Problem 26 · 2009 Math Kangaroo
Stretch
Number Theory
perfect-squarecasework
A square is cut into 2009 smaller squares. The side length of each smaller square is a whole number. What is the minimum possible side length of the original square?
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Answer: B — 45
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Hint 1 of 2
A side-n square split into unit squares gives n^2 pieces; merging blocks lowers the count.
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Hint 2 of 2
You need n^2 at least 2009 and must hit exactly 2009 by merging.
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Approach: bound n then adjust the piece count
- Side 44 allows at most 44^2 = 1936 < 2009 pieces, so 44 is too small.
- Side 45 starts at 2025 unit squares; replacing a 3x3 block by one square removes 8 pieces, and doing it twice removes 16 to reach exactly 2009.
- So the minimum side length is 45.
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