Problem 11 · 2024 Math Kangaroo
Hard
Spatial & Visual Reasoning
cube-viewscareful-counting
John has black and white unit cubes and wants to use 27 of them to build a 3×3×3 cube. He wants to make sure that the surface is exactly half white and half black. What is the minimum number of black cubes that he needs?
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Answer: E — another number
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Hint 1 of 2
Corner cubes show 3 faces, edge cubes 2, face-centre cubes 1 — use the ones showing the most faces first.
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Hint 2 of 2
The surface has 54 little faces; half is 27 black faces, so cover 27 of them with as few cubes as possible.
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Approach: maximise black faces per cube
- The 3×3×3 surface has 6×9 = 54 little faces; half of them, 27, must be black.
- A corner cube shows 3 faces, an edge cube 2, a face-centre cube 1.
- All 8 corners give 24 black faces; one edge (2) plus one face-centre (1) adds the last 3.
- That is 8 + 1 + 1 = 10 cubes — not 11/12/13/14, so the answer is another number.
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