Difference between revisions of "1LLL"

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* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12815 3x3 last layer: number of positions]
 
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12815 3x3 last layer: number of positions]
 
* Speedsolving.com: [http://www.speedsolving.com/forum/showpost.php?p=440250&postcount=31 Derivation of unique PLL cases]
 
* Speedsolving.com: [http://www.speedsolving.com/forum/showpost.php?p=440250&postcount=31 Derivation of unique PLL cases]
* Speedsolving.com: [https://www.speedsolving.com/forum/showthread.php?53675-Hierarchy-of-Last-Layer-Sub-Steps-Subsets-of-OLLCP-and-ZBLL]
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* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=53675 Hierarchy of Last Layer Sub-Steps, Subsets of OLLCP and ZBLL]
  
 
[[Category:3x3x3 methods]]
 
[[Category:3x3x3 methods]]

Revision as of 20:17, 24 February 2016

1LLL method
1LLL.jpg
Information about the method
Proposer(s): Bernard Helmstetter
Proposed: 2000
Alt Names:
Variants: ZBLL

M-CELL

No. Steps: 1
No. Algs: 3915
Avg Moves: 12.58 HTM
Purpose(s):
  • Not useful in practice.


1 Look Last Layer means completing the last layer using only one look. On the 3x3x3, currently ZBLL is the only known practical system for achieving a 1-look last layer, but other experimental approaches such as 1-look 2-alg (such as M-CELL) may have potential.


Number of 1LLL Cases

Assuming the last layer is an outer layer, the number of cases is calculated as:

corner orientations * edge orientations * corner permutations * edge permutations / 2

With the 2 at the end because of odd-even parity.

Numerically this is:

3^3 * 2^3 * 4! * 4! / 2 = 62208
  • The 3^3 at the beginning is because there are 4 corners on the last layer while the final one depends on the remaining three.
  • The 2^3 is because there are 4 edges on the last layer while the final one depends on the remaining three.

Treating cases which are the same, but rotated by 90, 180 or 270 degrees as the same case, the number of cases becomes:

62208/4 = 15552

Although these cases may be regarded as unique, some of them can be solved by applying the same algorithm from a different angle. For cases with no rotational symmetry, there are 3 equivalent cases which may be solved with the same alg. For cases with 180 degree rotational symmetry, there is 1 equivalent case which may be solved with the same alg. Cases with 90, 180 and 270 degree rotational symmetry are unique. By solving multiple cases with the same algorithm from different angles, the number of cases becomes:

3916 cases (including the solved state / AUF states and subsets such as PLL, ELL, 2GLL and L4C) which can be solved with 3915 algorithms

As well as rotationally symmetrical cases, there are also reflectively symmetrical cases. These may be solved by applying a reflection of the algorithm. Finally, some cases are inversions of others. These can be solved by reversing (inverting) the algorithm solving the original case. Exactly which cases are mirrors of each other requires a case-by-case analysis. Work by Bernard Helmstetter established that the number unique 1LLL cases (excluding mirrors and inverses) is:

1212 cases (including the solved / AUF states) which can be solved with 1211 algorithms

Since most speed solvers regard reflective symmetry and inversions as different cases (e.g. PLL is thought of as 21 algorithms rather than 13 algorithms plus mirrors and inverses), 1LLL should be thought of as 3915 algorithms.

PLL, ELL, 2GLL and L4C are all subsets of 1LLL, albeit a small proportion of the algorithms.

See also

External links