A cube painted blue on all faces is cut into 1331 cubes of equal size. How many cubes are painted on one face only?.

To solve this problem, we first need to understand the structure of the cube.

1. A cube has 6 faces, and if it's painted on all faces, then when it's cut into smaller cubes, the smaller cubes will have paint on them depending on their location.

2. The cubes that are inside the big cube will have no paint on them. The cubes that are on the edges will have paint on 1, 2, or 3 faces.

3. The number of smaller cubes can be found by taking the cube root of 1331, which is 11. This means the big cube was cut into 11 cubes along each edge.

4. The cubes with paint on only one face are the ones that are not on an edge. This means they are not on the top or bottom face (2 faces) and not on the four sides (4 faces).

5. So, the cubes with one face painted are the ones on the four sides, not counting the ones on the top and bottom.

6. Each side has (11-2) * (11-2) = 9 * 9 = 81 cubes with one face painted.

7. Since there are 4 sides, the total number of cubes with one face painted is 4 * 81 = 324.

So, there are 324 smaller cubes that are painted on one face only.