Problem 23 · 2018 Math Kangaroo
Stretch
Logic & Word Problems
Four brothers with the harmonious names A, B, C and D are all of different heights. They make the following claims:
A: I am neither the tallest nor the smallest. B: I am not the smallest. C: I am the tallest. D: I am the smallest.
Exactly one of them lies. Who is the tallest brother?
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Answer: B — B
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Hint 1 of 2
Two of the boys cannot both be telling the truth about being smallest/tallest at once.
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Hint 2 of 2
Suppose each boy in turn is the liar and check for consistency.
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Approach: assume one liar at a time and eliminate contradictions
- If C ('I am tallest') lies, then A (middle), B (not smallest) and D (smallest) are all true and consistent.
- That makes the tallest B.
- Every other choice of liar produces a contradiction (e.g. two boys both smallest, or two truthful tallest/smallest claims).
- So the tallest brother is B.
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