# Virtual Cubes

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### Fellay's Twelve Color Cube

Fellay's Twelve Color Cube is divided into twelve areas of distinct colors. Each edge has a uniform color, while the corners are split into three and the centers into four colors. Compared to a regular Rubik's cube the edges no longer need to be oriented but it is required for the side parts.

The layout of Fellay's Twelve Color Cube was created in 2017 by Fabien Fellay. It is a color variation of the Twelve Color Cube published by Douglas R. Hofstadter in the Scientific American issue of March 1981. In it's article Hofstadter also thought about alternate ways in coloring the cube:

«The potential of the 3 × 3 × 3 cube is not close to be exhausted. One rich area of unexplored terrain is that of alternate colorings. The idea was mentioned to me by various M.I.T. cube hackers. One can color the cubies in a variety of ways. Each new coloring presents a different kind of unscrambling problem. In one variant coloring, edge-cubie orientations become irrelevant and center-cubie orientations take on a vital importance. In an other variant, corner-cubie orientations are irrelevant and centers matter.» (Excerpt from Scientific American, March 1981)

A different coloring can make it more complicate solving a cube but can also simplify it. This variant coloring uses twelve colors in a way to focus more on the corners and side parts but no longer care about edge orientations.

#### The number of possible positions

The number of possible positions of the Twelve Color Cube is 8! × 37 × 12! / 2 × 46 / 2 = 43'252'003'274'489'856'000 ≈ 43.252 × 1018 which is about 43.3 quintillion on the short scale or 43.3 trillion on the long scale. The Twelve Color Cube has exactly the same amount of possible positions than a regular Rubik's Cube.

The Twelve Color Cube has eight corners, twelve edges and 6 marked centers. There are 8! (40'320) ways to arrange the corners. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 37 (2'187) possibilities. There are 12! / 2 (239'500'800) ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well. There are 46 / 2 (2'048) ways to orient the centers, since an even permutation of the corners implies an even number of quarter turns of the centres as well.

#### References

Douglas R. Hofstadter, «METAMAGICAL THEMAS: The Magic Cube's cubies are twiddled by cubists and solved by cubemeisters», Scientific American, vol. 244. #3, March 1, 1981, pp. 20-39.

Douglas R. Hofstadter, «MATHEMATISCHE SPIELEREIEN: Vom Zauber des Zauberwürfels», Spektrum der Wissenschaft, Heidelberg, May 1981, pp. 16-29, (German edition of the Scientific American, March 1981).