So I just started learning Abstract Algebra and I've recently appreciated an introduction to group theory. Among the interesting avenues of group theory is found in tackling a Rubik's Cube.
I've never solved any cube, not even the 3x3x3 cube, and many years ago someone taught me a single sequence of moves to interchange some edge faces. But other than that, I have not received any external assistance whatsoever (other people, online resources, etc.) in learning about the cube.
I want to learn more about the cube and discover things about group theory, but I don't want to spoil my own recreation with the cube. That is, there seems to be an endless stream of helpful and insightful resources (from both mathematical and recreational perspectives) online for understanding/solving the cube, but I still want to experience the reward of finding solutions myself.
My question: Am I holding myself back (mathematically) if I don't learn what others have discovered about the cube by waiting until I can solve it for myself without help? Is it realistic to solve a cube without help in a reasonable amount of time if I practice it consistently on a regular basis?
I am excited to learn all about how group theory applies to Rubik's Cubes but I'm unsure if the process of solving one by myself will take too long.
No, not really. Mathematics is not a sport, solving Rubik's cube faster than others is. I suppose mathletes would disagree with me on the former and football players on the latter. But my point is that mathematics is about reflection, meditation and contemplation.
A theologist does not solve the riddle of God on his own, but reads what others have written, consults with other theologists, and thinks about it. (I doubt God exists, but that's another can of worms).
So go ahead and read the little booklet that came with your cube. Solve it at your pace availing yourself of the techniques others have discovered. Try to invent your own techniques.
Besides, Rubik's cube is not an important application of group theory. Go up to a university professor who specializes in group theory and tell him "I am a genius at group theory, I can solve Rubik's cube in one minute." Watch how unimpressed he is by this.
Read this: http://www.math.uconn.edu/~kconrad/math216/whygroups.html