The Fake Coin

[From an unknown source. Wording by Sam Rebelsky]

Melvin and Myra Miser have just received nine coins. They know that one is a fake and that it has a different weight than the other coins. A local scientist is willing to sell them time on his balance (which allows them to compare the weights of two groups of coins), but they must reserve and pay for the time in advance.

Given their miserly outlook, they want to reserve the shortest time they can. What is the minimum number of comparisons they have to do to guarantee that they've found the fake coin? How can they determine which coin is fake in that number of weighings?

How does your answer change if they have twelve coins? How does your answer change if they know that the fake coin is heavier than the other coins? Lighter?

I've found ways to do nine and twelve coins in three weighings and nine coins with a known difference in two weighings. Can you match my numbers? Can you beat them?

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