Problem 13 · 2011 Math Kangaroo
Hard
Geometry & Measurement
perimeterarea
Nina made a wall around a square area, using 36 identical cubes. A section of the wall is shown in the picture. How many cubes will she now need to completely fill the square area?
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Answer: C — 64
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Hint 1 of 2
The 36 cubes form just the border of a square, one cube thick.
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Hint 2 of 2
Find the side of that square, then how many cubes fill the whole inside.
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Approach: the wall is the border; find the side, then fill the inside
- The 36 cubes are just the outer ring of a square, one cube thick.
- A square that has 10 cubes along each side uses 10 + 10 + 8 + 8 = 36 cubes around the edge (the corners are counted once), so each side is 10 cubes long.
- The whole 10-by-10 area holds 10 × 10 = 100 cubes, and 36 are already the wall, so she still needs 100 − 36 = 64.
Shortcut with a formula
The border of an \(n \times n\) square uses \(4n - 4\) cubes; \(4n - 4 = 36\) gives \(n = 10\), so the fill is \(100 - 36 = 64\).
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