The correct answer is Option A: 12.
To solve this problem, we need to understand how the cube is divided and how the painting affects the smaller cubes.
     1. Initial Setup:
            ○ We have a solid cube with a side length of 3 cm.
            ○ This cube is painted on all its faces.
     2. Cutting the Cube:
            ○ The cube is cut into smaller cubes, each with a side length of 1 cm.
            ○ Since the original cube is 3 cm on each side, it will be divided into \(3 \times 3 \times 3 = 27\) smaller cubes.
     3. Analyzing the Smaller Cubes:
        ● Corner Cubes: These are the cubes at the corners of the original cube. Each corner cube will have 3 painted faces. There are 8 corner cubes in total.  
        ● Edge Cubes: These are the cubes along the edges of the original cube, excluding the corners. Each edge of the original cube has 3 cubes, but the corner cubes are not included in this count. Therefore, each edge has 1 cube with exactly 2 painted faces. Since the original cube has 12 edges, there are \(12 \times 1 = 12\) such cubes.  
        ● Face Cubes: These are the cubes on the faces of the original cube, excluding the edges. Each face of the original cube has a \(3 \times 3\) grid of cubes, but the edge cubes are not included in this count. Therefore, each face has 1 cube with exactly 1 painted face. Since the original cube has 6 faces, there are \(6 \times 1 = 6\) such cubes.  
        ● Inner Cubes: These are the cubes inside the original cube, not touching any face. There is only 1 such cube, and it has no painted faces.  
     4. Conclusion:
            ○ The number of smaller cubes with exactly two painted faces is 12, which corresponds to the edge cubes.
     Therefore, the correct answer is Option A: 12.