# A Curiosity

Playing around with an optimal slice turn solver of mine, I found two turn sequences which are identical except for two turns which are swapped:

```R U2 R MU2 MF L' MU' L2 B2 R MU2 MF L U2 R MU2
R U2 R MU2 MF L' MU' L2 B2 R MF MU2 L U2 R MU2```

The first gives superflip and the other gives the 26 q-turn hermit position—superflip composed with a four spot pattern. I don't know what if any significance this has, but I find it remarkable.

## Comment viewing options

### A Curiosity

It may not be quite as remarkable as you think.

Consider the 4-spot pattern MF MU2 MF MU2, and call that P. This permutes only the centres, so this move sequence commutes with any normal move like R, U2, L', etc. In other words R P = P R or R P R' = P, and the same for any other face move.

P almost commutes with any middle slice move, in the sense that it remains a 4-spot pattern, but possibly oriented differently. This is because the reorientation CR P CR' equals (R MR L') P (R' MR' L) which due to commutivity of R, L and P equals MR P MR'.

What this means is that you can insert MF MU2 MF' MU2 into any pattern's move sequence at any point to get a move sequence for that pattern plus 4-spot. In this case it just so happens that your superflip sequence has consecutive quarter and half turn middle slice moves. Inserting the 4-spot sequence there results in cancellation and a net reversal of those two slice moves. You could do it just as well with the other MU2 MF in the sequence.

Jaap
Jaap's Puzzle Page: http://www.jaapsch.net/puzzles/

### Excellent. Nice analysis.

Excellent. Nice analysis.

Following your suggestion, all three symmetry equivalent hermit positions may be obtained by transposing the MF/MU2 turns of the superflip sequence.

```Superflip  R U2 R MU2 MF L' MU' L2 B2 R MU2 MF L U2 R MU2
Hermit_y   R U2 R MU2 MF L' MU' L2 B2 R MF MU2 L U2 R MU2
Hermit_x   R U2 R MF MU2 L' MU' L2 B2 R MF MU2 L U2 R MU2
Hermit_z   R U2 R MF MU2 L' MU' L2 B2 R MU2 MF L U2 R MU2
```

The transpositions change the orientation of the four spot pattern applied to superflip as your analysis would predict.