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 large cube has a side of 3 cm and is cut into small cubes of 1 cm side. This means the large cube is divided into 3x3x3 = 27 smaller cubes.

Cubes with exactly two painted faces are located on the edges of the original large cube, but not at the corners.

1. A cube has 12 edges.
2. Each edge of the 3 cm cube contains 3 small cubes (since 3 cm / 1 cm = 3).
3. The two cubes at the ends of each edge are corner cubes, which have three painted faces.
4. Therefore, for each edge, the number of cubes with exactly two painted faces is (Total cubes along an edge) - (2 corner cubes) = 3 - 2 = 1 cube.
5. Since there are 12 edges, the total number of small cubes with exactly two painted faces is 12 edges * 1 cube/edge = 12.

Analyzing the options: A) 12: This matches our calculation for cubes with exactly two painted faces. B) 8: This is the number of corner cubes, which have exactly three painted faces. C) 6: This is the number of cubes with exactly one painted face (1 cube per face * 6 faces). D) 4: This is not a direct category for a 3x3x3 cube.

The final answer is A.