Carrie's First Cubes

1. To get three of its faces coloured, the small cubes have to be on the corners of the large cube. There are 8 corners to a cube, so 8 of the smaller cubes have three faces painted red.
2. To get two of its faces coloured, a small cube will have to be on the middle of an edge. In fact there is one per edge. As there are 12 edges of the large cube, there are 12 small cubes that have two faces coloured.
3. The cubes with one coloured face are in the middle of a face. There are 6 of these.
4. Those cubes that haven’t seen any paint must be in the centre of the large cube. There is only one such cube. This can be seen by first noting that there are 9 small cubes to a face. So there are 9 cubes on each of the three levels of the big cube to give a total of 27. Two opposite and parallel faces use up 18 different small cubes. The remaining faces on the outside of the cube are 3 + 1 + 3 + 1 around the ‘middle’ between the two opposite faces. Hence there are 18 + 8 = 26 small cubes on the outside of the large cube. Then there are 27 – 26 = 1 on the inside.