# What is God's Number for Solving Just Edges on the 3x3x3?

#### Christopher Mowla

We already know what God's Number is for just solving corners because we already know God's Number for the 2x2x2. I know the superflip is at minimum 20 moves (as is God's Number for the entire 3x3x3), but would it require fewer moves if we didn't worry about solving back corners? How about the other 3x3x3 states which require 12-20 htm? (God's number is 11 htm for just solving corners/the entire 2x2x2, for those who don't already know).
If we can see a table of data for just the edges, as is on cube20.org, we can possibly use this table to calculate a new upperbound for the edge pairing process of 3x3x3 reduction on the nxnxn cube, considering that with reduction we perform outer layer turns to position composite edges in a specific way to then apply inner layer turns. Of course "inner layer turns" means either a conjugate, commutator, or a piece of a commutator.

If we somehow in the future could calculate the optimal 3x3x3 composite edge move setups for a single commutator, piece of commutator or conjugate, then we would have an edge pairing solution (for one orbit of wing edges) of the form:
minimum number 3x3x3 turns to position composite edges + 1 algorithm which uses inner layer turns.

For example, if we needed to do just a 12 2-cycle of wing edges such as the following after executing the minimum number of outer layer turns to set this case up, we would be able to solving it using the conjugate below.
[Rw2 F2 U2 z' Rw2 F2 U2 B' L R u2 R' u2 f2 U: M2 S2 E2] (43,31)

Represented on the 7x7x7 (nxnxn cube representation),
[3r2 F2 U2 z' 3r2 F2 U2 B' L R 2-3u2 R' 2-3u2 2-3f2 U: 3R2 3L2 3F2 3B2 3D2 3U2] (58,34)
Note that if we include all inner layer R,L,F,B,D, and U slices, we have an algorithm which does a 12 2-cycle to all orbits of big cube parts on the nxnxn supercube.
[3r2 F2 U2 z' 3r2 F2 U2 B' L R 2-3u2 R' 2-3u2 2-3f2 U: 2-3r2 2-3l2 2-3f2 2-3b2 2-3d2 2-3u2] (58,34)
If we were to solve the 12-cycle of wing edges case above using a product of 3-cycles, the minimum number of 3-cycles we can use would be 12. That's probably at least 100 btm when you include setup moves, whereas, as you can see with the conjugate above, it's only 34 btm (and it preserves the middle edges and the corners).

Even if we were able to use the optimal 2 2-cycle algorithm [r2 F2 U2: r2], assuming that no setup moves would be needed (which is far from the truth), that's 6(7) = 42 btm >34 btm.

And, of course, instead of using conjugates or commutators which preserve the colors of the centers, we can also use a Cage Method approach if we use my (9,7) 2-cycle of wing edges algorithm, l' F' R2 F2 u' F' l and apply that 12 times with setup moves to solve a 12 2-cycle, for example.
If nothing else, I do think that if God's number for edges on the 3x3x3 isn't known yet, then it should be.

EDIT:
Maybe God's number is more than 11 htm to solve corners on the 3x3x3 because we have fixed centers present as well.

Last edited:

#### Carrot

##### Member
hmm... I'd guess that it's probably about 14 htm. More pieces than corners, but only 2 orientations instead of 3.

Cmowla, trying to lower the upper bound for nxnxn puzzles, for n>3?

#### CarlBrannen

##### Member
Information content of the edges = 11! x 2^11/2 = 40874803200. Number of different moves = 18 but 3 are trivial, so log_15(40874803200) = 9.022

So I'm guessing 10 moves.

#### Ickathu

##### Member
From http://cubezzz.dyndns.org/drupal/text/fullcube.txt: 3x3x3 edges only has a God's number of 14 moves (ftm).
HA!

Anyway, ran some stuff with cube explorer - generated a random cube, cleared corners, then generated a solve. I did 9 (not many, but whatev) and the highest with 12moves.
Code:
Number of Cubes: 9
Cubes not solved yet: 0
---------------------------------------------------------------
Cubes solved optimally: 9

9f*:      1
10f*:      1
11f*:      5
12f*:      2

#### Christopher Mowla

HA!

Anyway, ran some stuff with cube explorer - generated a random cube, cleared corners, then generated a solve. I did 9 (not many, but whatev) and the highest with 12moves.
Code:
Number of Cubes: 9
Cubes not solved yet: 0
---------------------------------------------------------------
Cubes solved optimally: 9

9f*:      1
10f*:      1
11f*:      5
12f*:      2
Lucky guess of 14f, but I also could see where odder was coming from. Anyway, you would probably have to be talented (or lucky) to find a 14f position at random since there are only 248 such positions by the table. I wonder what those positions are!
Code:
      Analysis of 3x3x3 edges only using q+h turns
--------------------------------------------

Dist   Positions          Unique mod M    Unique mod M+inv
----   ---------------    -------------   ----------------
0                  1               1                    1
1                 18               2                    2
2                243               9                    8
3              3,240              75                   48
4             42,807             925                  505
5            555,866          11,684                6,018
6          7,070,103         147,680               74,618
7         87,801,812       1,830,601              918,432
8      1,050,559,626      21,890,847           10,960,057
9     11,588,911,021     241,449,652          120,788,522
10   110,409,721,989   2,300,251,615        1,150,428,080
11   552,734,197,682  11,515,452,614        5,759,027,817
12   304,786,076,626   6,349,914,756        3,176,487,580
13       330,335,518       6,896,891            3,500,434
14               248              24                   24
It also looks like 11 ftm is indeed the maximum for the corners, even with the presence of fixed centers.

#### cuBerBruce

##### Member
I've previously posted the antipodes of the edge group (FTM) on the Domain of the Cube forum. (http://cubezzz.duckdns.org/drupal/?q=node/view/147)

For convenience I include the list below. These are representatives of each symmetry/antisymmetry class.
Code:
Representative move sequence               elements  order
U  D  L  F  D  R  F  B' L' F' B  D  B  L       1       2
U  D  L  F2 D2 L  F' B' D  B2 D2 L' R  B       3       2
U  D  F' L' D  B  U' D2 L2 R' B  L  U  F2     12       4
U  D2 F  R' U' R' F2 B2 R' U' R' B' U' F2      6       2
U  D  L' F2 U2 L' F  B  U  L' R  B2 D2 F       3       2
U  D' F  U2 F2 L  R' U  L2 D2 F' B  R' B2     12       4
U  D  L2 F' L' U2 B2 D2 L' U2 D' B' R' F'     24       4
U  D  F  L  R  D' R2 U' F' B  R  U2 F2 L      12       4
U  D  L  F' B' R' U' F' U2 R2 F2 D  R2 F'     12       4
U  D  F2 B' L2 B2 D' L' B' L  B2 U' L  R'     12       4
U  D  R' F2 D' F  D' B' U2 R  F2 R2 F' L      24       4
U  D  F' L  U' R' B2 L' B  R2 D' L  F  B2      6       2
U  D2 L' F2 U' R  B' L' D  F2 R  U2 D' L2      6       2
U  D  L  F  U  F' B  D  F2 B  R' U' D' L'      6       2
U  D  R2 F' L' D2 F' R  U  L' B  D2 R  F'      6       2
U  D  L' U  F  R  B' D' L  F2 U2 R2 B  D2     24       6
U  D  L' U  F' D' L2 F' R' U  L' F2 U2 D      24       4
U  D' L2 F  L' U  R  B2 U  F  L' B  U2 L'     12       4
U  D  F  D2 L' F2 U  F' R' B' R2 D' L  D'      6       4
U  D' L  D2 B  R  F' U' B2 L' D2 R  F  L2      1       2
U  D  L' F2 U' D  R2 B' L  R  D' R2 B2 U       6       2
U  D  L' U2 B2 L' U  D  B  D2 B2 L  R' U       6       2
U  D' F' L  R  D2 B2 U  F2 B  L' F  D2 R      16       6
U  D  L  U' D  B2 L2 F  L' R' U  L2 F2 D       8       6

Last edited:

#### CarlBrannen

##### Member
From http://cubezzz.dyndns.org/drupal/text/fullcube.txt : 3x3x3 edges only has a God's number of 14 moves (ftm).
One part of the difference between my estimate and his calculation is that we're using different numbers for the number of states:

"Total number of positions on edges-only 3x3x3:
(2 ^ 12 / 2 ) * 12! = 980,995,276,800"

That is, he kept the centers stationary while moving the edges. Thus he has 12x as many positions as I analyzed as I assumed that we were considering only edges. Thus I keep one edge constant and have only (2^12/2)*11! = 81,749,606,400 positions. In doing this he rejects an overall rotation as a "free move".

#### qqwref

##### Member
That is, he kept the centers stationary while moving the edges.
As we should. There's no real point in solving the edges if the centers are wrong. It's different from the corners because when the corners are solved you can still do single moves that change the centers around without affecting the corners.

I've always wondered why we don't use T2. It stands for Top, which is the same as Up (so it's a U2)
Because Up/Down. If we used T for Top we would have to use B for Bottom and B is already used for Back.

#### CarlBrannen

##### Member
Yeah, I regularly get confused by the up/down notation because of quark notation. In addition to the up/down isodoublet there's also a top/bottom. So sometimes I do "D" when I'm supposed to do "B". Good thing there's no confusion with the charm/strange.

#### Herbert Kociemba

##### Member
HA!

Anyway, ran some stuff with cube explorer - generated a random cube, cleared corners, then generated a solve. I did 9 (not many, but whatev) and the highest with 12moves.
Obvious candidate for an antipode is superflip=all12 edges flipped. And indeed it is 14 moves.

#### Christopher Mowla

What's going on?

At this link, http://cubezzz.dyndns.org/drupal/text/fullcube.txt, the results for "Analysis of 3x3x3 edges only" has changed? Work in progress?
Code:
Analysis of 3x3x3 edges only
----------------------------

Distance    Number of  Branching    Number of  Branching
from   M-Conjugate   Factor    M-Conjugate   Factor
Start     Classes                 Classes
Without                  With
Centers                 Centers

0            1                       1
1            1       1.00            1         1.00
2            5       5.00            5         5.00
3           25       5.00           25         5.00
4          215       8.60          215         8.60
5        1,860       8.65        1,886         8.77
6       16,481       8.86       16,902         8.96
7      144,334       8.76      150,442         8.90
8    1,242,992       8.61    1,326,326         8.81
9   10,324,847       8.31   11,505,339         8.67
10   76,993,295       7.46   96,755,918         8.40
11  371,975,385       4.83  750,089,528         7.75
12  382,690,120       1.03      ....
13    8,235,392       0.02      work
14           54       0.00       in
15            1       0.02    progress
Before it was the following:

Code:
      Analysis of 3x3x3 edges only using q+h turns
--------------------------------------------

Dist   Positions          Unique mod M    Unique mod M+inv
----   ---------------    -------------   ----------------
0                  1               1                    1
1                 18               2                    2
2                243               9                    8
3              3,240              75                   48
4             42,807             925                  505
5            555,866          11,684                6,018
6          7,070,103         147,680               74,618
7         87,801,812       1,830,601              918,432
8      1,050,559,626      21,890,847           10,960,057
9     11,588,911,021     241,449,652          120,788,522
10   110,409,721,989   2,300,251,615        1,150,428,080
11   552,734,197,682  11,515,452,614        5,759,027,817
12   304,786,076,626   6,349,914,756        3,176,487,580
13       330,335,518       6,896,891            3,500,434
14               248              24                   24
I have two more questions, so I will number them.

[1] Does this mean that they are all of a sudden unsure that it's 14 htm, or is it still 14 htm for all edge cases?

[2] Does this mean that, if we assume all corners are solved, we can solve any position of the edges (without affecting the corners) in 14 htm or less,...or does this mean that we are allowed to mess up corners in solving the edges in 14 htm or less?

EDIT:
Nevermind about question (2). For some reason I didn't think about the superflip. So it's 20 for edges (for not moving corners). Sorry.

Last edited:

#### cuBerBruce

##### Member
What's going on?

At this link, http://cubezzz.dyndns.org/drupal/text/fullcube.txt, the results for "Analysis of 3x3x3 edges only" has changed? Work in progress?
It looks like the face turn metric results are missing from that page now. I don't know why.

[1] Does this mean that they are all of a sudden unsure that it's 14 htm, or is it still 14 htm for all edge cases?
The 248 antipodes of distance 14 have been independently confirmed by Richard Korf as well as myself. I would say there is little doubt as to the correctness of that.