Problem 25 · 2013 Math Kangaroo
Stretch
Counting & Probability
Logic & Word Problems
caseworkcareful-counting
A gardener wants to plant a row of 20 trees (lindens and oaks) in a park. There must never be exactly three trees between any two oak trees. What is the greatest number of the 20 trees that could be oaks?
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Answer: C — 12
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Hint 1 of 2
‘Exactly three trees between two oaks’ means two oaks four positions apart — that is forbidden.
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Hint 2 of 2
Split the 20 spots into groups that are 4 apart and pick as many as possible from each.
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Approach: forbid distance-4 pairs, maximise selection
- Two oaks may not sit 4 positions apart (that leaves exactly three trees between).
- Group positions by remainder mod 4: each group is a chain like 1,5,9,13,17 of length 5.
- In a chain of 5 where neighbours are forbidden, at most 3 can be oaks.
- Four chains × 3 = 12 oaks maximum.
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