Are there any unanswered questions regarding math behind Rubik's cubes?

Question

I'm a student who has to do either a capstone(as in release a product of some sort) or research project as part of graduating. Right now I'm planning on doing something related to computer science and Rubik's cubes, or at least math and Rubik's cubes. I'm wondering if there are any questions that might be unknown, or not have been looked at very much regarding them.

Answer 1

Given the open endedness of this question, I hope you don't mind if I answer with an anecdote.

When I was an undergraduate at MIT, I took abstract algebra. The professor was the legendary Michael Artin. At the time I was reading Adventures in Group Theory by David Joyner, and in the book, it stated that if you take any two random elements from the Rubik's Cube group, there is about a 50% probability that those two elements will generate the whole group. But the book didn't really offer a proof of that. So, I thought to myself, surely Professor Artin would have some insight here!

I went to one of his 1-1 office hours. I had never gone to one before, so this was the first time talking to him directly. When it was my turn, before I could introduce myself, he said "Wait! Let me see if I can recall your name. You're timidpueo!" I said yes, but how did you know? Then he pulled out a ring of flashcards with every undergraduate student's name and picture in it. "I try to remember all my students," he said.

After some more small talk, I told him about the book I was reading and asked him my question, "How do you prove that the probability is around 50%?" He thought for a minute, then put his hands up in the air. "I have no idea!" he exclaimed.

So yeah, if someone like Prof Artin doesn't have much insight here, this could be an area of research. And I think computers could be of help. If you're curious, the part of the book that makes the claim is Example 10.3.2.