Example problem 1

This problem requires only brainpower to solve, no higher math. We consider this to be a hard problem, worth 25 points.

The case of the debased currency

Long ago, coins were made by artisans out of gold supplied by the king or queen. The weight of coins returned by the artisan had better be the same as that of the gold or the royal displeasure would land on the artisan like a a ton of bricks. Some clever, dishonest artisans would substitute a like weight of lead for a small part of the gold, make coins of lead and gold, and then keep the extra gold. These part-lead coins were the same size as other coins, but slightly lighter (lead is less dense than gold). These coins are said to be debased.


(thanks to wikimedia commons)

A merchant has nine gold coins, one of which is debased. He also has a balance, like the one shown above, with two pans. The merchant challenges his customers to find the debased coin by using the smallest possible number of weighings with the balance. While the balance can be used to find the weight of objects, the merchant demands that his customers only use the balance to compare coins or groups of coins.

Question: what is the smallest number of weighings that is certain to find the debased coin? The answer should be in the form of directions for using the balance to find the debased coin. You should also explain why no smaller number of weighings will do.

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