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On the table, there is a balance with two pans. The display on the balance tells the difference between the weight in the left ban and the weight in the right pan (measured in gram).

There are also $2016$ coins on the table. Cosmo tells Fredo: "There are exactly $99$ fake coins among these $2016$ coins. All genuine coins have the same weight. Some fake coins weigh one gram less than the genuine coins, and the other fake coins all weigh one gram more than the genuine coins."

Cosmo points at the leftmost coin $x$ and asks Fredo to determine whether $x$ is fake.

Question: Can Fredo decide whether coin $x$ is fake by using the balance at most twice?

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up vote 13 down vote accepted

Fredo should weigh the single coin against nothing, then all 2016 coins together against nothing. If the coin is real, then 2016 times its weight will be within 99 grams of the total weight. Otherwise, it will be at least 1917 grams off.

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The answer is:

Yes, he can.

Determined as follows:

Fredo puts all 2016 coins on the left pan and notes the "difference". This is the total weight of all coins, real and fake.

Next:

He puts coin x and the left pan and notes the difference. If this number times 2016 is within 99 grams of the original number then x is genuine. Otherwise it is a fake. A fake coin's weight times 2016 will be off by almost 2,000.

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