metric, there is still much yet to be discovered. The diameter in the QTM
and the STM are unproved (although they are almost certainly 26 and 18,
respectively). The exact count of positions at distance 16, 17, 18, 19,
and 20 in the half-turn metric is unknown. This note reports some progress
on an estimate for the count of 20's in the half-turn metric.
It is fascinating to me how problems of distinctly different difficulty
exist around the 3x3x3 cube in the half-turn metric. Initially, back in
the early days, we could solve individual positions non-optimally.
A corner cubie may be moved to any of the eight corner cubicles in three different ways; untwisted, with clockwise twist or with counterclockwise twist. The standard convention is to assign the twist with reference the orientation of the cubie's U/D facelet vis-a-vis the cubicle's U/D face. If the cubie's U/D facelet is on the cubicle's U/D face the cubie is untwisted. If the cubie's U/D facelet is rotated 120° clockwise from the cubicle's U/D face the cubie has clockwise twist and vice versa. The disturbing thing about this convention is that it is unsymmetrical. Under this definition the 12 q-turns have different effects on the twist of the cubies turned. Turns of the U and D faces have no effect on the twist of the cubies while a q-turn of any of the other four faces twist two cubies clockwise and two cubies counterclockwise. Since all the faces are symmetry equivalent it has always seemed to me that there ought to be a way of defining corner cubie orientation which preserves this equivalence.
Analysis of ( U2, D2, F2, B2 ) ------------------------------ Level Number of Positions 0 1 1 4 2 10 3 24 4 53 5 64 6 31 7 4 8 1 --- 192Everyone agree? I'm not sure if this has been pointed out before.
Analysis of the 3x3x3 squares group ----------------------------------- branching Moves Deep arrangements (h only) factor loc max (h only) 0 1 -- 0 1 6 6 0 2 27 4.5 0 3 120 4.444 0 4 519 4.325 0 5 1,932 3.722 0
If we drop some moves we end up with some subgroups. The subgroups are:
1) I [the identity] 2) U 3) U,D 4) U,F 5) U,D,F 6) U,F,R 7) U,D,F,B 8) U,D,F,R 9) U,D,F,B,R
I know the depth table for subgroups 1) 2) 3) and 4):
The subgroup 1) generated by "no move", has the following obvious table:
Moves Deep arrangements (q only) 0 1 ------ 1The subgroup 2) generated only by the move U, has the following table:
Having repeatedly shot myself in the foot by mishandling the symmetry reduction of coset spaces, I finally sat down, laid out the math and put together a set of notes on the matter. These notes follow.
Solving Rubik's cube either manually or by computer usually involves dealing with coset spaces. A group may be partitioned into cosets of a subgroup of the group:
g * SUB where g is an element of the parent group and SUB is a subgroup of the parent group
I recently bought a new computer and wanted to put it through its paces. I dusted off my RUF three face coset solver and spruced it up a bit. Since I now have three iMacs in my household connected on an airport network, I rewrote the program using a server–client model. With this I can have all three computers working on a problem in parallel with as many as 14 cores. With these tools I have extended the enumeration of the three face group out to twenty q–turns:
Three Face Enumerator Client Fixed cubies in subgroup: UF, UR, UB, UL, DF, DR, FR, FL, BR. 92,897,280 cosets of size 1,837,080 Server Status: Three Face Group Enumerator Sequential coset iteration Enumeration to depth: 20 Snapshot: Friday, February 22, 2013 9:28:02 PM Central Standard Time Depth Reduced Elements 0 1 1 1 1 6 2 4 27 3 12 120 4 51 534 5 213 2,376 6 914 10,560 7 4,038 46,920 8 17,639 208,296 9 78,234 923,586 10 344,175 4,091,739 11 1,524,115 18,115,506 12 6,722,358 80,156,049 13 29,739,437 354,422,371 14 131,158,304 1,565,753,405 15 578,971,538 6,908,670,589 16 2,546,820,524 30,422,422,304 17 11,174,670,698 133,437,351,006 18 48,528,827,222 579,929,251,620 19 205,901,170,504 2,459,821,160,421 20 814,027,054,726 9,731,195,124,049 Sum 1,082,927,104,708 12,943,737,711,485 92,897,280 of 92,897,280 cosets solved
You can imagine my horror when I realized just how much work would be involved in salvaging the forum and make it usable again. I thought all I could do is make the drupal mysql file available to the web and figure out a way of upgrading later.
Finally as a last ditch effort I remembered the Ultimate Boot CD which has a hard drive cloning program and it was able to copy all the sectors still readable to another hard drive. The fact that the critical files were readable and there were multiple kernels bootable on the old failing hard drive was enough to get the server to at least boot, and I was able to restore the last missing files from another backup.