If you're primarily interested in speed then obviously memorize whatever the best method is, looks like CFOP (if you're willing to memorize tons of algorithms, the noob version of CFOP I learned was much slower)
I think Human Thistlethwaite is the coolest algorithm-based method since it makes the cube overall better and better oriented instead of making layers or blocks sequentially
However, most interesting to me is to solve without algorithms/memorization using commutators
https://web.archive.org/web/20231207094142/https://www.ryanheise.com/cube/commutators.html
I can solve most of a cube without any special technique at all, and then the last few pieces can be solved in this way. Usually one of the steps is a single face rotation. So I might rotate a corner completely messing up the cube, then rotate a single layer to place a different corner there, undo the steps that completely messed up the cube, then rotate that single layer again. The result is that only the two corners I rotated are altered but everything else is unaffected.
The value of this method is that it makes it fairly trivial to solve arbitrarily large cubes (the main thing to adjust to is that the even-numbered cubes don't have fixed center squares). I've solved 4x4 and 5x5 without memorization.
Interesting, I might check that commutators method out at some point. I have a 4x4 that I solved once guide in hand.
Everything was the same as 3x3 beginners method except a bunch of algorithms to memorize for the edges on the last layer being possible to be more jumbled because there are more of them.
I decided it would be more interesting to me to get faster at the 3x3 and left it at that.
I like the idea of knowing more ways so that you ultimately understand the cubes mechanics better.
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