Article,
Automorphisms of S-6 and the color cubes puzzle
Australasian Journal of Combinatorics, 68 (1): 71--93 (2017)
Abstract
Given six colors, a color cube is one where each face is single-colored and each color appears on some face. The Color Cubes puzzle is a variation of a classic problem due to P. MacMahon: one starts with an arbitrary collection of color cubes of unit length and tries to find a subset that can be arranged into an n x n x n cube where each face is a single color. In this paper we determine the minimum size of a set of cubes that, regardless of its composition, guarantees the construction of an n x n x n cube's frame, its corners and edges. We do this for all n, and find that for n >= 4 one has the best possible result, that as long as there are enough cubes to build a frame it can always be done. Part of our analysis involves the S-6 action on the set of color cubes. In addition to the problem simplification it provides, this action also gives another way to visualize the outer automorphism of S-6.
