How many 2x2 states with all colours unique on all sides?

[abunickabhi]

I know there are definitely more than one such case / pattern.

I am guessing a few thousand cases, but want an exact number.

FYI, the total number of 2x2 possibilities are 3,674,160 or ~3.6 million cases.

 

[OtaMota]

I know there are definitely more than one such case / pattern.

I am guessing a few thousand cases, but want an exact number.

FYI, the total number of 2x2 possibilities are 3,674,160 or ~3.6 million cases.

Could you elaborate further, I am not sure I understand the question. If I think I am understanding correctly though, the answer is 153,090

 

[abunickabhi]

Could you elaborate further, I am not sure I understand the question. If I think I am understanding correctly though, the answer is 153,090

How did the answer of 153,090 derive or get calculated?

 

[OtaMota]

How did the answer of 153,090 derive or get calculated?

3.6 M divided by 24

 

[OtaMota]

Oh I understand the question now, I do not know the answer to it

 

[abunickabhi]

Oh I understand the question now, I do not know the answer to it

How do we compute the states of all such unique colors on each side using group theory?

 

[qwr]

How do we compute the states of all such unique colors on each side using group theory?

Group theory may not be needed. 2x2 only has one invariant so you can place 7 cubies and the 8th is determined. Actually if you want to distinguish cube rotations as the same you may need Burnside's Lemma.

 

[bcube]

I am guessing a few thousand cases, but want an exact number.

17 004 according to Bruce.

 

[abunickabhi]

17 004 according to Bruce.

He did not mention how he derived 17,004 cases in that thread. I guess there must be a brute force algorithm.

 

[qwr]

He did not mention how he derived 17,004 cases in that thread. I guess there must be a brute force algorithm.

3 million is very much brute forcable. in fact you could print out a list of every possible case and optimal solution and fit it into one book.

 

[Deleted member 36445]

To answer that question, we have to talk about parallel universes

 

[Averageof12]

To answer that question, we have to talk about parallel universes

maybe the quantum realm too....

 

[OtaMota]

3 million is very much brute forcable. in fact you could print out a list of every possible case and optimal solution and fit it into one book.

You could do a whole 2 hour video
Edit: bc of rotational symmetry, you could reduce this video by 24, and still learn full 2x2, so less than 6 minutes.

 

[abunickabhi]

You could do a whole 2 hour video
Edit: bc of rotational symmetry, you could reduce this video by 24, and still learn full 2x2, so less than 6 minutes.

Off topic and already known to many cubers and impractical.

anyway I want to know if there is a way to compute 17,004 cases.

 

[qwr]

anyway I want to know if there is a way to compute 17,004 cases.

It would go something like this:
fix one corner. try to place the 3 other cubies in one face. Place 3 more cubies on the last layer and the last one is determined by invariant.
I don't see any way to account for symmetry other than Burnside's lemma and a bunch of cases.

 

[bcube]

anyway I want to know if there is a way to compute 17,004 cases.

I don't know anything about these kinds of stuff, but wouldn't be analyses of his Hamiltonian cicruit helpful in some way?

For instance, I can immediately see first 29 159 moves of his move sequence are preserving the DFL and DBL corners (so those positions can be already excluded).