Problem 15 · 2009 Math Kangaroo
Stretch
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
Every person from the 2nd on calls the person in front a liar, so neighbours must be opposite types.
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Hint 2 of 2
That forces a strict alternation — then check the first person's claim.
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Approach: force alternating types, then fix the start
- Each person (from the 2nd) calls the one in front a liar, so any two neighbours have opposite types: types alternate.
- If person 1 were a noble, his claim 'all behind are liars' would fail (alternation puts nobles behind him).
- So person 1 is a liar; then positions 1,3,5,…,25 are liars and the even positions are nobles.
- That is 13 odd positions, so there are 13 liars.
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