Thread starter
#1

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).

Spoiler: Motivation
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.

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)
Spoiler
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 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)

[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)

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.

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: Dec 13, 2012