Been working on some megaminx stuff. Mostly a proposed LL system, which I haven’t learned yet, but I intend to learn it and use it, and which I have put a lot of thought into. It provides a feasible 2-look system for LL (sort of). First, I would like to propose a new notation for megaminx, which can easily define any twist. Spoiler tags will make this post shorter, and easier to find things.
So now for the LL system, although it is actually more for last 2 faces, but I think of it as being for LL. First, solve all but two adjacent faces. Then, go Petrus style, and orient edges while solving one of those faces, minus the pieces which are in the other face, and one corner/edge slot (like an F2L slot). Solve the pair in such a way that you end up with at least 2 adjacent corners on the last face oriented. This can be done intuitively, I recommend just messing around with stuff until you can do this, the same way you would with blockbuilding. You have a fair bit of freedom to do this, but if you are struggling, I developed a backup system which you can use. I’ve not learned this yet and didn’t spend long on it, but it will do as it is only a backup, I don’t recommend relying on it (though it may be useful sometimes). Actually, you can do any way of solving up to LL plus one slot which leaves you with remaining edges oriented, but that’s how I solve it.
Slot + 2 adjacent oriented corners in LL:
So, we have LL with edges oriented and two adjacent corners oriented, without too much effort. We now have to solve CO (with only 6 unsolved cases), EP, and CP. Instead of doing this is 3 steps (which is fairly easy, and essentially how I used to solve except I didn’t control corner orientation), it is now feasible to do it in 2, by splitting up the EP step.
First step: solve CO, and influence EP. You want to end up with (at least) two edges from non-adjacent faces to be solved relative to each other. This gives either the EP case which can be solved with R2 U2’ R2’ U’ R2 U2’ R2’ (by far the easiest and fastest EPLL, up to mirrors/inverses of this depending on your preference), or EP skip.
Second step: You now have a subset of PLL, which is feasible 1-look. You either have CPLL (which any serious megaminx solver should know already, 15 cases, 9 without mirrors/inverses), or horseface PLLs (60 cases on the site, I haven’t yet looked into how it reduces but it’s less than 30 non-isomorphic cases anyway). This step is documented already, so I won’t cover it here. Until you learn the PLL subset, you can 2-look this using the fastest EPLL for megaminx, which is easier to recognise than normal since you would usually have more possible EPLL cases to distinguish it from, then CPLL (or just CPLL for EPLL skip).
The first step is split up into the OCLL cases. There may be a lot of cases to recognise (6*20=120), but the system is easy enough to follow, and you don’t need many algs, as each one solves multiple cases.
Recognise uses certain pair of edges each time, which are from non-adjacent faces. Edge positions are numbers from 1-5 (given later for each case individually). Note the number of the first edge, then the number of the edge which goes 2 places clockwise of it (eg. UR then UL). Look up the row of the first number, and the column of the second, to get the number of an alg. Then look up the alg from the given list. For each case I have tried to give the best algorithm which will solve it. However, if you decide you don’t like one of the algs I’ve given (they are numbered in order of how good I think they are, although I didn’t spend too much time on fine tuning the list. High numbers mean a less preferable alg) you can switch which edges you use to recognise. I recommend using the second edge you normally use as the first edge, and the appropriate other edge as the new second edge. Following from earlier, if you started with UR->UL, switch to UL-UB. This might also provide an advantage other than a better alg, obviously at the expense of longer recognition time: you know which EPLL case you will have. Why? Well, you are switching to a better alg, which will not solve the first 2 edges you looked at, or it would be the given alg for that case, so you have the 3-cycle case. Sometimes, this may well be better than using the algorithm I’ve listed in the table.
If you get good enough, you can check alternative pair of edges as you start to solve this step, which can give you an indication of which EPLL you will get (or you can just try tracking pieces). You can also recognise using multiple pairs of edges if you like: check several pairs and solve the best case you see. With practice, it might even be reasonably quick to check all 5 pairs, especially since the second edge in each pair is the first edge in the next pair, and the first edge should be the easiest edge to spot (I have a blue LL colour, and my first edge is blue/white). This will eliminate the need for some of the less good algs here, although I’m not sure how many it will eliminate.
AUF matches how you solve the first listed alg, edge numbers are listed by case. 0 is used to fill out the table for the diagonal entries (you can’t have 2 edges in the same place).
OCLL with EPLL control:
For an example: you get the chameleon case. First edge (blue white for me) is between the two twisted corners, second (blue yellow) is 3 places clockwise of this. Then, our recognition is 1,4. The number in the first row and fourth column is 2. Alg 2 is listed as R U R U2 R' U' R U' R' U' R'. We also have the AUF (U), which means to AUF correctly you need to place the twisted corners where the first alg would solve them, which is UFL and UFR. So, place them there, do a U turn, then do R U R U2 R' U' R U' R' U' R'. The two edges you recognised with should now be solved relative to each other, and all corners are oriented.
So, that’s my (sort of) 2LLL system for megaminx. I’ve sometimes been thinking of it as lolLL, so maybe you can call it that. Anyone think this is a good system? Any improvements you can think of? I hope to get some feedback, although I’ve already been working on this for a while and I think I’ve improved it as much as I can. Also feel free to point out mistakes as this is a long post and I will probably have made at least one, or better algs for any of the cases. Now, I just need to learn the damn thing along with all the other cube stuff on my to-do list, in between work and university.
To try and answer some criticisms before they are asked:
I am well aware that my recognition can be seen as excessive. Why not just pick one edge position for each case, and learn all the different pairs of edges you need to recognise? Then, you only have 4 cases. Actually, that is how I suggest you learn to start with. That is, pick the best row for each case, and learn it. 6*4=24 cases to learn to get the same effect. Then, you can start learning the rest of the table, starting with the easy cases (go through the algs in order of speed, and learn what they all solve), to end up with more efficient solutions on average. Also, this version of recognition could be useful later if anyone extends this system, which leads me to the next point.
Alternative method (not documented) is to solve corner orientation and edge permutation in one go. Recognition would be similar to the beginner system, except with two edges to look at. Pick a certain edge position, and look for the edge which would be solved 2 placed CW of it, then the edge which would be solved 2 places CW of that one. You would need to learn to recognise which ones to look for, which are in order: UR, UL, UB, UF, UD#. This is actually pretty easy to do. There are only 12 cases for each OCLL case to learn, which is 12*6=72 cases all in. This is certainly feasible to learn. The main reason I didn’t go straight to this system is because it took long enough to find enough algs for this system, since I did it by hand. I did try and download a megaminx LL solver, but it didn’t run for me, and I didn’t get any reply after contacting the site. It would probably be necessary to generate all the required algs in a reasonable amount of time.
Actually, wouldn't it be fun if this was another surprise challenge thread? Now, go do gigaminx OH . Not really, but it would be fun to see the reactions.