Mathematics.

# Mathematics.

### Groups.

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group. Groups can be denoted explicitly, but when the order of the group becomes large, it is sometimes useful to denote the group as a linear combination of some elementary operations, called generators. Since the number of possible permutations of the Rubik's Cube Group is enormous and the generators are clearly the six face permutations, one denotes the Rubik's Cube Group as G=< F, B, R, U, L, D > , a subgroup of Sym(48). We have to check that this Group satisfies the group axioms:

*Closure: any two moves from {F, B, R, U, D, B} results in a Rubik' Cube state.

*Associativity: it doesn't matter if you first apply move sequence AB, and then C; or first apply A, and then BC.

*Identity: the identity element of the group is the cube with all faces of a solid color, thus the solved cube is the identity element.

*Invertibility: for any given cube, there is a move sequence that brings the scrambled cube back to the solved state, which is the identity.

Recall that a group doens't have to be abelian, which means that one cannot automatically invert a move sequence AB to BA. The Rubik's Cube Group is in particular not abelian, even though F and B commute (but for instance U and F do not).

Now we can use all sorts of theorems from group theory for our problem to determine God's Number. We will especially use this knowledge for optimal solutions and in all mathematic papers.

### Graphs.

If you don't fully understand what is said above, then maybe the Rubik's Cube Graph can help you visualize the Rubik's Cube Group.

Suppose you have the solved cube written on a piece of paper (we usually denote it as (1)). Write down all positions which have distance 1 to the solved state, like R, U', D2, etc, and connect these to the solved state. Now, write down all new positions which have distance 1 to some allready written position and connect positions if they differ only one elementary operation. At this point you should realize that this will allready create an enormous amount of positions and arcs connecting all kinds of positions. You should also realize that it is impossible to do this without crossing any arcs, making it more confusing and chaotic. And this is only up to distance 2 from the origin. To avoid this, we could imagine a 3-D world where all possible cubes are present, and all arcs connecting two cubes if they have distance 1 to each other. A picture of this is called a graph, and in this case, it would look extremely complicated.

God's Number can now be determined easily if you have enough time. For any given state, calculate its distance to the origin, and determine the greatest distance. However, this would take a tremendous amount of time, because first there are so many different positions. And second, calculating a distance is also a very complicated job, thus taking some time. We will have to come up with some very smart idea to determine God's Number.