cube files
Indexed Cube Lovers Archive
Permutations of the corners and edges: FTM
Submitted by Bruce Norskog on Tue, 02/21/2006 - 17:00.
U D B' U2 D2 B2 R' U B U2 R2 D B' L'
U D F U' F2 L' B L' U F' D2 F' D' B2
U D F L U D2 B' D' L' B D2 F2 U' B
U D F B L2 U D R' F' B' U D F' B'
U D2 F2 U2 L U2 B' U R F2 L2 D' F U2
U D F D R B' L B U2 R F B L D'
U D F' U' R U' D B L' F D R2 L' F'
U D' R L' D2 R F' B' R2 U2 R' L' B' L2
U D R U D2 F B2 U R U2 R L U R'
U D2 B D' B2 U R L2 B R2 F2 R2 U' B
U D B2 R F' U F U F' R2 D2 F' D' R
U D2 F B U' F2 B2 D R2 L2 U' R L D'
U D R2 F' R' L F2 R2 F B R' B2 U2 R2
U D2 R' U2 F2 D R B L D' F L2 F' B2
U D B U' L D B' L' F2 L' B2 R U B2
U F U R2 F D F' L2 U2 D R2 L' U B2
U F' D' R2 U' F' D2 F2 U' L2 F2 L2 D B
- Bruce Norskog
I have finally completed the face-turn metric version of the analysis of the permutations of the corners and edges of the 3x3x3 Rubik's Cube. That is, I have generated a table of the Cayley graph distances for the positions where the permutation of the corner cubies and the permutation of the edge cubies are considered, but not the orientation of either the edge or corner cubies. This is a set of 8!*12!/2 or 9,656,672,256,000 positions. As with the quarter-turn metric analysis, I used symmetry in the corner permutations to reduce the number of stored distances to 984*12!/2 or 235,668,787,200.
I also plan to compute the distribution of the distances in terms of unique positions with respect to symmetries of the cube, and also the distribution where antisymmetry is also used. The reduced counts in the table below are upper bounds on the values for the number of positions that are unique with respect to symmetries of the cube.
Permutations of the corners and edges of the
3x3x3 Rubik's Cube, ignoring orientation
Distance Positions "Reduced count"
-------- --------- ---------------
0f 1 1
1f 18 2
2f 243 9
3f 3,240 98
4f 43,239 1,251
5f 574,308 15,546
6f 7,599,538 198,547
7f 100,336,092 2,558,710
8f 1,320,811,484 33,223,104
9f 17,277,892,377 430,571,941
10f 220,284,799,718 5,453,945,549
11f 2,346,113,524,904 57,687,613,279
12f 6,808,973,777,807 165,731,116,833
13f 262,592,892,783 6,329,542,290
14f 248 40
----------------- ---------------
9,656,672,256,000 235,668,787,200
There are 18 antipodes when counting positions that are unique with respect to symmetries of the cube. Move sequences for generating representatives of the antipodes are:
U D R' B' D' L2 D F B D2 F D2 L BU D B' U2 D2 B2 R' U B U2 R2 D B' L'
U D F U' F2 L' B L' U F' D2 F' D' B2
U D F L U D2 B' D' L' B D2 F2 U' B
U D F B L2 U D R' F' B' U D F' B'
U D2 F2 U2 L U2 B' U R F2 L2 D' F U2
U D F D R B' L B U2 R F B L D'
U D F' U' R U' D B L' F D R2 L' F'
U D' R L' D2 R F' B' R2 U2 R' L' B' L2
U D R U D2 F B2 U R U2 R L U R'
U D2 B D' B2 U R L2 B R2 F2 R2 U' B
U D B2 R F' U F U F' R2 D2 F' D' R
U D2 F B U' F2 B2 D R2 L2 U' R L D'
U D R2 F' R' L F2 R2 F B R' B2 U2 R2
U D2 R' U2 F2 D R B L D' F L2 F' B2
U D B U' L D B' L' F2 L' B2 R U B2
U F U R2 F D F' L2 U2 D R2 L' U B2
U F' D' R2 U' F' D2 F2 U' L2 F2 L2 D B
- Bruce Norskog
