> Some unit cubes are put together and form a larger cube, and then > some of the faces of this cube are painted. After the paint dries, > then the larger cube is disassembled into the unit cubes and there are > 24 of these have no paint on any of their faces. > Question: How many faces of the larger cube were painted?
If all the outer cubes of a 5x5x5 were removed there would be 27 unpainted cubes inside . . . so clearly the large cube has side less than 5. If none of the faces of a 2x2x2 cube were painted then only 8 cubes would be unpainted. So the large cube has either side 4 or side 3. If even one side of a 3x3x3 were painted then only 18 unpainted cubes would remain.
Therefore, some faces of a 4x4x4 cube were painted.
A 4x4x4 has a core of 8 unpainted cubes with 56 cubes lying on the surface and we must color all but 16 of them. So the problem is: how to paint 40 of the face cubes of a 4x4x4 cube.
The solution: paint any three faces which do not meet at a corner.