Problem:
Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true.
Bill is the oldest.
Amy is not the oldest.
Celine is not the youngest.
Rank the friends from the oldest to the youngest.
Answer Choices:
A. Bill, Amy, Celine
B. Amy, Bill, Celine
C. Celine, Amy, Bill
D. Celine, Bill, Amy
E. Amy, Celine, Bill
Solution:
Bill is not the oldest, because if he were, the first two statements would be true. Celine is not the oldest, because if she were, the last two statements would be true. Therefore, Amy is the oldest. So the first two statements are false. The last statement must be true. This means that Celine is not the youngest, so Bill is the youngest. The correct order from oldest to youngest is Amy, Celine, Bill.
The possible cases for the three statements are
Case leads to a contradiction of statements and .
Case means Celine is youngest and neither Bill nor Amy is oldest, but one must be oldest.
Only case is possible. The correct order from oldest to youngest is Amy, Celine, Bill.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions