Problem 26 · 2021 Math Kangaroo
Stretch
Number Theory
Logic & Word Problems
divisibilitycasework
2021 coloured kangaroos are arranged in a row and are numbered from 1 to 2021. Each kangaroo is coloured either red, grey or blue. Amongst any three consecutive kangaroos, there are always kangaroos of all three colours. Bruce guesses the colours of five kangaroos. These are his guesses: Kangaroo 2 is grey; Kangaroo 20 is blue; Kangaroo 202 is red; Kangaroo 1002 is blue; Kangaroo 2021 is grey. Only one of his guesses is wrong. What is the number of the kangaroo whose colour he guessed incorrectly?
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Answer: B — 20
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Hint 1 of 2
'Any three in a row use all three colours' forces the colouring to repeat with period 3.
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Hint 2 of 2
So a kangaroo's colour depends only on its position modulo 3; compare the guesses at equal residues.
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Approach: use the period-3 structure
- With every three consecutive kangaroos all different, the colour pattern repeats every 3 positions.
- Positions 2, 20 and 2021 are all 2 (mod 3), so they must share one colour.
- Guesses say k2 = grey, k20 = blue, k2021 = grey; the lone disagreement (k20) must be the wrong one.
- So the answer is B.
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