Three coins, one fake, is one weighing ever enough?

Question

You have three coins. One of them is fake, and the fake is either heavier or lighter than the other two, but you are not told which. You have a two-pan balance and nothing else. Two questions: 1. Can a single weighing, however cleverly chosen, always identify the fake coin and tell you whether it is heavy or light? 2. What is the smallest number of weighings that always works?

Accepted Answer

One weighing is never enough; two always are. Why one fails. There are three candidate coins and two possible directions, so 6 states of the world: A heavy, A light, B heavy, B light, C heavy, C light. A single weighing has only 3 possible outcomes. Three signals cannot label six different states, so whatever you put on the pans, at least two states produce the same result and you cannot tell them apart. This is true even if you only ask which coin is fake without the direction: try A against B. If the pans balance you know C is fake, but if A drops you cannot tell "A heavy" from "B light", a…

Three coins, one fake, is one weighing ever enough?

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