Problem 19 · 2023 Math Kangaroo
Stretch
Number Theory
divisibilitycareful-counting
A rabbit, a beaver and a kangaroo have a race. They all start at the same time from “Start” and hop in the same direction around the loop. With each jump the beaver moves forward 1 position, the rabbit moves forward 2 positions, and the kangaroo moves forward 3 positions. Whoever needs the fewest jumps to land exactly on the position marked “Ziel” wins. Who wins the race?
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Answer: E — Kangaroo and beaver
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Hint 1 of 2
Ziel sits exactly opposite Start, so its distance from Start is half the number of positions around the loop.
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Hint 2 of 2
Check whether each animal's jump size can ever add up exactly to that distance, and compare how many jumps it takes.
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Approach: compare each step size against the half-way distance to Ziel
- Ziel is directly opposite Start, so reaching it means covering exactly half the loop, an odd number of positions.
- The rabbit moves 2 at a time, so its total is always even and it can never land exactly on the odd-distance Ziel.
- The beaver (step 1) and the kangaroo (step 3) both reach Ziel, and they need the same number of jumps to first land there.
- Tied for the fewest jumps are the kangaroo and the beaver, answer E.
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