FMC minimal parity challenge

[AvGalen]

Interesting links. I never saw them before. I guessed 11 because of the 2x2x2, but I figured that because of the centers it could (in theory) be higher.

* Solve F2L, without caring about the last layer? Guess = 15, for the answer 695,280,402,432,000 should be analysed
* Swap 2 corners and 2 edges without caring about their orientation, but without disturbing the rest of the cube? Guess = 10 for minimum amount of moves needed. Proof is getting stronger, but not full yet.
* Solve all 8 corners, without caring about the edges? Answer = 11
* Solve all 12 edges, without caring about the corners? Answer = 14

 

[Kenneth]

Kenneth gave cases for solving the 8 corners, without disturbing the rest of the cube, but I think he was wrong about the two cases (H and pi) being equally bad.
R' U2 R U2 R2 B' D' R' F R2 F' D B U' R' U' (16f*) (I think this is H)
R U2 L2 D L' U2 L2 D' L R' U R2 B2 R2 U' (15f*) (I think this is pi)

Yes, you are right. What I wrote first is what I remember from running those cases in cubex (pretty long ago) but when I checked the page you are linking to I found it was only the H case that is 16 HTM.

But it was not a 8 corner case I was thinking of, it's a LL case

 

[Stefan]

* Swap 2 corners and 2 edges without caring about their orientation, but without disturbing the rest of the cube?

If people weren't so obsessed with a messed up understanding of orientation, the bold part would be completely unnecessary.

I have a feeling that I once checked that no two corners can be swapped in less than ten moves without disturbing the other corners, i.e., like Kenneth proposed. But I'm not sure. Anyway, this can be reduced further. You just need to check nine cases. Fix some corner as the first one to be swapped. For the second corner you have three possible distances times three possible orientations.

 

[Kenneth]

I'm also pretty sure it is not possible to swap two corners on a 2x2x2 in less than 10 turns. Arnaud is talking about A-PLL and yes, A-PLL is 9 turns but does a 3-cycle, to make it a 2-cycle you also have to AUF.

The AUF is also the reason I belive it is 10 or more turns, To create a parity, just do U or U', that's all you need to get a LL you cannot solve using solemny commutators, at some point you have to restore that U-turn...

So a parity alg is at least 1 AUF and that leaves 8 turns to do the rest and that's not enough

 

[ooveehoo]

Hahaaa! L' U' L U L F' L' F U' = R-perm (ignoring orientation) and NINE moves!

Really I agree Kenneth, I think it isn't possible to get swap two corners in 2x2x2 under 10 moves without changing the orientation (because F R U R' U' F doesn't seem to count)..

 

[cuBerBruce]

Hahaaa! L' U' L U L F' L' F U' = R-perm (ignoring orientation) and NINE moves!

To my understanding, the idea was to find a 9-move (or less) 2x2x2 alg that swapped two corner pieces without any side effects on the other six corners. L' U' L U L F' L' F U' twists the UBR corner, so it has such a side effect.

If no such alg exists, it would mean no 3x3x3 alg could exist with less than 10 moves that only swaps 2 corners and two edges, without side effects on the other pieces.

I've now used my 2x2x2 solver code to verify that in fact there is no such alg.

 

[mrCage]

Hahaaa! L' U' L U L F' L' F U' = R-perm (ignoring orientation) and NINE moves!

To my understanding, the idea was to find a 9-move (or less) 2x2x2 alg that swapped two corner pieces without any side effects on the other six corners. L' U' L U L F' L' F U' twists the UBR corner, so it has such a side effect.

If no such alg exists, it would mean no 3x3x3 alg could exist with less than 10 moves that only swaps 2 corners and two edges, without side effects on the other pieces.

I've now used my 2x2x2 solver code to verify that in fact there is no such alg.

Hi

Even if a 9-turn corner swap solution was found on a 2x2x2 cube it would be no evidence that it is possible to swap 2 corners and 2 edges on a 3x3x3 (in 9 turns). It just confirms it would be POSSIBLE.

Cheers!!

- Per