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  1. MHB Craftsman
    mathmaniac's Avatar
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    #1
    There are 101 coins out of which 50 are fake and are 1 gram lighter than the original ones.One coin is chosen randomly.How can you find whether it is fake or original by a machine that shows the difference in weights of what is placed on the two pans?

  2. MHB Journeyman
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    #2
    Request for clarification:
    - How many weightings are we allowed to perform?
    - When you say the machine shows the difference in weights, does it also give which one is lighter, or just the absolute difference in weight?

  3. MHB Craftsman
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    #3 Thread Author
    Oh,sorry...We are allowed only one weighing and the machine does not show which is heavier or lighter,it just gives the difference....

    Looks like you have something in your mind...

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    #5
    Could you post the answer now Mathmaniac? It's been over a week and I don't think anyone is going to be answering, and I frankly have no idea but I am interested in the solution.

  6. MHB Apprentice
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    #6
    Partial Answer

    Let x be the weight of a lighter coin, then the total weight of all coins is 50x + 51(x + 1) = 101x + 51

    Suppose you pick a light coin, place it on 1 side , place all remaining coins on the other side so now on one side you have 49x + 51(x + 1) and on the other side you have x. The difference is

    49x + 51(x + 1) - x = 48x + 51(x + 1) = 99x + 51

    Suppose you pick a heavier coin, place it on 1 side, place all remaining coins on the other side so now on one side you have 50x + 50(x + 1) and on the other side you have (x + 1). The difference is

    50x + 50(x + 1) - (x + 1) = 50x + 49(x + 1) = 99x + 49


    So... When the machine gives you the difference. subtract 49 then divide by 99. If you get a whole number then you picked the heavier coin, otherwise you picked the lighter coin.

    Note*This method works pretty well only if the coins have whole number weights. Further consideration of what may happen if the coins don't have whole number weights is giving me a headache so i stop here.


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    #7 Thread Author
    Another very useful hint:




    Parity seems to be very simple but it is very useful in many problems like this.

    I wonder why the MHB fails to give the answer.
    Last edited by mathmaniac; April 8th, 2013 at 23:19.

  8. MHB Apprentice
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    #8
    Quote Originally Posted by mathmaniac View Post
    Another very useful hint:




    Parity seems to be very simple but it is very useful in many problems like this.

    I wonder why the MHB fails to give the answer.
    I think i got it using the hint. The weight of the coin is inconsequential, what matters is the +1 difference in weight. The scale will always read integer so if you read even integer you have heavy coin set aside, if you read odd integer you have light coin set aside.




    I wouldn't be surprised if someone gets an easier way.

    Last edited by agentmulder; April 9th, 2013 at 02:20.

  9. MHB Craftsman
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    #9 Thread Author
    ‚ÄčAgent,you made it look more difficult than it really is.

    My solution



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