Beyer-Hardwick Method

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Beyer-Hardwick method
Information about the method
Proposer(s): Chris Hardwick, Daniel Beyer
Proposed: ~2009
Alt Names: 3-Style
Variants: Algorithms
No. Steps: 2
No. Algs: 0-818 (Intuitive) (378 for corners 440 for edges)
Avg Moves: ???

The Beyer-Hardwick Method, abbreviated to BH, commonly known as 3-Style, is an advanced blind-solving 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') and (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.


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)

- 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