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You have just been hired as the quality-control engineer for a company that makes coins. The coins must have identical weight.

You have just been hired as the quality-control engineer for a company
that makes coins. The coins must have identical weight. You are given a set of
n coins, and are told that at most one (possibly none) of the n coins is either too
heavy or too light (but you do not know which). Your task is to develop an ecient
test procedure to determine which of the n coins is defective, or report that none is
defective. To do this test you have a scale. For each measurement you place some
of the coins on the left side of the scale and some of the coins on the right side.
The scale indicates either (1) the left side is heavier, (2) the right side is heavier,
or (3) both subsets have the same weight. It does not indicate how much heavier
or lighter.
(a) Prove that in the worst-case the minimum number of measurements using the
scale is at least log3(1 + 2n). (Hint: Use a decision tree argument.)
(b) Present a method to determine the defective coin using at most (log3(1 + 2n)+ c) scale measurements, where c is a constant (independent of n). Try to make
c as small as possible. Explain your algorithm's correctness. (If you cannot
succeed in this, then try to get at least c log3(1 + 2n), for a small constant
c.)
To simplify your arguments, you may assume, for example, that 1 + 2n is a power
of 3 or that n is a power of 3.

This question was asked on Mar 16, 2010.

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