hoho, I think I have a general method for constructing repeating processes for the 6 X order 3 pattern. It is easy: Start with a process that generates 6 X order 3 combine it with any process of order 2 next repeat these combined processes twice Does it result in the 6 X order 3 pattern? If so then... calculate the optimized process for the sequence without repeating Call this P1 verify that (P1)2 results in the 6 X order 3 pattern if so then that is the solution The 2nd step of "combine it with any process of order 2" is the key to finding an intermediate state on the way to 6 X order 3. This formalizes a general method for constructing repeating solutions of this type. FUD45: (F U D)45 FUD45: (F U2 L2 U3 D3 L2 D2)45 = 315 total moves (3.971 s) p142a corner/edge pair (F2 B U2 R U3 B2 R3 F2 B)5 p142a corner/edge pair (U3 B U R2 F2 R U3 B3 U F2)3 C14 bottom edges: (F2 B R L3 F3 D F2 D2 R3)7 = 63 total moves (4107.651 s) C14 bottom edges: (2L D 2L3 D3 2L D3 2L3 D3 2L D3 2L3 D2 2L D 2L3)1 = 15 total moves (89.782 s) C14 bottom edges: (R F D F3 D F3 R F R3 D2 R3)5 = 55 total moves (459.516 s) Double swap edges: (2L2 D3 2L D2 2L2 D2 2L D 2L2)1 = 9 total moves (0.171 s) Cube in a Cube: (U2 F R' F2 R F2 R F2 R' F U2 Fv Rv)2 (note that full cube rotations were required) Cube in a cube in a cube: (U2 F R' F2 R F2 R F2 R' F U2 Fv2' B' (R3 F1 U2 F3 R1 D2)2 B Fv2 Fv Rv)2 Mark Pattern 1: (F' L2 D' B D2 B2 R' U L' B' U F' U F2 L U' Fv' Lv')2 Mark Pattern 1: (R3 F3 B2 U2 F L2 B2 R3 F3 R D2)8 (2U' 2F')4 (two stage repeating solving) Mark Pattern 1: ((2F 2U)4 R3 F3 B2 U2 F L2 B2 R3 F3 R D2)8 (a true repeating solution) Mark Pattern 1: ((2F 2R 2F' 2R') R3 F3 B2 U2 F L2 B2 R3 F3 R D2)8 (improvement) Mark Pattern 1: ((2R 2F 2R' 2F') D L D L' B2 U2 L' F2 L U' R U L' R' B2 R')2 (GAP + twsearch) Mark Pattern 1: (B' R' F2 B' L' D2 L' B 2L U2 B2 D 2F U D B U2)2 (GAP + twsearch) ML Edge Pattern p180: (F1 B1 R1 D1)12 (found by hand) ML Edge Pattern p180: (F B2 D F2 B2 D R2 B2 R2 B2 D' F B' D2 F2 D2 R B)2 twsearch solution of (F B R D)6 ML Edge Pattern p180: (F B U D U3 L R L3)12 = 96 total moves (337.856 s) (GAP search using full group) ML Edge Pattern p180: (F B3 D3 U2 L U2 B2 D2 R2)4 = 36 total moves (8503.43 s) (GAP search) 4 Cross order 4: (U 2L D3 2L3)7 (GAP search) 4 Cross order 4: (D' 2F2 U 2F2)3 4 Cross order 4: (L U D B2 L2 F' B' L' R' U2 F2 U Uv Lv)3 4 Cross order 4: (L2 U R2 L2 D2 U3 D2 B2)3 (GAP search) 4 Cross order 4: (U3 R2 B2 D2 R2 B2 L2 D2 R2 B2)2 (GAP search ) 4 cross o4 2 X: (U R2 F2 L2 D2 F2 R2 B2 D)3 (GAP search ) 6 Cross order 2: (L F B L2 R2 F L2 R2 B2 U2 D2 F' L R2)2 6 Cross order 3: ((D' 2F2 U 2F2)3 Lv')2 6 Cross order 3: ((L U D B2 L2 F' B' L' R' U2 F2 U Rv3 Uv2)2) 24 moves 2 X order 2: (U2 F2 U2 F2 D2 B2 L2 R2)2 (Gap search) 16 moves 4 X order 2: ((F2 D2)3 (U2 B2)3 Uv)2 6 X order 2: (L3 R U3 D)3 also known as the pons asinorum, 12 moves 6 X order 2: (U2 D2 Fv Uv')3 6 moves 6 X order 2: (2L 2F 2U)2 6 slice moves 6 X order 2: (L R' Lv F B' Fv U D' Uv)2 12 moves 6 X order 2: (L R' U D' L R' Uv2 Fv')2 12 moves 6 X order 3: (L D U B2 L2 B' F' R' L' U2 F2 U)2 24 moves 6 X order 3: (F' R' D' B' L2 D' F' D' R' B2 D' L' U' R' B2 D' B Uv2)2 34 moves 6 X order 3: (F U L2 B' U' R' L D U' L F U2 L' B' D2 U2 Uv2)2 32 moves 6 X order 3: (2D2 Lv' U 2F 2D' 2F U D 2F2 D' 2F2 D2)2 6 X order 3: (U1 L2 F1 L1 R2 U2 D2 R1 D1)8 72 moves (GAP search) 2301.874 seconds 6 X order 3: (F' D' B' U L F2 R B' L U' F' U R' F L2 F2 R)2 34 moves (GAP + twsearch) 6 X order 3: (D2 R2 U3 R3 U R F R F L2 U2)8 88 moves (GAP search) (47307.449 s) uses subgroup 6 X order 3: (D2 R1 B2 U3 R1 D2 B3 U3 B2 L2 D2)8 88 moves (GAP search) (28438.347 s, 7.89 hours) uses subgroup 6 x order 3: (D2 R1 B2 U3 R1 D2 B3 U3 B2 L2 R1 B2 U3 R1 D2 B3 U3 B2 L2 D2)4 80 moves, derived from above 6 X order 3: (F3 D2 U2 B3 R2 L2 B2 Uv Lv)20 searching for order 6 edge processes 6 X order 3: (U F B2 U3 D L2 R3 D3 Fv' Uv)35 generated from one edge 3 cycle 6 x order 6: cycle structure (uf bl ur db fr dl) (ul ub br dr df fl) [6] 6 X order 6: (F R L3 U2 F2 B2 U3 D3 F B3 L3 U R2 L2 U2 F2 B2 D F2 U2 F2 U2 F2 U2 Fv Lv')2 48 moves 6 X order 6: (R2 F B' L D2 U2 F2 B2 L' U L2 R2 F2 B2 U F B' Fv Lv')2 34 moves 6 X order 6: (R2 F B' L D2 U2 F2 B2 L' U L2 R2 F2 B2 U F B' Uv Fv)2 34 moves 6 X order 6: (U 2D 2F3 R 2D 2F3 2L2 U 2L3 F)14 = 140 total moves (159085.904 s) searching in [States explored so far: 55933009637] 6 X order 6: (2F2 U 2F2 U2 2L2 U R 2D2 R2 2F2 R 2D 2F2)2 twsearch optimized (p1)^7 2 X 2 H : (U2 L2 F2 L2 F2 R2 B2 D2)2 16 moves 2 X 2 H : (L2 F2 U D3 B2 R2)2 12 moves 4 diagonals (L R F B)3 4 Z (F B2 R3 F3 L2 F R F)6 48 total moves (35.974 s) 4 Z (F2 R2 F2 B3 L3 B3 R2 B L B)3 = 30 total moves (5287.991 s) 4 dot: (L R 2U R L)2 4 dot: (L R' F2 B2 Lv')2 Not using slices 4 dot: (2F D2)6 6 dot: (2F 2U)4 6 dot: (F B' U' D L' R F B' L2 R2)2 face turns only, 20 moves 6 dot: (F B' U D' F2 B2 L' R F B')2 face turns only, 20 moves 2 Tetrads (2F 2L 2F' 2L' U2 F' R F2 R' F2 R' F2 R F' U2 Fv Rv)2 30 slice moves 2 X (R2 U2 L2 D2)3 12 moves 4 X (F2 U2 B2 2L)2 8 moves 2 twist (R2 F U2 F' R D2)2 in D face opposite (dlf)- (drb)+ 2 flip ((U F R')30 R2)2 2 flip (R' B' F U' B2)6 flips two adjacent edges in B 2 flip (F U F2 L' B' U' B2 L)3 flips two adjacent edges in U 24 moves (uf,ul) [2] 2 flip (U1 R1 U3 R3 B1 R2 B2 R2 B3)2 = 18 total moves (183.417 s) (ub, ur) [2] 4 flip in L (F1 B1 L1 D1 U1)6 30 moves 4 flip in L (F B R2 L D U)6 36 moves (GAP search) 4 flip in L (F B R2 F R L3 D U2)4 = 32 total moves (318.766 s) GAP search 4 flip in L (F R3 B2 U3 F3 R L3 B2 D2)3 = 27 total moves (10953.561 s) GAP search 10 flip: (F R3 B3 L D U3 B)10 = 70 total moves (15.317 s) 10 flip: (F U2 L F D R2 B)6 = 42 total moves (21.483 s) 12 flip: (R' L' D2 L' B L F2 L2 U)3 27 moves 12 flip: ((2F U)4 Fv Uv')3 12 flip: (F1 R1 L3 U2 F2 B2 U3 D3 F1 B3 L3 U1 R2 L2 U2 F2 B2 D1 (B3 U2 B2 U1 B3 U3 B3 U2 F1 R1 B1 R3 F3))6 6xo6 plus 2-flip generates 12 flip 12 flip: ((R U2 F' )7 Lv Uv')9 12 flip: (U D2 L2 R F)18 = superflip 12 flip: (L3 R B U3 F B3 L D3 Uv Lv')6 48 moves 3-cycle edges (F2 U R2 F2 R2 U F2)2 14 moves p161 (F2 B2 D2 F2 B2 U2 (F1 R1 L1 B1)3 Uv D1 U3 L2 R2 D1 U3 F2 B2)17 p161 (U2 F U2 F B D2 F D2 F3 B)2 20 moves I don't claim these are optimal repeating solutions, merely that they are the best known repeating solutions. special cases using C_X, C_UB 12 flip (R3 U2 B1 L3 F1 U3 B1 D2 F2 U1 D3 C_X)2 expansion: (R3 U2 B1 L3 F1 U3 B1 D F U1 D3) (L D2 F' R B' D F' U' B' D' U) 6 X order 3 pattern stated as a commutator: [R2 L3 D1 F2 R2 D3 F1 B3, C_UB] C_UB is a rotation about an edge axis Thus the expansion of that would be: R2 L3 D1 F2 R3 D3 F1 B3 U1 D3 F1 L1 D2 F3 R1 L2