# How many 8 Color Cube solutions are there?



## Waran (Nov 25, 2018)

The '8 Color Cube' is a Rubik's Cube variant with 8 different colors / numbers. 
In its solved state all colors and numbers only represented once per face.

A virtual model of this cube can be found on this page:

A total of 21 additional solutions (apart from the initial state of the cube) have been found so far. 
But we are convinced that there are many more ... so therefore, our question is how many?

It has proved to be very difficult to understand the cube on a more fundamental level and thus to get a full picture of the possibilities inherent in it. 
We would be very interested to know if you can find additional valid solutions ... or even all of them.

The following images are showing the frontside and backside of all known 8 Color Cube solutions.


Peter Tchamitch and Walter Randelshofer


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## Waran (Nov 25, 2018)

Initial state of the '8 Color Cube':


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## Waran (Nov 25, 2018)

Additional valid '8 Color Cube' solutions:


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## Waran (Nov 25, 2018)

Additional '8 Color Cube' solutions:


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## Waran (Nov 25, 2018)

Addition '8 Color Cube' solution:


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## Waran (Dec 15, 2018)

Of course we only count solutions reachable through standard 3x3x3 moves. 
This means no solutions that violate parity or twist restrictions for the 3x3x3 group.

A first step to approach the problem we can try to figure out more about this Rubik's Cube variant. 
Since we don't have to care about corner orientations, the number of possible cube positions is much smaller compared to a regular Rubik's Cube.

I tried to compute the number of possible positions of the 8 Color Cube:

The 8 Color Cube has eight corners and twelve edges. 
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 2^11 (2'048) possibilities.

8! × 12! / 2 × 2^11 = 19'776'864'780'288'000 ≈ 19.776 × 10^15
Could someone please double check if this number is correct?

It might be probably easier to figure out how many valid solutions there are, regardless if it's a possible cube position or not.
This would give us a first upper bound and a rough idea in which range solutions can be expected.

By the way, you can create an '8 Color Cube' easily by yourself. All you need is a felt marker. Just add the numbers 1 to 8 on each face like on the '8 Color Cube' layout.


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## xyzzy (Dec 15, 2018)

Since the centres are all identical, we can fix a corner instead of the centres. With the centres freely rearrangeable, we can now get odd permutations of the edges as well.

7! corner permutations × 12! edge permutations × 2^11 edge orientations = 4944216195072000 ~ 4.9 × 10^15


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