Cube painted in red is cut in to 729 identical pieces and from that, corner cubes are removed again the cube?

Question

a cube painted in red is cut into 729 identical pieces and from that corner cubes are removed.now again the cube is painted in black.

then answer the following

1)what is the diff b/w cubes with one face painted in the original one and in tha present one?

2)what is diff in no.of cubes with no faces painted in the original one and the one painted in black?

3)what is the diff in no.of cubes with 2 faces painted in the original one and the one painted in blabk?

4)what is the diff in no.of cubes with 3 faces painted in the original one and the one painted in black?

Answer 1

so the cube is 9 cubes on a side.

the edges take up one cube on each side, so the sides with just one face are 6 * 7 * 7  or 294 originally

1. no change. once the corners are removed, the new exposed faces are just on  the edges, so the one face cubes aren't affected.  still 294.

2. no faces painted doesn't change either.

3. two face cubes are edges but not corners. There are 12 edges, 7 pieces (not corners) on each.  so originally there are  84 of these.  once the corners are removed,  and the cube repainted, the cubes at the ends of the rows now are 3 surfaces. so the 2 sides pieces are 12 * 5 or 60 vs. 84.

4. 84 - 60 = 24  cubes with 3 faces black.  the original red cube had 12 cubes with 3 faces red.