A Russian coin-weighing puzzle:
You have 101 coins, and you know that 50 of them are counterfeit. Every true coin has the same weight, an unknown integer, and every false coin has the same weight, which differs from that of a true coin by 1 gram. You also have a two-pan pointer scale that will show you the difference in weight between the contents of each pan. You choose one coin. Can you tell in a single weighing whether it’s true or false?
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Yes. Put the coin in one pan, and put all 100 remaining coins in the other. Now suppose a true coin weighs t grams and a fake coin weighs f grams. If the coin is fake, then the difference between the weights will be
51t + 49f – f = 51t + 48f = 51t + 48(t ± 1) = 99t ± 48,
which is always divisible by 3. If the coin is true, then the difference will be
50t + 50f – t = 49t + 50f = 49t + 50(t ± 1) = 99t ± 50,
which is not divisible by 3.
Another solution is to put aside the chosen coin and divide the remaining 100 coins into two piles. If the difference in weight between these piles is even, then the chosen coin is genuine; if it’s odd then it’s false.
(From Sergei Aleksandrovich Genkin, Dmitrii Vladimirovich Fomin, and Ilia Itenberg, Mathematical Circles: (Russian Experience), 1996.)
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