$3\times 3$ Rubik's cube

In this section we shall, for the most part, simply introduce enough notation (due to Singmaster [Si]) to allow us to check that the puzzle is in fact a permutation puzzle. We shall also introduce a two-person game which is easier to play and learn than solving the cube. The Rubik's cube has 6 sides, or ``faces", each of which has $3\cdot 3 = 9$ ``facets", for a total of $54$ facets. We label these facets $1, 2, ..., 54$ as follows:

The generators, corresponding to the six faces of the cube, may be written in disjoint cycle notation as:

$$\begin{array}{c} F= (17, 19, 24, 22)(18, 21, 23, 20)(6, 25, ... ...14, 22, 30, 38) (15, 23, 31, 39)(16, 24, 32, 40). \end{array}$$ (4.1)

The notation for the facets will be similar to the notation used for the $2\times 2$ Rubik's cube. The corner facets will have the same notation and the edge facets will be denoted by xy, where x is the face the facet lives on and y is the face the facet borders to. In clockwise order, starting with the upper right-hand corner of each face:

  
                front face:  fru, fr, frd, fd, fld, fl, flu, fu 
                 back face:  blu, bl, bld, bd, brd, br, bru, bu 
                right face:  rbu, rb, rbd, rd, rfd, rf, rfu, ru 
                 left face:  lfu, lf, lfd, ld, lbd, lb, lbu. lu 
                   up face:  urb, ur, urf, uf, ulf, ul, ulb, ub
                 down face:  drf, dr, drb, db, dlb, dl, dlf, df

Exercise 4.8.2   Check that the cycles in (4.1) are correct. (It is helpful to xerox the above diagram, cut it out and tape together a paper cube for this exercise.)

Exercise 4.8.3   Verify that the properties of a permutation puzzle are satisfied for this puzzle.

The following exercises require a Rubik's cube.

Exercise 4.8.4   Verify that $(R^2U^2)^3$ is the product of $2$-cycles $(uf,ub)(fr,br)$ on the edges.

Exercise 4.8.5   Let $M=FD^{-1}R$. Verify that $(M^{-1}D^2MU^2)^2$ is the product of $2$-cycles $(ufr,ufl)(ubr,ubl)$ on the corners.

Exercise 4.8.6   Let $M=R^{-1}D^{-1}R$. Verify that $M^{-1}UMU^{-1}$ is the $3$-cycle $(bru,dlf,urf)$ on the corners.

Exercise 4.8.7   Verify that $R^2UFB^{-1}R^2F^{-1}BUR^2$ is the $3$-cycle $(uf,ub,ur)$ on the edges.

Exercise 4.8.8   Verify that $(R^{-1}D^2RB^{-1}U^2B)^2$ twists the ufr corner clockwise and the dbl corner counterclockwise.