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? (A) The number of smaller cubes with two surfaces painted is 36. (B) The number of smaller cubes with one surface painted is 24. (C) The number of smaller cubes with no surface painted is 8. (D) The number of smaller cubes with at least two surfaces painted is 30.
Solution
✅ Correct Option: 3
First, let's understand what we're working with: The original cube is colored red on all faces, cut into $64$ smaller cubes, making it a $4 \times 4 \times 4$ cube. Let's analyze each type: Cubes with three painted faces: These are the corner pieces. Number of corners = $8$ cubes Cubes with two painted faces: These are the edge pieces (but not corners). On a $4 \times 4 \times 4$ cube, each edge has $2$ cubes between corners. Number of edges = $12$. Total = $12 \times 2 = 24$ cubes Cubes with one painted face: These are center pieces on each face (not edges/corners). Each face is $4 \times 4$ with interior $2 \times 2$. Each face has $4$ center pieces. With $6$ faces, total = $6 \times 4 = 24$ cubes Cubes with no painted faces: These are completely interior cubes, forming a $2 \times 2 \times 2$ interior cube = $8$ cubes Let's check each statement: (A) "Two surfaces painted = $36$" False, it's $24$ cubes (B) "One surface painted = $24$" True (C) "No surface painted = $8$" True (D) "At least two surfaces painted = $30$" False, it's $8$ (three faces) + $24$ (two faces) = $32$ cubes Therefore, only statements (B) and (C) are correct.