TITLE: Can RUBIK's cube be solved using group theory?
QUESTION [13 upvotes]: Can RUBIK's cube be solved using group theory? If yes, how can we use it to solve a $2\times2$ Rubiks Cube?

REPLY [7 votes]: I cannot allow this question to go by without mentioning: Morwen Thistlewaite.  This is a British mathematician who used group theory to find the smallest known bounds on how many face twists it takes to reach the solved state from an arbitrary state, along with some knowledge of which twists to take (a.k.a 'God's Algorithm').  Instead of positioning individual cubies in sequence, his is a more holistic approach called 'descent through nested sub-groups'.  It works something like this:
A) Begin with the whole Cube group: G.
B) Pick a set of face twists generating a proper sub-group: H [example: the sub-group generated by twisting faces L, R, F, B, but NOT U, D].  For best results pick H such that |G|/|H| is as small as possible.
C) For each coset of H, find the shortest sequence of twists which will transform any member of the coset to a member of the sub-group [each coset is of the form aH; if b is in aH, then bH=aH; find the element b in aH generated by the fewest twists, and take the twist sequence for the inverse of b].
D) If H=E (i.e H is the trivial sub-group), stop.  Otherwise replace G with H and return to (B).
To solve a scrambled Cube with this approach, one must determine which coset of the top-most sub-group contains the Cube's starting state, and make the sequence of face-twists calculated in (C) for that coset; this will change the cube to a scrambled state within the top-most sub-group.  Repeat for each successive nested sub-group until you reach E; at that point the Cube has been unscrambled.
With this approach Thistlewaite was able initially to show that unscrambling any scrambled state takes no more than 26 (?) twists [source: 'Metamagical Themas' by Hofstadter, in Scientific American (I forget which exact issue)].  Since then he has been able to reduce this number enough to determine the theoretical maximum number of twists required.