## Presentation for Rubik's cube

Submitted by jaap on Fri, 03/12/2010 - 03:32.I found the following short presentation for the miniature 2x2x2 Rubik's cube of order 3674160: < a,b,c | a^4 = b^4 = c^4 = 1, ababa = babab, bcbcb = cbcbc, abcba = bcbac, bcacb = cacba, cabac = abacb, (ac)^2 (ab)^3 (cb)^2 = 1 > See the following link for more info as to why

## 1,000,000 cubes optimally solved in both QTM and FTM

Submitted by rokicki on Sun, 03/07/2010 - 19:43.12h 13h 14h 15h 16h 17h 18h 19h sum 15q 1 1 3 2 - - - - 7 16q - 2 18 48 35 - - - 103 17q - 3 23 143 347 354 - - 870 18q - 5 40 305 1713 4520 2034 - 8617 19q - 1 40 505 5190 29711 33363 474 69284 20q - 2 39 674 9932 100164 212466 7213 330490 21q - - 9 345 7697 104052 301668 16371 430142 22q - - - 41 1533 28173 120449 9720 159916

## 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