# Introducing a variation for Fewest Moves



## campos20 (Dec 23, 2017)

I’ve been considering a variant of domino reduction (or HTA) feasible for Fewest Moves. It solves the cube into some steps:

orient edges in multiple axis (let’s say FB and LR);
solve all edges along with some corners;
complete the cube with corner commutators.
To solve edges, one can build a 2x2x3 and then use the 2 remaining layers to do Edge Permutation or freely use blocks.
I’ve gotten some good results in training. It’s pretty easy to get some skip during the solve after the double EO.

*Additional comments*

The multiple axis orientation can be done in 2 sub steps: orient axis 1 then axis 2 (you can even orient the third axis if you wish).
To have a chance to solve a corner, it needs to be oriented when all the edges get oriented.
You may try double EO in different ways to skip the corner orientation part.
You can do some stuff between the EOs.
To build a 2x2x3, we can build a (lower or upper) 1x2x3 (which I call corner line), place 2 corresponding edges, align 2 more edges (edge line), then build the 2x2x3. This preserves EOs.
To complete edge permutation, we may stumble with 2 permuted edges. We can use Sune to solve this.
For FM, you may want to avoid the Sune need and finish with AB5C or AB3C.
As consequence of double EO, the E edges are in the E layer.
The core of it is to orient in multiple axis.
*Good examples*
Scramble: R' U' F L2 U2 B' L2 F' U2 B L2 B R' B2 L U B L2 B' D' F2 R F R' U' F


Spoiler



D' R' U' F // EO FB
L R2 U R // EO LR
D F2 D2 B2 // CORNER LINE+ EDGE LINE
U2 // 123
L2 // LINE FOR 223
D R2 U // AB5C (17)



The next examples were taken from the Facebook Weekly Challenge from Fewest Moves group, week 17.

1. R' U' F L2 D2 B2 U B2 D' B2 U B' D2 R' D L2 B2 R' U' F' R' D2 R' U' F


Spoiler



L' U R F // EO FB
(R D R') // EO LR
B2 D' B2 // CORNER LINE
L2 D B2 D U' L2 // EDGE LINE
D' U F2 // BLOCKS
U2 F2 D' F2 // AB3C (23)



2.- R' U' F D R2 U B2 D2 U' R2 F' U' F2 L' D R D F' U F' D' U2 R' U' F


Spoiler



(R B) // EO FB
(D2 L) // EO LR
(D L2 D2 L2) // CORNER LINE
R2 U D2 B2 D2 // EDGE LINE
B2 U' B2
U2 B2 U' // AB5C (19)



3.- R' U' F D2 R U2 F2 L B2 R' D2 B2 F U2 R B' D B R' F2 L' D B2 R' U' F


Spoiler



(U2 L R F') // EO
(U' L2 R D') // EO
D2 L F2 B2 // 123
L B2 R F2 // EDGE LINE
L' F2 L2 // 223
R' F2 // AB5C (21)



*Example by Anderson Pavia in Weekly FMC*
Scramble: R L' D U' F2 B' R2 B R U' B2 F U2 R2 D B2 U' D' R B2


Spoiler



F' L2 D' B' // EO+SQUARE
F2 U' L // EO+PAIR
U2 L2 U // SQUARE
U2 L' R B2 L R D // AB5C (16)



*(Almost) Complete example:*
Scramble: D2 F2 R2 U B2 F2 D2 R2 B2 R2 F2 L' B' D R F2 U2 F U L2 B'


Spoiler



U B R F2 D F // EO
R B2 D L // EO
L' D' L D' L' D2 L // Sune for corner orientation
L2 F2 B2 // 1x2x3
R2 D2 U F2 // Align edges+Edge Line
D' B2 D2 // 2x2x3
B2 U2 B2 U2 B2 U' // Permuted edges case
U2 B U B' U B U2 B' // Sune to fix this
U' // AB6C



*Comments on the naming*


Spoiler



Why didn't I call it Double EO and name the steps EO1 / EO2 or EO / new EO

1: When you do the 2nd EO, you must not disturb EO1, thus you must avoid 2 moves. This is not true when you are doing pure EO.

2: In a solve, you can do EO1, EO2 in such a way that EO2 destroys EO1 and still have a good solution. Changing the name, this would never be the case.

3: Domino reduction is already out there, but is basically found with brute force these days, not very human friendly. I was quite inspired by domino reduction when I thought about this new method, so naming after it seemed right.

4: In this new method, the number of oriented corners is very, very important. If you have 3 good corners after EO2, the best skeleton you can get is AB5C without breaking any of the EO. If you have 4 good corners, you can end up with AB4C (or AB5C). If you have 5, 6 good corners, AB3C. 8 good corners is full domino reduction. Less than 3 good corners is usually not worth it (although we can have good AB6C with 2 good corners, but it cancels badly). In my point of view, this is a new way of thinking and it's not true for traditional EO (we usually don't care about the numbers of good corners). That's the main reason why I decided to name it some way else.

So, I don't use
moves // EO
stuff // EO

I prefer
moves // EO
stuff // Partial Domino Reduction or PDR.

Sometimes I even indicate the number of good corners after PDR.

I'm not saying everyone should name like that, I'm just saying my point of view.



Matěj Grohmann kindly provided more (and very good) examples!
Using his idea, I decided to gather more examples.


----------



## shadowslice e (Dec 23, 2017)

This looks really interesting! Do you think there is a good way to combine EO wrt 2 of the axises in a reasonably quick way? Or alternatively would it be reasonably easy way to orient the corners along with the edges wrt the second axis?


----------



## campos20 (Dec 23, 2017)

shadowslice e said:


> This looks really interesting! Do you think there is a good way to combine EO wrt 2 of the axises in a reasonably quick way? Or alternatively would it be reasonably easy way to orient the corners along with the edges wrt the second axis?



Sure we can combine 2 axis EO, but we would need some algorithms, the same goes for corners along axis 2. To be honest, orient corners with axis 2 seems a good idea! I haven't considered it before! Perhaps this can let full domino reduction human friendly.


----------



## shadowslice e (Dec 23, 2017)

campos20 said:


> Sure we can combine 2 axis EO, but we would need some algorithms, the same goes for corners along axis 2. To be honest, orient corners with axis 2 seems a good idea! I haven't considered it before! Perhaps this can let full domino reduction human friendly.


I was thinking of sort of combining the ideas with SSC where you orient the corners while placing some edges. I think it could fix the essential problem of SSC which is that the method hasn't really got a very good "last phase". Perhaps we could instead orient the edges wrt a second axis while orienting some (or perhaps even all) the edges.


----------



## campos20 (Dec 23, 2017)

shadowslice e said:


> I was thinking of sort of combining the ideas with SSC where you orient the corners while placing some edges. I think it could fix the essential problem of SSC which is that the method hasn't really got a very good "last phase". Perhaps we could instead orient the edges wrt a second axis while orienting some (or perhaps even all) the edges.


I read a bit about your SSC method, it looks interesting! I'll get used to it in the next days to see if we can think of a way to combine it.


----------



## AlphaSheep (Dec 23, 2017)

shadowslice e said:


> This looks really interesting! Do you think there is a good way to combine EO wrt 2 of the axises in a reasonably quick way? Or alternatively would it be reasonably easy way to orient the corners along with the edges wrt the second axis?


Instead of thinking of it as orienting on 2 axes, I'd think of it as planning EO on one axis, (say wrt F and B as in ZZ), then you combine it with moving the 4 edges that belong on the E slice into the E slice. It's easier to think of it like that rather than as EO on 2 axes.


----------



## campos20 (Dec 23, 2017)

AlphaSheep said:


> Instead of thinking of it as orienting on 2 axes, I'd think of it as planning EO on one axis, (say wrt F and B as in ZZ), then you combine it with moving the 4 edges that belong on the E slice into the E slice. It's easier to think of it like that rather than as EO on 2 axes.


IMHO, orienting 2 axis is easy. It places E edges into the E layer automatically. I'll add this to the comments in the description.


----------



## Pyjam (Dec 23, 2017)

Interesting approach.

If I understand what you mean by edge orientation in 2 axis, I think there's a mistake in this example :



campos20 said:


> 1. R' U' F L2 D2 B2 U B2 D' B2 U B' D2 R' D L2 B2 R' U' F' R' D2 R' U' F



After:
L' U R F // EO FB
(R D R') // EO LR

...FL and BR seems misoriented. In the end, we don't have AB3C.


----------



## DGCubes (Dec 23, 2017)

Pyjam said:


> Interesting approach.
> 
> If I understand what you mean by edge orientation in 2 axis, I think there's a mistake in this example :
> 
> ...



Moves in parentheses mean they are done on the inverse scramble.

(Do the inverse of the moves on the normal scramble: F' R' U' L, then the inverse scramble, then R D R'.)


----------



## xyzzy (Dec 23, 2017)

AlphaSheep said:


> Instead of thinking of it as orienting on 2 axes, I'd think of it as planning EO on one axis, (say wrt F and B as in ZZ), then you combine it with moving the 4 edges that belong on the E slice into the E slice. It's easier to think of it like that rather than as EO on 2 axes.



Yet another way of thinking about it: solving EOline on both the top and the bottom faces. (Except you can freely permute the four edges used for the lines, of course.)


----------



## shadowslice e (Dec 23, 2017)

AlphaSheep said:


> Instead of thinking of it as orienting on 2 axes, I'd think of it as planning EO on one axis, (say wrt F and B as in ZZ), then you combine it with moving the 4 edges that belong on the E slice into the E slice. It's easier to think of it like that rather than as EO on 2 axes.


That' sounds more like SSC already though :^)


----------



## campos20 (Dec 24, 2017)

Pyjam said:


> Interesting approach.
> 
> If I understand what you mean by edge orientation in 2 axis, I think there's a mistake in this example :
> 
> ...


Well, sorry about that. This notation is getting pretty common among FMers.

Just like @DGCubes pointed, the moves between parenthesis must be done in the inverse. An approach to follow the solve if to do R D' R' before starting the scramble, the skip this part in the normal solve.


----------



## campos20 (Jan 19, 2018)

Example by Marlon Marques

Scramble: R' U' F U' L2 F2 L2 B2 U B2 U L' D' B D R F' L' B L2 U L B2 R' U' F



Spoiler



All on inverse
L2 B' // EO FB
L' D' U R // EO RL + square (partial domino reduction)
U2 R2 U' L2 // 2x2x3
B2 D' // good AB6C (12)

Optimal insertion gives 24.


----------



## Bertus (Jan 23, 2018)

So I have tried that and it really looks promissing, but most of the time i stumbled into the problem that i had all the top and bottom edges in the right spot but two of the E-layer edges were swapped. 
Ich understand that this is kind of a 2E2C cycle, but is there any other possibility to avoid that other than try an other way for 2-axis EO?


----------



## campos20 (Jan 23, 2018)

Bertus said:


> So I have tried that and it really looks promissing, but most of the time i stumbled into the problem that i had all the top and bottom edges in the right spot but two of the E-layer edges were swapped.
> Ich understand that this is kind of a 2E2C cycle, but is there any other possibility to avoid that other than try an other way for 2-axis EO?


You can put the 2 misplaced edges on top and perform a Sune or R2 U' R2 U' R2 U2 R2 (super sune). For FMC, that's adding 7 moves, which is not that great. This problem occurs in any blockbuilding method (petrus, heise, ZZ...).


----------



## Pyjam (Jan 23, 2018)

You might try to insert a J-Perm in your solution, hoping for cancellations. It's 10 move long, so not so much longer than a 3 corner cycle.


----------



## campos20 (Feb 27, 2018)

We usually say EO first, but we can do some stuff before actually doing EO and that can still be called EO first somehow. With multiple EO, you can split the EOs and do some stuff between them. This was pointed to me by Kim Vianna. He said he was using PDR like this, leaving final EO for later. As I clearly remembered his words in the following solve, I decided to give him the credits.

https://speedcube.de/forum/showthread.php?tid=5795&pid=206905#pid206905

1. R' U' F U2 B L2 F R2 D2 F' L2 F L' D L D2 R' B L B' U2 F2 U F' R' U' F

U2 F D L' // EO
U' F2 L2 F2 // SQs
D2 B' // 222
L2 // P223
U F' // PARTIAL DOMINO REDUCTION (3 GOOD CORNERS)
U' F2 U' L2 // GOOD AB6C (17)


----------



## campos20 (Mar 14, 2018)

Example by Matěj Grohmann.

Scramble: R' U' F R B2 R2 U2 R' F2 R' U2 R2 D' L B' L R' B L F2 D' F2 R' U' F

F' B' R / EO
(F D B' D2 B) / PDR 6C (renamed, this actually means Partial Domino Reduction with 6 good corners)
B2 U' D2 F2 R2 U / 5C

Skeleton: F' B' R B2 U' D2 F2 R2 U % B' D2 B D' & F' (14)

% U B' D2 B U' B' D2 B / 7 move cancel
& D R2 D' L D R2 D' L' / 2 move cancel

Solution: F' B' R B2 U' D2 F2 R2 U2 B' D2 B U' R2 D' L D R2 D' L' F' // 21


----------

