Starting off: The notion of a group

In this series of posts exploring the Rubik’s Cube, the notion of a group is going to be central. A set S and an operation @ (like +, *, – ) are said to form a group if they satisfy the following conditions.

Screen Shot 2017-04-26 at 7.23.14 PM.png

(The text above looks different because I had to first type it in Tex and then take a screenshot since typing math in WordPress is a pain!)

Note here that an important property is missing; it doesn’t require the operation @ to be commutative(a@b = b@a)( groups which have this additional property are said to be abelian). Now it may seem curious to be suddenly defining a structure with some arbitrary conditions. This curiosity is not misplaced. In fact, several structures can be defined by considering conditions different from the ones stated above, but it turns out that this set of conditions yields a whole lot of nice mathematics; being able to recognize which concepts are the most useful is often required to good mathematics. And at first the definition of a group may seem to be entirely abstract, but that is precisely the purpose of these posts on the Rubik’s cube: to prove that Group theory is something applicable in the real world!

Now for a few examples. Consider the set of integers mod 4 denoted by \mathbb{Z}/4\mathbb{Z} : {0, 1, 2, 3}. Now consider the operation “+”  on this set. Then (\mathbb{Z}/4\mathbb{Z}, +) form a group. One way to see this is to draw an operation table like the one below:

Screen Shot 2017-04-26 at 7.48.43 PM.png

Operation table for {0,1,2,3} with +

The first the two conditions are clearly satisfied. Then for the third condition, 0 is our identity element. And each element has an inverse (the element that will “neutralize” it); for example the inverse of 1 is 3 and for 2 it is 2 itself.

Now let’s talk about the cube for a bit. First of all we need an effective way to refer to each cubie(the mini cubes) in the cube. We will follow the notation to refer to the faces as show in the picture below: up (u), down(d), left(l), right(r), front(f), back(b); we will use lower case letters to denote faces and reserve the upper case ones for the face twists.


To refer to the corner cubie, we will list the visible faces in clockwise order. So in the cube above, to refer to the corner cubie in the upper right we will say urf (or rfu or fur). The one in the upper left corner will be luf. We will adopt a similar method to name the edge and center cubes. So the edge cube on the front face to the left will be lf and the center cubie on the front face will simply be f. 

We will denote a clockwise twist to a face by X. So a clockwise twist to will be denoted by F. Finally we will need to sometimes talk about the space in which the cubies exist; let’s call them cubicles. Note that it is the cubies that move around while the cubicles stay where they are.

Let’s see the challenge that the cube poses with a few calculations. Two things are quite clear: one, a cubie that was at the corner in the solved configuration will always occupy a corner cubicle and a cubie initially at the edge will always occupy a edge cubicle subsequently. So now there are in total 8 corner cubicles and 8 corner cubies. These corner cubies can be filled in 8! ways (the first cubicle may be filled in by 8 cubies, the second cubicle then can be filled in with any of the remaining 7 cubies, and so on). And also in each cubicle, each cubie can have three different orientations. Therefore effectively and theoretically the corner cubicles can be filled in 3^8.8! ways. A similar calculation for the 12 edge cubicles gives 2^12.12!  ways to fill them in. Therefore theoretically the Rubik’s cube can have 3^8.8!.2^{12}.12! (\text{approximately } 5.19\times10^{20} ) different configurations. But not all of these are practically possible; that is there exist several configurations in these that cannot be reached using any sequence of moves from an original solved configuration. In fact, we will later figure out, using Group theory, how many of these are valid configurations.





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