enough from each other by multiplication for further examination.
  4. Feb 10, 2013 #3
    Hey guys!

    I had no idea it was so simple! I was looking into it too much. Thanks for the help.
  5. Feb 15, 2013 #4
    This also depends upon the value incorporated --In case one is choosing (7 ,77 ,93) as the three value -it may lead to come with 3 guesses .
    Finite length numbers can be taken as simple guesses.Comes handy only with vectors
  6. Feb 15, 2013 #5
    I'm probably way off here but don't you get a single equation with 3 unknowns? (and a restricted domain)
  7. Feb 15, 2013 #6

    D H

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    Staff Emeritus
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    That's one additional guess, not one guess. The conversation would go like this:

    Person A: I've picked three numbers from 0-99. Can you guess what they are, in the order in which I picked them? As a hint, I'll tell you the inner product of my numbers and your guess if your guess is wrong.
    Person B: OK. Here's my first guess: b1, b2, and b3.
    Person A: Hey! That's cheating! It's also wrong. But since I didn't make my rules clear enough, I guess I'll have to tell you that the inner product is c.
    Person B: OK! Here's my second guess: a1, a2, and a3.
    Person A: Correct.

    Two guesses, not one. The puzzle is how to frame the first guess so that the second guess will inevitably be correct.

    No. There is a way (there are an infinite number of ways) to formulate the initial guess so that the second guess will always be correct.
  8. Feb 15, 2013 #7
    How? :confused:
  9. Feb 16, 2013 #8


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    Homework Helper

    Read Ferramentarius' clue again, and note that the guesses for B are not restricted to the range 0-99.
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