A cube made from a block of white pine is then painted blue on the outside. The cube is 6 centimeters on each side. After the paint dries, the cube is cut into 27 small cubes. However, after making one saw cut, you can rearrange the pieces and stack them anyway you like before you make the next cut. If you employ this idea, can you make the 27 smaller cubes in fewer than 6 cuts?
Solution
All but one of the small cubes has at least one face painted blue, that is, one face comes from the outside of the original cube. However, all six faces of the central, small cube are white because all six of its faces are internal to the original large cube. Each of those six small white faces requires a separate cut so six cuts are necessary to make the 27 small cubes.