# 4x4x4 Corner Exchange Search

#### reThinking the Cube

##### Member
I have been searching without luck for a good algorithm that will exchange just 2 corner pieces:

(UFL<->DRB) on a 4x4x4, and leaves everything else intact.

I already have the very good edge "parity" exchange [r2 U2 r2 (Uu)2 r2 u2] + PLL's, but this is NOT what I am looking for. My solve method already has all 4 LL edges and 2 LL corners solved before these final 2 corners - therefore the permutation issue does not even manifest itself until the very end.

A good CX (corner exchange) algorithm would be an awesome tool useful for blindsolve 4x4x4 as well. The cross diagonal positioning makes for the easiest corner conjugations.

All the best,

reThinker

#### Faz

##### Former Clock NR Holder
r2 U2 r2 Uw2 r2 u2

R2 y' T permutation (R U R' U' R' F R2 U' R' U' R U R' F')

B2

*Note u2 is inner slice only, and Uw2 is both the inner and outer slices.

#### reThinking the Cube

##### Member
I already have the very good 4x4x4 edge "parity" exchange [r2 U2 r2 (Uu)2 r2 u2] + PLL's(T-perm), but this is NOT what I am looking for.

Needs to be just a single corner exchange (ULF)<->(DBR).

Yes - the edge exchange "parity" algorithm can be used to transform the corner exchange "parity", but this is not very efficient. How many qturns and grip changes in all that? To put this into perspective - CubeEx will always solve whole cubes in many less turns! And here there are only 2 pieces left to be switched, and it should not take more turns than for a whole 3x3x3 cube solution. I also do not discover that the corner swap is even needed until after the LL is basically done. There is nothing wrong with this, as long as a good algorithm can be found to do the swap. Still looking for a better way.

reThinker

#### reThinking the Cube

##### Member
Obviously T-perm (with setup moves) can be used to eventually solve ANY cube position. (i.e. OldPochman Blindsolve)

This might be considered efficient in terms of the mental effort required to learn and apply algorithms, but you would all rupture yourselves laughing at me if I recommended using nothing but conjugated T-perms to do the whole Last Layer. Nevertheless, it CAN be done this way.

So why should we accept this approach (edgex+setup+T-perm) for this specific 4x4x4 cube position:

Corner exchange 2 pieces: (ULF)<->(DBR) leaving everything else unchanged.

There just has to be a better way to do this case, and I am seriously interested in any ideas or suggestions that would help me find it.

reThinker

#### nigtv

##### Member
I didn't know this situation was possible (although, I'm still a few weeks away from my first 4x4x4 arriving...). I may be way off here in making assumptions, but if you used reduction to get to this point, and didn't have some other sort of weirdness (like the two flipped edges, treating them as one would mean this cannot happen, as you're thinking about it as a 3x3?), you would need to cross out of the reduced state and back into it again in order to fix the problem?

I may be way off though...

#### Jake Gouldon

##### Member
The parity you are encountering is really a swapping of two edges, but manifesting itself as a swapping of two corners. Unless someone has developed something like cubeexplorer for 4x4, a better alg (probably) will not be found.

#### rahulkadukar

##### Member
I didn't know this situation was possible (although, I'm still a few weeks away from my first 4x4x4 arriving...). I may be way off here in making assumptions, but if you used reduction to get to this point, and didn't have some other sort of weirdness (like the two flipped edges, treating them as one would mean this cannot happen, as you're thinking about it as a 3x3?), you would need to cross out of the reduced state and back into it again in order to fix the problem?

I may be way off though...
It is possible

#### reThinking the Cube

##### Member
The parity you are encountering is really a swapping of two edges, but manifesting itself as a swapping of two corners. Unless someone has developed something like cubeexplorer for 4x4, a better alg (probably) will not be found.
The new 4x4x4 edge swap algorithm (r2 U2 r2 (Uu)2 r2 u2)! replaced some older ones that were awful. I think that maybe this was found with Acube by setting it up so that the R & L faces of the 3x3x3 are pictured like the r & l slices of the 4x4x4. This makes the corners solve like 4x4x4 edge pieces. Then by only allowing any R and L turns but only U2 D2 F2 B2 some algorithms can be produced using 3x3x3 that also correctly cycle 4x4x4 edge pieces. After playing around a little and throwing in a couple of well placed u slices - Eureka! I am just guessing here, and maybe somebody else can confirm the history behind the discovery of this algorithm - which is probably the most important 4x4x4 one ever found. Anyway, it is true that this edge swap ends up manifesting itself as a corner swap. Since a beautiful algorithm has now been found for this edge swap, maybe a nice one can also be found for its sister - the corner swap.

reThinker

#### reThinking the Cube

##### Member
(Rr)2 f2 U2 (Ff)2 U' (Rr)2 U2 (Ff)2 U (Ff)2 R2 U2 F2 (Rr)2 switches URF and ULB

This is interesting. Modifying for (ULF)<->(DBR) could be~

U r2 f2 U2 (Ff)2 U' (Rr)2 U2 (Ff)2 U (Ff)2 R2 U2 F2 (Rr)2 U R2 U'

More direct than I have seen before, but still a bit cumbersome to execute.

How did you find this?

reThinker

#### Tortin

##### Member
It was on Dan Harris' site. He has a couple other parity algs for annoying cases, like for adjacent edge switch.

#### reThinking the Cube

##### Member
Thanks for the pointer. I had been on Dan's site not too long ago and did not see some of that good stuff. Mostly he is trying to efficiently combine the standard parity fixing algs with LL perms. Too bad I did not see any breakthrough algs for the odd-parity corner swap (ULF<->DBR) position.

After racking my brains on this problem I have come up with the following insights:

Once the 4x4x4 has been reduced to a psuedo 3x3x3 by edge pairing - dedge parity (+odd) is easy to change by simply swapping 2 of the 3x3x3 dedges (r2 U2 r2 (Uu)2 r2 u2). The reason this algorithm is so simple, is that swapping 2 - 3x3x3 DEDGE pieces is actually a dbl exchange involving 4 -4x4x4 EDGE pieces - so this odd edge permutation parity fix is paradoxically an EVEN parity operation for the actual 4x4x4!

Trying to fix the permutation parity problem by swapping 2 corners does not work as well since the parity (+odd) of the corners would be a single swap of corner pieces. Easy single exchanges (only possible on even cubes) leaving everything else solved are tough to get. Double exchange algs (possible on all cubes) are not so tough to get. The true 4x4x4 edge parity that shows up as the flipped dedge OLL of the psuedo 3x3x3 is harder to solve for the same reason since it too is a swap of just 2 pieces.

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