Beyer-Hardwick Method

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Beyer-Hardwick method
Information about the method
Proposer(s): Chris Hardwick , Daniel Beyer
Proposed: ~2009
Alt Names: none
Variants: Algorithms
No. Steps: 2
No. Algs: 0-819 (Intuitive) (378 for corners 441 for edges)
Avg Moves: ???
Purpose(s):


Beyer-Hardwick 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)
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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

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