[GAP Forum] Finding OLL algorithms for the 2x2x2 Rubik's cube

[Bill Allombert]

On Sun, Sep 27, 2020 at 08:43:29PM +0100, rain1 at airmail.cc wrote:
> How would I come up with 2x2x2 Rubik's cube OLL algorithms that perform
> certain operations using GAP?
> 
> Here is an example of some OLL algorithms: https://jperm.net/algs/2x2/oll
> They leave the bottom half of the cube fixed, and they move the yellow
> squares onto the top of the cube. They are free to permute the grey squares
> however they like.
> 
> I have represented the 2x2x2 Rubik's cube as the following group in GAP:
> 
> U:=(513,523,524,514)*(153,351,352,253,254,452,451,154)^2;
> D:=(613,614,624,623)*(163,164,461,462,264,263,362,361)^2;
> F:=(163,153,154,164)*(513,514,451,461,614,613,361,351)^2;
> B:=(253,263,264,254)*(523,352,362,623,624,462,452,524)^2;
> L:=(351,361,362,352)*(513,153,163,613,623,263,253,523)^2;
> R:=(451,452,462,461)*(514,524,254,264,624,614,164,154)^2;
> 
> f:=FreeGroup("U","D","F","B","L","R");
> G:=Group([U,D,F,B,L,R]);
> hom:=GroupHomomorphismByImages(f,G,[f.1,f.2,f.3,f.4,f.5,f.6],[U,D,F,B,L,R]);
> SG:=Stabilizer(G,[613,614,624,623,163,164,461,462,264,263,362,361],OnTuples);
> 
> I was able to find large words by applying the PreImagesRepresentative to
> random elements of SG.
> 
> I implemented a brute force search for short words and I was able to find 2
> of the OLL algorithms.
> 
> I found that Factorization will calculate the shortest word, I'm running
> that now but it seems to be taking a very long time.
> 
> What I would really like to do is calculate very short words that extend a
> specific action, like:
> 
> word:=RepresentativeAction(G, [523, 524, 513, 514,
> 613,614,624,623,163,164,461,462,264,263,362,361], [523, 524, 153, 154,
> 613,614,624,623,163,164,461,462,264,263,362,361], OnTuples);
> 
> or
> 
> word:=RepresentativeAction(SG, [523, 524, 513, 514], [523, 524, 153, 154],
> OnTuples);
> 
> I sampled from this randomly and found long words that achieve the OLL
> algorithms, but I can't work out how to find short words.
> 
> Any advice and suggestions would be welcome!

If you want to know how the optimal sequences have been found, see
Bernard Helmstetter web site:
http://www.ai.univ-paris8.fr/~bh/cube/

Cheers,
Bill.