Problem 26 · 2024 Math Kangaroo
Stretch
Geometry & Measurement
areacasework
Jill has some black and some white unit cubes. She uses 27 of them to build a 3×3×3 cube, and she wants exactly one third of the surface to be black. If A is the smallest possible number of black cubes she can use and B the largest, what is the value of \(B - A\)?
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Answer: D — 7
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Hint 1 of 2
The 3x3x3 cube has 54 unit faces on its surface; one third is 18 black faces.
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Hint 2 of 2
Corner cubes show 3 faces, edges 2, face-centres 1, and the single hidden centre cube shows 0; use that for the maximum.
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Approach: count black faces by cube position to find A and B
- One third of the 54 surface faces is 18 black faces.
- Fewest black cubes: use 6 corner cubes (3 faces each) = 18 black faces, so A = 6.
- Most black cubes: cover 18 faces with cubes showing as few faces as possible, then add the fully hidden centre cube as a free black cube; this gives B = 13, so B - A = 7.
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