Discussions on the mathematics of the cube

15f* = 91365146187124313

The number of positions at distance 15 in the face turn metric is 91,365,146,187,124,313.

This result is from a collaboration between Morley Davidson, John Dethridge, Herbert Kociemba, and Tomas Rokicki.

More details will be forthcoming in a future announcement.

Relation between positions and positions mod M in FTM

Tom proposed that I give the relation between the number Nm of positions mod M and the number of positions N in the way N = 48*Nm - constant C. This is indeed possible, because the symmetric positions of the cube are completely analyzed.

God's Algorithm out to 17q*

Well it's done. Here are the results in the Quarter Turn Metric for positions at exactly that distance:
 d  (mod M + inv)*   mod M + inv       mod M             positions
-- --------------- --------------- --------------- -----------------
 0               1               1               1                 1
 1               1               1               1                12
 2               5               5               5               114
 3              17              17              25              1068
 4             135             130             219             10011
 5            1065            1031            1978             93840
 6            9650            9393           18395            878880
 7           88036           86183          171529           8221632
 8          817224          802788         1601725          76843595
 9         7576845         7482382        14956266         717789576
10        70551288        69833772       139629194        6701836858
11       657234617       651613601      1303138445       62549615248
12      6127729821      6079089087     12157779067      583570100997
13     57102780138     56691773613    113382522382     5442351625028
14    532228377080    528436196526   1056867697737    50729620202582
15   4955060840390   4921838392506   9843661720634   472495678811004
16  46080486036498  45766398977082  91532722388023  4393570406220123
17 426192982714390 423418744794278 846837132071729 40648181519827392

God's Algorithm out to 16q*

So switching to edge cube positions as cosets (instead of corner cube positions and twist) made a huge impact on the calculation, but more on that later. So my results for up to 16q* for positions at exactly that distance are:
 d (mod M + inv)*     positions
-- -------------- ----------------
 0              1                1
 1              1               12
 2              5              114
 3             17             1068
 4            135            10011
 5           1065            93840
 6           9650           878880
 7          88036          8221632
 8         817224         76843595
 9        7576845        717789576
10       70551288       6701836858
11      657234617      62549615248
12     6127729821     583570100997
13    57102780138    5442351625028
14   532228377080   50729620202582
15  4955060840390  472495678811004
16 46080486036498 4393570406220123
* The (mod M + inv) column means symmetry reduced postions but only considering the edge cube positions. I just included it because I had those numbers. It is not comparable to earlier calculations because it may include duplicate positions of the full cube and is dependent on what cosets are used in the calculation.

even and odd cube positions

After calculating 14f* and 15f* being at bit out of range, I started calculations in the quarter turn metric and just for the heck of it I switched to edge cube positions as cosets. I am in the process of verifying 15q*, which will take about 24 hours on a total of 120 cores. In the process I noticed something strange in my output. For coset 0 I get:
0 11 1 0 0
0 12 1 121477 121477
0 13 1 0 0
0 14 1 2981152 2981152
0 15 1 0 0
so I only have positions for coset 0 at even depths (12 and 14) and none at odd depths (11, 13 and 15). I get the same for coset 1:
1 11 12 4284 51408

God's Algorithm out to 14f*

Here are the results for postions at exactly that distance:

 d   mod M + inv          mod M       positions
-- -------------- --------------- ----------------
 9      183339529       366611212      17596479795
10     2419418798      4838564147     232248063316
11    31909900767     63818720716    3063288809012
12   420569653153    841134600018   40374425656248
13  5538068321152  11076115427897  531653418284628
14 72805484795034 145610854487909 6989320578825358
I had the good fortune to run a few test calculations on a brand new Cray XT6m with over 4000 Opteron cores. As I have done in the past I tried my own program for rubiks cube calculation. Results looked promising and after some optimisations I was able to complete the calculation out to 14 moves in the full turn metric in a about nine days. If I had been able to use the entire machine alone it should have taken a bit over a day.

3-face subgroup of the 3x3x3 cube

Somewhat surprisingly I was unable to find any tables for the number of positions of various lengths in ascending order for this order 170659735142400 group w.r.t. three mutually orthogonal face generators such as U,F and R, i.e. This is seemingly beyond the reach of conventional software tools to solve, so I am appealing to the seemingly large number of contributors who have developed their own bespoke utilities. I'd be grateful if a table could be posted here. In particular is the diameter of the Cayley graph for this known?

free rubiks cubes

to get free cubes and more click on this banner
Cube Lagoon

All the Syllables, Corners Group, Quarter Turn Metric

My recent posting on |EndsWith| values for the corners of the 3x3x3 in the quarter turn metric was really secondary to another project I was working on.  Namely, I was working on determining all the syllables for the corners of the 3x3x3 in the quarter turn metric.  I discovered many more syllables than I was expecting.  As a part of determining why there were so many syllables, I discovered that the |EndsWith| values were much higher than I was expecting.  To a certain extent, one can think of the higher |EndsWith| values as being the "cause" of there being a larger number of syllables than expected.

EndsWith Values, Corners Group, Quarter Turn Metric

My calculations of God's algorithm have generally included an analysis of the distribution of EndsWith values.  My best God's algorithm results for the quarter turn metric are out to 13q. Tom Rokicki has since calculated out to 15q.  Out to 13q, it is the case that for the vast majority of positions x we have |EndsWith(x)| = 1.  Tom may be able to speak to the 14q and 15q cases, but I cannot.

Given the predominance of |EndsWith(x)| = 1 out to 13q, I have assumed that for the vast majority all of cube space it's probably the case that |EndsWith(x)| = 1.  I no longer believe that assumption is correct.  Which is to say, I now have an EndsWith distribution for the entirety of the corners group in the quarter turn metric.  Beyond a certain distance from Start in the corners group, there are no instances of |EndsWith(x)| = 1 whatsoever, and overall the |EndsWith(x)| = 1 cases are very much in the minority.  It now seems to me that the same is probably true for the complete cube group including corners and edges.  I suspect the reason we have not seen this effect for the complete cube group is simply that we have not yet been able to calculate out far enough from Start.