God's numbers for partial solves

[irontwig]

We all know that (by now) that God's number for the entire 3x3x3 is 20. Also God's number is known for LL (16) and tripod finish (15). What about e.g. LS+LL or "2x2x3 finish"? Also does there exist a position with all the edges solved that still requires 20 moves?

 

[Kirjava]

"2x2x3 finish"

wat

 

[Jaysammey777]

I thought it was 18 not 20

 

[irontwig]

wat

Petrus steps 3+4+5+6+7.

 

[Kirjava]

Petrus steps 3+4+5+6+7.

I thought you meant a 2x2x3 finish, not 2x2x3 then finish.

I was like O_O

 

[cuBerBruce]

Silviu Radu proved that all the configurations with all edges solved require at most 22 quarter turns (http://cubezzz.duckdns.org/drupal/?q=node/view/33). I'll try to take a look at the cases that might require 20 face turns. (I suspect it may still be a big task to prove all of these cases to be < 20f*).

 

[qqwref]

Hm, if there are about 44 million cases for the corners, it should actually be doable to get the optimal quarter and half turn distribution for the entire set... A lot of work, but doable.

 

[cuBerBruce]

Hm, if there are about 44 million cases for the corners, it should actually be doable to get the optimal quarter and half turn distribution for the entire set... A lot of work, but doable.

From Radu's data, I counted 438101 cases not trivially less than 20f*.

 

[uberCuber]

God's number for the cross is commonly known. Are any of these known?

2x2x2
XCross
2x2x3
1x2x3
EOLine

 

[rokicki]

Hm, if there are about 44 million cases for the corners, it should actually be doable to get the optimal quarter and half turn distribution for the entire set... A lot of work, but doable.

Actually very little work. I have a coset solver for this subgroup (and all of its cosets) and it runs very fast; this was the first coset solver I wrote.

There are some results already on the cubelovers list; I'll dig out pointers this evening. The trivial coset was one of the first I ran.

 

[rokicki]

Actually, didn't take long at all; here's the results from 2006:

http://cubezzz.dyndns.org/drupal/?q=node/view/41

When all edges are solved the worst case is 19 moves remaining. There are 216 such positions (out of the 44 million).

 

[cuBerBruce]

Building a specific 2x2x3 can require 12 moves. That's the worst case.

For color neutral 2x2x3, the worst case is 11 moves.

EDIT 1: For a specific 2x2x2 block, the worst case is 8 moves. It's also known to be 8 for the color neutral case.

EDIT 2: Building a specific Xcross can require up to 10 moves (face turns). See [post]110398[/post]. I have independently confirmed this.

Building Xcross from the superflip position requires 9 moves, so God's number for the color neutral case is either 9 or 10.

EDIT 3: I see this post ( [post]185213[/post] ) contains all the answers (face turn metric) to all the cases uberCuber asked about.

EDIT 4: God's number for Petrus steps 3-7 is at least 18.

Example: B' R2 B2 R2 U' L2 R' F R' B F2 D2 L2 B2 U F R' U' (18f*)

If anyone cares, God's number for "2x2x3 finish" is 17.
U' R U' F U' B2 U' F2 L2 D B F L2 B U R' U' (17f*)
R2 U B F2 D' U' L' F' L' U2 L2 U2 F' D B' R' U' (17f*)

 

[cuBerBruce]

I thought I would list a few more God's numbers for partial solves.

ZBLL, without AUF: 15
ZBLL, with AUF: 16
ELL (following CLL), without AUF: 13
ELL (following CLL), with AUF: 14
CLL (following ELL), without AUF: 15
CLL (following ELL), with AUF: 16
LL with a 1x2x2 block formed, without AUF: 14
LL with a 1x2x2 block formed, with AUF: 14

(For LL with a 1x2x2 block formed, the only position where all four AUF cases are 14f* is V-Perm.)

Roux Step 4 (L6E and last 4 centers - 46080 positions): 16
<M, U> group (but solved optimally in terms of face turns, 184320 positions): 16

 

[irontwig]

If I'm not mistaken LL with a solved pair also requires at most 14 htm.

Any lower bounds for S3-S7 with 0, 2 or 4 bad edges?

 

[cubernya]

2x2 is also well known, 14q 11f

 

[Cubenovice]

To check my (in)efficiency at FMC I would like to know God’s number for the following sub steps:
- F2L- slot with no edges oriented, to “leaving 3 corners”
- F2L- slot with two edges oriented, to “leaving 3 corners”
- F2L- slot with all edges oriented, to “leaving 3 corners”

Same for “leaving 5 corners”

To keep things “simple” let’s assume the remaining corners should not be twisted in place.

 

[cuBerBruce]

To check my (in)efficiency at FMC I would like to know God’s number for the following sub steps:
- F2L- slot with no edges oriented, to “leaving 3 corners”
- F2L- slot with two edges oriented, to “leaving 3 corners”
- F2L- slot with all edges oriented, to “leaving 3 corners”

You need to define what you mean by edge orientation. If a last layer edge is in the slot, do you define that edge to be oriented?

 

[Cubenovice]

I guess calling it "neutral" as per Heise's definition doesn't really help ;-)

So yes: an LL edge in the slot should be considered oriented.

 

[cuBerBruce]

If I'm not mistaken LL with a solved pair also requires at most 14 htm.

That's only true if you're only concerned with solving the last layer with respect to itself. An additional AUF move may be required.
Examples:
B' R D B L' B L2 U2 B L' D' R B2 R2 U2 (15f*)
R U R2 F2 D' R2 B L' B' R2 D F' R F' U2 (15f*)
B F' D U' F U F' L D' B' L U' L' F U2 (15f*)
L2 D R F2 R F' D' F2 L D2 R2 B D2 L U' (15f*)
L' U B2 R' B2 R' F R B2 F' R B2 U' L U' (15f*)
R2 D' R U' R' D R' B U B' U' R2 U2 R U' (15f*)
R' U' F' U L' U2 L F2 R2 B' R2 F' R B U' (15f*)

LL with at least 1 oriented CE pair, without AUF: 14
LL with at least 1 oriented CE pair, with AUF: 15

 

[Athefre]

A couple of things I've been interested in:

- Average for a non-matching layer (2x2x2, completely color neutral). For example do R, R', or R2. This versus the average for a matching layer.
- Average for non-matching EG. An example setup is F2RU2RUR'UR.