Problem 23 · 2023 Math Kangaroo
Stretch
Algebra & Patterns
Starting with the four numbers 2, 0, 2, 3 the kangaroo-machine creates numbers according to the following rule: the next number is always the smallest non-negative integer that is different from the four directly previous numbers. Which number is in position 2023?
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Answer: C — 2
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Hint 1 of 2
Generate a few terms: the next number is the smallest non-negative integer not among the last four.
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Hint 2 of 2
After a short start the sequence becomes periodic — find the period and reduce 2023.
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Approach: find the eventual period
- Starting 2, 0, 2, 3 the rule gives 1, 4, 0, 2, 3, 1, 4, 0, 2, 3, …
- From the 5th term on it cycles with period 5: 1, 4, 0, 2, 3.
- Position 2023 is (2023 − 5) = 2018 steps past the cycle start; 2018 mod 5 = 3.
- That lands on the 4th cycle value, which is 2.
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