Problem 15 · 2009 Math Kangaroo
Medium
Logic & Word Problems
work-backwardcasework
On the island of the truth-tellers and the liars, there are 25 people standing in a line. The person at the front claims that everybody standing behind him is a liar. Everybody else claims that the person standing in front of them is a liar. How many liars are standing in the line? (Truth-tellers always tell the truth and liars always lie.)
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Answer: C — 13
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Hint 1 of 2
Suppose the front person tells the truth and check whether the chain stays consistent.
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Hint 2 of 2
Once the front person must be a liar, the labels alternate down the line.
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Approach: test the front person, then alternate
- If person 1 were truthful, everyone behind would be a liar; but then liar person 3 calling liar person 2 a liar would be true - contradiction. So person 1 lies.
- Person 2 truthfully calls person 1 a liar, so person 2 tells the truth; the labels then alternate, putting liars at the odd positions.
- Among 25 people the odd positions 1, 3, ..., 25 give 13 liars.
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