Terry Tao has a really great post where he depicts group actions (every Rubik's cube move sequence is a group action) as arrows in a graph, and with this picture, commutators can be seen as arrows that make a relatively small change: https://terrytao.wordpress.com/2012/05/11/cayley-graphs-and-... (see figure 14).

If you want to take the arrows very literally, you can think of the commutator as measuring the amount of curvature in the space the arrows lie in. The less of a difference the commutator makes, the less curved - with a commutator that doesn't do anything meaning the space is flat. Expressed in these terms, the two-part algorithm technique amounts to finding places in Rubik's cube algorithm space that are almost – but not quite – flat.

This is how one of my professors described it to me back in college, inspiring me to derive my own solution to the Rubik's cube. (And later for the "Square One" cube.) And I think the general concept is easy to describe to anyone: do something to the top and mess up the bottom, rotate the top and then do the exact opposite.

To me, it's much more satisfying than memorizing/learning existing solutions, even though I'll never be as fast as those that do leverage those solutions.

Either way, though, I'd recommend buying an inexpensive "speed cube" from Amazon, they are much less frustrating than the original design. (It doesn't bind up in the middle of a transformation.)

It had a bit of the opposite effect on me as a lazy kid.

It showed me that intuition only goes so far, beyond a point one has to put in some serious hard work (memorize patterns/algos etc) to see how deep one can go in any field.