Problem 14 · 2009 Math Kangaroo
Medium
Geometry & Measurement
area-fractionsymmetry
What fraction of the largest square is grey?
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Answer: A — 14
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Hint 1 of 2
Don't measure the curves one by one - look for symmetry that pairs grey bulges with white bites.
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Hint 2 of 2
The circular bulges and bites cancel, leaving a clean fraction of the square.
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Approach: use symmetry to cancel curved pieces
- The figure is built from circles, a tilted square and a central square with full four-fold symmetry.
- By symmetry every curved piece sticking out of the grey is matched by an equal curved bite taken from it, so the grey equals a plain straight-edged region.
- That region is exactly 1/4 of the large square.
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