Problem:
Initially an urn contains black marbles and white marbles. Repeatedly, three marbles are removed from the urn and replaced from a pile outside the urn as follows:
Which of the following sets of marbles could be the contents of the urn after repeated applications of this procedure?
Answer Choices:
A. black marbles
B. white marbles
C. black marble
D. black and white marble
E. white marble
Solution:
Since the number of white marbles is either unchanged or decreases by after each replacement, the number of white marbles remains even. Since every set removed that includes at least one white marble is replaced by a set containing at least one white marble, the number of white marbles can never be zero. Note that is the only choice including at least two white marbles. We can attain this result in many ways. One way is to remove white marbles times to arrive at black and white marbles, and then remove black and white marbles times.