Problem 24 · 2019 Math Kangaroo
Stretch
Fractions, Decimals & Percents
area-fractionspatial-reasoning
Linus builds a 4 × 4 × 4 cube made up of 32 white and 32 black 1 × 1 × 1 cubes. He arranges the small cubes so that the surface of the big cube shows as much white as possible. Which fraction of the surface is white?
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Answer: A — 34
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Hint 1 of 3
First count the whole surface: the big cube has 6 faces, each split into 16 little squares.
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Hint 2 of 3
To show the most white, place white cubes where they expose the most faces — corner cubes show 3, edge cubes show 2.
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Hint 3 of 3
Count how many white faces those favorable spots give you, then compare with the total.
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Approach: place white cubes to expose the most faces
- The surface has 6 × 16 = 96 unit faces. Corner cubes (8) show 3 faces each, edge cubes (24) show 2 each.
- Filling the 8 corners and 24 edges with white uses all 32 white cubes and exposes 8×3 + 24×2 = 72 white faces.
- So the white fraction of the surface is 72⁄96 = 3⁄4 (A).
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