Problem 10 · 2019 Math Kangaroo
Medium
Logic & Word Problems
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Every day the three kangaroos Alex, Bob and Carl go for a walk. If Alex does not wear a hat, then Bob wears a hat. If Bob does not wear a hat, then Carl wears a hat. Today Carl does not wear a hat. Which kangaroos can we be sure are wearing a hat today?
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Answer: E — only Bob
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Hint 1 of 2
Carl wears no hat. Use the rule 'if Bob has no hat then Carl wears one' in reverse.
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Hint 2 of 2
Since Carl has no hat, Bob must have one; then check whether Alex is forced.
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Approach: contrapositive reasoning from Carl
- Rule: if Bob has no hat, Carl wears one. Carl has no hat, so Bob must have a hat.
- Rule: if Alex has no hat, Bob wears one — already satisfied, so Alex is not forced either way.
- Thus for certain only Bob is wearing a hat.
- Answer (E) only Bob.
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