Thirteen suspects, one reference coin, three weighings

Question

You have thirteen coins numbered 1 to 13. Exactly one is counterfeit, and it may be heavier or lighter than a genuine coin; you do not know which. You also have a fourteenth coin, G, known to be genuine. Using a two-pan balance three times, identify the fake coin and whether it is heavy or light. Without the reference coin, three weighings are known to handle only twelve suspects. Show how the genuine coin lets you handle a thirteenth.

Accepted Answer

Weighing 1: coins 1, 2, 3, 4 and G on the left against coins 5, 6, 7, 8, 9 on the right. Branch A: balanced. Coins 1 to 9 are all genuine. The fake is among 10, 11, 12, 13, heavy or light: 8 states, two weighings left (9 outcomes). - Weighing 2: 10 and 11 on the left against 12 and G on the right. - Balanced: 13 is the fake. Weighing 3: 13 against G tells heavy or light. - Left heavy: 10 heavy, 11 heavy, or 12 light. Weighing 3: 10 against 11. If 10 drops, 10 is heavy; if 11 drops, 11 is heavy; if balanced, 12 is light. - Left light: 10 light, 11 light, or 12 heavy. Weighing 3: 10 against 11.…

Thirteen suspects, one reference coin, three weighings

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