Conjugates and commutators for twisty puzzles -- so what? This question isn't just rhetorical. I want to know what I'm missing.
Twisty puzzle tutorials keep talking about how useful conjugates (operation sequences of the form ${XYX}^{-1}$) and commutators (${XYX}^{-1}Y^{-1}$) are. I understand the concepts but I just don't see how the concepts are actually useful.


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*Early in a solve, when there's plenty of space to work, I work out simple sequences on the fly like anyone would. They don't really take any brain power, so it's pretty clear these concepts don't help in this case.

*Later on things get more challenging. I need to think harder to get results. Maybe it would be useful to employ the concepts in this case, but it doesn't turn out that way. Even though the sequences I find often turn out to be conjugates, I'm not thinking "${XYX}^{-1}$", I'm just getting stuff out of the way, putting something where I want it, and then putting the stuff back where it was. It's like one person tells you rotating the whole puzzle won't scramble it, and you see the truth of that, then someone else comes along and says the symmetry group of these-and-such rotations is a normal subgroup of yada yada yada. Thanks, but I already got the gist.

*Let's say you're deliberately inventing a commutator or conjugate. The paradigm doesn't seem to save you any brain power because you still need to visualize what's happening. Just because they're easy to talk about, and due to their structure there are some constraints on their effects, doesn't mean they're easy to devise. You want me to square a 4-digit number in my head? Yeah OK, polynomials, but it's still going to strain my short term memory and be quite prone to error.

*Finally, let's take a look at some essential sequences for Rubik's Cube provided by Lars Petrus. None of them are conjugates or commutators. (I don't see how to refactor them, anyway.) (Here is a notation reference.)


$${R U R}' {U R U}^2 R' U^2$$
$${L U}' R' {U L}' U' {R U}^2$$
$$F^2 {U L R}' F^2 {R L}' {U F}^2 U^2$$
So sometimes a simple passing sequence happens to be a conjugate; thinking in terms of conjugates and commutators doesn't seem to simplify the harder parts; the most useful sequences seem to ignore the paradigm.
So why is this paradigm useful?
 A: My two cents:
Conjugates are useful because they let you take an 'operator' that does one thing, and adapt it to do a different but related thing.  Example: Suppose you have an operator that, say, twists two corner cubies on opposite corners of a face.  You can use this to twist two adjacent corner cubies by conjugation: (a) quarter-turn a face to turn the two adjacent cubies into opposite cubies; (b) apply the opposite twist operator; (c) un-quarter-turn.
Commutators are useful precisely for building operators.  I picture this as follows: Think of some face-turn sequence as 'an arrow' from the current cube state to some other state, and its inverse as 'an arrow' in the reverse direction.  A conjugate of the sequence is simply 'an arrow' in a somewhat different direction.  Now a commutator is simply a sequence followed by a conjugate of its inverse.  The result is to take an arrow in one direction, followed by a back-arrow 'at an angle', which leaves a residual arrow representing the net change.  If you plan ahead, you can make the residual arrow quite short, i.e leave only a few things changed.
I happen to have read Hofstadter's article in Scientific American ['Metamagical Themas'] on Rubik's Cube back in the 80's.  I was struck by his example of a 'monoflip': Consider a sequence of turns that leaves the top layer unchanged except for a single flipped edge cubie; the rest of the cube may be scrambled as you wish.  It is easy to turn this into an operator to flip two edge cubies: Simply commute the 'monoflip' with a turn of the top face!  It is possible to apply this insight to create as many operators as needed to solve the Cube.
A: In the days when I used to play with the Rubik Cube, I used conjugates a lot, because they reduce the number of sequences that you need to remember.
Suppose, for example, you know a sequence of moves $R$ that perform a $3$-cycle on three specific edge cubies $E_1,E_2,E_3$, perhaps the three surrounding a corner.
Then, if you want to do a $3$-cycle on any other set of $3$ edge cubies $E_4,E_5,E_6$, you first perform a sequence $X$ that moves $E_4 \to E_1$, $E_5 \to E_2$, $E_6 \to E_3$. Then you do $R$. Then you do $X^{-1}$ to move the three you wanted move back to their original positions - but cycled in the way you want. So, you have carried out the conjugate $XRX^{-1}$.
Of course, you have to work out $X$ on the fly, and remember it while you are doing $R$, but I found with practice that I could manage that. And you won't break any speed records using these techniques, but they do enable you to solve the cube while minimizing the amount of memorizing required.
I found commutators also useful, but that was more for specific sequences $R$.
