# Optimal Void Cube Up Face Odd Parity Maneuvers

Optimal Void Cube Up Face Odd Parity Maneuvers

When solving the void cube one encounters odd position parity positions as the cube approaches being solved. The last step for me is to solve the Up face corners and this is where I encounter odd permutations. Corners first solvers may encounter an odd permutation of the Up face edges. The question is what is the shortest maneuver which will convert an odd permutation of the Up face corners into an even permutation of the Up face corners leaving everything else unchanged. This is regardless of the effect on the corner twist. Likewise, what is the shortest maneuver which will convert an odd parity up face edge permutation to an even parity permutation regardless of the edge flip?

There are (4! x 33) / 2 = 324 odd parity Up face corner position and (4! x 23) / 2 = 96 odd parity Up face edge positions. These reduce by C4v symmetry to 53 symmetry unique corner positions and 21 symmetry unique edge positions. I have found optimal solutions for these positions and determined that odd parity edge positions may be corrected in a minimum of 12 q-turns or 8 f-turns. Odd parity corner permutations may be corrected in a minimum of 15 q-turns or 13 f- turns.

1. Edge Permutations
1. q-turn metric (12 turns)
1. F R' L D F' R L' U F' R L' U' (standard)
F R' TR F U' R TR' U F' R TR' F' (void)

2. R R L2 D' R L' F R R L2 (standard)
R R TR TR U' R TR' U R R TR TR (void)

3. R F B' D R' F B' D L' F' B D' (standard)
R F TF' R U' F TF' U R' F' TF R' (void)

2. f-turn metric (8 turns)
1. R2 L2 D' R L' F R2 L2 (standard)
R2 TR2 U' R TR' U R2 TR2 (void)

2. Corner Permutations
1. q-turn metric (15 turns)
1. R U' R' U' F' D L D' L' U R U F L' F' (standard)
R U' R' U' F' TU TF TR' U' R TU F TR TF' TU' (void)

2. R F R F' B L' U' F' B L D L' B' U' L (standard)
R F R F' TF U' R' F' TF R U R' TF' R' U (void)

3. R' U L B L B' R F R' U' F B' D F' U' (standard)
R' U TR TU TF TR' TU F TU' TF' U TU' R U' TR' (void)

4. R U' L' U L B' R' D' R' D L' U' L B D (standard)
R U' TR' F TR TF' U' R' U' R TU' F' TU TF TU (void)

5. L U' L' U' B' D U R D' R' L U B R' B' (standard)
TR TF' TU' TR' U' R TR U R' U' TU TF TU F' TU' (void)

6. U R' F' R L' B' R' B' F' D' R L U' F' B' (standard)
U R' F' R TR' U' R' U' TU' TR' U TU F' R' TR' (void)

7. R F' L2 D' B' U R U' B D L2 F L' (standard)
R F' TR TR U' F' TU F TU' F U TR TR F TR' (void)

8. U R' L' D' F B U' L' R' B' L' F' B R' B' (standard)
U R' TR' TF' R TR U' F' TF' TU' R' U' TU TF' R' (void)

9. R U R L' D L B U D F' L' R' B' D2 (standard)
R U R TR' TF U R F TF U' R' TR' TF' TR TR (void)

10. R F U' F' D B' D' B' R' F U F' U' B D (standard)
R F U' F' TU R' TU' TF' U' F TR U' TR' TF TU (void)

11. R U F B' D' B U D R' U' F' U' F D' B' (standard)
R U F TF' R' TF U TU F' U' TR' F' TR TU' TF' (void)

2. f-turn metric (13 turns)
1. R D' B' D2 L' F' U L D2 B R F' U (standard)
R TU' TR' TF2 U' TR' TU TR TF2 TU TF R' U (void)

2. F' U' R B L' B' U R2 F D L' F' R2 (standard)
F' U' R TF U' TF' U R2 F TU TF' TU' R2 (void)

3. R F' L2 D' B' U R U' B D L2 F L' (standard)
R F' TR2 U' F' TU F TU' F U TR2 F TR' (void)

4. B2 D' F' R D B2 L U' F' U B L' D' (standard)
TF2 U' F' TR TF TR2 U TF' U' TF TR TF' TR' (void)

Two turn sequences are given for each maneuver above. The first in the standard < R U F L D B > turn set and the second in the equivalent < R U F TR TU TF > turn set. In the second, tier turns are substituted for the L D B turns. Rather than turning say the L face one turns the MR slice and the R face in the opposite direction. As discussed elsewhere, in the second system turn sequences behave as elements of a mathematical group. This is highlighted by inverting the turn sequence. Inverting the void cube sequence has the expected result—the inverse permutation is applied to the Up face. Inverting the standard sequence has unexpected results. The standard sequences of necessity give the void cube rotated by an odd number of 90° whole cube rotations. In order for the inverse standard turn sequence to give the expected result one must rotate the void cube by this rotation prior to applying the inverse turn sequence.

The optimal edge maneuvers are all figure eight four cycle swaps with different edge flip. The single optimal f-turn maneuver gives the same permutation as the second q-turn maneuver. The corner maneuvers include both single swaps and four cycle swaps. The second q-turn maneuver, equivalent to the first f-turn maneuver, is of particular interest. This is a clockwise four cycle of the Up face corners with no concomitant twist.

## Comment viewing options

### Nice

I used CubeExplorer and asked it: what is the shortest move sequence that will take a void cube with *any* arrangement of top cubies in an odd permutation, and give a void cube with *any* arrangement of top cubies in an even permutation. CubeExplorer told me 8f*. So requiring that it preserve up corners or up edges does not increase the length of the minimal move sequence, at least not in the face turn metric.

### Minimum Solutions for Any Odd Up Face Position

I performed a search of all odd parity Up face positions and can confirm your CubeExplorer search—eight f-turns is the minimum. Furthermore, the one position (and its symmetry conjugates) listed above is the only depth 8 position. I did the q-turn search and although I found 12 more depth 12 positions, I found no shorter positions. All the positions were odd parity edge positions. So, the minimum odd parity corner + even parity edge position is 13 since adding an U turn to a distance 12 odd parity edge position gives an odd parity corner position

### That is interesting and it co

That is interesting and it confirms the above results. Since I can put the Up layer edges in order faster than I can mentally calculate the parity of the Up layer as a whole, a shorter non-specific Up layer odd parity maneuver wouldn't do me much good anyway. Adding middle slice turns to the generators, the optimal numbers are 5 for the edge permutation and 12 for the corner permutation:
1. Edge Maneuver (5 turns)
1. MR2 U' MR' U MR2

2. Corner Maneuvers (12 turns)
1. R F MR TU F R F' TU' MR' F' R' MU'
2. F R2 MU' R2 F TU2 F2 MR' F TU2 F2 MR'

The edge maneuver is exactly the same as the f-turn maneuver since R' TR = MR. The first corner maneuver is the clockwise four cycle with no twist and the second is the clockwise four cycle with all four corners twisted.

### Fun with Conjugation

Still playing around with my void cube solver, the cw four cycle corner swap above is very interesting. Changing the sense of only two turns converts the maneuver into its inverse—a ccw four cycle corner swap:

```   R F MR TU F R  F' TU' MR' F' R' MU' cw corner four cycle
R F MR TU F R' F' TU' MR' F' R' MU  ccw corner four cycle```

Inspecting the turn sequence reveals that the sequence contains a similarity transform:

`   P R P'  where P = R F MR TU F`

This similarity transform converts the R turn, two cw four cycles of the right face edge and corner cubies, into a cw four cycle of the MU slice edge cubies and a cw four cycle of the Up face corner cubies. The final MU' turn puts the middle slice edges back in place leaving a cw rotation of the up face corners. Inverting the sense of rotation of the right face turn inverts the sense of rotation of its conjugate.

Since a cw four cycle of the corners can be converted to a ccw four cycle of the edges with a turn of the Up face, this basic maneuver is easily modified to give the edge cubie four cycle:

```   R F MR TU F R  F' TU' MR' F' R' TU' ccw edge four cycle
R F MR TU F R' F' TU' MR' F' R' TU  cw edge four cycle```

So, with this maneuver you get four for the price of one.