Virtual Cubes

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

The layout of Fellay's Eight Color Cube was created in 2017 by Fabien Fellay. It is a color variation of the Eight 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 eight colors in a way to focus more on the edges and side parts but no longer care about corner orientations.

A classic cube in new colors

Fabien Fellay devised the same layout by imagining an octahedron (its 'Dual Polyhedron') inside the cube. The eight colored faces of the octahedron are then projected on the enclosing cube. The color concept is based on the additive and subtractive color model and brings the colors in relation to each other.

The black color origin, the product of the three subtractive primary colors cyan, magenta and yellow, is located on the frontside of the cube, at the intersection of faces U (Up), R (Right) and F (Front).

The white color origin, the product of the three additive primary colors red, green and blue, is located on the backside of the cube, at the intersections of faces D (Down), L (Left) and (B) Back.

The product of two subtractive prime colors is located in between the corresponding two colors: adding yellow to cyan yields green, adding cyan to magenta yields blue and adding magenta to yellow yields red.

Vise versa the same is applied to the product of two additive prime colors: adding blue to green yields cyan, adding green to red yields yellow and adding red to blue yields magenta.

Dual Polyhedron

In geometry, polyhedra are associated into pairs called 'duals', where the vertices of one correspond to the faces of the other. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

Each Platonic solid has a dual Platonic solid. If a midpoint (center) of each face in the Platonic solid is joined to the midpoint of each adjacent face, another Platonic solid is created within the first.

The illustration shows, that the dual of the cube is the octahedron.

The number of possible positions

The number of possible positions of the Eight Color Cube is 8! × 12! / 2 × 2¹¹ × 4⁶ / 2 = 40'503'019'070'029'824'000 ≈ 40.503 × 10¹⁸ which is about 40.5 quintillion on the short scale or 40.5 trillion on the long scale. Compared to a regular Rubik's Cube the Eight Color Cube has about 93.6% of its possible positions.

The Eight Color Cube has eight corners, twelve edges and 6 marked centers. There are 8! (40'320) ways to arrange the corners. 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. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 2¹¹ (2'048) possibilities. 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).