🇺🇸 AMC 8 ⇄ switch contest
2004 AMC 8

Problem 13

Problem 13 · 2004 AMC 8 Medium
Logic & Word Problems exactly-one-true

Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true. I. Bill is the oldest. II. Amy is not the oldest. III. Celine is not the youngest. Rank the friends from the oldest to the youngest.

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Answer: E — Amy, Celine, Bill.
Show hints
Hint 1 of 3
The hidden lever is 'exactly one is true', which means two are false. Hunt for statements that drag each other along: if assuming one true forces a second to be true, that pair can't be the single true one — so that assumption is dead.
Still stuck? Show hint 2 →
Hint 2 of 3
This is the exactly-one-true elimination: test each statement as 'the true one' and reject any that would make a second statement true as well. Watch for one claim logically implying another — that's what cracks it.
Still stuck? Show hint 3 →
Hint 3 of 3
Once a statement is known false, flip it to its opposite — a false 'Amy is not oldest' means Amy is oldest. The two falsehoods, read backwards, pin the whole order.
Show solution
Approach: test each statement as the lone truth
  1. Try I (Bill oldest) as the true one: but Bill being oldest also makes II (Amy not oldest) true. Two trues — forbidden ⇒ I is false.
  2. Try II (Amy not oldest) as the true one: with Bill already not oldest, the oldest would have to be Celine — but that makes III (Celine not youngest) true as well. Two trues again ⇒ II is false.
  3. By elimination III is the lone truth, so I and II are false. Negate them: I false ⇒ Bill is not oldest; II false ⇒ Amy is the oldest. III true ⇒ Celine is not the youngest, so Celine is in the middle and Bill is youngest.
  4. Order oldest→youngest: Amy, Celine, Bill.
  5. The reusable idea: in 'exactly one true' puzzles, look for statements that force another to be true — those can never be the unique truth, and ruling them out usually leaves a single survivor.
Another way — check the answer choices directly:
  1. For each ordering, count how many of I, II, III come out true; keep the one with exactly one true.
  2. Amy, Celine, Bill: I (Bill oldest)? No. II (Amy not oldest)? No. III (Celine not youngest)? Yes — exactly one true.
  3. Every other listed order makes zero or two statements true, so the answer is Amy, Celine, Bill.
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