3x3x3 explored out to 12 in FTM

[Johannes91]

This tells you that considering M and inv, after 2 moves there are 8 different possibilities:

Cubes equivalent to R L
Cubes equivalent to R L'
Cubes equivalent to R L2
Cubes equivalent to R2 L2
Cubes equivalent to R U
Cubes equivalent to R U'
Cubes equivalent to R U2
Cubes equivalent to R2 U2

A silly question? If you only consider M, what is the 9th possibility?

R2 U

 

[Lucas Garron]

These tables tell us how many different cubes we can reach with a certain number of moves, starting from the solved cube. You maybe think you can reach 18 different cubes with one turn - choose from 6 sides and between 3 degrees (90/180/270), and 6*3=18.

But is the cube you get with R U2 really different from the one you get with F L2? Well, they sure look different at first sight, but you really did the same thing twice, just from different angles. A clockwise quarter turn on one side, then a double turn on an adjacent side. So the question is, how many really different cubes do we get? For that you must specify what exactly you mean with different. Thinking positively, we instead specify what we consider equivalent. And what we consider equivalent will only be counted once.

That's where the symmetries come into play. The M symmetry considers equivalent what just differs by whole cube rotation and mirroring, for example R U2 is considered equivalent to F' L2. The inv symmetry considers equivalent what just differs by inversing, for example L U2 is considered equivalent to U2 L'. Take both symmetries together and you also consider L U2 equivalent to let's say F2 R.

Example:

d  mod M + inv
-- ------------
 0            1
 1            2
 [B]2[/B]            [B]8[/B]

This tells you that considering M and inv, after 2 moves there are 8 different possibilities:

Cubes equivalent to R L
Cubes equivalent to R L'
Cubes equivalent to R L2
Cubes equivalent to R2 L2
Cubes equivalent to R U
Cubes equivalent to R U'
Cubes equivalent to R U2
Cubes equivalent to R2 U2

A silly question? If you only consider M, what is the 9th possibility?

R2 U, I think.

 

[Stefan]

Yes, cubes equivalent to R2 U. Considering inv in addition to M, it's equivalent to R U2.

You might wonder why not R2 L. That's because it's equivalent to R L2 already considering M alone.

 

[Herbert Kociemba]

A silly question? If you only consider M, what is the 9th possibility?

For example R2 U.

Edit: Did not see that Lucas and Stefan already gave the answer

 

[Rune]

A silly question? If you only consider M, what is the 9th possibility?

For example R2 U.

Edit: Did not see that Lucas and Stefan already gave the answer

For example?

 

[Herbert Kociemba]

A silly question? If you only consider M, what is the 9th possibility?

For example R2 U.

Edit: Did not see that Lucas and Stefan already gave the answer

For example?

Yes, for example. Because you also may take R2 U' or U2 R etc. for the 9th possibility.