## Relation between positions and positions mod M in FTM

Submitted by kociemba on Thu, 07/15/2010 - 05:10.## God's Algorithm out to 17q*

Submitted by tscheunemann on Wed, 07/14/2010 - 15:25.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*

Submitted by tscheunemann on Fri, 07/09/2010 - 05:06.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

Submitted by tscheunemann on Tue, 07/06/2010 - 12:00.0 11 1 0 0 0 12 1 121477 121477 0 13 1 0 0 0 14 1 2981152 2981152 0 15 1 0 0so 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*

Submitted by tscheunemann on Wed, 06/23/2010 - 15:13.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 6989320578825358I 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

Submitted by secondmouse on Fri, 05/07/2010 - 08:36.__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

Submitted by whatsupdog1 on Thu, 05/06/2010 - 21:17.## All the Syllables, Corners Group, Quarter Turn Metric

Submitted by Jerry Bryan on Thu, 04/22/2010 - 22:03.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

Submitted by Jerry Bryan on Sun, 04/11/2010 - 11:19.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.

## algorithm for generating permutations for the rubik cube

Submitted by nooneimportant on Fri, 04/02/2010 - 11:13.let's say I want to distribute permutation checking over say 10 computers.

so if there are a total of n permutations the first machine will check n/10.

the second will check from n/10 to 2n/10

third will check from 2n/10 to 3n/10.

and so forth.

the algorithm that generates permtuations needs to generate the ith permutation in O(1)

so that I can efficiently start each machine's work.

what algos for generating permtuations do you know that can do this ?

thanks