Alas, no antipodes yet, but some interesting results nonetheless.
Process UR8 improves on the best known process for a certain
quad-twist in the U layer at 20 q turns. Table 3 in Winning Ways
gives a 22 q turn process.
The following results should be particularly interesting to
the physical cube solver as it is easier to execute a sequence
of 2 adjacent sides compared to a sequence using 3 or more
sides, which may require some re-orienting of the cube. I will
measure the "face index" of a process by the number of different
sides used in a certain cube sequence. Such a measure could be
used to evaluate the relative elegance of two equally long
processes with respect to their face indices.
Jerry Bryan mentions:
> Also, the global maxima are of length 25.
> Does this tell us anything about the Q-turn length of the global
> maxima for the full cube group?
Well, that reminds me of one of the hardest patterns that Dik Winter
tried to find an optimal sequence for:
p141a alternate method F1 R1 L2 U3 R2 L3 U3 D2 R2 F1 D1 B1 D1 F2 U3 of Superfliptwist + 6 X R3 D3 F2 D2 L2 **This process was one of the hardest ever to reduce to 20 moves, requiring over 19 hours on an SGI R4K Indigo, 28 q turns**
My own $.02 worth is that an antipode for the full group of the 3x3x3
cube is probably deeper than an antipode for the < U, R > group.
Optimal Sequences for < U, R > group elements (positions) ---------------------------------------------------------
Edge 3-cycle UR1 = U3 R1 U2 (R1 U1)^2 R2 U3 R3 U3 R2 U1 (16 q, 13 h) Double adjacent edge swap UR2 = U3 R3 U3 R2 U1 R1 U1 R3 U3 R1 U1 (R1 U3)^2 R3 U3 (18 q, 17 h) Diagonal Corner twist UR3 = U1 R1 U3 R1 U3 R2 U1 R1 U1 R3 (U3 R1)^2 U2 R3 U3 R3 (20 q, 18 h) Double opposite edge swap, also in sq group 24 q, 12 h UR4 = R2 U2 R3 (U2 R2)^2 U2 R3 U2 R2 (20 q, 11 h) Edge 7-cycle, equivalent to (U1 R1)^15 UR5 = U3 R1 U3 R3 U3 R1 U2 R3 U1 R3 U2 R1 U3 R3 (U3 R1)^2 (20 q, 18 h) Corner Tri-Twist UR6 = (U3 R3)^2 U1 R1 U3 R3 U3 R2 U1 R2 U3 R3 U3 R1 U1 R3 (20 q, 18 h) Corner Quad-Twist, Flat style UR7 = R1 U3 (R1 U1)^2 (R3 U3)^2 R2 U3 R1 U1 R3 U3 R1 U3 R3 (20 q, 19 h) Corner Quad-Twist, Arms & Legs style (20 q, 20 h) UR8 = R1 U1 R3 U1 R3 U3 R1 U1 R1 (U3 R3)^2 (U1 R1)^2 U3 R3 U3 ML Doodle Position UR9 = (U2 R2)^2 U2 R3 U1 R2 (U3 R2)^2 U1 R1 (22 q, 14 h) Same position found by hand: (a non-optimal 24 q, 15 h) (U2 R2)^3 U1 R1 (U2 R3)^2 U2 R1 U1 4 Opp Corner Swap, also in sq group at 26 q, 13 h UR10 = U3 R3 (U1 R1)^2 U2 R3 U1 R1 (U2 R2)^2 U1 R3 U1 (22 q, 17 h)
Other Subgroups within reach ----------------------------
11. |<U, R2, L2>| = 2^12 3^4 5^2 7 = 58060800 12. |<U2, R, L2>| = 2^12 3^4 5^2 7 = 58060800 17. |<U, R2, F2>| = 2^8 3^5 5^2 7 = 10886400 21. |<U, R2, L2, D2>| = 2^13 3^4 5^2 7 = 116121600 22. |<U, R2, L2, D>| = 2^15 3^4 5^2 7^2 = 3251404800
I welcome any proposed < U, R > group antipodes. I haven't really
looked for anything exotic like < U, R > positions which are
shift invariant, or even if such a beast is possible!
Of course I already mentioned that...
U2 R2 U2 R2 U2 R2 = R2 U2 R2 U2 R2 U2
...but aside from that nothing comes to mind.
Generally when there are elements which occur in both the square's
group AND the < U, R > group the latter is the shorter in q turns.
-> Mark <-