perhaps i should give the figures i obtained when getting my upper
bound of 39 face / 56 quarter turns in case i also have an error.
recall the method: we work in three stages:
1. from <U, F, R, L, B, D> to <U, R, F> 2. from <U, R, F> to <U, R2, F2> 3. from <U, R2, F2> to START
where we're only allowed to use moves that keep us within the
given subgroup. the number of positions that must be considered
in each stage are 253440, 15676416, and 10886400 respectively.
stage 1: using face turns: using quarter turns: distance number distance number 0 1 0 1 1 9 1 6 2 90 2 39 3 852 3 276 4 7169 4 1899 5 44182 5 11245 6 131636 6 49412 7 68940 7 117221 8 561 8 70767 9 2574 stage 2: using face turns: using quarter turns: distance number distance number 0 1 0 1 1 2 1 2 2 12 2 8 3 72 3 36 4 420 4 158 5 2410 5 694 6 13752 6 2980 7 75796 7 12744 8 390421 8 53646 9 1735771 9 216354 10 5351383 10 799868 11 6696700 11 2477802 12 1399195 12 5310848 13 10481 13 5419046 14 1356020 15 26192 16 17 stage 3: using face turns: using quarter turns: distance number distance number 0 1 0 1 1 5 1 2 2 14 2 3 3 44 3 8 4 128 4 14 5 392 5 24 6 1157 6 52 7 3458 7 96 8 10057 8 176 9 29286 9 352 10 82814 10 664 11 228621 11 1248 12 591704 12 2409 13 1362497 13 4516 14 2545752 14 8519 15 3272940 15 16100 16 2260555 16 30171 17 484818 17 56140 18 12133 18 102981 19 24 19 186728 20 331234 21 563985 22 912719 23 1365051 24 1812011 25 2044832 26 1783956 27 1105488 28 450322 29 97881 30 7958 31 745 32 10 33 4
it would be nice if someone could confirm these figures. i have done
some testing, but not extensively.
mike