Question

A cube is painted on all sides. It is then cut into 64 smaller cubes of equal size. How many smaller cubes have exactly two painted faces?

• A. 8
• B. 12
• C. 24
• D. 36

Hint:

For painted cube problems, focus on the position of smaller cubes (corners, edges, faces, or center). Subtract corner cubes from edge counts for two painted faces.

Answer 1

The Correct Option is C

To solve the problem, we need to determine how many smaller cubes have exactly two painted faces after a large painted cube is cut into smaller cubes.

1. Understanding the Concepts:

- The original cube is painted on all six sides.
- It is cut into 64 smaller cubes of equal size.
- Since \( 64 = 4^3 \), the cube is cut into 4 equal parts along each edge.
- Smaller cubes can have 0, 1, 2, or 3 painted faces depending on their position in the original cube.
- We need to find how many smaller cubes have exactly two painted faces.

2. Positions of smaller cubes based on painted faces:

- Cubes with 3 painted faces are the corner cubes.
- Cubes with 2 painted faces are the edge cubes excluding the corners.
- Cubes with 1 painted face are the face cubes excluding edges and corners.
- Cubes with 0 painted faces are the interior cubes not touching any face.

3. Calculating the number of smaller cubes with exactly two painted faces:

- Each edge of the cube has 4 smaller cubes.
- The 2 cubes at the ends of each edge are corners (3 painted faces).
- The cubes in between corners on an edge have exactly 2 painted faces.
- Number of such cubes per edge = \( 4 - 2 = 2 \).

- The cube has 12 edges.
- Total number of smaller cubes with exactly two painted faces = \( 12 \times 2 = 24 \).

Final Answer:

The number of smaller cubes that have exactly two painted faces is 24.