If we represent each of the small cubes in each face of a cube by a base 6 number (one of the 6 colors) , can we then represent the configuration of a cube by a 54 digit number (9 * 6) ?
Interestingly, the configurations could also be represented in terms of a base 6 number (representing the 6 operations ). Of course, in our case, things are ot commutative . How could this be related to diagonalization ?? The fully solved case reminds one of Jordan's normal form and Jordans block.
Could the above be related to problems in coding theory ?
One idea i suppose would be to look at certain subsets / slices / planes as eigenvectors of a particular operation (F, L U etc . i.e rotation of a particular face / plane counterclockwise by 90 deg). As compared to the usual convention, here the eigenvectors would be those that are typically affected by the operation. Conversely, one could also at the sets of cubelets left invariant by the particular operation as eigenvectors as usual, the problem of course being that there are so many of them. One could look at the above in terms of intersection of planes
I'm not sure, but I'd suggest looking at https://en.wikipedia.org/wiki/Rubik%27s_Cube_group
Dear Ramchandran
There exist mathematics approach ( see above link ), but I think, you want algorithm to solve Rubiks cube ?
best regards
One could look at the scrambled version of the cube as a redistribution of the cubelets and the colors. Each color would be distributed among all the faces of the cube. Could one fit a plane , either directly or in a least squares sense to the "facelets" of each color , giving us 6 planes ?? Will these planes be non-orthogonal to one another ?? If so, can we apply gram schmidt's orthogonalization process to reconstruct the original cube or one that can be easily transformed to the original cube (by simple rotations of the cube as a whole ) ??
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