9. The Counterfeit Coins

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IN RECENT YEARS a number of clever coin-weighing or ball-weighing problems
have aroused widespread interest. Here is a new and charmingly simple
variation. You have 10 stacks of coins, each consisting of 10 half-dollars. One
entire stack is counterfeit, but you do not know which one. You do know the
weight of a genuine half-dollar and you are also told that each counterfeit coin
weighs one gram more than it should. You may weigh the coins on a pointer
scale. What is the smallest number of weighings necessary to determine which
stack is counterfeit?

The answer

The counterfeit stack can be identified by a single weighing of coins. You take
one coin from the first stack, two from the second, three from the third and so on
to the entire 10 coins of the tenth stack. You then weigh the whole sample
collection on the pointer scale. The excess weight of this collection, in number
of grams, corresponds to the number of the counterfeit stack. For example, if the
group of coins weighs seven grams more than it should, then the counterfeit
stack must be the seventh one, from which you took seven coins (each weighing
one gram more than a genuine half-dollar). Even if there had been an eleventh
stack of ten coins, the procedure just described would still work, for no excess
weight would indicate that the one remaining stack was counterfeit.
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