How many 'families' of possible states are there for a 3x3x3?

[Tim Vermeulen]

It is well known that if a 3x3x3 cube is assembled with the pieces on random positions and in random orientations, the chance is very small it is in a solvable state. It is very likely that that position is part of another 'family' of states. I consider a family of states to be a group in which any state can be changed to any other in that family by only turning sides, rather than taking the cube apart. The family all our cubes (most of them, at least) are in, is what I call the main family. This main family contains the solved state. For example, the solved state except for one edge which is flipped is not part of the main family, because it can not be solved without taking the cube apart. Also, just one rotated corner cannot be solved.

There are three cases for corner orientation:

1. an equal number of clockwise corner rotations as anti-clockwise rotations can orient all cornerns properly (the main state is one of those)
2. after an equal number of clockwise corner rotations as anti-clockwise corner rotations are done, one corner is yet to be rotated clockwise for all corners to be oriented properly
3. after an equal number of clockwise corner rotations as anti-clockwise corner rotations are done, one corner is yet to be rotated counter-clockwise for all corners to be oriented properly

There are two cases for edge orientation: an even number of edges are oriented properly, or an odd number of edges are oriented properly. So there are at least 2 * 3 = 6 different families.

However, I do not know about how many cases there are for edge and corner permutations, apart from that there are different cases for those as well. As an example, if you (in the solved state) swap the green-white edge with the blue-white edge, the cube becomes unsolvable (thus becomes part of another family of states) while all edges and corners are oriented correctly. My questions is: how many possible families are there? In other words, if the chance of a randomly assembled cube to be in the main family of possible states is equal to 1/k, what is the value of k?

And how about other cubes? I'm pretty sure the 2x2x2 has less families and the bigger cubes have more, but how much do they have? I'm pretty sure it has been figured out for the 3x3x3 already, but how about the others?

 

[Stefan]

12

 

[Florian]

2 for Edge orientation
3 for Corner-Orientation
2 for permutation

2*3*2=12

 

[cmhardw]

And how about other cubes? I'm pretty sure the 2x2x2 has less families and the bigger cubes have more, but how much do they have? I'm pretty sure it has been figured out for the 3x3x3 already, but how about the others?

For the n x n x n cube the number of families is:

Spoiler

12 for all odd n x n x n cubes and 3 for all even n x n x n.

Spoiler

For all n > 3 the centers come in indistinct groups, so their overall permutation has no concept of "even" or "odd" parity in the supercube sense. For the puproses of creating new families, these pieces can be ignored.

For the wing edges (NOT including the centralmost edge of odd n x n x n) each orbit can end up having either even or odd permutation parity if you randomly assign all pieces to a location. The parities of the wing edges are independent of the parities of the corners on any sized cube and centralmost edges of odd sized cubes. The parities of each wing orbit are independent of every other wing edge orbit. In the supercube sense, knowing the parity of every wing edge orbit, as well as the parity of the corner orbit, will uniquely determine the parity of every center orbit. However, on a regular cube the center orbits don't have the concept of parity, so the wing orbit parities do not affect the parity of any other piece type, therefore these pieces can also be ignored for determining how many families there are on the n x n x n cube.

This leaves us with only corners and central most edges on the n x n x n cube to determine the number of families. We are assuming that the centers of an odd n x n x n cube are fixed in location to create the definition of the solved state. For even n x n x n cube we will fix the location and orientation of one corner to determine the solved state.

So for an even n x n x n cube the permutation of the pieces may be either even or odd following cube laws. However the orientation of the corners can create 3 different possible families.

For odd n x n x n cubes, there are 12 families as listed in the posts above.

 

[irontwig]

And how about other cubes? I'm pretty sure the 2x2x2 has less families and the bigger cubes have more, but how much do they have? I'm pretty sure it has been figured out for the 3x3x3 already, but how about the others?

Even cubes: 3 Odd cubes: 12