"I think we're going to have a problem here..."

Rubiks cubes are group theory examples, not mechanics examples.

I think this person is trying to say the Rubik's cubes are made out of group theories. I think the fact that the puzzles, like any puzzle, are physical and have mechanical properties like any puzzle, so they are computational examples, and also

*mechanical* examples. I haven't been able to find any references, during my enforced leave, to any examples of cubes or anything else, built out of a group theory except another group theory.

Personally I find this statement mind-numbing and about as naive as you get from people's grasp of problem spaces, what a theory is vs what the theory is based on.

But as we say in the Gastrology Dept. "let's move on, shall we?"

Now most people who have managed (unlike

*some* of us) to solve a scrambled cube, maybe more than once, know about the twist. The twist re-orients pieces but leaves them in the correct place.

A place, is just a "coordinate location" for a piece; there are different sets of pieces (fairly obvious) although each square looks like the side of a smaller cube, this is (obviously) an external.

Then finding a solution, using detective strategies, rather than stepping on a cube, we find that having at least two (doubling the problem) is more helpful than having only one (because, if you scramble the only example you have, you leave the identity behind, or "hide" it in a jumbled-up pattern). Then with two cubes, you can scramble one of them (or get the kids to), Now you can try to solve one, and you can slowly scramble the other and try to make them look "the same".

This means finding a node in a graph where the doubled cube has equivalent color patterns (an isomorphism/isometry). The location of this node is a number of steps from either the scrambled or unscrambled one of the pair, so they meet and you see one is "redundant". It is most definitely about developing algorithms (I have several).

The algorithms are more interesting than the cube because different words in the operator set can get to the same node, taking different paths. Most of these details are obvious, you can figure all this out and stay close to the solved state (i.e. experiment with a handful of moves, and back to observe results). The twist is a fundamental "parity function", which acts to restrict cycles.

There is:

generators {g,g1,g2,...,gn} are coordinators; relations {r,r1,r2,...,rn} are coordinates. F and Q are the full-face and quadratic 'root' (restricted to M1) phase spaces. F is in M4. MnMn is a subgroup of Mn (clearly?!).

Want to relate m,n to M: M -> Mn (?) Have T(m,n) map for K the corners with 3 facets and the twist. K is a divisor of G at N > 2. Restrict K in M (ignore 3-facets, use a 2-facet translation algebra coordinatized to the 1-facets which have rotational symmetry in M, from 0,..,4 2-facets align (pivot on a 1-facet). Restrict the centers (desticker them) and coordinatize by choosing a single facet color as the pivot. <G|R> is a map of coordinates and a set of transforms between.

So M is just a symbol that you can take to mean something like "an M-dimensional space", and the subscripts k to mean "k numbers in a M-dimensional space" or "a k-dimensional number in M".

The $$ M_4 $$ are obvious - there are slice groups in the original which vanish in the N=2 version (the pocket cube). These have 4 centers and 4 edges (1-facets, and 2-facets) around an axis of rotation. In this case a middle-layer slice is obviously in $$ M_4 $$.

p.s. the cube solving robot is an example of a mechanical solution to a mechanical problem -

*awesome* eh?