Problem 14 · 2009 Math Kangaroo
Hard
Logic & Word Problems
casework
On the island of nobles and liars, 25 people are standing in a queue. The first person in the line claims that everybody behind him is a liar. Each of the other people claims that the person in front of him is a liar. How many liars are actually in the queue? (Nobles always tell the truth and liars always lie.)
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Answer: C — 13
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Hint 1 of 2
Each person’s claim is about the one right in front, which forces neighbours to be of opposite type.
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Hint 2 of 2
Decide the first person’s type by testing his claim about everyone behind him.
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Approach: force an alternating pattern, then fix the start
- A claim ‘the person in front is a liar’ makes each pair of neighbours opposite types, so the line strictly alternates.
- If the first were a noble, all 24 behind would be liars—impossible under alternation—so the first is a liar.
- Then liars sit in the 13 odd positions: 13 liars.
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