Problem 28 · 2013 Math Kangaroo
Stretch
Logic & Word Problems
casework
On an island live only Truthtellers (who always tell the truth) and Liars (who never tell the truth). I met two inhabitants and asked the taller one whether they were both Truthtellers. From his answer I could not tell which group they belonged to. So I asked the shorter one whether the taller one is a Truthteller. After his answer, I knew the type of both. Which statement is correct?
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Answer: D — The taller one was a Liar and the shorter one a Truthteller.
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Hint 1 of 2
Decode the first answer: when is 'are you both truthtellers?' ambiguous to the asker?
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Hint 2 of 2
Then see which second answer (about the taller) lets you pin both types down.
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Approach: rule out by what stays ambiguous
- A 'no' to the first question would reveal taller=truthteller, smaller=liar, so the answer was an ambiguous 'yes'.
- Asking the smaller about the taller resolves it only if that answer was 'no'.
- 'No' means the taller is a Liar and the smaller a Truthteller: D.
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