Problem 23 · 2022 Math Kangaroo
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Logic & Word Problems
casework
30 people are sitting around a round table. Some of them are wearing a hat. People without a hat must tell the truth; people with a hat may either tell the truth or lie. Each person claims: “At least one of my two neighbours is wearing a hat.” What is the largest number of people who can be without a hat?
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Answer: D — 20
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Hint 1 of 2
A truth-teller (no hat) says truly that a neighbour wears a hat, so a no-hat person cannot sit between two no-hat people.
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Hint 2 of 2
That means no three no-hat people in a row - fit as many no-hat people as that allows around 30 seats.
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Approach: forbid three hatless people in a row, then pack the most
- A hatless person always tells the truth, so their claim 'a neighbour wears a hat' must be true; thus no hatless person sits between two hatless people.
- So at most two hatless people can sit consecutively, in a pattern like (no-hat, no-hat, hat) repeated.
- Around 30 seats that gives 2 out of every 3, i.e. 20 hatless people.
- So the answer is D.
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