Columns first algorithms

[qqwref]

You only need (at most) one adjacent swap, before doing other moves. That is, if you swap the correct two adjacent pieces, you can finish it with 2gen.

Proof:

Spoiler

Let's take some 2x2x2 scramble with DLB and DLF solved. We want to make it solvable with 2gen, so let's only consider permutation (since any valid orientation can be applied 2gen). Given the two pieces at DRB and DRF, there is exactly one position (up to AUF) of the top four pieces that is solvable with 2gen only. There are three cases:
- The four pieces are correct. We're done.
- The four pieces are an adjacent swap away from being correct. So do that swap.
- The four pieces are an opposite swap away from being correct. So swap DRF and DRB. To see that this makes the position solvable with 2gen, after that swap, execute R' U R U R' to return DRF and DRB to their proper places and do an opposite swap on top.

 

[cuBerBruce]

Nice proof, qqwref. In fact, you've proved something slightly stronger than my claim. Of the 7 possible adjacent swaps, only a particular 5 need be considered. In fact, from a given configuration, those five swaps will each reach a different one of the five other cosets.

I also note that from a specified configuration, doing any of the swaps (DRF,DBR), (UFR,UBL), or (ULF,URB) will reach the same coset. Note that there are 15 total possible swaps among 6 objects. These 15 swaps can be partitioned into 5 sets of 3, such that within a given set of 3 swaps, those three swaps result in reaching the same coset.

I further note that the set of pairs {DRF,DBR}, {UFR,UBL}, and {ULF,URB} matches one of the patterns given on Jaap's Puzzle Page's Two-generator corners group page. Indeed all 5 of the sets of three pairs mentioned in the previous paragraph correspond to patterns on Jaap's page.

Now for a way of recognizing whether or not a given configuration is in the 2-gen compatible coset. Consider the pairs of corner cubies {{DRF,DBR},{UFR,UBL},{ULF,URB}}. If the pairs of cubies in this set are at locations matching one of the five patterns given on Jaap's page, then the configuration must be in one of two cosets, one of which is the 2-gen compatible coset. We still must distinguish whether it's in the right coset or the wrong one. We can do this by considering the pairs of corner cubies {{UFR,UBL},{UFR,DBR},{URB,DRF}}. If it's also true that the pairs of cubies in this set are at locations matching one of the five patterns on Jaap's page, then we can safely conclude the position is 2-gen solvable.

(I'm not sure if this recognition method has been discussed before, or if it can be done fast enough to be practical speedsolving.)