Problem 13 · 2020 Math Kangaroo
Hard
Logic & Word Problems
caseworksum-constraint
On a distant island, 2020 kangaroos hold hands in a large circle. Each kangaroo is either brown (and always tells the truth) or grey (and always lies). Every one of them says, “One of my neighbours is brown and the other is grey.” How many of the kangaroos are brown?
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Answer: A — 0
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Hint 1 of 2
Suppose a brown (truthful) kangaroo exists: its statement about its neighbours would have to hold.
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Hint 2 of 2
Test whether any mix of brown and grey can sit in a circle when all say the same sentence - it collapses to one case.
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Approach: check the statement's consistency around the circle
- A brown kangaroo tells the truth, so its two neighbours would be one brown and one grey.
- Following that around the circle leads to a contradiction, so no truthful (brown) kangaroo can exist.
- Every kangaroo is therefore grey and lying - consistent, since the statement is then false for each.
- The number of brown kangaroos is 0.
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