## Largest coset solved yet of actual Rubik's positions

Submitted by mdlazreg on Sun, 02/28/2010 - 23:32.Here is the distribution table:

0 1 1 4 2 10 3 36 4 123 5 368 6 1336 7 4928 8 16839 9 63920 10 257888 11 1019992 12 4317941 13 20240924 14 102343680 15 568081384 16 3458261494 17 22676234692 18 153062896516

## Positions with the same distance in both QTM and FTM

Submitted by mdlazreg on Sun, 02/14/2010 - 07:53.do nothing (0q* , 0f*) U (1q* , 1f*) U R (2q* , 2f*) F B U D R L F B U D R L (12q*, 12f*) F B U D R L F' B' U' D' R' L' (12q*, 12f*) F' B' R' L' F B U D R' L' U' D' (12q*, 12f*) F B R' L' F B U' D' R L U' D' (12q*, 12f*) F U F' R B U D' L' D' R U R L' F' D' B L' B' (18q*, 18f*)

## Rubik Xcode Project

Submitted by B MacKenzie on Fri, 01/29/2010 - 19:46.I have put together some source code demonstrating my approach to modeling the Rubik's cube puzzle. I have made an attempt to make the code clear, understandable and well commented. The language is Objective C and makes much use of the Mac OS Foundation and Application kit class libraries. So it is pretty Mac specific although C++ programmers may wish to browse the source files for ideas. Although the syntax is different, as object oriented languages C++ and Objective C bear many similarities.

Those interested may download the Rubik Xcode Project from my web site.

## 1,000,000 cubes optimally solved

Submitted by rokicki on Sat, 01/23/2010 - 20:57.12f*: 1 13f*: 14 14f*: 172 15f*: 2063 16f*: 26448 17f*: 267027 18f*: 670407 19f*: 33868No 20f* cube was encountered, which is as expected. No symmetrical or anti-symmetrical positions were encountered.

These results are very close to Kociemba's results for 100,000 cubes; much closer to his overall predictions than those extrapolated from the 250 cosets I ran completely. This seems to indicate that running random cubes may be a more effective way to get a distribution estimate than by running many fewer random cosets (but which contain collectively many more individual positions).

## Void cube diameter at least 20 (face turn metric)

Submitted by rokicki on Tue, 01/19/2010 - 20:15.The "superflip" void position has a distance of 20. This can be shown by computing the optimal solution for all 12 axis insertions in the 3x3 cube; this yields only three unique positions (mod M), and all three have a distance of 20.

U1F1U2F1L2B1U2F1L3R3F2D1R2U2L2B1F3L1F2D1 (20f*) //superflip

## An analysis of the corner and edge orientations of the 3x3x3 cube

Submitted by mdlazreg on Wed, 01/13/2010 - 13:49.Distance Positions -------- --------- 0q 1 1q 12 2q 114 3q 1,068 4q 10,011 5q 93,840 6q 877,956 7q 8,197,896 8q 76,405,543 9q 710,142,108 10q 6,565,779,580 11q 59,762,006,092 12q 506,821,901,799

## An interesting conjugate class

Submitted by mdlazreg on Fri, 01/01/2010 - 11:51.Unfortunately as far as I know there is no fast way to calculate the optimal distances distribution for a chosen conjugacy class. The only way is to search for the optimal distance of each position one by one. That is what I did for the following conjugacy class:

CE x CC : 1_1_1_1_1_1_1_1_4 x 1_1_1_1_4

which has 495*6*2^3*70*6*3^3 = 269438400 positions and which include the 12 cube generators.

I did this search however only for one orientation which reduced the number to 495*6*70*6 = 1247400 positions. Here is the optimal distribution:

## New estimate for 20f*: 300,000,000

Submitted by rokicki on Sat, 11/28/2009 - 15:15.Of these 250 random cosets, 232 had absolutely no distance-20

## Syllables and Graphs

Submitted by Jerry Bryan on Sat, 11/14/2009 - 14:27.I've been reading a little bit about graph theory, and I wish I knew more about it. So pardon what may perhaps be a bit of naivetÃ© about graphs on my part.

I suspect that everyone who reads this forum is familiar with Cayley graphs and how they relate to Rubik's cube space. One of the best references in this regard may be found at http://www.jaapsch.net/puzzles/cayley.htm I also suspect who everyone reads this forum is familiar with the concept of doing a breadth first search while storing all the results as a way of investigating a space such as the Cayley graph for Rubik's cube.

## Fast solver for arbitrary target groups

Submitted by mdlazreg on Thu, 11/12/2009 - 09:06.My question is can this be applied to arbitrary target groups whose elements share something in common? for example let's say my target group is some conjugacy class of the cube.