When one disassembles a standard 3X3X3 Rubik's Cube one discovers that it is just a fixed
3D cross connecting the center squares, and a bunch of edge and corner pieces that move
around the stationary cross. Consider the state diagram for the conguration space of all
assembled cubes, that is one state for each and every possible way of putting the cube back
together. In order not to count a state multiple times (eg placing it upside down) we will x
the orientation of the center squares (ie the red center square will always be up and the blue
center square will always be to the right).
(a) Write down an expression for the number of nodes/states of this diagram, explaining the
reasoning for each term. How many decimal digits does this number have? Is it feasible
to keep track of all the nodes?
(b) Consider all of the obvious legal (without disassembly) basic minimal moves/actions/edges
for going from one assembled state to another (while preserving our fixed orientation of
the center squares). What are these moves? How many arrows/edges are there leading
out of each node? and how many are leading in? [Note: the diagram is not connected
since one can assemble unsolvable cubes.]
(c) For these actions how many times does one have to repeat such an action to get back to
the original state?