Problem 23 · 2017 Math Kangaroo
Stretch
Number Theory
careful-counting
Diana adds either 2 or 5 to every whole number from 1 to 9. She wants to achieve as few different sums as possible. What is the minimum number of different values she obtains?
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Answer: B — 6
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Hint 1 of 2
Each number 1–9 becomes either n+2 or n+5; you choose to make sums coincide.
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Hint 2 of 2
The reachable sums run from 3 to 14 — choose so as few distinct values appear as possible.
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Approach: overlap the +2 and +5 results
- Adding 2 gives values 3..11; adding 5 gives 6..14; since n+5 = (n+3)+2, results three apart can be merged.
- Choosing cleverly, the distinct sums collapse to just 6, 7, 8, 9, 10, 11.
- The minimum number of different sums is 6.
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