Problem 14 · 2011 Math Kangaroo
Hard
Algebra & Patterns
sequence-of-figurescareful-counting
Black and white tiles can be laid on square floors as shown in the pictures. We can see floors with 4 black and 9 black tiles respectively. In each corner there is a black tile, and each black tile touches only white tiles. How many white tiles would there be on a floor that had 25 black tiles?
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Answer: D — 56
Show hints
Hint 1 of 2
Look at the small floors: 4 black tiles sit on a 3-by-3 floor, and 9 black tiles sit on a 5-by-5 floor.
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Hint 2 of 2
Spot the pattern in the floor sizes, then count the white tiles as total minus black.
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Approach: see how the floor grows with the black count, then subtract
- Watch the pattern: 4 black tiles go on a 3-by-3 floor, and 9 black tiles go on a 5-by-5 floor — the side jumps by 2 each time.
- Following the pattern, 25 black tiles go on a 9-by-9 floor, which is 9 × 9 = 81 tiles in all.
- White tiles = 81 − 25 = 56.
Why the pattern works
With \(n^2\) black tiles the floor is \((2n-1) \times (2n-1)\); for \(n = 5\) that is \(9 \times 9 = 81\), so white \(= 81 - 25 = 56\).
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