A more challenging version of the Counterfeit Coin puzzle from 2011:
You have 12 coins, one of which has been replaced with a counterfeit. The false coin differs in weight from the true ones, but you don’t know whether it’s heavier or lighter. How can find it using three weighings in a pan balance?
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Put four coins in each pan. If they balance, then all these coins are good. Weigh three of them against three of the four remaining unknown coins. If they balance then the remaining untested coin is bad; if not then we’ve identified three suspected-bad coins and we’ve discovered whether the counterfeit coin is heavy or light. Weighing two of the three suspected coins against one another will reveal the bad one.
If the first weighing does not balance, then the four untested coins are good and we have four suspected-heavy coins and four suspected-light ones. Call these three groups G, H, and L. In the second weighing, put HHHL in the left pan and HGGG in the right:
- If the left pan sinks, then one of its suspected-heavy coins is a heavy counterfeit, and we can find it by trying to balance two of the three, as above.
- If the right pan sinks, then the culprit is either the suspected-light coin in the left pan or the suspected-heavy one in the right pan. To decide between them, weigh one of these against a known-good coin in the third weighing.
- If the two pans balance, then the bad coin is one of the three suspected-light coins that are presently off the scale. Weighing two of these against one another will identify the bad coin, as above.
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