From:

~~~ ~~~ Subject:

With the help of a suggestion from Dan Hoey, I can now provide processes

for the 27 antipodal positions of <U,R> that are unique up to

W-conjugancy. (It is the positions that are unique up to W-conjugancy,

not necessarily the processes.) Without further ado, here are processes

which generate each of the 27 positions.

1 R U U R R U' R U' R U' R U R' U R U R' U' R R U' R' U R U 2 R' U R' U R' U R R U' R' U' R' U' R U' R' R' U' R R U' R' U R U 3 R R U R' U R U' R' U R' R' U R R U' U' R R U R R U R R U' 4 U' R U R' U R U' R' U' R' R' U R' U' R U U R' U R' U' R' U R U' 5 R' R' U U R U R U' R U' U' R' U R R U' R R U' R' U R' U R' U'

6 U' R R U R' U R' R' U' R' U U R U' R' U R R U R R U R' U' R 7 U R' U R' U R U U R' U' R R U R' U' R' U R R U R' U R U' R 8 R' U R' U' U' R U' R U' U' R' U R' U R' R' U' R U R U R U' R U 9 U R U' R U R R U' R U' R' U R U' R' R' U U R' R' U' R U' R R 10 U' R R U R U U R' U' R' U R U' R' R' U' U' R' U' R' U' R' U R U' 11 R' U R U R' U' R U R' U' U' R' U' R R U U R' U R U R U' R' R' 12 R U U R' U U R' U R U' R' U R' U' R U' R' U' R' U R U' R R U 13 R R U R R U' R U' U' R U' R R U' R' R' U' R R U' R U' R R U 14 U' R' U' R' U' R' U R R U' R U' R U' R U' R' U R U' U' R U U R 15 R R U' R' R' U' R U U R' U U R' R' U' R' U' U' R' R' U' R U' U' R 16 U R' U R' R' U' R U' R U U R R U' R U R' U' R U' R' U' R R U' 17 U' R U' U' R' R' U' R U' R U' R U' R R U R' R' U' R' R' U' R' R' U' 18 R U U R U R U' R' U R' U R U' R R U' R' U' R U U R U' U' R' 19 R' U R' U' U' R U' R U' R' R' U U R U' R' U R U' R U R U U R' 20 U R R U' R R U' R' U' R' U U R' U R' U R U U R' U R U' R R 21 R' U R' U' R' U R U U R' U R' U R' U' R U' R' U R U R' U R R 22 U R U U R U' R U R' U' R U U R U' R U' R' R' U R R U R U 23 R' U U R U R' U R' U R U R' U' R R U' R U' U' R' U R R U R 24 R' U R' U' R R U U R' U R U' R' R' U U R' U R' U' U' R U U R 25 U R' R' U U R U R U' U' R U' R U U R' U R' U' U' R' U R R U 26 U' R' U R U' R' U U R R U R' U R' U R' U' U' R R U' R' U' R' R' 27 R U' R' U' U' R R U' R' U' R U' R U' U' R' U' R' R' U' R U' R R U'

The 27 processes are in the same order as the 27 positions I posted on

11 November this year. However, I want to repost the 27 positions anyway.

I found a formatting inconsistency in that posting. Generally speaking,

when you unfold the cube for printing with the Back above, you can choose

to print the Back face right-side-up or up-side-down, and up-side-down

makes more sense to me. All my screen displays work that way, and it

makes the cubies move smoothly under repeated applications of the R or L

operators. However, I discovered that the print program I used for the

11 November posting printed the edge facelets of the Back face right-side-

up. That wouldn't have been so bad, I suppose, but at the same time I

printed the corner facelets correctly as up-side-down. So herewith

I reprint the 27 antipodal positions with all the Back faces correctly

up-side-down.

BBL BBL BBU BBU BBU BBF LRF RRF LRR UDR FDU UBD BUU BUU DUU FUB FUR UUFFRD RFR DRU URD RFB URL FRB LFD RLB LLL FFF RRB LLL FFF RRB LLL FFU RRR LLL FFB RLU LLL FFD RLU LLL FFF UBRDDU DDB DDR DDU DDU DDU DDB DDB DDB---------------------------------------------------------------BBU BBR BBU BBF BBU BBF DRR BRB FBD FDU RDL LUB UUB FUB UUU BUR RUL LUFRBR DFB URF URU FFF URU ULU BFD RRR LLL FFU LRR LLL FFU LRR LLL FFB RRR LLL FFL BRL LLL FFR BBF LLL FFU RRRDDU DDD DDF DDU DDU DDD DDF DDD DDB----------------------------------------------------------------BBR BBU BBB BBU BBU BBU LRD RLF FRF UDB DUR UFD UUF FUD DUU UUR BUU UURFRB LFF DRR FRR DFR BRU RRL FLB URR LLL FFB RRB LLL FFB RRR LLL FFU FRB LLL FFU RLB LLL FFU LBL LLL FFB URRDDF DDB DDL DDU DDU DDB DDU DDF DDD----------------------------------------------------------------BBR BBR BBL BBF BBU BBU FFR URU DRF LUB FFF FFR UUU DUU DUU RBR BBB DBRULD BRU FRU LRL URU RRR RRB RRB URU LLL FFU BRR LLL FFU BRF LLL FFU BRF LLL FFB URF LLL FFR BLF LLL FFU LLFDDL DDD DDB DDD DDU DDU DDD DDD DDU------------------------------------------------------------------BBR BBF BBF BBU BBU BBU URU FRF FRF FFF RDR RDR UUU FUU FUU BBB DBR BBRLRL URU RLR DRB RRB URU DRR DRB URU LLL FFU BRF LLL FFU BRL LLL FFU BRL LLL FFR BRF LLL FFU LFU LLL FFL BFUDDD DDB DDU DDD DDU DDU DDD DDL DDL--------------------------------------------------------------BBF BBR BBR BBU BBU BBU DRL URR FRD RFU FFB RDB UUD DUU FUU RBR RBF UBLBRF DRB URB LRD BRR UBU DRB LRF URR LLL FFU BRL LLL FFU RRL LLL FFU FRB LLL FFU RFU LLL FFB UFF LLL FFR ULUDDF DDL DDB DDU DDU DDU DDL DDD DDF--------------------------------------------------------------BBB BBU BBU BBU BBU BBU FRL RRF BRR RDF BFR UUU FUU DUU DUF UBB FBU UBDDRF RRL URU DRD RRL FLU LRL FRR FRF LLL FFU FRB LLL FFU FRB LLL FFU FRB LLL FFD RLU LLL FFB URR LLL FFD RLRDDB DDL DDB DDU DDU DDU DDR DDB DDB---------------------------------------------------------------BBB BBB BBD BBU BBF BBU URU UFD LRF BUF RUR UBR DUF DUU DUU UBU UBR UFRRRB LRL FRR FRL FRB URF FRB LRB URU LLL FFU FRB LLL FFU LRR LLL FFU FRB LLL FFF RLR LLL FFB UBR LLL FFD RLRDDD DDL DDB DDU DDU DDU DDD DDD DDF----------------------------------------------------------------BBD BBF BBR BBF BBF BBF BFB UFF LFR RUL FUR UUD UUU UUU UUU RUL BUR UUDURU FBF ULU LRL UBB ULU FRB LBR FLB LLL FFB RRR LLL FFB RRR LLL FFB RRR LLL FFD RRR LLL FFB DRD LLL FFR FRBDDB DDR DDU DDD DDD DDD DDF DDR DDU------------------------------------------------------------

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU