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1LLL

2018-07-12T14:23:43Z

Master luigi: added info about birdflu.lar5.com

{{Method Infobox
|name=1LLL
|image=1LLL.jpg
|proposers=[[Bernard Helmstetter]]
|year=2000
|anames= OLLCP-EP
|variants=[[ZBLL]]
[[M-CELL]]
|steps=1
|algs='''3915'''
|moves=12.58 [[HTM]]
|purpose=
* [[Speedsolving]], however the multitude of algorithms makes it much less useful in whole
* [[FMC]]
}}

'''1''' '''L'''ook '''L'''ast '''L'''ayer 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.

A non-optimised database for all 1LLL algorithms & cases can be found at http://birdflu.lar5.com

== 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]], [[L4C]] and [[ZBLL]]) 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]], and [[ZBLL]] are all subsets of 1LLL, albeit a small proportion of the algorithms. Knowing some of these subsets means less algorithms need to be learnt for a 1LLL:

*If the solver already knows all 21 [[PLL]] cases, which is a subset of [[ZBLL]], there would be '''3894''' algorithms to learn.
*If the solver already knows all 493 [[ZBLL]] cases, there would be '''3422''' algorithms to learn.
*If the solver already knows all 29 [[ELL]] cases, there would be '''3886''' algorithms to learn.
*If the solver already knows all 518 [[ELL]] and [[ZBLL]] cases (4 [[EPLL]] cases appear in both sets), there would be '''3397''' algorithms to learn.

It should be noted that additional subsets of 1LLL also exist such as [[L3C]] (subset of L4C, which is a subset of ZBLL) , Pure OLL (orientation without affecting permutation), [[ZZLL]] (subset of ZBLL), [[ZZ-Blah]] (subset of ZBLL), [[Tripod]] LL and more.

== List of 1LLL subsets ==
* [[ZBLL]] (includes [[PLL]])
* [[ELL]]
* [[Tripod]] LL
* [[Line]] LL
* [[Pure OLL]]
{{work}}

== See also ==
* [[Last Layer]]
* [[:Category:3x3x3 last layer methods]]
* [[2LLL]], [[3LLL]], [[4LLL]]
* [[ZBLL]]
* [[Orientation]]
* [[Permutation]]

== External links ==
* [http://www.ai.univ-paris8.fr/~bh/cube/ Bernard Helmstetter's last layer algorithms]
* [http://www.speedcubing.com/docs/uppertable.zip 1211 last layer algorithms(1MB zipped)]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=51999 1LLL/OLLCP-EP algorithms]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=55518 BindeDSA's Algorithms]
* 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/showthread.php?t=53675 Hierarchy of Last Layer Sub-Steps, Subsets of OLLCP and ZBLL]

[[Category:3x3x3 methods]]

1LLL

2018-07-12T14:09:37Z

Master luigi:

{{Method Infobox
|name=1LLL
|image=1LLL.jpg
|proposers=[[Bernard Helmstetter]]
|year=2000
|anames= OLLCP-EP
|variants=[[ZBLL]]
[[M-CELL]]
|steps=1
|algs='''3915'''
|moves=12.58 [[HTM]]
|purpose=
* [[Speedsolving]], however the multitude of algorithms makes it much less useful in whole
* [[FMC]]
}}

'''1''' '''L'''ook '''L'''ast '''L'''ayer 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.

A non-optimised database for all 1LLL Algorithms & Cases can be found at http://birdflu.lar5.com/

== 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]], [[L4C]] and [[ZBLL]]) 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]], and [[ZBLL]] are all subsets of 1LLL, albeit a small proportion of the algorithms. Knowing some of these subsets means less algorithms need to be learnt for a 1LLL:

*If the solver already knows all 21 [[PLL]] cases, which is a subset of [[ZBLL]], there would be '''3894''' algorithms to learn.
*If the solver already knows all 493 [[ZBLL]] cases, there would be '''3422''' algorithms to learn.
*If the solver already knows all 29 [[ELL]] cases, there would be '''3886''' algorithms to learn.
*If the solver already knows all 518 [[ELL]] and [[ZBLL]] cases (4 [[EPLL]] cases appear in both sets), there would be '''3397''' algorithms to learn.

It should be noted that additional subsets of 1LLL also exist such as [[L3C]] (subset of L4C, which is a subset of ZBLL) , Pure OLL (orientation without affecting permutation), [[ZZLL]] (subset of ZBLL), [[ZZ-Blah]] (subset of ZBLL), [[Tripod]] LL and more.

== List of 1LLL subsets ==
* [[ZBLL]] (includes [[PLL]])
* [[ELL]]
* [[Tripod]] LL
* [[Line]] LL
* [[Pure OLL]]
{{work}}

== See also ==
* [[Last Layer]]
* [[:Category:3x3x3 last layer methods]]
* [[2LLL]], [[3LLL]], [[4LLL]]
* [[ZBLL]]
* [[Orientation]]
* [[Permutation]]

== External links ==
* [http://www.ai.univ-paris8.fr/~bh/cube/ Bernard Helmstetter's last layer algorithms]
* [http://www.speedcubing.com/docs/uppertable.zip 1211 last layer algorithms(1MB zipped)]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=51999 1LLL/OLLCP-EP algorithms]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=55518 BindeDSA's Algorithms]
* 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/showthread.php?t=53675 Hierarchy of Last Layer Sub-Steps, Subsets of OLLCP and ZBLL]

[[Category:3x3x3 methods]]

1LLL

2018-07-12T08:54:01Z

Master luigi:

{{Method Infobox
|name=1LLL
|image=1LLL.jpg
|proposers=[[Bernard Helmstetter]]
|year=2000
|anames= OLLCP-EP
|variants=[[ZBLL]]
[[M-CELL]]
|steps=1
|algs='''3915'''
|moves=12.58 [[HTM]]
|purpose=
* [[Speedsolving]], however the multitude of algorithms makes it much less useful in whole
* [[FMC]]
}}

'''1''' '''L'''ook '''L'''ast '''L'''ayer 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.

A database for all 1LLL (Algorithms & Cases) can be found at http://birdflu.lar5.com/

== 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]], [[L4C]] and [[ZBLL]]) 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]], and [[ZBLL]] are all subsets of 1LLL, albeit a small proportion of the algorithms. Knowing some of these subsets means less algorithms need to be learnt for a 1LLL:

*If the solver already knows all 21 [[PLL]] cases, which is a subset of [[ZBLL]], there would be '''3894''' algorithms to learn.
*If the solver already knows all 493 [[ZBLL]] cases, there would be '''3422''' algorithms to learn.
*If the solver already knows all 29 [[ELL]] cases, there would be '''3886''' algorithms to learn.
*If the solver already knows all 518 [[ELL]] and [[ZBLL]] cases (4 [[EPLL]] cases appear in both sets), there would be '''3397''' algorithms to learn.

It should be noted that additional subsets of 1LLL also exist such as [[L3C]] (subset of L4C, which is a subset of ZBLL) , Pure OLL (orientation without affecting permutation), [[ZZLL]] (subset of ZBLL), [[ZZ-Blah]] (subset of ZBLL), [[Tripod]] LL and more.

== List of 1LLL subsets ==
* [[ZBLL]] (includes [[PLL]])
* [[ELL]]
* [[Tripod]] LL
* [[Line]] LL
* [[Pure OLL]]
{{work}}

== See also ==
* [[Last Layer]]
* [[:Category:3x3x3 last layer methods]]
* [[2LLL]], [[3LLL]], [[4LLL]]
* [[ZBLL]]
* [[Orientation]]
* [[Permutation]]

== External links ==
* [http://www.ai.univ-paris8.fr/~bh/cube/ Bernard Helmstetter's last layer algorithms]
* [http://www.speedcubing.com/docs/uppertable.zip 1211 last layer algorithms(1MB zipped)]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=51999 1LLL/OLLCP-EP algorithms]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=55518 BindeDSA's Algorithms]
* 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/showthread.php?t=53675 Hierarchy of Last Layer Sub-Steps, Subsets of OLLCP and ZBLL]

[[Category:3x3x3 methods]]