Four suspects, one genuine coin, two weighings

Question

You have four coins, A, B, C and D. Exactly one of them is counterfeit, and the counterfeit could be either heavier or lighter than a real coin. You do not know which. You also have a fifth coin, G, that you know for certain is genuine. Using a two-pan balance, find the fake coin and determine whether it is heavy or light in only two weighings.

Accepted Answer

Two weighings suffice. Here is a procedure. Weighing 1: A + B against C + G. Case 1: balanced. Every coin on the scale is genuine, so the fake is D. Weighing 2: put D against G. If D's pan drops, D is heavy; if it rises, D is light. Case 2: the left pan (A + B) drops. The pans contain A, B, C and the genuine G. The only ways the left side can be heavier are: A is heavy, or B is heavy, or C is light. Three cases remain, and one weighing has three outcomes. Weighing 2: put A against B. If A drops, A is the heavy fake. If B drops, B is the heavy fake. If they balance, neither is fake, so C is th…

Four suspects, one genuine coin, two weighings

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