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Counterfeit Coin Revisited

Okay, as some noticed, we're really recycling an older puzzle in a harder version, but the solution was already available on the site. But hopefully now that the 12 coin version is a puzzle in its own right, we won't get as much email asking us to solve it. There have been several wordings of the solution, but the clearest written was written for our old FAQ (now obsolete) by "James", a Washington University student:

First, weigh 4 coins againt 4 coins. If the balance tips, we have 4 coins we know are correct, 4 coins which might be too heavy but cannot be too light, and 4 coins which might be too light but cannot be too heavy.

Second, (assuming we tipped the balance on the first weighing) place 2 possibly light coins and 1 possibly heavy coin on each side. If the balance tips, we have 9 known correct coins, 2 possibly light coins, and 1 possibly heavy coin.

Third, weigh the two possibly light coins againt each other. If the balance tips, the lighter coin is counterfeit and too light. Otherwise the possibly heavy coin is counterfeit and is too heavy.

If the balance in the second weighing does not tip, we have 10 known correct coins and 2 possibly heavy coins. Weigh those two coins against each other, the heavier coin is counterfeit and too heavy.

If the balance in the first weighing does not tip, we have 8 known correct coins and 4 possibly heavy or light coins. For the second weighing, weigh 3 of the possibly bad coins againt 3 known correct coins. If the balance does not tip, we have 11 known correct coins and one possibly light or heavy coin.

Third, weigh the counterfeit coin against a correct coin to determine if the counterfeit is too heavy or light.

If the balance in this second weighing tips, we have 9 known correct coins and either 3 possibly heavy coins or 3 possibly light coins. For the third weighing, weigh one of the possibly counterfeit coins against another, and from that we know which coin is counterfeit. (From the last weighing we already knew whether it would be too heavy or light.)

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