# Tripod Finish - optimal FTM analysis

I have used Cube Explorer to solve all "tripod finish" cube configurations. All positions were solved in no more than 15 face turns. I calculated an average distance of about 12.746 face turns per position.

A "tripod finish" configuration has all cubies solved except four corners and three edges that make a configuration resembling a tripod. For example, corners URF, UFL, UBR, and DFR along with edges UF, UR, and FR make up one representative tripod configuration. For the above tripod configuration (or any equivalent one), there are a total of 7776 possible legal arrangements of the cubies. These 7776 arrangements can be reduced by symmetry to 1317 cases.

I wrote a GAP program to generate a maneuver for each of the 1317 tripod finish positions that are unique with respect to the symmetry of the tripod configuration. I then loaded these positions into Cube Explorer to generate optimal solutions for each one. (Technically speaking, I loaded only 1316 positions, and optimally solved the identity position manually.) I then checked the results of 36 symmetrical cases (excluding the identity position) in order to derive the results for the full 7776 tripod finish configurations.

The following is a summary of the results. The Cube Explorer execution time was less than 2.5 hours.

```Tripod Finish - FTM Analysis

Distance Positions Symmetry-reduced
-------- --------- ----------------
0f*         1          1
4f*         6          1
7f*         6          1
8f*        66         11
9f*        84         14
10f*       282         49
11f*       714        119
12f*      1556        265
13f*      2725        459
14f*      2047        347
15f*       289         50
----       ----
Total       7776       1317
```

For those interested in the deepest positions, the following table gives generators for the 15f* positions.

```15f* Tripod Finish positions
----------------------------
D' U2 L2 D' B' D' B2 D2 L2 U' F D U2 R2 U  (15f*)
R2 U2 R2 F2 U' F2 D R2 U B U' B' D' R2 U2  (15f*)
U R2 B U B' R2 U' R2 U' R2 U R2 U' R2 U  (15f*)
F2 R U2 L' R' U2 B' U2 B L2 F2 L' U F U'  (15f*)
U F' L F2 D2 F D2 L B' L2 F2 L' B L U'  (15f*)
F' U' B' U' B U' L2 F' L2 F L F' L' F2 U'  (15f*)
D' L2 B' U B L2 D U' R' F' R F U2 F' U'  (15f*)
U' B' U2 L D' F2 D L' U' R' U' R' U' B U2  (15f*)
U L2 F2 L' F' L F' L' F' L' U' F' U F U'  (15f*)
F R2 B2 D' B D' B' D B2 R2 F' R U R' U'  (15f*)
D R F2 R' D' U2 L' U' F D' L D L F U'  (15f*)
F' U F R U R' F2 R2 U' L' U L R2 F2 U'  (15f*)
R U R2 U' R2 F R' F' R2 F' U F2 R F' U'  (15f*)
B' U' B L R F U F' U2 L' B2 R B2 R2 U'  (15f*)
U R2 B U B' U' R' U R' F' L F' L' F2 U'  (15f*)
R' B' F D' R2 D B F2 U F R' B U' B' U'  (15f*)
U' B' R2 B' D2 L2 F2 D' L2 D' B2 R' U R' U'  (15f*)
L D B2 U2 F' U' F U' L U2 L' B2 D' L' R  (15f*)
U F' U' F2 U R U' F' U' F' U F U2 R' U'  (15f*)
U2 B' R' F' U' L F2 R F L' F U' B F U'  (15f*)
U2 B' D' U2 L' F2 L D2 R' D' R2 U' R2 B U  (15f*)
F2 R' F R D' F U F' D U L' U' L F2 U'  (15f*)
U R B2 D' F R' F' D B2 U L U L' R' U'  (15f*)
R F2 D' F U L2 F U2 F D2 B D' F' R' U  (15f*)
U' R2 B2 L' B' L B' R' U' R' U R U' R' U'  (15f*)
U R2 U' F' L2 D' B2 D L2 F U2 R' U2 R' U'  (15f*)
D U L F' L' D' R2 B L2 U' L2 U B' R2 U'  (15f*)
R2 D B2 F L2 F' D2 B D U B U' B R2 U'  (15f*)
U' B' D' R' D' L' D R' D' L D2 B2 U' B' U2  (15f*)
D' F U L2 D F D2 F' L2 F D F2 D F2 U'  (15f*)
F R' U' F2 U F2 R F' U F' U2 F2 U2 F' U'  (15f*)
F' R' U L U' R U L2 F R F' L F R' U'  (15f*)
F2 D' F U L2 F R B' U2 B R' F2 D F2 U'  (15f*)
U2 R2 B' R' B L' U' B' U B' R B2 L R2 U2  (15f*)
F' L' F2 D' B L2 B' D F U2 F L U F U'  (15f*)
F2 U' R2 U R2 U F2 R F2 L F L' F R' U'  (15f*)
U2 B2 F2 D L D' L' B2 R2 B L B' R2 F2 U2  (15f*)
F D B R B' D' R F' U2 L' U R' U' L U2  (15f*)
U F R B L U2 B2 R D2 R F R' B R2 U'  (15f*)
R B2 U' F' U L2 B' L U L B' U' F R' U'  (15f*)
U L F' L F2 U B L2 B' U2 F U F L2 U'  (15f*)
F' U2 F R B L2 U2 B L' B' U2 L2 B' R' U2  (15f*)
R U L U' R' U F L F L' F L F2 L2 U'  (15f*)
R2 F' R2 B U2 F L F' L' B' U2 F2 U2 F2 U2  (15f*)
U R2 B L2 D2 R' F R D2 L B' L U2 R2 U'  (15f*)
F' U' F U2 B L2 D F' D' F' L' F L' B' U'  (15f*)
D' F U F' U' F2 U F D R2 U2 R' U2 R' U'  (15f*)
U' L' R' D2 L' F' L' F2 L2 D2 R' B L R2 U'  (15f*)
U F R F R2 F2 U2 F' U2 F' R2 F2 U2 R' U'  (15f*)
U R2 B L U2 L' R' F' R B' R' U2 F R' U'  (15f*)
```