Problem 26 · 2020 Math Kangaroo
Stretch
Counting & Probability
careful-countingcomplementary-counting
Pedro assembled a cube using 64 little white equal cubes and then painted the cube red. Then he dismantled the cube and reassembled it so that all of its faces were white, and painted the cube red again. How many white faces of little cubes remained white?
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Answer: E — 192
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Hint 1 of 2
Count all the little-cube faces, then subtract the ones that get painted.
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Hint 2 of 2
After the first paint 96 faces are red; on reassembly another 96 (white ones turned outward) get painted.
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Approach: track which faces stay unpainted through both paintings
- There are 64·6 = 384 little faces; the first painting reds 96 of them.
- That leaves 288 white faces.
- Reassembling white-out and repainting reds 96 more of those white faces.
- Faces still white = 288 − 96 = 192.
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