Bags of cash

I have three bags, each containing 100 coins. I know the coins in two of the bags weigh 10 grams each, but the coins in the third bag weigh 11 grams each.

I have some scales (not a pair of balances) and have one weigh available to me. How do I determine the bag that contains the heavier coins?

Thanks to Lieutenant Columbo for sharing this nice little puzzle.


6 Responses to “Bags of cash”

  1. Sam Lloyd on July 6th, 2008 22:56

    Give up, 42?

  2. Nick Robinson on July 7th, 2008 14:33

    Might have to disagree with your use of “one weigh” – perhaps intended as the singular of “weights” or possibly a typo?

    Also, I’ve got confused by your use of the term scales.

    Isn’t a scale used to DETERMINE the mass of an object (or set of objects), whilst a balance is a pivoting mechanical device used to COMPARE the mass of two sets of objects?

    If you have a scale, couldn’t you just put each bag on it in turn and see which one weighs the most? (Assuming that each bag was at rest as you measure its mass and therefore acceleration – other than gravity, which we can risk assuming is constant – doesn’t play a part).

    Now if you had a balance, you might have to put your weight on one side and add coins from the first bag (call it B1) onto the other side until it balances. You might then remove one coin and replace it with one from another bag (call that B2). If the balance still balances you have identified the two bags containing 10g coins. If the coins go up (i.e. are now lighter than your weight), then B1 contained the 11g coins. If the coins go down (i.e. are now heavier than your weight), then B2 contained the 11g coins.
    Or perhaps there is an interesting sampling technique I’m missing?

  3. Dan on July 7th, 2008 15:31

    “Weigh” is used as a verb. I.e. you can put one thing or set of things on the scale, read the weight and then you need to give your answer as to which bag contains the heavier coins. It is indeed a scale, not a balance. There are no weights apart from the coins themselves.

    During the weigh, you cannot add or remove any objects from the weighing pan.

  4. Kurt Gödel on July 7th, 2008 21:38

    I am just not sure about this.

  5. Dan on July 7th, 2008 22:29

    The way to do this is to take a different number of coins from each of the three bags, and weigh them together.

    Take 1 coin from Bag A
    Take 2 coins from Bag B
    And take 3 coins from Bag C.

    If the total weight is 60.1g, then there is only one heavier coin in your six, so Bag A contains the heavier coins. A weight of 60.2g means Bag B contains the heavier coins. And a weight of 60.3g means Bag C contains the heavier coins.

  6. Paul Erdös on July 8th, 2008 23:12

    Is it six?

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