Problem 24 · 2014 Math Kangaroo
Stretch
Algebra & Patterns
arithmetic-seriessum-constraint
Grandma gives 180 marbles to her ten grandchildren. No two children get the same number of marbles. Anna gets the most. What is the smallest number of marbles Anna could get?
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Answer: E — 23
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Hint 1 of 2
To make Anna's share as small as possible, the other nine children should get as much as they can while still all being different and less than Anna's amount.
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Hint 2 of 2
If Anna gets A, the most the group can total is A + (A−1) + ... + (A−9); make that at least 180.
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Approach: make the other nine as large as possible just below Anna, then see how small Anna can be
- To let Anna take as little as possible, the other nine children should grab as much as they can while still all being different and below Anna.
- So they take the nine amounts right below Anna's: if Anna has 23, the rest are 22, 21, 20, ..., 14.
- Those ten amounts 14 + 15 + ... + 23 add up to exactly 180, the whole bag, so Anna = 23 works.
- If Anna had only 22, even the biggest allowed amounts (22, 21, ..., 13) total just 175, too few to reach 180.
- So the smallest Anna could get is 23.
The same idea with a quick formula
The largest ten distinct amounts ending at Anna's value A total 10A − (1+2+⋯+9) = 10A − 45, and this must reach 180, so 10A ≥ 225 and A ≥ 23.
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