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  1. #1
    filling some space UnitOfPopulation's Avatar
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    Default difficult-ish weighing puzzle

    There are 12 balls, 11 of equal and one of different weight. You have a beam balance which you may use to weigh two groups of balls to find out which one is heavier, or if they are of equal weight. You have 3 weighings. How should you arrange the weighings to find out which of the balls is of different weight, and whether it is heavier or lighter than the rest?

    Edit: please put your solution in between spoiler alert, like this
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  2. #2
    RETIRED CzeCze's Avatar
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    Wouldn't you just keep on halving things until the last attempt?

    1st Attempt
    6 left ---- 6 right

    whichever is heavier you weigh again

    2nd Attempt
    3 left --- 3 right

    whichever is heavier you weigh again

    3rd Attempt
    You take 2 of the 3 balls at random.

    You onlyweigh these 2 out of the 3 remaining balls

    Final Answer
    If one that you weigh is heavier than the other, you found the odd ball.
    If the two you weigh are equal, the third you didn't weigh is the odd ball.

    *curtsies*

    And people say us ENFPs can't think straight.

    ...

    (I'm right, right?)

  3. #3
    filling some space UnitOfPopulation's Avatar
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    That's good thinking! That would find the heavier ball, if the ball of different weight would indeed be heavier. What would you do if you didn't know whether it was heavier or lighter?

    When you find the correct solution, mark the solution with spoiler warning, or color the solution with a background color like this (like this)

  4. #4
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    I've figured out a method that will determine the correct ball and whether it is heavier or lighter, but 1/12 of the time I will not know if the ball is heavier or lighter. The devil is always in the details. If I figure out how to determine the relative weight of that 12th ball then I'll post a solution.
    My wife and I made a game to teach kids about nutrition. Please try our game and vote for us to win. (Voting period: July 14 - August 14)
    http://www.revoltingvegetables.com

  5. #5
    insert random title here Randomnity's Avatar
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    I did a crappy job explaining this, but it works out on paper I think, unless I'm forgetting something.

    1) weigh 4 v 4 balls. If they balance, you have 8 "same" and 4 "heavier or lighter" balls. go to 2a. If not, you have a "heavier" group and a "lighter" group, plus 4 "same". go to 2b.
    2a) take 3 of the 4 unweighed "heavier or lighter" balls and weigh them against 3 of the weighed "same" balls. If they balance, they are all the same and you have eliminated all but one ball. go to 3a. If not, you have determined whether the ball will be light or heavy. go to 3b.
    2b) weigh 2 balls from the heavier side + 1 ball from the lighter side on both sides of the scale. If they balance, you have eliminated these 6 balls. go to 3c. If not, go to 3d.
    3a) take the remaining unweighed ball (which is the odd one) and weigh it against a weighed one to see if it's lighter or heavier.
    3b) If the balls are lighter than the weighted "same" balls, weigh 2 of the 3 against each other. If they balance, the third one is lighter. Otherwise it's the lighter one. Reverse if the balls are heavier than the previously weighed balls.
    3c) take the remaining two unweighed balls from the "lighter" group. weigh them against each other. The ball is the lighter one.
    3d) take the "winning" light and heavy balls aside (both sides HHL, so you're left with 1 ball from the light side and 2 from the heavy, regardless). Weigh the winning light side ball + one of the winning heavy side balls, against 2 balls which had been previously weighed ("same"). If the side of interest is lighter/heavier, it's the lighter/heavier ball. If the sides balance, it's the heavier ball in reserve.

    maybe this is more clear:

    1) 4 vs 4 -->(4H + 4L + 4S) OR (8S + 4?)
    2) [2H+1L vs 2H+1L) --> (6S + 2L + 4S) OR (2H+1L + 3S + 4 S)] OR
    [3? vs 3S --> (3S+8S+1?) OR (3H + 1S + 8S) OR (3L + 1S + 8S)]
    3) 1? vs 1S --> odd ball is lighter or heavier according to results.
    1H vs 1H --> heaviest ball, or remaining H ball if balanced is odd one out
    1L vs 1L--> lightest ball, or remaining L ball is odd one out
    1L+1H vs 2S --> if lighter, light ball is odd. if heavier, heavy ball is odd. if balanced, remaining heavy ball wins.



    It's kinda sad how long it took me to solve, and that I didn't give up. Haha
    Last edited by Randomnity; 11-09-2007 at 11:50 AM.

  6. #6
    filling some space UnitOfPopulation's Avatar
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    Quote Originally Posted by Randomnity View Post
    I did a crappy job explaining this, but it works out on paper I think, unless I'm forgetting something.
    You explained well, it took only a few minutes to examine your answer.

    My comment on Randomnity's solution:

    The first explanation is otherwise correct, but there is a typo in one section, where you tell to go from 2b to 3b, when you must have ment 3d.

    The symbolic expression shows that you have it solved, and it was evident from the explanation as well.

    Well done!

    Quote Originally Posted by Randomnity View Post
    It's kinda sad how long it took me to solve, and that I didn't give up. Haha
    Nah, it's fun to do puzzles once in a while

  7. #7
    insert random title here Randomnity's Avatar
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    Quote Originally Posted by Santtu View Post
    you must have ment 3d.
    Oh haha, yes I did. Good catch. Fixed.

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