# Permutations of the corners and edges: FTM

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