Tom was building with 20 red cubes and 7 white ones. All his cubes were of the same size. He made a larger cube using all of these smaller cubes. Part of the surface of the larger cube was red and part was white. What is the smallest fraction that could be white?
Solution
Tom needs to get as few of the white cubes on the surface of the bigger 3 × 3 × 3 cube as possible. Now it is possible to ‘hide’ one of the cubes in the middle of the 3 × 3 × 3 cube. Then there are six left. It is possible to make sure that only one of the faces of the white cubes is on the surface of the bigger cube. This can be done by putting the white cubes in the middle of each face.
So we have 6 white faces exposed on a 3 × 3 × 3 cube. Such a cube has 6 × 3 × 3 little cube faces exposed. This is a total of 54. So the smallest fraction of white that Tom can make is 6/54 = 1/9.
Extension
How many white cube faces can be exposed in the final 3 × 3 cube? What are all of the possibilities?