Problem 19 · 2013 AMC 8
Medium
Logic & Word Problems
logic-puzzle
Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest score?
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Answer: D — Cassie, Hannah, Bridget.
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Hint 1 of 2
The hidden fact is that Cassie and Bridget have seen exactly one score besides their own: Hannah's. For either to be certain of her claim, the comparison with Hannah must settle it — so each statement secretly compares that girl to Hannah.
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Hint 2 of 2
In these "what can they deduce" puzzles, ask what information each speaker actually has. A confident claim is only justified if their limited view forces it — that turns each sentence into one inequality.
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Approach: convert each confident statement into a comparison with Hannah
- Both girls saw only Hannah's score. Cassie says "I'm not lowest" with certainty — she can only be sure if she beat the one score she saw, so Cassie > Hannah.
- Bridget says "I'm not highest" with certainty — she can only be sure if she scored below Hannah, so Hannah > Bridget.
- Chain them: Cassie > Hannah > Bridget, i.e. Cassie, Hannah, Bridget.
- Why this transfers: the trick is always "a certain statement reveals what the speaker can see" — the knowledge each person has is the real data, not just the words.
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