ZZ-reduction, also called ZZ-r, is a variant of the ZZ method for 3x3x3 proposed by Adrian Currier in 2014. It focuses on a reduction of PLL cases with the added benefit of faster recognition and more frequent PLL skips. It has the lowest algorithm count of any 2LLL method with a total algorithm count of 16.
After EOLine is completed, Phasing is employed during the insertion of the final block pair in F2L, with LL edges resulting in opposite colours (eg Blue/Green or Orange/Red) being placed opposite each other. Either they are SOLVED or there is PARITY, which means that adjacent edges are not correct with respect to each other. (An easy way to distinguish between SOLVED and PARITY is to attempt aligning the edges by rotating the U-layer. If its only possible to align two then it is the PARITY case.)
See OLC Algorithms
OLC (Orient Last Corners) is a subset of OLL that orients corners only and also preserves two opposite or all four edge permutations.
Of the 7 orientation cases in ZZ, 4 do not commonly preserve edge permutation (Sune, Antisune, Pi, Double-Sune). The other 3 commonly preserve edges (Headlights, Chameleon, Triple-Sune).
See PLL Algorithms
Since the phased edges were preserved in OLC, you should end up with only 9 possible PLL cases, down from 21 in full PLL. These are Aa/b, E, F, H, Na/b, T, Z.
Alternatively, you can limit PLL to a different set of 15 cases (Aa/b, E, Ga/b/c/d, H, Ja/b, Ra/b, U, V, Y) by antiphasing. That is, instead of placing LL opposite edge colors when completing F2L, placing adjacent edge colors.
The probabilities for the limited PLL set in ZZ-r are different than for full PLL. The probability of skipping PLL is 1/24, while the probabilities of getting each case are as follows:
- H 1/24
- Z 1/12
- A 1/3
- E 1/12
- F 1/6
- N 1/12
- T 1/6
- Reduced algorithm count: Only 16 algs are needed for full ZZ-r (7 for orientation and 9 for permutation), lower than any other 2LLL method.
- Faster recognition: Because there are so few last layer cases, recognition is very quick.
- Frequent skips: The low number of last layer cases make for more frequent skip cases. Skip probabilities are as follows: orientation 1/27; permutation 1/24; skip both (LL solved) 1/648.
- Phasing - Although not a great hindrance to move count, the phasing step adds an average of 2 7/12 moves. This may be offset by the increase in PLL skip probability and faster recognition.