From now we will use Rubik's Cube for some of our examples of permutations. For that reason, we need a reasonable notation to describe the moves that can be made. Here we are talking only about the standard cube, although much of what we do can easily be applied to other versions.
The cube has six faces, each of a different color, but different cubes have different coloring patterns, so it is not useful to refer to the colors.
Here is a good method to describe a general move. Imagine that you hold the cube in front of you looking directly at the center of one face, and with the top and bottom faces parallel to the ground. There are six faces--the front and back, the up and down, and the left and right. Conveniently, the first letters of these words are all different: , , , , , . Rearrange them as ``'', and it reminds you of the English word, ``befuddler'', which is also quite useful for describing some aspects of the cube.
There are six primitive moves that can be made--any of the six faces can be turned 1/4 turn clockwise. Obviously, if you want to turn a face counter-clockwise, that's what you would do, but to keep the description mathematically simple, remember that a single twist counter-clockwise is the same as three clockwise twists.
By ``clockwise'' is meant that if the face in question is grasped in the right hand, it is turned in the direction pointed to by the right thumb.
We will use the befuddler letters as names for these primitive moves. Thus the move ``'' means that the front face is turned 1/4 turn clockwise, et cetera. We can combine letters as well. ``'' means first twist the front face clockwise, then twist the up face, then the back face. All twists are 1/4 turn clockwise. To turn the front face by 1/2 turn or 3/4 turn (3/4 turn clockwise = 1/4 turn counter-clockwise), use the notation or . Note that , so we could write as equally well. We will tend to use the form here.
As a final example, means to turn the front face a half turn, then twist the up face 1/4 turn counter-clockwise, followed by a 1/4 clockwise twist of the top, back, and down faces.
If you think of the entire cube as being composed of a bunch of smaller ``cubies'', the befuddler notation gives a good method to name the individual cubies. The cubies in the corners are identified by the three faces they share. The cube on the up right front can be called , and so on. The edge cubies are identified by the two faces it lies on, so the one on the up and front faces would be called . But in order to distinguish between the cubie and the transformation that is a rotation about the up face followed by a rotation about the front face, we will put boxes around the cubie names: and for the cubes just mentioned.
With this cubie notation, we can describe (using our permutation cycle notation) certain results that transformations may achieve. For example, refers to an operation that cycles the left-down, front-down, and right-down cubies. The left-down cubie moves into front-down postion, et cetera.
The notation still isn't perfect. You'll find that when you solve Rubik's cube that sometimes a cubie will be in the right place in the cube, but rotated (if it's a corner cubie) or flipped (if it's an edge cubie). But we can describe it as follows. Suppose there is an operation that leaves everything fixed, but flips and in place. We can write this as: , , .