What do I need to learn to solve the math behind a Rubiks cube?

Question

I stopped studying math at precalc. What should I study to be able to understand how to solve a Rubics cube using mathematics? I don't think this is an opinion question, and I really want to learn this on my own, but I don't know where to start. I saw some of the other questions related to this, but I think I need to get to where most of them are and would appreciate a direction.

Answer 1

The math of the Rubik's Cube is rooted in an area of math called abstract algebra, specifically group theory. Groups are a mathematical construct which is a generalization on the algebraic relations we are familiar with in the real numbers. A group $G$ is a set of elements $$G=\{g_1,g_2,...,g_n\}$$ combined with an operation which I will denote here with $\oplus$ with which takes two elements and then gives a third: $$g_i\oplus g_j=g_k.$$ (Notice that I assumed that the group has a finite number of elements. This is not always true, but it is true for the Rubik's group, so I'll stick with finite groups). This operation $\oplus$ has some properties that adding in the real numbers has:

• $g_i\oplus(g_j\oplus g_k)=(g_i\oplus g_j)\oplus g_k$. This is called associativity.
• One of the things in $G$, which I will call $e$ has the property that $e\oplus g_i=g_i$ and $g_i\oplus e=g_i$. So combining an group element with $e$ does not change it. We call $e$ the identity, and you can see that it acts like $0$ in the real numbers since we can add $0$ to any number $x$ and the result is $x$.
• For every element $g_i$ in $G$, there is an other element of $G$ which I will call $g_i^{-1}$ which has the property that $g_i\oplus g_i^{-1}=g_i^{-1}\oplus g_i=e$. The element $g^{-1}_i$ is called the inverse of $g_i$, and it "cancels out" $ g_i$, just like if $x$ is a positive number, we can add $-x$ to get $x+(-x)=0$. Note that the inverse of $g_i^{-1}$ is $g_i$.

Notice that missing from this list is the property from real numbers that $x+y=y+x$, called commutivity. Since we can get some interesting results from studying groups where $g_i\oplus g_j\neq g_j\oplus g_i$, we do not require this. In fact, the group corresponding to the Rubik's Cube is not commutative.

There are lots and lots of groups which all have the above properties that have application all over math and science. Consequently, group theory is a very deep topic, and some people spend their entire career doing research in this field.

The group corresponding to the Rubik's Cube is called the Rubik's Cube group. The elements of the group correspond to a series of twists (and a series could consist of just one twist) such as twisting the top level to the left one quarter turn, or twisting the right side halfway around and then twisting the bottom to the left a quarter turn. The identity element $e$ corresponds to just doing nothing. The inverse element to a twist is just reversing the twist.

David Singmaster and others have done research on how the elements of the Rubik's Cube group relate to each other, i.e., when one series of twists gives the same result as another. There are lots of great results, such as there being over $43$ sextillion elements of the Rubik's Cube group. This means that there are over $43$ billion billions of ways to arrange a Rubik's Cube. Another great result is that any one of these can be made from 20 or fewer basic one-step twists, so you can solve a Rubik's Cube in 20 or fewer steps from any starting position! In general, figuring out which 20 moves those are is a hard problem, but maybe you'll figure out a solution!

For more info on group theory, see this, and for more information on the Rubik's Cube Group, see here.