## 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 facets", for a total of facets. We label these facets as follows:

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

 (4.1)

The notation for the facets will be similar to the notation used for the 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 is the product of -cycles on the edges.

Exercise 4.8.5   Let . Verify that is the product of -cycles on the corners.

Exercise 4.8.6   Let . Verify that is the -cycle on the corners.

Exercise 4.8.7   Verify that is the -cycle on the edges.

Exercise 4.8.8   Verify that twists the ufr corner clockwise and the dbl corner counterclockwise.

David Joyner 2007-09-03