A solid cube of 3 cm side, painted on all its faces, is cut up into small cubes of 1 cm side. How many of the small cubes will have exactly two painted faces?

The correct option is 12.

To understand why, let us break down the geometry of the cube and the process of cutting it into smaller cubes step-by-step:

1. Understanding the Dimensions:
We start with a large solid cube where each side length is 3 cm.
This cube is painted on all six of its outer faces.
It is then cut into smaller cubes, each having a side length of 1 cm.

2. Total Number of Small Cubes:
The volume of the large cube is:
$3^3=27{\text{cm}}^3$
Since each small cube has a volume of $1{\text{cm}}^3$, there are exactly 27 small cubes in total.

3. Analyzing the Painted Faces:
Depending on where a small cube is located within the original large cube, the number of painted faces it has will vary:
• Three painted faces: These are the corner cubes. A cube has 8 corners, so there are 8 such cubes.
• Exactly two painted faces: These are the cubes located along the edges of the large cube, excluding the corner cubes.
• Exactly one painted face: These are the cubes in the center of each face.
• No painted faces: These are the completely internal cubes (the core).

4. Calculating Cubes with Exactly Two Painted Faces:
A cube has 12 edges.
Along any edge of length 3 cm, there are 3 small cubes of 1 cm.
The two cubes at the ends of each edge are corner cubes (which have 3 painted faces).
Therefore, the number of cubes with exactly two painted faces on each edge is:
$3-2=1$
Since there are 12 edges on a cube, the total number of small cubes with exactly two painted faces is:
$12\times 1=12$