Hello guys,
I have a question. I was wondering how many possible cases does the Last Layer have? So if I wanted to solve the LL with one algorithm always (just in theory ofc.) how many algorithms did I have to learn? (something tells me that the answer is not just simply 21*57 (+1 solved case).
I'm pretty sure that this is how it's done
51 OLL cases with no symmetry, 51*72= 3672
5 OLL cases with U2 symmetry, 5*36= 180
2 OLL cases with 4-way symmetry, 2*21= 42
3672+180+42=3894 possible algs to learn. The 1212 figure excludes mirrors and inverses (I think, maybe something else too), so it's not really accurate. You might as well say that full ZBLL is 177 algs.
Easy way to figure out things like this:
So I got the 72, 36, and 21 by finding the number of possible PLL cases for each. Technically, PLL is 21 algs because the top face is solved, so you can simply perform an alg from a different angle because a solved face has what's known as symmetry.
A good way to think of symmetry is if you can do a U/U'/U2 and perform the alg (and it solves the case), the case has total symmetry, if you can only do a U2 the case has U2 symmetry, and if you can't do any, the case has no symmetry. Keep in mind that when I say case, I mean orientation cases and permutation cases separately.
So for PLL, there are 4 cases with total symmetry, (H, N, N, solved) 2 cases with U2 symmetry (Z and E) and 16 with no symmetry. From that, there's 4 cases for each no symmetry case, 2 cases for each U2 symmetry case, and 1 case for each total symmetry case. So (4*1)+(2*2)+(16*4)=72, so 72 total PLL cases.
Now if we do the same for OLL, we get 51 no symmetry cases, 5 U2 symmetry cases, and 2 total symmetry cases. For the U2 symmetry cases, we have to halve the number of possible PLL cases, so we get 36, and for the total symmetry cases (pure 4-flip, and solved), we simply take the number of PLL cases as we see them, so 21. Then we just do the math above, and theres the answer.