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How does one figure out how to solve a puzzle?

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I am new to cubing in general. Most of the cubes I own solve like a 3 x 3 or something pretty similar. The only cube I have that has a fairly different structure is the Skewb. I did use a tutorial to start solving the 3 x 3 puzzles. After that, I started making my own algorithms in the form of a, b, a', but that is all I can do. The algorithms are inefficient, compared to ones used in popular methods, where doing the sequence backwards isn't necessary and that form only works for puzzles with a similar structure to a 3 x 3. I would like to be able to learn how to create algorithms to solve any given puzzle and how to analyze its structure. What do I need to know? Where do I start? Do I need to know group theory or any other form of mathematics (If so, I have a good amount of knowledge until Calculus and know a bit of set theory)? If so, where can I learn or what do I need to learn to be able to understand the subject? etc.

Thank you
 
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#2
Yes, the field of mathematics that you will want to learn is group theory.

What you've got there (A B A') is called a conjugate, and it's a handy trick for changing which pieces B has an effect on. There's a related concept called a commutator, which has the form A B A' B'. Basically if you want to solve a piece (or group of pieces), then derive a sequence of moves, A, that solves those pieces, without worrying what happens to the rest of the puzzle. Then you derive a sequence of moves B that isolates those solved pieces from any region that A changes. After you undo A, and then undo B, the only pieces affected will be the pieces that you solved, the pieces that were originally in the place of the solved pieces, and the pieces that were originally in the position that B moved the solved pieces to after you had solved them. If you derive your moves carefully, and conjugate your commutator, you can get it so that three groups of pieces are simultaneously cycled into their correct places.

You can solve a 3x3, and any related cubic puzzle using nothing else apart from this concept. As far as I'm aware, this works on all twisty puzzles, even the Skewb.

That's probably tricky to follow in text, but there are many resources on commutators and conjugates around if you do a search. Also, here is an excellent video that explains how to come up with your own commutators.
 
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cmhardw

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#3
I really like AlphaSheep's post on conjugates and commutators, those are definitely useful for learning to solve a new puzzle.

In addition to conjugates and commutators, I use the order of a permutation very often when trying to discover algorithms on a new puzzle.

Try moves like "sexy move" and "sledgehammer" on the puzzle over and over until you return to the state you started from. Let's say it takes n repetitions of your alg to do this.

If n is divisible by 2, then n/2 may be a useful alg. If n is divisible by 3 then n/3 (and by extension 2n/3) may be a useful alg. If n is divisible by 6 then treat it like both cases above.

The "sexy move" on 3x3 is R U R' U' but in general the sexy move is a commutator of the form:
[turn a side clockwise, turn an adjacent side clockwise]

The "sledgehammer" on 3x3 is R' F R F' but in general it is a commutator of the form:
[turn a side counter-clockwise, turn an adjacent side clockwise]

You can even make these more general by saying that sexy turns the sides in the same direction in the first half of the commutator, and sledge turns the sides in opposite directions in the first half of the commutator.

Using the orders of sexy and sledge commutators is almost always helpful on a puzzle, or a slight variation on the sexy and sledge commutators.

Also try commutators of the form:
[A, B C B'] I have found those to be very helpful on the helicopter cube.

I'll try to think of more strategies that I use, and I'll post about them here if I do. With these and what AlphaSheep said about commutators and conjugates, you should at least have a good start to solving any Rubik-style puzzle.

Good luck, and have fun!
 
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#4
I used a tutorial for Skewb (Kind of? More like a mix of different ideas from different tutorials.) and for finishing the last centers I have 1 algorithm I learned online for fixing 3 centers that I use a conjugate to fix any set of 3 unfixed centers into thay case, I have 1 algorithm for a 4 centers undone (Double 2 centers swap, like how an E-perm swaps double 2 corners) and use a commutator to set it up into that case, and I just use the 4 centers case to get the top center fixed when I have too many centers to fix. I know how I'd use conjugates for the last 4 corners too, but it'd be a long and pointless amount of moves and it'd be too hard to remember how to reverse the set up.
 
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