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.
Description
Automorphisms of S-6 and the color cubes puzzle // Lafayette Digital Repository
%0 Journal Article
%1 berkove2017automorphisms
%A Berkove, Ethan
%A Katz, R.
%A Condon, D.
%A Nava, D. C.
%D 2017
%J Australasian Journal of Combinatorics
%K automorphism cube group macmahon math puzzle
%N 1
%P 71--93
%T Automorphisms of S-6 and the color cubes puzzle
%U http://hdl.handle.net/10385/2208
%V 68
%X 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.
@article{berkove2017automorphisms,
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.},
added-at = {2022-01-27T21:19:25.000+0100},
author = {Berkove, Ethan and Katz, R. and Condon, D. and Nava, D. C.},
biburl = {https://www.bibsonomy.org/bibtex/28ef8b898bfd73c7bcadd94d0166330d7/jaeschke},
description = {Automorphisms of S-6 and the color cubes puzzle // Lafayette Digital Repository},
interhash = {4546e03dd640c34be9c73119f57df2a9},
intrahash = {8ef8b898bfd73c7bcadd94d0166330d7},
journal = {Australasian Journal of Combinatorics},
keywords = {automorphism cube group macmahon math puzzle},
number = 1,
pages = {71--93},
timestamp = {2022-01-27T21:19:25.000+0100},
title = {Automorphisms of S-6 and the color cubes puzzle},
url = {http://hdl.handle.net/10385/2208},
volume = 68,
year = 2017
}