22LL Finished

Link: 22LL page

I finally got around to finishing the 22LL page. I started it a long time ago, but stopped after filling out the A set and the E set (minus EBD and EBE). I was very recently sent two PMs on SpeedSolving about 22LL, so I decided to put in the tables and pictures for the rest of the cases, and I changed the page layout the match the rest of my pages. I was soon sent algs for all 12 R set cases by Robert Yau, so I added those in, and I also added in the I and D set cases which were easy setups to PLL. Today I generated the last 10 or so algorithms, and finished the list. There are still some cases which I’m not totally happy with. Cases such as AAI where the algorithm is followed by (?) indicate that I think there’s a better algorithm, but I don’t have it. (In the case of AAI, I know that there is a better algorithm, but every time I ask Kirjava for it, I forget to write it down)

What is 22LL?

22LL is a subset of last layer algorithms. Each case consists of a 2-cycle of corners and a 2-cycle of edges. So, some PLLs such as T perm are 22LL cases. Z perm is not a 22LL case because while it has 2 2-cycles, they are both edge cycles.

How many cases are in 22LL?

22LL is only 56 cases. There are 5 sets, based on the corner 2-cycle, each of which represents a separate CLL case. The A and D sets contain adjacent and diagonal corner swap PLLs, respectively. The A set contains T, J, L, F, and both R perms, and the D set contains Y, V, and both N perms. The D set contains less algorithms than the other 4 sets because there were 4 cases which were reducible by AUF, which I didn’t notice until I made the case pictures for them. Additionally, there is no F set ( for the 2-cycle (UBR LUF) ) because it is identical to the I set with a U2 AUF.

Why make 22LL?

Originally, I planned to learn how to solve each 2×2-cycle for (UBR x) (DF y), where x is an edge and y is a corner, by using setups to PLLs or ZBLLs. I soon realised that it would be much simpler if I allowed edges to be unoriented. For this purpose, the entirety of 22LL would not be used. Rather, a slightly different setup would be used to reach a better 22LL case. For example, (UBR ULB)(DF UR). M’ setup could be used to reach ABI, or F2 setup could be used to reach ABC, which is a much better case (ABC is right R perm).

I also suspect that if 22LL is used by people for blindsolving, setups would be a lot simpler if the edge and corner buffers are adjacent. For example, URF and UF buffers. In my case, UBR and DF, this makes setups very awkward, although I can start by doing M2 to set DF next to UBR.

But I digress, 22LL is also a useful LL subset for speedsolving. Recognition is simple (only 5 CLL cases, and half of the pieces are solved) and the case number is low. In the past when I’ve seen 22LL cases I’ve thought to myself “that looked like a really simple LL case, why don’t I know how to solve it in 1 look?”

Example solves

Speedsolve « Show »

U’ B’ R’ D B R’ D2 B2 L’ F U’ F2 U B2 D F2 U2 L2 U L2 D’ F2

z2 y // inspection

U2 L R U R F B’ D // cross
U’ R’ U’ R // F2L #1
U’ R U’ R’ U’ L’ U’ L // F2L #2
R U’ R’ U y’ R’ U’ R // F2L #3
y’ R U2 R’ // F2L #4
U F U R’ U’ R D’ R2 U R’ U’ R2 D F’ U’ // 22LL

BLD « Show »

F’ R’ D’ L D’ F’ R’ L B D F2 U R2 U2 L2 U B2 F2 D2 F’

// memo

// corners
D L’ U2 L D’ L’ U2 L // UBR->LDB->FDL
R’ U2 R’ D’ R U2 R’ D R2 // UBR->URF->ULB
y’ R2 D2 R U2 R’ D2 R U2 R y // UBR->RBD->RFU
U2 L D’ L’ U2 L D L’ // UBR->ULF->RDF

// edges
U2 M’ U L U’ M U L’ U // DF->BU->LB
M U2 M U M’ U2 M’ U’ // DF->UF->UR
R2 u M’ U L U’ M U L’ U’ u’ R2 // DF->RD->FL
z L’ U M’ U’ L U M U’ z’ // DF->DL->LU

(M2) R2 U R U R’ U’ R2 F’ U F R’ F’ U’ F R2 U’ R2 (M2) // 22LL parity

z2 M’ U M’ U M’ U2 M U M U M U2 // flip edges

 

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