Edit: this is a work in progress, I failed and hit submit thread before I meant to. Bear with me.
So a few months back I started solving the square-1. Unlike the cubic twisty puzzles, I became very interested in the theory behind this puzzle. However, as I tried to learn/discover as much as I could, I soon found that the theory behind this puzzle remains vastly unexplored (or at least very hard to find). Quite luckily I had some very valuable resources available to me in the forms of some very experienced sq1 solvers like Andrew Nelson, Tomas Kristiansson, and Dan Cohen. Through talks with them and some good old-fashioned experimentation I was able to learn quite a bit, and I would like to share it here.
Some things I will talk about:
1. Beyond Vandenbergh: a few modifications of the Vandenbergh method that are undeveloped or rarely used. (You LBL people are on your own)
2. Turning Styles: I've noticed almost every solver uses one of 3 distinct turning styles, and I'd like to explain them and give examples. I know its not technically theory, but I think its very interesting and didn't want to make another thread.
3. Parity: An explanation of what parity is, how it works on the square-1, and why the parity algorithms work.
1. Beyond Vandenbergh
1.1 Parity CP
Parity CP is method where parity is determined along with corner permutation (instead of during EP), and then parity and corner permutation are solved at the same time. The main advantage of this method is that it only requires 8 additional CP algs, while eliminating the 50 parity EPs. Andrew Nelson is the only person known to use this, and I'd say he's been rather successful with it. His explanation and the algs he generated can be found here.
EOCP does exactly what it sounds like- orients edges and permutes corners at the same time.
The idea that serves as a sort of precursor to this and eventually led to me thinking of this method is looking ahead to CP. A lot of good sq1 solvers do this, and I recently switched all of my EO algs to ones the preserve CP in order to make this easier. The algs themselves are actually faster than the ones I was using before, and having the extra lookahead helps a lot. But this got me thinking "If certain EO algs switch corners, and other don't, then certainly you should be able to solve EO and CP at the same time."
Now that I had this idea I had to determine whether or not it was actually practical. However there are lot of possible cases:
6*6 CP in each layer *6 orientation cases = 216 cases
However I soon realized that every case except for adjacent-adjacent can either be reduced to a single edge swap or solve EO with a simple M2 ((1,0)/(-1,-1)/). So with and M2 preface you would only need to learn 72 algs plus the 8 regular CPs.
1.2.c The Algs
Unfortunately most of the algs for this do not exist. A few of them do, in the form of the various EO algs floating around the internet, but the rest would need to be generated. The problem is that Jaap's solver has no was to ignore stickers, so to find the best algs 72 cases* 100 EPs = 7200 cases would have to be tested. Someone proposed writing a program that would determine and generate the inputs for these cases, but I have no idea how to do that. If an experienced programmer who is familiar with Jaap's Solver thinks they can help it would be much appreciated.
2. Turning Style
The only person who really falls outside these 3 groups his Kazuhito Iimura. His turning style is unique and rather difficult to emulate, so I won't really go into it here.
2.1 Beginner Style
This is characterized by lots of wrist turning of the D layer, and turning the U and D layers separately. Most notably / moves are always sliced clockwise with a lot of wrist movement. Most people will begin to properly fingertrick the U and D layers as they get faster, but the wristy / can stay with them well into the low 20 averages. I'm not going to bother finding an example, youtube is filled with videos of this. Chances are very good that you do this now or did so in the past.
2.2 Polish Style
This style is not exclusive to Polish square-1 solvers, but it originated from and is extremely popular there. The U and D layers are fingertricked normally, but whats notable is the / turn. All turns are still clockwise, but the wrist isn't used at all (it often doesn't move at all!) Rather, its turned with some combination of the thumb, ring, and pinky fingers that I have yet to fully figure out. The advantage of this is that there is very little regrip time, the right hand stays in its place allowing you to turn the U/D layers without a delay. Some good examples are Grzegorz Prusak and Piotr Padlewski of Poland and Lee-Seung Woo of Korea.
2.3 Western Style
This is most common among Americans and some western Europeans. It is characterized by alternating clockwise and counterclockwise / turns, with varying sets of U/D fingertricks based on whether a cw or ccw / turn has just occurred. There is zero regrip time when you are alternating, which means some algs can be very fast. However, regrips are sometimes forced, like when a clockwise / is followed by (4,0) or something similar. In cases like these its common for only clockwise / turns to be used, but they can still be quite fast. Some good examples of this style are Dan Cohen, Stephanie Chow, all of my CPs, and Dan's Adj Parity.
3.1 What parity is
(most of what I'm about to say is rather incorrect, I should have known better than to talk about parity on 4x4) Parity occurs when there is an odd number of total swaps on the puzzle. Most people are familiar with parity as it occurs in the reduction method on 4x4. Oll parity is simply 2 edges that need to be swapped, and occurs when there has been an odd number of slice moves perform on the cube, hence the name parity.
3.2 Parity on Square-1
Parity on square-1 is the same thing- an odd number of swaps. However, the way it occurs is different from on cubic puzzles. You may have noticed that solving parity is the only step that requires you to leave cubeshape. This is because while in cubeshape, the square-1 more or less follows the laws of the 3x3 (i.e. no single swaps allowed!) However, when not in cubeshape, there is a lot more freedom and "The Laws of The Cube" do not apply (because its not a cube any more). I'll show you an example using one of the simpler parity algorithms, one of Andrews parity CPs:
The bolded red / is where the parity of the puzzle is changed, lets evaluate why. The first thing you may notice is that the shape of the puzzle does not change. This is usually a very good indicator of the point where parity changes. Now if you look closely, you can see that you are actually swapping three corners in the U layer with 3 corners in the D layer- which is three(an odd number!) of 2-swaps. More specifically you are swapping UFR-DBR, UR-DR, UBR-DFR. Then you solve cubeshape by undoing the setup moves you performed earlier. Once its back in cubeshape we can clearly see the results: 1 diagonal corner swap in the U layer and 2 diagonal corner swaps in the D layer- 3 swaps! Almost every parity algorithm generally follows ABA', where A leaves cubeshape, B changes parity without changing cubeshape, and A' solves cubeshape again. Sometimes orientation is disturbed, so I guess you could say they follow ABA'C, where C solves orientation if needed. Here are a few more examples, broken down into their components:
Opp-H: /(-3,-3)/(3,0)/(-3,-3)/(2,0)/( -4,2)/(4,-2)/(1,0)/(-3,-3)/
A: /(-3,-3)/(3,0)/(-3,-3)/(2,0)/( -4,2)
You probably noticed that A and A' are not inverses of one another. This is because we have taken the alg from earlier, and inserted and Nperm into A. As a result, some moves have been cancelled and A is inherently different from A' (it would be counter-productive to insert the same N perm back into A')
Here is another example, where B is face turns instead of a / move:
ccw O: /(3,3)/(1,0)/(-2,-2)/(2,0)/(2,2)/(-1,0)/(-3,-3)/(0,2)/(-2,-2)/
In this case, B is a clockwise 6-cycle, which is 5 swaps(think of BLD solving with Classic Pochmann, solving a 6 cycle uses 5 swaps) When back in cubeshape, you can see that 5 swaps have been performed: cw O (3 swaps) and H perm (2 swaps) resulting in a ccw O perm.
This was the first parity algorithm I learned, and I used it almost exclusively for a very long time. I noticed that this alg followed the ABA' formula and decided that the (2,0) in the middle of the alg had to be the secret behind parity. On a whim, I decided to do (-2,0) instead and discovered my very own clockwise O-perm without having to mirror the whole alg! (note that C changes to (1,0)/(2,2)/) At the time, I had no idea why it worked, but at least I figured out how parity algorithms seemed to work. When I brought this up with Tomas, he pointed out that an odd number of swaps was wholly possible while out of cubeshape, and the final piece fell into place.
I am still not sure how some algs like adjacent parity work, I think there is an insertion somewhere that cancels with B. I'll look into it some more later.