Wetten, dass..? vom 5. November

Have a look at the above picture. See how the white layer isn't quite solved? Think that's bad? Well, let me show you the advantages compared to solving it correctly...

A well-known method for solving the 2x2x2 in two steps is to solve one layer as the first step and then solve the other layer as the second step. This second step is called CLL and has 43 cases averaging 8.46 moves. However... what happens if in the first step, you only solve a "rotten" first layer? By this I mean arranging the first layer pieces into one layer, but not solved relative to each other. For example like the "white layer" in the pictures above. As it turns out, you can actually save moves by intentionally making "mistakes" this way.

Here I analyze 18 variations, the DBL and DRB corners being solved and acting as a reference, but the DFR and DLF corners being wrong (like in the above picture). There are 18=3*3*2 variations because DFR and DLF can each be oriented three ways and they can be swapped. One of these 18 of course actually isn't rotten but a correctly solved first layer. The following table shows the number of moves necessary to fix the first layer and solve the second layer.

 DFR_DLFholds ... HTM QTM min wavg max min wavg max RDF_FDL 5 7.506 9 5 9.012 11 FRD_LFD 5 7.506 9 5 9.012 11 LFD_FRD 5 7.605 9 5 9.031 11 FDL_RDF 5 7.605 9 5 9.031 11 FDL_FRD 5 7.611 10 6 9.031 11 DLF_RDF 4 7.679 9 4 9.068 12 LFD_DFR 4 7.679 9 4 9.068 12 FDL_DFR 6 7.716 9 8 9.630 11 DLF_FRD 6 7.716 9 8 9.630 11 FRD_FDL 5 7.747 9 8 9.179 12 RDF_LFD 5 7.747 9 8 9.179 12 LFD_RDF 5 7.753 9 7 9.062 12 DFR_LFD 6 7.870 10 6 9.154 12 DFR_FDL 6 7.870 10 6 9.154 12 FRD_DLF 6 7.870 10 6 9.154 12 RDF_DLF 6 7.870 10 6 9.154 12 DLF_DFR 6 8.037 9 8 9.827 12 DFR_DLF 0 8.463 11 0 9.907 13

The table lists the 18 variations sorted by average HTM. The variation names refer to what's at DFR and DLF, so for example "FDL_RDF" means that the FDL corner is at position DFR and the RDF corner is at position DLF. The case "DFR_DLF" thus refers to the variation where the first layer is correct, not rotten.

For each variation I have computed a full set of length-optimal algorithms covering all cases, click on an average to see them. Min and max tell the length of the shortest and longest algorithm, and "wavg" is the weighted average (i.e., taking into account the probability of each case to occur in a solve).

Interestingly, all 18 variations have the same number of cases, namely 43. Since the "correct first layer" variation has the "solved cube" case, it needs 42 algorithms. The other variations each need 43.

More interesting however is that the "correct first layer" actually needs the most moves on average and maximum and in both HTM and QTM! And yes, that does include the "solved cube" case which gives the "correct first layer" a little boost of zero moves.

The best variations are RDF_FDL and FRD_LFD, that's the two where DFR and DLF are at the right place but misoriented the same way. Which means that the last layer is misoriented as well, i.e., its corner orientations aren't a multiple of 3 and thus the LL cases look different from normal CLL. If you do want them to look like normal CLL, then the best variation is FDL_FRD.  pictures thanks to...

Stefan Pochmann