a cube painted on its six faces and then it is cut into 4913 smaller cubes of equal size

how many cubes are painted only on one face
hw many cubes are painted only 2faces

4913 is 17 cubed. There are 17^2 = 289 small squares visible on each face.

6x15^2 = 1350 are painted on one side only.

Fifteen cubes along each of the 12 edges are painted on two sides. That makes 180

The eight at corners are painted on three sides.

4913 - 1350 - 180 - 8 = 3375 are not painted at all. That happens to be 15 cubed.

(4913)^(1/3) = 17

so this big cube is 17 of the little ones on a edge. That means 6*17*17 = 1734 are on the outside and would be painted on one side except that the ones on the edges are painted on at least two sides so it is really 6*15 *15 = 1350

Only the ones along the 12 edges (excluding the 8 corner cubes which are painted on three face) are painted on two faces. Therefore 15*12

To determine the number of cubes that are painted on only one face, we need to consider the structure of the cube. The given cube has 6 faces, and when it is divided into smaller cubes, we can observe that only the cubes on the outer layer will have some of their faces painted.

The outer layer of the cube consists of the face cubes on each face (which would be the perimeter cubes) and the corner cubes (which would have three faces painted). The remaining cubes on the inner layers will have their faces hidden and will not be painted on any face.

Now, since the cube consists of 4913 smaller cubes, we know that it is a 17x17x17 cube, as 17^3 = 4913. Therefore, the outer layer will be a 15x15x15 cube, as the first and last layers on each axis will be composed of corner cubes.

To determine the number of cubes painted on only one face, we need to calculate the number of cubes in the outer layer. The number of cubes on each face of the outer layer will be (15x15) - the number of corner cubes.

The number of corner cubes on each face is 8, as there are 8 corners in total. So, the number of cubes painted on only one face per face is (15x15) - 8.

Since there are 6 faces on the cube, the total number of cubes painted on only one face is:
(15x15 - 8) x 6 = 1980.

Now, let's determine the number of cubes that are painted on two faces. We know that the corner cubes have three faces painted, so they are not part of the cubes painted only on two faces. Therefore, we need to exclude the corner cubes from the total number of cubes in the outer layer.

The number of corner cubes is 8, so the number of cubes painted on two faces in the outer layer will be (total cubes in the outer layer) - 8.

Hence, the number of cubes painted on two faces per face is (15x15) - 8.

Since there are 6 faces on the cube, the total number of cubes painted on two faces is:
[(15x15) - 8] x 6 = 1692.

Therefore, there are 1980 cubes painted on only one face and 1692 cubes painted on two faces in the given cube.

Whew, we agree :)