$2\times 2$ Rubik's cube

The ``pocket" Rubik's cube has 6 sides, or ``faces", each of which has $2\cdot 2 = 4$ ``facets", for a total of 24 facets. Fix an orientation of the Rubik's cube in space. Therefore, we may label the 6 sides as f, b, l, r, u, d, as in the picture. It has 8 subcubes. Each face of the cube is associated to a ``slice" of 4 subcubes which share a facet with the face. The face, along with all of the 4 cubes in the ``slice", can be rotated by 90 degrees clockwise. We denote this move by the upper case letter associated to the lower case letter denoting the face. For example, F denotes the move which rotates the front face by 90 degrees to clockwise. We label the 24 facets of the $2\times 2$ Rubik's cube as in Exercise 4.4.18. The 24 facets will be denoted by xyz where x is the face on which the facet lives and y, z (or z, y - it doesn't matter) indicate the 2 edges of the facet. Written in clockwise order:

   
                front face:  fru, frd, fld, flu 
                 back face:  blu, bld, brd, bru 
                right face:  rbu, rbd, rfd, rfu 
                 left face:  lfu, lfd, lbd, lbu 
                   up face:  urb, urf, ulf, ulb
                 down face:  drf, drb, dlb, dlf

For future reference, we call this system of notation (which we will also use for the $3\times 3$ Rubik's cube) the Singmaster notation.

Exercise 4.8.1   Verify that the properties of a permutation puzzle are satisfied for this puzzle.