Problem 19 · 2017 Math Kangaroo
Stretch
Counting & Probability
careful-counting
Eight kangaroos stand in a row, facing the directions shown in the picture. Whenever two kangaroos that are next to each other are facing each other, they swap places by hopping past one another. This continues until no more hops are possible. How many times did a swap take place?
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Answer: D — 13
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Hint 1 of 2
A swap happens only where a right-facing kangaroo is directly in front of a left-facing one (they face each other).
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Hint 2 of 2
Each such facing pair eventually passes through every opposing kangaroo — count the total crossings.
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Approach: count the head-on pairs that must pass each other
- Two neighbours swap only when a right-facing kangaroo is immediately in front of a left-facing one, so they meet head-on.
- Over the whole process, each right-facing kangaroo ends up passing every left-facing kangaroo that started to its right — exactly one swap per such pair.
- Reading the picture, the facing pattern gives 13 such right-then-left pairs.
- So 13 swaps happen (D).
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