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If a cube is coloured red on all faces and cut into 64 smaller cubes of equal size, then which of the following are correct?

A. Sixteen small cubes have no face coloured.

B. Twenty four small cubes have only one face coloured.

C. None of the cubes can have its opposite faces coloured red.

D. There are eight cubes which have three faces coloured.

Solution

✅ Correct Option: 4

Since 64=4364=4^3, the large cube is divided 4×4×44\times4\times4.

  • No face coloured: (4−2)3=8(4-2)^3=8, so A is false.
  • Exactly one face coloured: 6(4−2)2=246(4-2)^2=24, so B is true.
  • No small cube reaches two opposite outer faces, so C is true.
  • Three faces coloured: the 8 corner cubes, so D is true.

Therefore B, C and D are correct.

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