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A cube is coloured red on all of its faces. It is then cut into 64 smaller cubes of equal size. Which of the following statements are correct? (I) The number of smaller cubes with two surfaces…

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A cube is coloured red on all of its faces. It is then cut into 64 smaller cubes of equal size. Which of the following statements are correct?

(I) The number of smaller cubes with two surfaces painted is 36.

(II) The number of smaller cubes with one surface painted is 24.

(III) The number of smaller cubes with no surface painted is 8.

(IV) The number of smaller cubes with at least two surfaces painted is 30.
  1. (I), (II) and (IV) only
  2. (II), (III) and (IV) only
  3. (II) and (III) only
  4. (I) and (IV) only

Answer and explanation

Correct answer: C - (II) and (III) only
Cutting into 64 cubes gives a 4 × 4 × 4 arrangement. Cubes with two faces painted sit on the 12 edges, 2 per edge, so there are 24 and statement I is false. One face painted: 4 per face × 6 = 24, so II is true. No face painted: the inner 2 × 2 × 2 = 8, so III is true. At least two faces painted: 24 edge cubes + 8 corners = 32, so IV is false. Hence (II) and (III) only.

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