Automorphisms of $S_6$ and the Colored Cubes Puzzle

Given a palette of six colors, a colored cube is a cube where each face is colored with exactly one color and each color appears on some face. Starting with an arbitrary collection of unit length colored cubes, one can try to arrange a subset of the collection into an $n \times n \times n$ cube where each face is a single color. This is the Colored Cubes Puzzle. In this paper, we determine minimum size sets of cubes required to complete an $n \times n \times n$ cube's frame, its corners and edges. We answer this problem for all $n$, and in particular show that for $n \geq4$ one has the best possible result, that as long as there are enough cubes to build a frame it can always be done, regardless of the cubes in the collection. Part of our analysis involves the set of $6$-colored cubes and its associated $S_6$ action. In addition to the problem simplification this action provides, it also gives another way to visualize the outer automorphism of $S_6$.