Problem 14 · 2016 Math Kangaroo
Medium
Number Theory
place-valuecareful-counting
Hansi writes the number 2581953764 on a strip of paper. Twice he cuts through the strip between two digits, getting three numbers which he adds. What is the smallest sum he can obtain in this way?
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Answer: B — 2975
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Hint 1 of 3
A long piece is worth a lot (thousands or more), so very long pieces make the sum big.
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Hint 2 of 3
The ten digits split into three pieces whose lengths add to 10, so keep every piece short — at most four digits.
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Hint 3 of 3
Among the short splits, choose the one whose biggest piece has the smallest leading digits.
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Approach: keep the pieces short, then shrink the leading digits
- Two cuts make three pieces, and a piece with 5 or more digits already passes every answer, so each piece should have at most 4 digits.
- That means the lengths are 3, 4, 3 in some order; the 4-digit piece dominates the sum, so we want it to start with the smallest digits.
- Cutting as \(258 + 1953 + 764\) makes the 4-digit piece start with 1, and the total is \(258 + 1953 + 764 = 2975\).
- No split beats this, so the smallest sum is 2975, choice (B).
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