Problem 3 · 2011 AMC 8
Easy
Geometry & Measurement
border-count
Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?
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Answer: D — 32 : 17.
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Hint 1 of 2
The border only adds black tiles — the 17 white tiles never change. So the only thing to find is how many black tiles the new ring adds.
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Hint 2 of 2
Find the new total by squaring: a square of 25 tiles is 5 × 5, and wrapping one ring around it makes it 7 × 7. New tiles = big square − old square.
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Approach: the white count is frozen; only count the new black ring
- The original 25 tiles form a 5 × 5 square. Adding one ring all around makes it 7 × 7 = 49 tiles (each side grows by 2: one tile at each end).
- The 49 − 25 = 24 new tiles are all black, so black becomes 8 + 24 = 32; white stays 17.
- Ratio: 32 : 17.
- Worth keeping: wrapping one ring around an n × n square turns it into (n+2) × (n+2), adding (n+2)2 − n2 = 4n + 4 tiles — here 4·5 + 4 = 24.
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