Problem 24 · 2010 Math Kangaroo
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Logic & Word Problems
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Six-legged, seven-legged and eight-legged octopuses serve Neptune, king of the sea. The seven-legged ones always lie, while the six-legged and eight-legged ones always tell the truth. One day four octopuses meet. The blue one says: “Together we have 28 legs.” The green one says: “Together we have 27 legs.” The yellow one says: “Together we have 26 legs.” The red one says: “Together we have 25 legs.” Which colour octopus is telling the truth?
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Answer: C — green
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Hint 1 of 2
Truth-tellers all know the real total, so if there were two of them they would say the same number.
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Hint 2 of 2
Since all four said different numbers, only one can be telling the truth, so the other three are 7-legged liars.
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Approach: find the total that makes exactly one honest
- Only one octopus can be honest (the others would otherwise repeat the same total), so the other three are 7-legged liars with \(3 \times 7 = 21\) legs.
- The honest one has 6 or 8 legs; \(21 + 6 = 27\) matches a claim, while \(21 + 8 = 29\) matches nobody, so the real total is 27.
- The octopus that truthfully says 27 is the green one — the answer is C.
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