Problem:
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
Answer Choices:
A. Aaron
B. Darren
C. Karen
D. Maren
E. Sharon
Solution:
We know that they must be arranged as such:
with each "__" representing people whose location we don't yet know. Note that Maren must be there since he sat in the last car. Since we know Sharon sat in front of Aaron, we can take cases of the possible locations of them both.
Case 1: . Since Darren sits in front of Aaron, we can't find a space for him, so this case can't work.
Case 2: . Since Darren sits in front of Aaron, we know the configuration must be Darren, Sharon, Aaron, __, Maren — leaving one spot for Karen. The final configuration is: .
Case 3: . This case would involve Darren and Karen being next to each other, which contradicts the condition that at least one person is between them.
Since the only valid configuration is Darren, Sharon, Aaron, Karen, Maren, we know Aaron is in the middle.
Thus, the correct answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions