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
partlead 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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