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Tripod Finish - optimal FTM analysis
Submitted by Bruce Norskog on Tue, 01/20/2009 - 15:51.
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*) |
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