Problem 23 · 2002 AMC 8
Stretch
Geometry & Measurement
area-fractionsymmetry
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Answer: B — 4/9.
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Hint 1 of 2
You'd never count the whole floor — and you don't have to. A pattern that repeats has a single *tile of repetition*; find the smallest block that, stamped over and over, rebuilds the floor.
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Hint 2 of 2
The whole floor's dark fraction equals the dark fraction inside that *one* block. Inside it, count dark area as whole squares plus triangles (two half-square triangles glue into one square).
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Approach: find the repeating block, then the dark fraction inside it
- Since the design tiles the plane, the dark fraction of the *whole* floor is the same as the dark fraction of one repeating block. The block here is 3 × 3 = 9 unit squares.
- Dark area in that block: 3 full squares, plus two corner triangles that together make a 4th square — 4 dark out of 9.
- Fraction dark = 4/9.
- *Why this transfers:* for any repeating pattern, shrink the question to its fundamental tile. One block's ratio *is* the whole floor's ratio — and edge bits that look like halves often pair up into whole units.
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