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.

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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