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Some more interesting groups

Submitted by rokicki on Mon, 01/30/2006 - 18:10.
Here are the results of an exploration of four groups, all of whom leave the edges in some M-symmetric class, for the face turn metric:

Fixes edges:

dposMM+inv
0111
8528116
936085
1025216343
11522011768
12720081541821
1321737445842378
1410846242274311690
15490589410241951896
1616944222353629178786
1720029566418360212304
18827386175029419
1921677
44089920920985467424
Pons Asinorums edges:
dposMM+inv
6422
7611
87833
911666
104621212
11570813078
1232556690373
1315161831941668
14896527187909605
15506562410581053641
1620437568426585215786
1717156739358427182222
1834284473314023
197044
44089920920985467424
Superflips edges:
dposMM+inv
14487210354
15512161071548
16539792112775727
17605345012634763806
1829140880608273307613
19829862917386189626
2010815350
44089920920985467424
Pons Asinorums and Superflips edges:
dposMM+inv
1413442814
1518216381200
1634532872213677
1744900049366847308
1830168301629581318152
19906641919008598053
203082120
44089920920985467424
These are the result of a program I wrote that attempts to solve a bunch of cubes at once, that share something in common (in this case, a given position of the edges). It's remarkable to me how much the peaks of the distributions are shifted between the four. If nothing else, this just shows that if there is a distance-21 cube, it does not have the edges in an M-symmetric position.
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Here's the results for the co

Submitted by rokicki on Mon, 02/13/2006 - 15:22.
Here's the results for the coset that has the edges in Reid's position (quarter turn metric this time): 
 d      pos        M    M+inv
14      864       55       28
16    48104     3016     1524
18  2003896   125521    63317
20 32050973  2006185  1010090
22  9986060   626478   318134
24       22        4        4
26        1        1        1
   44089920  2761260  1393098
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explain please

Submitted by WarrenSmith on Sun, 05/21/2006 - 15:47.
what is "Reid's position" and what is the unique 26q position
that you found? Some of us need these things explained to us...
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Reid's position is U2 D2

Submitted by silviu on Thu, 05/25/2006 - 06:19.
Reid's position is

U2 D2 L F2 U' D R2 B U' D' R L F2 R U D' R' L U F' B' (26q*, 21f)

It can be found on Reid's home page.
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There is also a notational co

Submitted by Jerry Bryan on Sat, 05/27/2006 - 12:26.
There is also a notational convention worth mentioning. The given maneuver is listed as 26q*. The 26q means that the maneuver is 26 quarter turns long (or we might also say that it's 26 moves long in the quarter turn metric), sort of like saying that an object is 3in. long for three inches or 5cm. long for five centimeters. The * says that the maneuver has been proven to be minimal in the given metric. The 21f means that this particular maneuver is 21 face turns long (or we might also say that it's 21 moves long in the face turn metric), and the absence of an * says that the maneuver has not been proven to be minimal in the given metric.

The q and f as units of measure originated in Cube-Lovers, and has been fairly widely adopted elsewhere -- including by Cube Explorer. I believe that the * to denote minimal maneuvers originated with Cube Explorer. Whether it originated there or not, it is certainly is used by Cube Explorer, and has been fairly widely adopted elsewhere.

As you watch Cube Explorer run, it displays progressively shorter maneuvers as it finds them. It is fairly common for the program to display (for example) an 18f maneuver and that a short time later the 18f becomes 18f*. The 18f says that an 18f maneuver has been found, but that it may or may not be minimal. When the 18f becomes 18f*, Cube Explorer has managed to prove that the maneuver is minimal. So the absence of the * does not mean that the maneuver is not minimal. It simply means that the maneuver may or may not be minimal, and if it's minimal it has not yet been so proven.

It turns out that the 21f maneuver given above for Reid's position is indeed not minimal in the face turn metric, but the same maneuver is minimal in the quarter turn metric at 26 quarter turns.
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Congratulations to this work!

Submitted by silviu on Tue, 01/31/2006 - 07:19.
Congratulations to this work! When are you doing the quarter turn metric? How long did it take to compute each table?

I also wanted to say that is better to call those tables for cosets rather then groups because if we call the group that fixes edges for H. Then you computed idH , g_1H , g_2H , g_3H. And only the first one is actually a group.
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Thanks! I'm doing the quarte

Submitted by rokicki on Tue, 01/31/2006 - 11:36.
Thanks! I'm doing the quarter turn metric now; it should be done soon. This work took several days (not sure exactly; some of the above were done in less than 24 hours but some took a few days, and I'm using four machines, including two 1999-era Celeron boxes so even that timing is suspect.)

After the quarter-turn metric, I'm thinking of trying the cosets (is that right?) that have UD symmetry; I think there are 28 of them that don't have also M-symmetry. I think if there is a distance-21f or distance-27q position, it is likely it has some symmetry, so these are fertile grounds to mine, and solving 44M cubes at a go helps.

I'm not a mathematician by a long shot. Thanks for helping with the precision!
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