Problem 23 · 2013 Math Kangaroo
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Logic & Word Problems
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2013 people live on an island; some are truth-tellers (who always tell the truth) and the rest are liars (who always lie). Each day one person says ‘When I have left the island, the number of truth-tellers will equal the number of liars,’ and then leaves. After 2013 days no one is left on the island. How many liars were living there to begin with?
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Answer: B — 1006
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Hint 1 of 3
Remember the rule: whatever a truth-teller says is really true, and whatever a liar says is really false.
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Hint 2 of 3
Look at the very first speaker: after he leaves there are 2012 people, and an even number can split into two equal halves.
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Hint 3 of 3
Decide whether that first speaker must be a truth-teller or a liar, and what that forces about the rest.
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Approach: reason about the first speaker, then balance the counts
- After the first person leaves, 2012 remain. The claim 'truth-tellers equal liars' would mean \(1006 = 1006\), which is possible, so the first speaker can be a truth-teller telling the truth.
- That makes the remaining 1006 truth-tellers and 1006 liars, and the very first speaker an extra truth-teller, giving \(1006 + 1 = 1007\) truth-tellers at the start.
- The rest are liars: \(2013 - 1007 = 1006\) liars to begin with, which is choice B.
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