Problem 17 · 2016 Math Kangaroo
Hard
Logic & Word Problems
casework
On the island of knights and liars everybody is either a knight (who only tells the truth) or a liar (who always lies). On your journey on the island you meet 7 people who are sitting in a circle around a bonfire. They all tell you “I am sitting between two liars!” How many liars are sitting around the bonfire?
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Answer: B — 4
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Hint 1 of 2
A knight's claim is true, so both its neighbours are liars; a liar's claim is false, so it has at least one knight neighbour.
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Hint 2 of 2
Place knights with no two adjacent around the circle of 7 and check consistency.
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Approach: truth/lie constraints around the circle
- A knight truly sits between two liars, so knights are never adjacent.
- A liar falsely claims this, so each liar has at least one knight neighbour.
- Seating knights at three alternating seats among the 7 satisfies everything, leaving 4 liars.
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