Notes for Thurs Feb 24

 * Rubik "twistiness" and "turniness" as conservation laws
 * final version of number of positions, 3x3x3
     - other sizes?
     - 2x2x2
     - 4x4x4
     - 5x5x5
     - other shapes?
 * cubelets, various approaches to solving the thing
     - layer by layer
     - corners, then edges
     - edges, then corners
     - back 2 corners, back 5 edges, front 2 faces
     - others
 * subgroups:
     * (RL') (UD') (FB') slice group
         - number of positions?
         - pretty patterns?
         - ways to "solve" it?
     * (R) (F) 2-side group
     * corners only group
     * edges only group
 * operators to find:
     * permute 3 corners, leave all edges and other corners alone
     * permute 3 edges, leave all corners and other edges alone
     * twist 2 corners in opposite directions 
         - without worrying about edges
         - leaving edges alone
     * twist 2 edges in opposite directions
     * 
 * Jim's method: 
     * edge's, via standard commutator and 3-cycle ignoring corners
     * 2 out of place => odd number of moves => make one move
     * 2 flipped => 2 commutators
     * corners, via 3-cycles and rotate 2 in place
 * One way to think about finding operators:
     Let "A" = (bunch of moves that disturb only one cubie on a layer)
     Let "B" = rotation of that layer
     Then A B A' does something interesting...
 * Testing your visualization of the Rubik's Cube:
     have a friend make up to 3 moves.  Can you undo those three
     exactly?  4 moves?  5??  6???
 
assign:
 * Continue working on solving the cube, in any way you like.
   See how many of the suggested algorithms you can find.
   How many moves can you backtrack?

 * How many positions are there in the (R) (F) generated 2-face group?

 * See if you can come up with a step-by-step 
   written algorithm for getting back
   to the solved position from any element, i.e.
     First step: use the () operator which does () to get () pieces solved.
     Second step: use the ...

 * These kinds of solution techniques are  
   called "nested subgroups" method of solution.  Why?