Think about that after every transfer you reorient the dice in order that the UR edge is once more on the prime proper place. Clearly this has the identical impact on the dice; the generated patterns are the identical, simply checked out from a unique course.
Because of this as an alternative of an U transfer, you flip the underside and center layer within the different course, which is denoted Dw, and equally instead of R you do Lw. As an alternative of getting the face centres as a hard and fast reference level, the UR edge is a hard and fast reference level whereas the centres transfer.
I’ve analysed such dice subgroups intimately, and this one is listed on that web page as entry 2i within the desk of teams involving huge strikes. Be aware that right here the Uw and Rw strikes are used, so that it’s the DL edge that’s mounted as an alternative of the UR edge, however that’s equal.
| Turbines | Mounted items | Dimension | Corners | Edges | Centres | Restrictions | ||||
|---|---|---|---|---|---|---|---|---|---|---|
| Uw, Rw | DBL DL DFL | $7151455567872000$ | = | $frac{6!}{6} frac{3^6}{3}$ | * | $frac{11!}{2} frac{2^{11}}{2}$ | * | $24$ | / | $2$ |
Clearly the 2 corners adjoining to the UR edge by no means separate from it. The opposite six corners transfer as within the two-generator group, so solely $5!=120$ permutations are reached as an alternative of $6!=720$. Solely even permutations of the sides are doable, as a result of a large transfer has two 4-cycles of edges which is even. All different piece actions are doable, topic to the standard restrictions on piece orientations and whole permutation parity.
To show this group dimension, I provide the transfer sequences beneath. In any blended place of this group, the 6 corners are simply solved first utilizing the usual two-generator strategies that many speedcubers know. The sides and centres can then be solved utilizing the next sequences and their conjugations.
(DF, UF, UB)
Uw Rw Uw’ Rw Uw2 Rw Uw’ Rw’ Uw Rw’ Uw’ Rw Uw2 Rw Uw’ Rw’ Uw Rw2 Uw’
UR+, UB+
Rw’ Uw Rw’ Uw’ Rw2 Uw2 Rw Uw2 Rw’ Uw2 Rw’ Uw’ Rw Uw2 Rw Uw’ Rw Uw Rw2 Uw2 Rw’ Uw2 Rw Uw2 Rw Uw Rw’ Uw2
(U,L,F)(D,R,B) Six-spot
Rw Uw2 Rw Uw’ Rw’ Uw2 Rw’ Uw2 Rw Uw Rw Uw Rw2 Uw’ Rw2 Uw’ Rw Uw Rw2 Uw Rw2 Uw’ Rw’ Uw’ Rw’ Uw2
To summarise, $7,151,455,567,872,000$ positions are reachable if the sting strikes anyplace, but when it has to return to its house location (i.e. centres additionally solved) then there are $7151455567872000/24=297,977,315,328,000$ positions.
