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The Gold Coins

In front of you are ten piles of ten gold coins each. One pile consists of under weight coins, although they look identical to the naked eye. A genuine coin weighs 10g whereas a fake coin weighs only 9.9g. You are allowed one use of an exact weighing machine, but it can only measure up to 500g. How do you determine which is the pile of fake coins?

What would 5 genuine coins weigh? What would five fake coins weigh?

For each fake coin in the collection you weigh, the scale will read 0.1g less than if all the coins were real. Therefore weighing 1 coin from pile one and 2 from pile two will tell you which pile, if either, is the fake one.

Number the piles from one to ten. Take 1 coin from pile one, 2 from pile two and so on until you take 9 coins from pile nine. Weigh this collection of 45 coins. (Remember, that weighing more than 50 coins will exceed the machine's limit of 500g.)

Weighing 45 genuine coins would give a weight of 450g. Therefore, if we do get a weight of 450g we know that all the coins in the collection described in the answer above are genuine and that the fake pile is the tenth one (from which we took no coins).

However, if the tenth pile is not the fake one, our collection will contain some fake coins and will therefore weigh less than 450g. How much less will depend on the number of fake coins and we can use the difference of the real weight from the ideal figure of 450g to tell us which pile is the fake one as we know that each fake coin is 0.1g less than a genuine one.

If, for example, the total weight was only 449.7g we would know that 3 fake coins were in the pile, and thus that the fake pile was pile 3 (due to our planned selection of coins). 449.8g would imply 2 fake coins and pile 2 and so on.

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