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Beyer-Hardwick Method

2017-05-30T05:18:06Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

Previewing Beyer-Hardwick Method
Save

Tell us what you changed (optional)
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BH Corners
These are the different types of commutators you will "see" while learning BH.

Pure Commutators (8 moves) - are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves: R U R' R U' R' R U2 R' All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

A9 (9 moves) - are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

Orthogonals (10 moves) - are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

Cyclic Shifts (11 moves) - they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse) Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

Column Cases (11 moves) - they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

Per Specials (12 moves) - a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
These are the types of edge commutators you'll "see" while learning BH edges.

'''Half Slice-Planes (4 moves)''' - A special type of commutator that makes its interchanges on the same outer layer.

Formula: [A, B] where a A is the interchange and B is the setup to the interchange.

'''Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves''' - Slice-Planes are conjugated Half Slice-Planes commutators.

'''Pure Commutators (8 moves)''' - Same as BH Pure Comms.

'''A9/B9 (9 moves)''' - Same as BH A9s except B9s exists.

'''Orthogonals (10 moves)''' - Same as BH Orthogonals.

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== Recognition ==

'''BH Corners'''

1. '''Pure Commutators''' 
- 2 corners in one layer and another corner in a different layer. (Ex. 2 corners at U and and the 3rd corner at D) 
- The 2 corners in one layer are interchangeable 
- There's a 3 move insert 

2. '''A9''' 
- Exactly like a Pure Commutator but with no 3 move insert (Not applicable to all A9s) 
- Use elimination. (Ex. If it's not a Pure Comm with no 3 move insert then check if it's one of the other BH Corner Cases, if not then it's an A9) 

3.'''Orthogonals''' 
- All 3 corners are opposite and are all AnI to each other 

4. ''' Cyclic Shifts ''' 
- All 3 corners are in the same layer are are all AnI to each other 

5. ''' Columns ''' 
- Like a Pure Comm with no 3 move insert but the interchange corners are interchangeable with a half turn (Don't be confused with A9s) 

6. ''' Per Specials ''' 
- All 3 corners are opposite 
- All 3 corners are mutually interchangeable 

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-30T05:17:14Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

Previewing Beyer-Hardwick Method
Save

Tell us what you changed (optional)
By saving changes, you agree to release your contribution under the license.
BH Corners
These are the different types of commutators you will "see" while learning BH.

Pure Commutators (8 moves) - are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves: R U R' R U' R' R U2 R' All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

A9 (9 moves) - are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

Orthogonals (10 moves) - are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

Cyclic Shifts (11 moves) - they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse) Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

Column Cases (11 moves) - they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

Per Specials (12 moves) - a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
These are the types of edge commutators you'll "see" while learning BH edges.

'''Half Slice-Planes (4 moves)''' - A special type of commutator that makes its interchanges on the same outer layer.

Formula: [A, B] where a A is the interchange and B is the setup to the interchange.

'''Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves''' - Slice-Planes are conjugated Half Slice-Planes commutators.

'''Pure Commutators (8 moves)''' - Same as BH Pure Comms.

'''A9/B9 (9 moves)''' - Same as BH A9s except B9s exists.

'''Orthogonals (10 moves)''' - Same as BH Orthogonals.

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== Recognition ==

'''BH Corners'''

1. '''Pure Commutators''' 
- 2 corners in one layer and another corner in a different layer. (Ex. 2 corners at U and and the 3rd corner at D) 
- The 2 corners in one layer are interchangeable 
- There's a 3 move insert 

2. '''A9''' 
- Exactly like a Pure Commutator but with no 3 move insert (Not applicable to all A9s) 
- Use elimination. (Ex. If it's not a Pure Comm with no 3 move insert check if it's one of the BH Corner Cases, if not then it's an A9) 

3.'''Orthogonals''' 
- All 3 corners are opposite and are all AnI to each other 

4. ''' Cyclic Shifts ''' 
- All 3 corners are in the same layer are are all AnI to each other 

5. ''' Columns ''' 
- Like a Pure Comm with no 3 move insert but the interchange corners are interchangeable with a half turn (Don't be confused with A9s) 

6. ''' Per Specials ''' 
- All 3 corners are opposite 
- All 3 corners are mutually interchangeable 

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-30T05:16:48Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

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BH Corners
These are the different types of commutators you will "see" while learning BH.

Pure Commutators (8 moves) - are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves: R U R' R U' R' R U2 R' All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

A9 (9 moves) - are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

Orthogonals (10 moves) - are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

Cyclic Shifts (11 moves) - they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse) Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

Column Cases (11 moves) - they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

Per Specials (12 moves) - a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
These are the types of edge commutators you'll "see" while learning BH edges.

'''Half Slice-Planes (4 moves)''' - A special type of commutator that makes its interchanges on the same outer layer.

Formula: [A, B] where a A is the interchange and B is the setup to the interchange.

'''Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves''' - Slice-Planes are conjugated Half Slice-Planes commutators.

'''Pure Commutators (8 moves)''' - Same as BH Pure Comms.

'''A9/B9 (9 moves)''' - Same as BH A9s except B9s exists.

'''Orthogonals (10 moves)''' - Same as BH Orthogonals.

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== Recognition ==

'''BH Corners'''

1. '''Pure Commutators''' 
- 2 corners in one layer and another corner in a different layer. (Ex. 2 corners at U and and the 3rd corner at D) 
- The 2 corners in one layer are interchangeable 
- There's a 3 move insert 

2. '''A9''' 
- Exactly like a Pure Commutator but with no 3 move insert (Not applicable to all A9s) 
- Use elimination. (Ex. If it's not a Pure Comm with no 3 move insert check if it's one of the BH Corner Cases, if not then it's an A9) 

3.'''Orthogonals''' 
- All 3 corners are opposite and are all AnI to each other 

4. ''' Cyclic Shifts ''' 
- All 3 corners are in the same layer are are all AnI to each other 

5. ''' Columns ''' 
- Like a Pure Comm with no 3 move insert but the interchange corners are interchangeable with a half turn (Don't be confused with A9s) <

6. ''' Per Specials ''' 
- All 3 corners are opposite 
- All 3 corners are mutually interchangeable 

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T21:19:34Z

LSLL:

Beyer-Hardwick Method

2017-05-24T21:18:40Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== Recognition ==

'''BH Corners'''

1. '''Pure Commutators''' 
- 2 corners in one layer and another corner in a different layer. (Ex. 2 corners at U and and the 3rd corner at D) 
- The 2 corners in one layer are interchangeable 
- There's a 3 move insert 

2. '''A9''' 
- Exactly like a Pure Commutator but with no 3 move insert (Not applicable to all A9s) 
- Use elimination. (Ex. If it's not a Pure Comm with no 3 move insert check if it's one of the BH Corner Cases, if not then it's an A9) 

3.'''Orthogonals''' 
- All 3 corners are opposite and are all AnI to each other 

4. ''' Cyclic Shifts ''' 
- All 3 corners are in the same layer are are all AnI to each other 

5. ''' Columns ''' 
- Like a Pure Comm with no 3 move insert but the interchange corners are interchangeable with a half turn (Don't be confused with A9s) <

6. ''' Per Specials ''' 
- All 3 corners are opposite 
- All 3 corners are mutually interchangeable 

'''BH Edges'''

1.''' Half Slice-Plane ''' 
- All 3 edges are in the same slice 
- All 3 edges are mutually interchangeable by a half turn 

2.''' Slice-Planes ''' 
''' SP5s ''' 
- All edges are in the same layer whether you rotate or not 
- 2 edges are in the same layer and are opposite to each other and the 3rd edge is in between in a slice layer 

'''SP7s''' 
''' 1st Variant ''' 
- 2 edges of the same group in the same layer opposite to each other and the 3rd edge is in between them 
''' 2nd Variant ''' 
- The first edge is in an outer layer and the 2nd edges in the slice not while not occupying the layer of the first edge and they're in opposite places (Ex. UL and DF) 
- All edges are in the same group whether you rotate or not 

3. ''' Pure Commutators '''
- 2 edges are in one layer/slice and are interchangeable and a there's a 3 move insert 
- There are SP7s that follows the same rules so check first if it's SP7 

4. ''' Orthogonals ''' 
- All 3 edges occupy the same space and all 3 can't be interchanged with each other in 2 moves. 
- Looks like a Half Slice-Plane but they're all interchangeable with a 1/4 turn instead

Edges don't read need confirmation

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T21:16:03Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== Recognition ==

'''BH Corners'''

1. '''Pure Commutators''' 
- 2 corners in one layer and another corner in a different layer. (Ex. 2 corners at U and and the 3rd corner at D) 
- The 2 corners in one layer are interchangeable 
- There's a 3 move insert 

2. '''A9''' 
- Exactly like a Pure Commutator but with no 3 move insert (Not applicable to all A9s) 
- Use elimination. (Ex. If it's not a Pure Comm with no 3 move insert check if it's one of the BH Corner Cases, if not then it's an A9) 

3.'''Orthogonals''' 
- All 3 corners are opposite and are all AnI to each other 

4. ''' Cyclic Shifts ''' 
- All 3 corners are in the same layer are are all AnI to each other 

5. ''' Columns ''' 
- Like a Pure Comm with no 3 move insert but the interchange corners are interchangeable with a half turn (Don't be confused with A9s) <

6. ''' Per Specials ''' 
- All 3 corners are opposite 
- All 3 corners are mutually interchangeable 

'''BH Edges'''

1.''' Half Slice-Plane ''' 
- All 3 edges are in the same slice 
- All 3 edges are mutually interchangeable by a half turn 

2.''' Slice-Planes ''' 
''' SP5s ''' 
- All edges are in the same layer whether you rotate or not 
- 2 edges are in the same layer and are opposite to each other and the 3rd edge is in between in a slice layer 

'''SP7s''' 
''' 1st Variant ''' 
- 2 edges of the same group in the same layer opposite to each other and the 3rd edge is in between them 
''' 2nd Variant ''' 
- The first edge is in an outer layer and the 2nd edges in the slice not while not occupying the layer of the first edge and they're in opposite places (Ex. UL and DF) 
- All edges are in the same group whether you rotate or not 

3. ''' Pure Commutators '''
- 2 edges are in one layer/slice and are interchangeable and a there's a 3 move insert 
- There are SP7s that follows the same rules so check first if it's SP7 

4. ''' Orthogonals ''' 
- All 3 edges occupy the same space and all 3 can't be interchanged with each other in 2 moves. 
- Looks like a Half Slice-Plane but they're all interchangeable with a 1/4 turn instead

Edges don't read need confirmation

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T21:03:02Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== Recognition ==

'''BH Corners'''

1. '''Pure Commutators''' 
- 2 corners in one layer and another corner in a different layer. (Ex. 2 corners at U and and the 3rd corner at D) 
- The 2 corners in one layer are interchangeable 
- There's a 3 move insert 

2. '''A9''' 
- Exactly like a Pure Commutator but with no 3 move insert (Not applicable to all A9s) 
- Use elimination. (Ex. If it's not a Pure Comm with no 3 move insert check if it's one of the BH Corner Cases, if not then it's an A9) 

3.'''Orthogonals''' 
- All 3 corners are opposite and are all AnI to each other 

4. ''' Cyclic Shifts ''' 
- All 3 corners are in the same layer are are all AnI to each other 

5. ''' Columns ''' 
- Like a Pure Comm with no 3 move insert but the interchange corners are interchangeable with a half turn (Don't be confused with A9s) <

6. ''' Per Specials ''' 
- All 3 corners are opposite 
- All 3 corners are mutually interchangeable 

'''BH Edges'''

1.''' Half Slice-Plane ''' 
- All 3 edges are in the same slice 
- All 3 edges are mutually interchangeable by a half turn 

2.''' Slice-Planes ''' 
''' SP5s ''' 
- All edges are in the same layer whether you rotate or not 
- 2 edges are in the same layer and are opposite to each other and the 3rd edge is in between in a slice layer 

'''SP7s''' 
''' 1st Variant ''' 
- 2 edges of the same group in the same layer opposite to each other and the 3rd edge is in between them 
''' 2nd Variant ''' 
- The first edge is in an outer layer and the 2nd edges in the slice not while not occupying the layer of the first edge and they're in opposite places (Ex. UL and DF) 
- All edges are in the same group whether you rotate or not 

3. ''' Pure Commutators '''
- 2 edges are in one layer/slice and are interchangeable and a there's a 3 move insert 
- There are SP7s that follows the same rules so check first if it's SP7 

4. ''' Orthogonals ''' 
- All 3 edges can't be interchanged with each other in 2 moves 
- Looks like a Half Slice-Plane but they're all interchangeable with a 1/4 turn instead

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T20:55:31Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== Recognition ==

'''BH Corners'''

1. '''Pure Commutators''' 
- 2 corners in one layer and another corner in a different layer. (Ex. 2 corners at U and and the 3rd corner at D) 
- The 2 corners in one layer are interchangeable 
- There's a 3 move insert 

2. '''A9''' 
- Exactly like a Pure Commutator but with no 3 move insert (Not applicable to all A9s) 
- Use elimination. (Ex. If it's not a Pure Comm with no 3 move insert check if it's one of the BH Corner Cases, if not then it's an A9) 

3.'''Orthogonals''' 
- All 3 corners are opposite and are all AnI to each other 

4. ''' Cyclic Shifts ''' 
- All 3 corners are in the same layer are are all AnI to each other 

5. ''' Columns ''' 
- Like a Pure Comm with no 3 move insert but the interchange corners are interchangeable with a half turn (Don't be confused with A9s) <

6. ''' Per Specials ''' 
- All 3 corners are opposite 
- All 3 corners are mutually interchangeable 

'''BH Edges'''

1.''' Half Slice-Plane ''' 
- All 3 edges are in the same slice 
- All 3 edges are mutually interchangeable by a half turn 

2.''' Slice-Planes ''' 
''' SP5s ''' 
- All edges are in the same layer whether you rotate or not 
- 2 edges are in the same layer and are opposite to each other and the 3rd edge is in between in a slice layer 

'''SP7s''' 
''' 1st Variant ''' 
- 2 edges of the same group in the same layer opposite to each other and the 3rd edges is in between them 
''' 2nd Variant ''' 
- The first edge is in an outer layer and the 2nd edges in the slice not while not occupying the layer of the first edge and they're in opposite places (Ex. UL and DF) 
- All edges are in the same group whether you rotate or not 

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T20:43:05Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== Recognition ==

'''BH Corners'''

1. '''Pure Commutators''' 
- 2 corners in one layer and another corner in a different layer. (Ex. 2 corners at U and and the 3rd corner at D) 
- The 2 corners in one layer are interchangeable 
- There's a 3 move insert 

2. '''A9''' 
- Exactly like a Pure Commutator but with no 3 move insert (Not applicable to all A9s) 
- Use elimination. (Ex. If it's not a Pure Comm with no 3 move insert check if it's one of the BH Corner Cases, if not then it's an A9) 

3.'''Orthogonals''' 
- All 3 corners are opposite and are all AnI to each other 

4. ''' Cyclic Shifts ''' 
- All 3 corners are in the same layer are are all AnI to each other 

5. ''' Columns ''' 
- Like a Pure Comm with no 3 move insert but the interchange corners are interchangeable with a half turn (Don't be confused with A9s) <

6. ''' Per Specials ''' 
- All 3 corners are opposite 
- All 3 corners are mutually interchangeable 

'''BH Edges'''

1.''' Half Slice-Plane ''' 
- All 3 edges are in the same slice 
- All 3 edges are mutually interchangeable by a half turn 

2.''' Slice-Planes '''
'''SP5s'''

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T20:25:06Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== Recognition ==

'''BH Corners'''

1. '''Pure Commutators'''
- 2 corners in one layer and another corner in a different layer. (Ex. 2 corners at U and and the 3rd corner at D)
- The 2 corners in one layer are interchangeable
- There's a 3 move insert

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T20:21:30Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== Recognition ==

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T20:21:05Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year= ~2009
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== Learning Approach ==

Learning BH is a large task whether you'll opt for more speed-optimised algs some where down the line for some cases knowing the basic structure of a commutator [A, B] will help whether you'll look up a list or create the algs yourself since shortening an alg to [A, B] is alot easier to learn.

Creating the algs on your own is sometimes suggested as it will help you remember them more since you created them but looking up a list and not creating the algs yourself is fine but don't rely it on too much since that person's list is probably speed-optimised for them so if an alg is too hard to do try and make one for your self.

Note that BH or any form of commutators are intuitive meaning you don't need to generate the alg, YOU make them.

Whether you make them or not there's 2 routes to learning them: On the fly or learn the entire list before applying it on the cube.

Just like how F2L can be solved on the fly as long you know how to recognize certain cases or have a good sense for commutators you can actually not list every single case amd learn them rather you can do it while solving with no prior knowledge about a certain case just like you don't know every single F2L case but you can still do them.

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T20:10:18Z

LSLL:

Beyer-Hardwick Method

2017-05-24T20:06:49Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] , [[Daniel Beyer]]
|year=???
|a names=
|variants= Algorithms
|steps= 2
|algs= 0-819 (Intuitive) (378 for corners 441 for edges)
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== Learning Approach ==

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T20:02:24Z

LSLL:

{{Method Infobox
|name= Beyer-Hardwick
|image=
|proposers=[[Chris Hardwick]] [[Daniel Beyer]]
|year=???
|a names=
|variants= Algorithms
|steps= 2
|algs=
|moves=???
|purpose=
* [[Blindsolving]]
* [[FMC]]
}}

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== Learning Approach ==

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T19:56:51Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== Learning Approach ==

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T19:56:28Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T19:56:01Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T12:07:03Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

I put to much info on this. I should have made a tutorial instead but the info here is not tutorial wothy yet.

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved witg either a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-24T12:06:32Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

I put to much info on this. I should have made a tutorial instead but the info here is not tutorial wothy yet.

-LSLL

== BH Edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- There are no 9 move cases in BH edges. Every 9 move cases can be solved eith with a Pure Comm or a Slice-Plane.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-23T23:24:53Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

I put to much info on this. I should have made a tutorial instead but the info here is not tutorial wothy yet.

-LSLL

== BH edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- like A9 Corners they're recognized by using elimination and/or having no 3 move insert that looks like a Pure Comm. Only difference now that there are cases that can only be solved optimally by having the setup move cancel with B.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves or they're in the same slice and are mutually interchangeable by a 1/4 slice turn (Half Slice-Planes are mutually interchangeable by a half turn). To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-23T22:40:25Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

I put to much info on this. I should have made a tutorial instead but the info here is not tutorial wothy yet.

-LSLL

== BH edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- like A9 Corners they're recognized by using elimination and/or having no 3 move insert that looks like a Pure Comm. Only difference now that there are cases that can only be solved optimally by having the setup move cancel with B.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 1 moves. To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-23T22:18:59Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

I put to much info on this. I should have made a tutorial instead but the info here is not tutorial wothy yet.

-LSLL

== BH edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- like A9 Corners they're recognized by using elimination and/or having no 3 move insert that looks like a Pure Comm. Only difference now that there are cases that can only be solved optimally by having the setup move cancel with B.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 2 moves. To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

-LSLL

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-23T22:18:42Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

I put to much info on this. I should have made a tutorial instead but the info here is not tutorial wothy yet.

-LSLL

== BH edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- like A9 Corners they're recognized by using elimination and/or having no 3 move insert that looks like a Pure Comm. Only difference now that there are cases that can only be solved optimally by having the setup move cancel with B.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 2 moves. To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-23T22:17:38Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH edges ==
Types of commutators you'll "see" while learning BH.

''' Half Slice-Plane Commutators (4 moves)'''
- These commutators are recognized by being on same slice and are mutually interchangeable. To do them you must interchange twice on the same outer layer. Unlike traditional [A, B] commutators, A is the insert move and B is the interchange move. But for Half-Slice Planes A is the interchange move and B is a just a setup to an interchange for A.

''' Slice-Planes (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Alongside Pure Comms most of the commutators come from Half Slice-Planes meaning Slice-Planes are conjugated Half Slice-Planes.

To recognize them 2 edges are of the same group (UF and UB are outer stickers while stickers like RF are inner stickers). To do them you must do a 1 move setup to either a Half Slice-Plane(SP5) or a SP5(SP7). SP5s' setup moves cancels with the Half Slice-Plane making the move count to 5 instead of 6.

SP9s are Chris's BS. They're SP5s made longer for no reason. Correct me if I'm but if do a SP9 I can do a SP5 instead. Wtf Chris?

''' Pure Commutators (8 moves)'''
- alongside with Half Slice-Planes all other commutators come from them. They are recognized by having 2 edges in one layer/slice and are interchangeable and the other corner in another just like how Pure Corner Comms are recognized (instead corners are on different layers) and there's a 3 move insert.

''' A9/B9 (9 moves)'''
- like A9 Corners they're recognized by using elimination and/or having no 3 move insert that looks like a Pure Comm. Only difference now that there are cases that can only be solved optimally by having the setup move cancel with B.

''' Orthogonals (10 moves)'''
- are recognized by having all the edges not interchangeable with each other in 2 moves. To do them you must do some sort of setup move and it turn into a Pure Comm, do it and undo the setup move. You actually need to choose the right S move for this.

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-23T21:59:38Z

LSLL:

Beyer-Hardwick Method

2017-05-23T16:54:51Z

LSLL:

Beyer-Hardwick Method

2017-05-23T15:18:18Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
- these are the type of edges you'll "see" while learning BH.

'''Half Slice-Plane (4 moves)'''
- they're the Per Specials of edge commutators by being recognized similarly to Per Specials and has a unique property. Per Specials need a 5 move insert while Half Slice-Plane to be optimal must interchange twice on the same outer layer making the commutator really small only needing 4 moves to do it.

It is recognized by having all the edges on the same slice and are mutually interchangeable.

To do them you must know what they're composed of: [A, B]

A is some sort of half turn of a outer layer where most of the edges are in
B is some sort 1/4 turn of a slice where all the edges reside and setups into the A layer for the interchange

Ex. UF ->DF -> UB

To do them figure out what's the middle piece of the cycle. UF is the middle piece and must go to DF. They must interchange on the A layer, so what's A? In the U layer 2/3 edges resides so A is a U2. How to we interchange it with DF with a U2?

That's where B comes in, B is the 1/4 of the slice where all edges reside so M is B. How do we know if we do M or M'? Well use logic, if we do M the UF and DF stickers will go to the D layer but we want it to go to the U layer so we do M'.

Final answer is [A=U2, B=M']

'''Slice-Plane (SP5, SP7) (5, 7 moves)'''
- Half Slice-Planes really are special as not all commutators come from Pure Comms instead some come from Half Slice-Planes.

Slice-Planes are conjugated Half Slice-Planes. They follow the formula: [S: [A,B]] where S is a one set up and [A, B] is the 1/2 Slice-Plane.

To recognize them is just to know all 1 move setups to a certain Slice-Plane:

5 movers are conjugated 1/2 Slice-Planes with a cancellation. There are no optimal cancelleation with S moves with Slice turns or Half turns of A so S moved must be a 1/4 turn of A.

SP7s are conjugated SP5s without cancellations.

''' Pure Commutators (8 moves)'''
- All other commutators come from these. They follow the formula: [A, B].

To recognize there's an interchange on the slices or layers and there is a either a layer insert or slice insert.
Ex.
M interchanges: UF -> DB (M2)
Layer Interchange: UR -> FR (R or R')
M insert: U M2 U' or [U: M2]
Layer insert: U R U' or [U: R]

They're exactly Pure Corner Comms so just do [A, B].

''' A9/B9 (9 moves)'''
- Exactly like A9 corner commutators except B9 exists meaning there's a optimal solution where the S cancels with B. To recognize there's no 3 move insert like A9 corner comms. There are no M S moves that cancels with B if B is also a M move.

''' Orthogonals (10 moves)'''
- To recognize all edges can't be interchange with each other in 1 move. To do them just do some sort of turn and it will setup to a Pure Comm. Formula: [S: [A, B]].

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-23T13:20:35Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
- these are the type of edges you'll "see" while learning BH.

'''Half Slice-Plane (4 moves)'''
- they're the Per Specials of edge commutators by being recognized similarly to Per Specials and has a unique property. Per Specials need a 5 move insert while Half Slice-Plane to be optimal must interchange twice on the same outer layer making the commutator really small only needing 4 moves to do it.

It is recognized by having all the edges on the same slice and are mutually interchangeable.

To do them you must know what they're composed of: [A, B]

A is some sort of half turn of a outer layer where most of the edges are in
B is some sort 1/4 turn of a slice where all the edges reside and setups into the A layer for the interchange

Ex. UF ->DF -> UB

To do them figure out what's the middle piece of the cycle. UF is the middle piece and must go to DF. They must interchange on the A layer, so what's A? In the U layer 2/3 edges resides so A is a U2. How to we interchange it with DF with a U2?

That's where B comes in, B is the 1/4 of the slice where all edges reside so M is B. How do we know if we do M or M'? Well use logic, if we do M the UF and DF stickers will go to the D layer but we want it to go to the U layer so we do M'.

Final answer is [A=U2, B=M']

'''Slice-Plane (SP5, SP7) (5, 7 moves)'''
- Half Slice-Planes really are special as not all commutators come from Pure Comms instead some come from Half Slice-Planes.

Slice-Planes are conjugated Half Slice-Planes. They follow the formula: [S: [A,B]] where S is a one set up and [A, B] is the 1/2 Slice-Plane.

To recognize them is just to know all 1 move setups to a certain Slice-Plane:

5 movers are conjugated 1/2 Slice-Planes with a cancellation. There are no optimal cancelleation with S moves with M turns or U2s so All S moved must be a U or U'

SP7s are conjugated SP5s without cancellations. Since there must be no U2 for Slice-Planes you can't use U or U' S moves instead you must do M, M', or M2 S moves.

''' Pure Commutators (8 moves)'''
- All other commutators come from these. They follow the formula: [A, B].

To recognize there's an interchange on the slices or layers and there is a either a layer insert or slice insert.
Ex.
M interchanges: UF -> DB (M2)
Layer Interchange: UR -> FR (R or R')
M insert: U M2 U' or [U: M2]
Layer insert: U R U' or [U: R]

They're exactly Pure Corner Comms so just do [A, B].

''' A9/B9 (9 moves)'''
- Exactly like A9 corner commutators except B9 exists meaning there's a optimal solution where the S cancels with B. To recognize there's no 3 move insert like A9 corner comms. There are no M S moves that cancels with B if B is also a M move.

''' Orthogonals (10 moves)'''
- To recognize all edges can't be interchange with each other in 1 move. To do them just do some sort of turn and it will setup to a Pure Comm. Formula: [S: [A, B]].

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-23T13:20:07Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
- these are the type of edges you'll "see" while learning BH.

'''Half Slice-Plane (4 moves)'''
- they're the Per Specials of edge commutators by being recognized similarly to Per Specials and has a unique property. Per Specials need a 5 move insert while Half Slice-Plane to be optimal must interchange twice on the same outer layer making the commutator really small only needing 4 moves to do it.

It is recognized by having all the edges on the same slice and are mutually interchangeable.

To do them you must know what they're composed of: [A, B]

A is some sort of half turn of a outer layer where most of the edges are in
B is some sort 1/4 turn of a slice where all the edges reside and setups into the A layer for the interchange

Ex. UF ->DF -> UB

To do them figure out what's the middle piece of the cycle. UF is the middle piece and must go to DF. They must interchange on the A layer, so what's A? In the U layer 2/3 edges resides so A is a U2. How to we interchange it with DF with a U2?

That's where B comes in, B is the 1/4 of the slice where all edges reside so M is B. How do we know if we do M or M'? Well use logic, if we do M the UF and DF stickers will go to the D layer but we want it to go to the U layer so we do M'.

Final answer is [A=U2, B=M']

'''Slice-Plane (SP5, SP7) (5, 7 moves)'''
- Half Slice-Planes really are special as not all commutators come from Pure Comms instead some come from Half Slice-Planes.

Slice-Planes are conjugated Half Slice-Planes. They follow the formula: [S: [A,B]] where S is a one set up and [A, B] is the 1/2 Slice-Plane.

To recognize them is just to know all 1 move setups to a certain Slice-Plane:

5 movers are conjugated 1/2 Slice-Planes with a cancellation. There are no optimal cancelleation with S moves with M turns or U2s so All S moved must be a U or U'

SP7s are conjugated SP5s without cancellations. Since there must be no U2 for Slice-Planes you can't use U or U' S moves instead you must do M, M', or M2 S moves.

''' Pure Commutators (8 moves)'''
- All other commutators come from these. They follow the formula: [A, B].

To recognize there's an interchange on the slices or layers and there is a either a layer insert or slice insert.
Ex.
M interchanges: UF -> DB (M2)
Layer Interchange: UR -> FR (R or R')
M insert: U M2 U' or [U: M2]
Layer insert: U R U' or [U: R]

They're exactly Pure Corner Comms so just do [A, B].

''' A9/B9 (9 moves)'''
- Exactly like A9 corner commutators except B9 exists meaning there's a optimal solution where the S cancels with B. To recognize there's no 3 move insert like A9 corner comms. There are no M S moves that cancels with B if B is also a M move.

''' Orthogonals (10 moves)'''
- To recognize all edges can't be interchange with each other in 1 move. To do them just do some sort of turn and it will setup to a Pure Comm. Formula: [S: [A, B]].

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}
Thanks to LexCubing it's no longer a stub. This is not a tutorial so if you want a better understanding and how to do them go look up Byu's unfinished BS but hey corners are done. I'll be making a tutorial someday so stay tune but you don't need to be optimal to 3-style just know how Pure Comms work and setup to them and everything will be okay.

Beyer-Hardwick Method

2017-05-23T13:03:27Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
- these are the type of edges you'll "see" while learning BH.

'''Half Slice-Plane (4 moves)'''
- they're the Per Specials of edge commutators by being recognized similarly to Per Specials and has a unique property. Per Specials need a 5 move insert while Half Slice-Plane to be optimal must interchange twice on the same outer layer making the commutator really small only needing 4 moves to do it.

It is recognized by having all the edges on the same slice and are mutually interchangeable.

To do them you must know what they're composed of: [A, B]

A is some sort of half turn of a outer layer where most of the edges are in
B is some sort 1/4 turn of a slice where all the edges reside and setups into the A layer for the interchange

Ex. UF ->DF -> UB

To do them figure out what's the middle piece of the cycle. UF is the middle piece and must go to DF. They must interchange on the A layer, so what's A? In the U layer 2/3 edges resides so A is a U2. How to we interchange it with DF with a U2?

That's where B comes in, B is the 1/4 of the slice where all edges reside so M is B. How do we know if we do M or M'? Well use logic, if we do M the UF and DF stickers will go to the D layer but we want it to go to the U layer so we do M'.

Final answer is [A=U2, B=M']

'''Slice-Plane (SP5, SP7, SP9) (5, 7, 9 moves)'''
- Half Slice-Planes really are special as not all commutators come from Pure Comms instead some come from Half Slice-Planes.

Slice-Planes are conjugated Half Slice-Planes. They follow the formula: [S: [A,B]] where S is a one set up and [A, B] is the 1/2 Slice-Plane.

To recognize them is just to know all 1 move setups to a certain Slice-Plane:

5 movers are conjugated 1/2 Slice-Planes with a cancellation. There are no optimal cancelleation with S moves with M turns or U2s so All S moved must be a U or U'

SP7s are conjugated SP5s without cancellations. Since there must be no U2 for Slice-Planes you can't use U or U' S moves instead you must do M, M', or M2 S moves.

SP9s are just SP5s made fingertrick friendly and are alternatives to some B9s.

''' Pure Commutators (8 moves)'''
- All other commutators come from these. They follow the formula: [A, B].

To recognize there's an interchange on the slices or layers and there is a either a layer insert or slice insert.
Ex.
M interchanges: UF -> DB (M2)
Layer Interchange: UR -> FR (R or R')
M insert: U M2 U' or [U: M2]
Layer insert: U R U' or [U: R]

They're exactly Pure Corner Comms so just do [A, B].

''' A9/B9 (9 moves)'''
- Exactly like A9 corner commutators except B9 exists meaning there's a optimal solution where the S cancels with B. To recognize there's no 3 move insert like A9 corner comms. There are no M S moves that cancels with B if B is also a M move.

''' Orthogonals (10 moves)'''
- To recognize all edges can't be interchange with each other in 1 move. To do them just do some sort of turn and it will setup to a Pure Comm. Formula: [S: [A, B]].

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}
Thanks to LexCubing it's no longer a stub. This is not a tutorial so if you want a better understanding and how to do them go look up Byu's unfinished BS but hey corners are done. I'll be making a tutorial someday so stay tune but you don't need to be optimal to 3-style just know how Pure Comms work and setup to them and everything will be okay.

Beyer-Hardwick Method

2017-05-23T12:42:01Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
- these are the type of edges you'll "see" while learning BH.

'''Half Slice-Plane (4 moves)'''
- they're the Per Specials of edge commutators by being recognized similarly to Per Specials and has a unique property. Per Specials need a 5 move insert while Half Slice-Plane to be optimal must interchange twice on the same outer layer making the commutator really small only needing 4 moves to do it.

It is recognized by having all the edges on the same slice and are mutually interchangeable.

To do them you must know what they're composed of: [A, B]

A is some sort of half turn of a outer layer where most of the edges are in
B is some sort 1/4 turn of a slice where all the edges reside and setups into the A layer for the interchange

Ex. UF ->DF -> UB

To do them figure out what's the middle piece of the cycle. UF is the middle piece and must go to DF. They must interchange on the A layer, so what's A? In the U layer 2/3 edges resides so A is a U2. How to we interchange it with DF with a U2?

That's where B comes in, B is the 1/4 of the slice where all edges reside so M is B. How do we know if we do M or M'? Well use logic, if we do M the UF and DF stickers will go to the D layer but we want it to go to the U layer so we do M'.

Final answer is [A=U2, B=M']

'''Slice-Plane (5 mover, S7, S9) (5, 7, 9 moves)'''
- Half Slice-Planes really are special as not all commutators come from Pure Comms instead some come from Half Slice-Planes.

Slice-Planes are conjugated Half Slice-Planes. They follow the formula: [S: [A,B]] where S is a one set up and [A, B] is the 1/2 Slice-Plane.

To recognize them is just to know all 1 move setups to a certain Slice-Plane:

5 movers are conjugated 1/2 Slice-Planes with a cancellation. There are no optimal cancelleation with S moves with M turns or U2s so All S moved must be a U or U'

S7s are conjugated 5 movers without cancellations. Since there must be no U2 for Slice-Planes you can't use U or U' S moves instead you must do M, M', or M2 S moves.

S9s are conjugated S7 with no cancellations. S9s are a different way to solve A9/B9 cases. Since S7 all start with M moves all the S must now me U moves.

''' Pure Commutators (8 moves)'''
- All other commutators come from these. They follow the formula: [A, B].

To recognize there's an interchange on the slices or layers and there is a either a layer insert or slice insert.
Ex.
M interchanges: UF -> DB (M2)
Layer Interchange: UR -> FR (R or R')
M insert: U M2 U' or [U: M2]
Layer insert: U R U' or [U: R]

They're exactly Pure Corner Comms so just do [A, B].

''' A9/B9 (9 moves)'''
- Exactly like A9 corner commutators except B9 exists meaning there's a optimal solution where the S cancels with B. To recognize there's no 3 move insert like A9 corner comms. There are no M S moves that cancels with B if B is also a M move.

''' Orthogonals (10 moves)'''
- To recognize all edges can't be interchange with each other in 1 move. To do them just do some sort of turn and it will setup to a Pure Comm. Formula: [S: [A, B]].

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}
Thanks to LexCubing it's no longer a stub. This is not a tutorial so if you want a better understanding and how to do them go look up Byu's unfinished BS but hey corners are done. I'll be making a tutorial someday so stay tune but you don't need to be optimal to 3-style just know how Pure Comms work and setup to them and everything will be okay.

Beyer-Hardwick Method

2017-05-23T09:14:12Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
- these are the type of edges you'll "see" while learning BH.

'''Half Slice-Plane (4 moves)'''
- they're the Per Specials of edge commutators by being recognized similarly to Per Specials and has a unique property. Per Specials need a 5 move insert while Half Slice-Plane can interchange twice on the same face making the commutator really small only needing 4 moves to do it. They follow the formula: [A, B]

It is recognized by having all the edges on the same slice and are mutually interchangeable. To do you do a 1/4 M move and do a half turn of the face where 2 of the 3 edges are in. Ex UF -> UB -> DF You do [U2, M']. This is not the usual insert-interchange style commutator. This commutator interchanges 2 edges on the U plane and the U slice. Read Chris Hardwick's explanation below on the External Link to better understand this commutator.

'''Slice-Plane (5 mover, S7, S9) (5, 7, 9 moves)'''
- Half Slice-Planes are really special as not all commutators come from Pure Comms instead some come from Half Slice-Planes.

Slice-Planes are conjugated Half Slice-Planes. They follow the formula: [S: [A,B]] where S is a one set up and [A, B] is the 1/2 Slice-Plane.

To recognize them is just to know all 1 move setups to a certain Slice-Plane:

5 movers are conjugated 1/2 Slice-Planes with a cancellation. There are no optimal cancelleation with S moves with M turns or U2s.

S7s are conjugated 5 movers without cancellations and S9s are conjugated S7 with no cancellations. S9s are a different way to solve A9/B9 cases.

''' Pure Commutators (8 moves)'''
- All other commutators come from these. They follow the formula: [A, B].

To recognize there's an interchange on the slices or layers and there is a either a layer insert or slice insert.
Ex.
M interchanges: UF -> DB (M2)
Layer Interchange: UR -> FR (R or R')
M insert: U M2 U' or [U: M2]
Layer insert: U R U' or [U: R]

They're exactly Pure Corner Comms so just do [A, B].

''' A9/B9 (9 moves)'''
- Exactly like A9 corner commutators except B9 exists meaning there's a optimal solution where the S cancels with B. To recognize there's no 3 move insert like A9 corner comms. There are no M S moves that cancels with B if B is also a M move.

''' Orthogonals (10 moves)'''
- To recognize all edges can't be interchange with each other in 1 move. To do them just do some sort of turn and it will setup to a Pure Comm. Formula: [S: [A, B]].

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}
Thanks to LexCubing it's no longer a stub. This is not a tutorial so if you want a better understanding and how to do them go look up Byu's unfinished BS but hey corners are done. I'll be making a tutorial someday so stay tune but you don't need to be optimal to 3-style just know how Pure Comms work and setup to them and everything will be okay.

Beyer-Hardwick Method

2017-05-22T19:27:33Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A (not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
- these are the type of edges you'll "see" while learning BH.

'''Half Slice-Plane (4 moves)'''
- they're the Per Specials of edge commutators by being recognized similarly to Per Specials and has a unique property. Per Specials need a 5 move insert while Half Slice-Plane can interchange twice on the same face making the commutator really small only needing 4 moves to do it. They follow the formula: [A, B]

It is recognized by having all the edges on the same slice and are mutually interchangeable. To do you do a 1/4 M move and do a half turn of the face where 2 of the 3 edges are in. Ex UF -> UB -> DF You do [U2, M']. This is not the usual insert-interchange style commutator. This commutator interchanges 2 edges on the U plane and the U slice. Read Chris Hardwick's explanation below on the External Link to better understand this commutator.

'''Slice-Plane (5 mover, S7, S9) (5, 7, 9 moves)'''
- Half Slice-Planes are really special as not all commutators come from Pure Comms instead some come from Half Slice-Planes.

Slice-Planes are conjugated Half Slice-Planes. They follow the formula: [S: [A,B]] where S is a one set up and [A, B] is the 1/2 Slice-Plane.

To recognize them is just to know all 1 move setups to a certain Slice-Plane:

5 movers are conjugated 1/2 Slice-Planes with a cancellation. There are no optimal cancelleation with S moves with M turns or U2s.

S7s are conjugated 5 movers without cancellations and S9s are conjugated S7 with no cancellations. S9s are a different way to solve A9/B9 cases.

''' Pure Commutators (8 moves)'''
- All other commutators come from these. They follow the formula: [A, B].

To recognize there's an interchange on the slices or layers and there is a either a layer insert or slice insert.
Ex.
M interchanges: UF -> DB (M2)
Layer Interchange: UR -> FR (R or R')
M insert: U M2 U' or [U: M2]
Layer insert: U R U' or [U: R]

They're exactly Pure Corner Comms so just do [A, B].

''' A9/B9 (9 moves)'''
- Exactly like A9 corner commutators except B9 exists meaning there's a optimal solution where the S cancels with B. To recognize there's no 3 move insert like A9 corner comms. There are no M S moves that cancels with B if B is also a M move.

''' Orthogonals (10 moves)'''
- To recognize all edges can't be interchange with each other in 1 move. To do them just do any turn and it will setup to a Pure Comm. Formula: [S: [A, B]].

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}
Thanks to LexCubing it's no longer a stub. This is not a tutorial so if you want a better understanding and how to do them go look up Byu's unfinished BS but hey corners are done. I'll be making a tutorial someday so stay tune but you don't need to be optimal to 3-style just know how Pure Comms work and setup to them and everything will be okay.

Beyer-Hardwick Method

2017-05-22T19:23:31Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A(not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
- these are the type of edges you'll "see" while learning BH.

'''Half Slice-Plane (4 moves)'''
- they're the Per Specials of edge commutators by being recognized similarly to Per Specials and has a unique property. Per Specials need a 5 move insert while Half Slice-Plane can interchange twice on the same face making the commutator really small only needing 4 moves to do it. They follow the formula: [A, B]

It is recognized by having all the edges on the same slice and are mutually interchangeable. To do you do a 1/4 M move and do a half turn of the face where 2 of the 3 edges are in. Ex UF -> UB -> DF You do [U2, M']. This is not the usual insert-interchange style commutator. This commutator interchanges 2 edges on the U plane and the U slice. Read Chris Hardwick's explanation below on the External Link to better understand this commutator.

'''Slice-Plane (5 mover, S7, S9) (5, 7, 9 moves)'''
- Half Slice-Planes are really special as not all commutators come from Pure Comms instead some come from Half Slice-Planes.

Slice-Planes are conjugated Half Slice-Planes. They follow the formula: [S: [A,B]] where S is a one set up and [A, B] is the 1/2 Slice-Plane.

To recognize them is just to know all 1 move setups to a certain Slice-Plane:

5 movers are conjugated 1/2 Slice-Planes with a cancellation. There are no optimal cancelleation with S moves with M turns or U2s.

S7s are conjugated 5 movers without cancellations and S9s are conjugated S7 with no cancellations. S9s are a different way to solve A9/B9 cases.

''' Pure Commutators (8 moves)'''
- All other commutators come from these. They follow the formula: [A, B].

To recognize there's an interchange on the slices or layers and there is a either a layer insert or slice insert.
Ex.
M interchanges: UF -> DB (M2)
Layer Interchange: UR -> FR (R or R')
M insert: U M2 U' or [U: M2]
Layer insert: U R U' or [U: R]

They're exactly Pure Corner Comms so just do [A, B].

''' A9/B9 (9 moves)'''
- Exactly like A9 corner commutators except B9 exists meaning there's a optimal solution where the S cancels with B. To recognize there's no 3 move insert like A9 corner comms. There are no M S moves that cancels with B if B is also a M move.

''' Orthogonals (10 moves)'''
- To recognize all edges can't be interchange with each other in 1 move. To do them just do any turn and it will setup to a Pure Comm. Formula: [S: [A, B]].

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}
Thanks to LexCubing it's no longer a stub. This is not a tutorial so if you want a better understanding and how to do them go look up Byu's unfinished BS but hey corners are done. I'll be making a tutorial someday so stay tune but you don't need to be optimal to 3-style just know how Pure Comms work and setup to them and everything will be okay.

Beyer-Hardwick Method

2017-05-22T19:20:45Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A(not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
- these are the type of edges you'll "see" while learning BH.

'''Half Slice-Plane (4 moves)'''
- they're the Per Specials of edge commutators by being recognized similarly to Per Specials and has a unique property. Per Specials need a 5 move insert while Half Slice-Plane can interchange twice on the same face making the commutator really small only needing 4 moves to do it. They follow the formula: [A, B]

It is recognized by having all the edges on the same slice and are mutually interchangeable. To do you do a 1/4 M move and do a half turn of the face where 2 of the 3 edges are in. Ex UF -> UB -> DF You do [U2, M']. This is not the usual insert-interchange style commutator. This commutator interchanges 2 edges on the U plane and the U slice. Read Chris Hardwick's explanation below on the External Link to better understand this commutator.

'''Slice-Plane (5 mover, S7, S9) (5, 7, 9 moves)'''
- Half Slice-Planes are really special as not all commutators come from Pure Comms instead some come from Half Slice-Planes.

Slice-Planes are conjugated Half Slice-Planes. They follow the formula: [S: [A,B]] where S is a one set up and [A, B] is the 1/2 Slice-Plane.

To recognize them is just to know all 1 move setups to a certain Slice-Plane:

5 movers are conjugated 1/2 Slice-Planes with a cancellation. There are no optimal cancelleation with S moves with M turns or U2s.

S7s are conjugated 5 movers without cancellations and S9s are conjugated S7 with no cancellations. S9s are a different way to solve A9/B9 cases.

''' Pure Commutators (8 moves)'''
- All other commutators come from these. They follow the formula: [A, B].

To recognize there's an interchange on the slices or layers and there is a either a layer insert or slice insert.
Ex.
M interchanges: UF -> DB (M2)
Layer Interchange: UR -> FR (R or R')
M insert: U M2 U' or [U: M2]
Layer insert: U R U' or [U: R]

They're exactly Pure Corner Comms so just do [A, B].

''' A9/B9 (9 moves)'''
- Exactly like A9 corner commutators except B9 exists meaning there's a optimal solution where the S cancels with B. To recognize there's no 3 move insert like A9 corner comms. There are no M S moves that cancels with B if B is also a M move.

''' Orthogonals (10 moves)'''
- To recognize all edges can't be interchange with each other in 1 move. To do them just do any turn and it will setup to a Pure Comm. Formula: [S: [A, B]].

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-22T15:12:46Z

LSLL:

Beyer-Hardwick Method

2017-05-22T01:01:39Z

LSLL:

'''B'''eyer-'''H'''ardwick Method. An advanced blindsolving method based on the use of 3-cycle algorithms. The fundamental idea behind the BH method is to use pre-memorized, move optimal, commutator 3-cycles for all possible 3-cycles starting from a fixed buffer location. For pieces where orientation is defined, the position and orientation of pieces are solved simultaneously. The use of commutators allows the BH method to be used for solving any sized cube blindfolded.

== BH Corners ==
These are the different types of commutators you will "see" while learning BH.

''' Pure Commutators (8 moves)'''
- are the purest and smallest form of commutators and all other commutators are just setups into them hence called Pure Commutators (except Cyclic Shifts and Per Specials). They follow this formula: ABA'B' or [A, B] for short.

A is always a 3 move insert
B is the interchange move and always 1 move and opposite from the middle move of A. Ex [R U R', D] U is the middle move of the A move that means the B move must be some form of D move

To recognize 1 corner is in 1 layer and the remaining 2 are in a different layer. The 2 corners in one layer must be interchangeable and there must be a 3 move insert between 1 corner in 1 layer and 1 corner in the different layer.

There are only 3 A moves:
R U R'
R U' R'
R U2 R'
All Pure Comms are composed from these 3 A moves meaning there's only 18 algs and everything else is just mirror/inverse/rotation from the 18 algs.

''' A9 (9 moves)'''
- are conjugated Pure Comms with a cancellation resulting in 9 moves instead of 10. They follow this formula: SABA'B'S' or [S: [A, B] ]where S is the setup move.

Ex. [R: RUR', D] becomes R R U R' D R U' R' D' R' or simply R2 U R' D R U' R' D' R'.

All setup moves or S moves must cancel with A. There are no B9s or a corner commutator with a S move that cancels with the B move. If you find 1 then that just means it's a Pure Comm. Edge commutators though have B9s.

To recognize them they're the same as Pure Comms but they don't have a 3 move insert. But that's not always the case so just use elimination to figure it out. The first rule always applies so if you see that then it's a A9.

''' Orthogonals (10 moves) '''
- are conjugated Pure Comms with no cancellations resulting in 10 moves instead of 9. They follow the same formula [S: [A, B]] as well but the S move doesn't cancel with A nor B.

To recognize them you must know 2 terms:

Opposites - when 2 corners can't go to each other's places with 1 quarter turn. Ex. UBL is opposite to DBR
Note: We're only talking permutation-wise. Meaning UBL being opposite to BDR is as the same as saying UBL is opposite to DBR.

AnI or nI (Adj Non-interchangeable or Non-interchangeable - nI is a better term since saying AnI sometimes doesn't make sense. Put it simply a sticker is not interchangeable with another if you can't do it 1 move.

To recognize all 3 corners are opposites and nI to each other. (See what I mean? Saying AnI here doesn't make sense when alll the corners are opposite)

To do them just do ANY quarter turn S move and it will setup into a Pure Comm and undo the S move.

''' Cyclic Shifts (11 moves)'''
- they don't come from Pure Comms and follow a different formula: [A: B] [C: B] or A B A' C B C'

A is always 2 moves which are comprised of 2 adjacent layers (Ex. F and R) and one is going clockwise and another going counterclockwise

B is always a half turn and is not a opposite layer to either A or C
Ex. A is R F so B must not be any form of B or L move

C is the reverse of A(not inverse)
Ex. A is R F so C must be F R

Ex. [R F: U2] [F R: U2]

To recognize all 3 corners are on the same layer and are nI.

To do them figure out the middle piece of the cycle. Ex. UBL -> LFU -> BRU The middle piece is UBL and must go to LFU. Do [R F: U2] to swap them and now the next piece must go to BRU so do [F R: U2] to swap them and then you're done.

If you don't understand any of these it's all right since all Cyclic Shifts are the same meaning just by learning the one alg above you just need to mirror/inverse/rotate to solve any Cyclic Shift.

'''Column Cases (11 moves)'''
- they're just 11 movers that can be done in 2 ways: a setup into an A9 with no cancellation or a setup into a Cyclic Shift with a cancellation.

To recognize there are 2 corners interchangeable by a half turn and the remaining corner is nI to them.

'''Per Specials (12 moves)'''
- a special type of commutators that follow the same [A, B] formula but the A part is a 5 move insert and B is just a half turn. There's 6 of them and they're all the same meaning you just need to learn 1 alg and just mirror/inverse/rotate it.

Ex. [L U2 R' U2 L, U2]

To recognize all 3 corners are opposite and mutually interchangeable. Ex. UBL -> DFL -> DBR

== BH Edges ==
- these are the type of edges you'll "see" while learning BH.

'''Half Slice or Slice Plane Moves (4,5,7,9 moves)'''
- These guys are the Per Specials of edge comms. Due to the properties of edges on a odd size cube a double interchange om the same layer is allowed making the smallest # of moves to solve edge comms is 4 (Note: Alongside with Pure Comms all commutators comes from them). The commutator [M, U2] is the basis of all slice/half slice plane moves. A 5 mover is a conjugated 4 mover with a cancellation, a 7 mover is a conjugated 5 mover, and a 9 mover is a conjugated 7 mover. Formula: [A, B](4) [S: [A,B]](5,7,9)

'''Pure Commutators (8 moves)'''
- alongside with 4 movers all other commutators comes from them. Everything else is a conjugated Pure/4movers commutator. Read BH Pure Comms for more info. Formula: [A, B].

''' A9/B9 (9 moves)'''
- same as BH A9s except B9s exist. S9(Slice/Half Slice 9 movers) are a different way of solving 9 moves cases. Formula: [S: [A, B]]

''' Orthogonals (10 moves)'''
- Read BH Corners for more info. I don't how to recognize them so good luck. All I know from Chris's post is that this is their formula. Formula: [S: [A, B]]

I'll update this once I know how to recognize them. Byu fucked up and never made a BH edges tutorial. But doing conjugated Pure Comms will still get you far. Besides move optimal is not always fast unless you're good at rotating. No one on SS Forums answered how to recognize BH edges so fuck you guys. :)

== External links ==
* [http://www.speedcubing.com/chris/bhcorners.html BH method for corners - Hardwick]
* [http://www.speedcubing.com/chris/bhedges.html BH method for 3x3 edges - Hardwick]
* [http://skarrie.se/edges/ BH method for 3x3 edges - Skarrie]
* [https://spreadsheets.google.com/pub?key=0AuQyS2mqoMAMdGxaamlSVFo5d0NhNkxUbWp0UGlLdXc&single=true&gid=0&output=html BH method for corners - Riffz]
* [https://docs.google.com/spreadsheet/pub?key=0AvKFSRYPmkKRdHY0QTVxQlRmV216SHJLRHlpWHBTakE&output=html BH method for corners - Jalocha]
* [http://www.speedsolving.com/forum/showthread.php?36014-The-edge-cycles-I-use-for-BLD&p=726087#post726087 edge cycles - lucarubik]
* [http://www.speedsolving.com/forum/showthread.php?36145-The-corner-cycles-I-use-for-BLD&p=728708#post728708 corner cycles - lucarubik]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=11909 Thoughts about the BH method]* [http://www.speedsolving.com/forum
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=12268 Brian Yu's tutorial for BH corners]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=310411 Chris Hardwick's explanation of BH 3x3 edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1601#post1601 Beginner BH ideas for big cube edges]
* Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?p=1643#post1643 Intermediate/Advanced BH ideas for big cube edges]

[[Category:Blindsolving methods]]

{{Stub}}

Beyer-Hardwick Method

2017-05-22T00:39:16Z

LSLL:

Beyer-Hardwick Method

2017-05-19T13:44:54Z

LSLL:

Beyer-Hardwick Method

2017-05-19T13:42:02Z

LSLL:

Beyer-Hardwick Method

2017-05-19T13:41:44Z

LSLL:

Beyer-Hardwick Method

2017-05-19T13:29:34Z

LSLL:

Beyer-Hardwick Method

2017-05-19T13:27:33Z

LSLL:

Beyer-Hardwick Method

2017-05-19T13:26:31Z

LSLL:

Beyer-Hardwick Method

2017-05-19T11:58:00Z

LSLL:

Beyer-Hardwick Method

2017-05-19T11:57:31Z

LSLL:

Beyer-Hardwick Method

2017-05-19T11:56:59Z

LSLL: :p

Beyer-Hardwick Method

2017-05-19T11:43:46Z

LSLL: Penis

Beyer-Hardwick Method

2017-05-19T11:38:15Z

LSLL: Sex