Problem 16 · 2023 Math Kangaroo
Medium
Logic & Word Problems
Counting & Probability
careful-countingcasework
Some kangaroos and three beavers are standing in a circle. No beaver stands directly next to another beaver. There are exactly three kangaroos that are standing next to another kangaroo. What is the biggest possible number of kangaroos in the circle?
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Answer: B — 5
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Hint 1 of 3
The 3 beavers split the circle into 3 blocks of kangaroos, and none of those blocks can be empty since no two beavers may touch.
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Hint 2 of 3
A kangaroo has a kangaroo neighbour exactly when it sits in a block of 2 or more; count how many such kangaroos a block of each size produces.
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Hint 3 of 3
You are allowed only 3 kangaroos-with-a-kangaroo-neighbour total, so spend that budget on one block and keep the rest as singles.
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Approach: beavers cut the circle into blocks; count kangaroos that touch a kangaroo
- The 3 beavers (no two adjacent) cut the circle into 3 non-empty blocks of kangaroos.
- In a block of size 1 the lone kangaroo touches only beavers, but every kangaroo in a block of size 2 or more touches another kangaroo, so a block of size k≥2 uses up k of the allowed 3.
- To keep the count at exactly 3, make one block of size 3 and the other two blocks size 1: arrangement B·KKK·B·K·B·K uses 3 + 1 + 1 = 5 kangaroos, and only the three in the KKK block touch a kangaroo.
- Any sixth kangaroo would enlarge another block to size 2+, pushing the touching-count above 3, so the maximum is 5 (B).
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