Reply To: !
@Gen_ruikt
I’m a big Rubik’s Cube fan! Here’s a cute bit of maths related to it. Every now and then, you see videos claiming that you can solve a Rubik’s Cube from any position by just repeating a certain sequence of moves over and over again. (For example, they might claim that if you move the right side up and then the upper side left and keep repeating this, you’ll eventually solve the cube.) Now, anyone who has solved a Rubik’s Cube knows intuitively that this is nonsense, but how can we prove it?
We note that the transformations of the Rubik’s Cube form a group (if you aren’t familiar with this concept, then I’m sure everyone will be happy to recommend their favourite algebra texts). In the language of group theory, the nonsense claim is simply that the Rubik’s Cube group is cyclic. So we just need to figure out if our group is cyclic.
One important property of cyclic groups is that they are abelian (the order in which you perform transformations doesn’t matter), but for the Rubik’s Cube, order does matter. To see this, compare moving the right side, then the top side, with moving the top side, then the right side. Thus, the Rubik’s cube group can’t be cyclic. If someone tries to tell you otherwise, you can chuckle at the confused person who believes in non-abelian cyclic groups.
You might be thinking that the above argument only works if we consider the sequence of moves as an indivisible unit that we somehow apply simultaneously (and you would be correct). If we allow the person performing the repeated action to stop mid-sequence, then such an “always solving sequence” can be found. Can you come up with one such sequence? [Hint: the easiest one to find is really trivial, but you won’t be able to write it down explicitly.]
In terms of puzzles, I mostly stick to the 3×3×3, but I’m a big fan of blind-solving. I find the techniques used for blind solving are much more mathematically interesting than the speed-solving methods. If you don’t yet know how to solve a Rubik’s Cube blindfolded, it may be a fun project to try to come up with your own method for doing so (some familiarity with algebra may help). However this is a hard problem and I certainly wasn’t mathematically mature enough to solve it when I first learnt blind-solving.