Summer of Math Exposition

I own a Rubik’s cube that I’ve never managed to solve and have always heard the “43 quintillion states” fact but never attempted to understand where that number comes from. I discovered these commutator operations on my own so it was interesting to see them named and given a mathematical notation here. The sections were broken up well and made it easy to keep reading. I think this post would be improved with better maths rendering and more clarity on some of the notation (specifically, cycles and transpositions). The section on Sune’s algorithm got very heavy in notation and became much harder to follow and I found myself skimming over the details

It was an enjoyable read, and it was definitely something interesting to read. However there were some short comings some of the text didn’t render correctly (probably not the authors fault). There were some parts that was missing explanations like the corner and edge constraints(would have like at least a link). some parts would have really benefited from having picture or animations(for example the cayley graph and example of parity). These factors I would say makes it hard of someone with little to not group theory knowledge a bit challenging to understand.

The entry was in layman’s terms. I feel it could be improved by introducing the use of notation and about how it works overall (For example the notation) if it is aimed at high school audience.

Loved the illustrations and it was quite readable. I learned group theory on my own in high school using Visual Group Theory and this feels a bit like it. It’s nice to give such a concrete example that people can play around with and I appreciate that you went into the combinatorics too. It feels like many math articles focus on either the group theory or the combinatorics making the Rubik’s cube just a method to discuss the “real” topic at hand, but yours places the Rubik’s cube itself at the forefront.

I know nothing about this topic, but the article held my attention the whole way. My only comment is that some of the symbols in your notation didnt render properly, making it difficult to follow in places. I’m on Chrome for android if you want to take a look at whats wrong. Great work!

Thank you very much for the contribution, your work has sparked my interest and will probably lead me to plan a teaching lesson for the math club!

From the perspective of SoME4, I think it might be worth to extend the article a bit in the future. Here are a few suggestions:

1. At the moment, the article is quite difficult for highschool students. I suggest that you change the target audience to undergrad/grad or to make an extra effort to simplify some of the things.
2. Tying in with the first point: although it is a lot of work, add more pictures! The Rubik’s Cube is one of those topics where it is difficult for uninitiated to follow the notation, but a few pictures would solve this problem.
3. You do introduce the meaning of most of the notation, but some things are left undefined which makes things harder to understand: When you define the notation R, L, U etc., you speak of the right face, the left face and so on. In the pictures, however, the cube is always shown with an edge in front. This makes it unclear, what you mean by front face and back face (and also there would be a FR and a BR?). My interpretation is that you actually think of the cube differently from what is shown in the picture. Another example is when you explain what a group is, it remains unclear what you mean by a concatenation of moves. Do you read them from left to right? from right to left? In mathematics, both conventions are used in different contexts. You actually do say it, but only later on in the section on the commutator.
4. Be careful how and when you introduce new words. I like the way in which you introduce what a group is. Unfortunately, you already use the word much earlier in the section on combinatorics. But at this point, it is not relevant that it is a group, instead you could simply refer to it as a set to avoid confusion.
5. It is always hard to find a balance between completeness and length. The important thing to remember is that you should very clearly state when you will assume something without proving it. If you do not, the reader might think that they are too stupid for not understanding it right away. In the section on combinatorics, I think you might want to clarify a bit more when you ask the reader to take a leap of faith instead of actually explaining something. This is particularly the case in “Contraint 2”, where you simply write “Here, we see that the sum of the flip values of all twelve edges must be an even number.”, the verb “see” implying that the reader should immediately understand why.
6. A final general advice: you do discuss a lot of things in the article. Unfortunately, that also means that it is not completely clear what the takeaway should be for me as a reader. For future works, it might be worthwhile to concentrate on one specific aspect. This does not only reduce the mental load for the reader who has to juggle with less concepts, it also allows you to take more time for fewer ideas, making them more accessible to an audience that is not accustomed to them.

I liked the section on counting the total number of possible states. I felt the section on commutators was a bit over-enthusiastic, and the Sune section could have been better motivated.