Suppose you’re a jester in a medieval court. You’ve just made a joke at the expense of the king’s new hairstyle, which you thought was hilarious and he did not.

At once, you are taken by the guards into the dungeons to await your fate.

Fortunately for you, the king decides to give you a chance to save your life, as long as you can pass a test.

The king has 80 coins. He puts them in a bag, along with one more coin – a counterfeit. You know that the counterfeit is slightly lighter than the other coins.

He shakes the bag around, and hands it to you, along with a set of balance scales.

His voice booms: “Jester – you must find the counterfeit coin if you are to save yourself. Use the scales at most four times or your fate is sealed.”

How can you identify the counterfeit coin?

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I really like this problem, particularly because I’ve been thinking about generalisations and these have distracted me from boring things like paying bills, etc. for a little bit.

Let’s write f(n) to denote the least number of weighings needed to determine the counterfeit coin in a set of n coins. Then, those who have solved the puzzle above could write f(81) = 4.

What about f(80)? And f(82)? It’s quite lovely that 81 is a power of three. In general, what else can we say about the funcion f(n)?

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