I certainly agree that confirming minimality is an important result.
But I can report that my search found 16 unique (*not* unique up to
conjugacy) half-way positions. I use the term "half-way" advisedly.
The "half-way" positions are 9q from Start and 8q from B or vice
versa. I guess you could say that the vice versa gives you a total
of 32=16+16 half-way positions, but the whole concept of "half-way"
is pretty slippery in this case anyway.
If I understand this, there are 16 positions at 9q from Start and 8q
from B, and there are 16 other positions at 8q from Start 9q from B.
Is each of the first bunch adjacent to exactly one of the other? And
vice versa? It would be good to get them reduced by Q2-conjugacy, as
> The four rotations are i, b, bb, and bbb, where we use lower case
> letters to simulate Frey and Singmaster's script notation for rotations.
> For example, b is the whole cube rotation consisting of grasping
> the Back face and rotating the whole cube (not just the Back face)
> clockwise by 90 degrees.
The reflections are similarly rrv, rrbv, ttv, and ttbv, where t and r
are the whole-cube rotations by the Top and Right faces, and v is the