One question is what should the mode be when multiple values occur the same number of times.

Like for example if the sample is: 7,7,8,9,9

Is the mode 7, 9, undefined, or both 7 and 9?

It seems to me that the way most statisticians define the mode is that it is not unique if multiple values occur the same number of times, but that it is still defined.

http://mathworld.wolfram.com/Mode.html
http://en.wikipedia.org/wiki/Mode_(statistics)

In Excel if the function Mode is applied to the list 7,7,8,9 the result will be 7.

In Mathematica if the function Mode is applied to the list 7,7,8,9 the result will be the set {7,9}.

Taking that as the definition consider three different possibilities:

8, 8 Mean, Median, and Mode are all 8

7, 7, 8, 9, 9 Mean=8, Median=8, Mode={7,9}

1, 1, 8, 15, 15 Mean=8, Median=8, Mode={1,15}

a. The two modes are also equal to 8.

>No. The mode doesn't have to be 8.

b. The two modes are equal, but their value is not 8.

>No. The modes do not have to be equal.

c. One mode is 8, and the other is not.

>No. It is possible for neither of the modes to be 8.

d. The two modes are different from one another, and none of them is 8.

>No. It is possible that one of the modes is 8.

e. There is not enough information to answer the question.

>Correct. There is not enough information to answer the question.

Another question could be: If the mean is equal to the median is it possible to have a unique mode that is not also equal to the mean and the median?

I don't believe it is possible to have a situation like this.

David