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Face-Turning Octahedron (FTO): algorithm sets

Edge 3-cycles (96)

Commutators that cycle three edges and change nothing else. Notation is the one cubing.js uses for the FTO.

Each algorithm is a commutator found by search and checked in a simulator to be a pure 3-cycle: three pieces of one type move, every other piece stays put and nothing is twisted. This is a sample of the shortest ones found, not a complete set.

8 moves

2BL 2D 2BL' D 2BL 2D' 2BL' D' — Case 1
2BL 2D 2BL' U' 2BL 2D' 2BL' U — Case 2
2BL 2F 2BL' B' 2BL 2F' 2BL' B — Case 3
2BL 2F 2BL' F 2BL 2F' 2BL' F' — Case 4
2BL 2F' 2BL' B 2BL 2F 2BL' B' — Case 5
2BL 2F' 2BL' F' 2BL 2F 2BL' F — Case 6
2BL 2L 2BL' BR' 2BL 2L' 2BL' BR — Case 7
2BL 2L 2BL' L 2BL 2L' 2BL' L' — Case 8
2BL B 2BL' 2F 2BL B' 2BL' 2F' — Case 9
2BL B 2BL' 2F' 2BL B' 2BL' 2F — Case 10
2BL B' 2BL' 2F 2BL B 2BL' 2F' — Case 11
2BL B' 2BL' 2F' 2BL B 2BL' 2F — Case 12
2BL BR 2BL' 2L 2BL BR' 2BL' 2L' — Case 13
2BL BR 2BL' 2L' 2BL BR' 2BL' 2L — Case 14
2BL BR' 2BL' 2L' 2BL BR 2BL' 2L — Case 15
2BL D 2BL' 2D 2BL D' 2BL' 2D' — Case 16
2BL D 2BL' 2D' 2BL D' 2BL' 2D — Case 17
2BL D' 2BL' 2D 2BL D 2BL' 2D' — Case 18
2BL D' 2BL' 2D' 2BL D 2BL' 2D — Case 19
2BL F 2BL' 2F 2BL F' 2BL' 2F' — Case 20
2BL F 2BL' 2F' 2BL F' 2BL' 2F — Case 21
2BL F' 2BL' 2F 2BL F 2BL' 2F' — Case 22
2BL F' 2BL' 2F' 2BL F 2BL' 2F — Case 23
2BL L 2BL' 2L' 2BL L' 2BL' 2L — Case 24
2BL L' 2BL' 2L 2BL L 2BL' 2L' — Case 25
2BL L' 2BL' 2L' 2BL L 2BL' 2L — Case 26
2BL U 2BL' 2D 2BL U' 2BL' 2D' — Case 27
2BL U 2BL' 2D' 2BL U' 2BL' 2D — Case 28
2BL U' 2BL' 2D 2BL U 2BL' 2D' — Case 29
2BL U' 2BL' 2D' 2BL U 2BL' 2D — Case 30
2BL' 2D 2BL D 2BL' 2D' 2BL D' — Case 31
2BL' 2D 2BL U' 2BL' 2D' 2BL U — Case 32
2BL' 2D' 2BL D' 2BL' 2D 2BL D — Case 33
2BL' 2D' 2BL U 2BL' 2D 2BL U' — Case 34
2BL' 2F 2BL B' 2BL' 2F' 2BL B — Case 35
2BL' 2F 2BL F 2BL' 2F' 2BL F' — Case 36
2BL' 2F' 2BL B 2BL' 2F 2BL B' — Case 37
2BL' 2F' 2BL F' 2BL' 2F 2BL F — Case 38
2BL' 2L' 2BL BR 2BL' 2L 2BL BR' — Case 39
2BL' 2L' 2BL L' 2BL' 2L 2BL L — Case 40
2BL' B 2BL 2F 2BL' B' 2BL 2F' — Case 41
2BL' B 2BL 2F' 2BL' B' 2BL 2F — Case 42
2BL' B' 2BL 2F 2BL' B 2BL 2F' — Case 43
2BL' B' 2BL 2F' 2BL' B 2BL 2F — Case 44
2BL' BR 2BL 2L 2BL' BR' 2BL 2L' — Case 45
2BL' BR' 2BL 2L 2BL' BR 2BL 2L' — Case 46
2BL' BR' 2BL 2L' 2BL' BR 2BL 2L — Case 47
2BL' D 2BL 2D 2BL' D' 2BL 2D' — Case 48
2BL' D 2BL 2D' 2BL' D' 2BL 2D — Case 49
2BL' D' 2BL 2D 2BL' D 2BL 2D' — Case 50
2BL' F 2BL 2F 2BL' F' 2BL 2F' — Case 51
2BL' F 2BL 2F' 2BL' F' 2BL 2F — Case 52
2BL' F' 2BL 2F 2BL' F 2BL 2F' — Case 53
2BL' F' 2BL 2F' 2BL' F 2BL 2F — Case 54
2BL' L 2BL 2L 2BL' L' 2BL 2L' — Case 55
2BL' L 2BL 2L' 2BL' L' 2BL 2L — Case 56
2BL' L' 2BL 2L 2BL' L 2BL 2L' — Case 57
2BL' U 2BL 2D 2BL' U' 2BL 2D' — Case 58
2BL' U' 2BL 2D 2BL' U 2BL 2D' — Case 59
2BL' U' 2BL 2D' 2BL' U 2BL 2D — Case 60
2D 2BL 2D' BL 2D 2BL' 2D' BL' — Case 61
2D 2BL 2D' BL' 2D 2BL' 2D' BL — Case 62
2D 2BL 2D' R 2D 2BL' 2D' R' — Case 63
2D 2BL 2D' R' 2D 2BL' 2D' R — Case 64
2D 2F 2D' B' 2D 2F' 2D' B — Case 65
2D 2F 2D' F 2D 2F' 2D' F' — Case 66
2D 2F' 2D' B 2D 2F 2D' B' — Case 67
2D 2F' 2D' B' 2D 2F 2D' B — Case 68
2D 2F' 2D' F 2D 2F 2D' F' — Case 69
2D 2F' 2D' F' 2D 2F 2D' F — Case 70
2D 2L 2D' BR 2D 2L' 2D' BR' — Case 71
2D 2L 2D' BR' 2D 2L' 2D' BR — Case 72
2D 2L 2D' L 2D 2L' 2D' L' — Case 73
2D 2L 2D' L' 2D 2L' 2D' L — Case 74
2D 2L' 2D' BR 2D 2L 2D' BR' — Case 75
2D 2L' 2D' BR' 2D 2L 2D' BR — Case 76
2D 2L' 2D' L 2D 2L 2D' L' — Case 77
2D 2L' 2D' L' 2D 2L 2D' L — Case 78
2D B 2D' 2F 2D B' 2D' 2F' — Case 79
2D B 2D' 2F' 2D B' 2D' 2F — Case 80
2D B' 2D' 2F 2D B 2D' 2F' — Case 81
2D B' 2D' 2F' 2D B 2D' 2F — Case 82
2D BL 2D' 2BL' 2D BL' 2D' 2BL — Case 83
2D BL' 2D' 2BL 2D BL 2D' 2BL' — Case 84
2D BL' 2D' 2BL' 2D BL 2D' 2BL — Case 85
2D BR 2D' 2L 2D BR' 2D' 2L' — Case 86
2D BR' 2D' 2L 2D BR 2D' 2L' — Case 87
2D BR' 2D' 2L' 2D BR 2D' 2L — Case 88
2D F 2D' 2F 2D F' 2D' 2F' — Case 89
2D F 2D' 2F' 2D F' 2D' 2F — Case 90
2D F' 2D' 2F 2D F 2D' 2F' — Case 91
2D F' 2D' 2F' 2D F 2D' 2F — Case 92
2D L 2D' 2L 2D L' 2D' 2L' — Case 93
2D L 2D' 2L' 2D L' 2D' 2L — Case 94
2D L' 2D' 2L 2D L 2D' 2L' — Case 95
2D R 2D' 2BL 2D R' 2D' 2BL' — Case 96

Centre 3-cycles (96)

Commutators that cycle three centre pieces and change nothing else.

Each algorithm is a commutator found by search and checked in a simulator to be a pure 3-cycle: three pieces of one type move, every other piece stays put and nothing is twisted. This is a sample of the shortest ones found, not a complete set.

8 moves

B 2BL B' R B 2BL' B' R' — Case 1
B 2BL' B' R B 2BL B' R' — Case 2
B 2BL' B' R' B 2BL B' R — Case 3
B 2D B' D B 2D' B' D' — Case 4
B 2D B' D' B 2D' B' D — Case 5
B 2D' B' D B 2D B' D' — Case 6
B 2D' B' D' B 2D B' D — Case 7
B 2L B' L B 2L' B' L' — Case 8
B 2L B' L' B 2L' B' L — Case 9
B 2L' B' L B 2L B' L' — Case 10
B 2L' B' L' B 2L B' L — Case 11
B D B' 2D B D' B' 2D' — Case 12
B D B' 2D' B D' B' 2D — Case 13
B D' B' 2D B D B' 2D' — Case 14
B D' B' 2D' B D B' 2D — Case 15
B L B' 2L' B L' B' 2L — Case 16
B L' B' 2L B L B' 2L' — Case 17
B L' B' 2L' B L B' 2L — Case 18
B R B' 2BL B R' B' 2BL' — Case 19
B R' B' 2BL B R B' 2BL' — Case 20
B R' B' 2BL' B R B' 2BL — Case 21
B' 2BL B R B' 2BL' B R' — Case 22
B' 2BL B R' B' 2BL' B R — Case 23
B' 2BL' B R' B' 2BL B R — Case 24
B' 2D B D B' 2D' B D' — Case 25
B' 2D B D' B' 2D' B D — Case 26
B' 2D' B D B' 2D B D' — Case 27
B' 2D' B D' B' 2D B D — Case 28
B' 2L B L' B' 2L' B L — Case 29
B' 2L' B L B' 2L B L' — Case 30
B' 2L' B L' B' 2L B L — Case 31
B' D B 2D B' D' B 2D' — Case 32
B' D B 2D' B' D' B 2D — Case 33
B' D' B 2D B' D B 2D' — Case 34
B' D' B 2D' B' D B 2D — Case 35
B' L B 2L B' L' B 2L' — Case 36
B' L B 2L' B' L' B 2L — Case 37
B' L' B 2L B' L B 2L' — Case 38
B' L' B 2L' B' L B 2L — Case 39
B' R B 2BL B' R' B 2BL' — Case 40
B' R B 2BL' B' R' B 2BL — Case 41
B' R' B 2BL' B' R B 2BL — Case 42
BL 2D BL' U BL 2D' BL' U' — Case 43
BL 2D BL' U' BL 2D' BL' U — Case 44
BL 2D' BL' U BL 2D BL' U' — Case 45
BL 2D' BL' U' BL 2D BL' U — Case 46
BL 2F BL' F BL 2F' BL' F' — Case 47
BL 2F BL' F' BL 2F' BL' F — Case 48
BL 2F' BL' F BL 2F BL' F' — Case 49
BL 2F' BL' F' BL 2F BL' F — Case 50
BL 2L BL' BR BL 2L' BL' BR' — Case 51
BL 2L' BL' BR BL 2L BL' BR' — Case 52
BL 2L' BL' BR' BL 2L BL' BR — Case 53
BL BR BL' 2L BL BR' BL' 2L' — Case 54
BL BR' BL' 2L BL BR BL' 2L' — Case 55
BL BR' BL' 2L' BL BR BL' 2L — Case 56
BL F BL' 2F BL F' BL' 2F' — Case 57
BL F BL' 2F' BL F' BL' 2F — Case 58
BL F' BL' 2F BL F BL' 2F' — Case 59
BL F' BL' 2F' BL F BL' 2F — Case 60
BL U BL' 2D BL U' BL' 2D' — Case 61
BL U' BL' 2D BL U BL' 2D' — Case 62
BL U' BL' 2D' BL U BL' 2D — Case 63
BL' 2D BL U BL' 2D' BL U' — Case 64
BL' 2D BL U' BL' 2D' BL U — Case 65
BL' 2D' BL U' BL' 2D BL U — Case 66
BL' 2F BL F BL' 2F' BL F' — Case 67
BL' 2F BL F' BL' 2F' BL F — Case 68
BL' 2F' BL F BL' 2F BL F' — Case 69
BL' 2F' BL F' BL' 2F BL F — Case 70
BL' 2L BL BR BL' 2L' BL BR' — Case 71
BL' 2L BL BR' BL' 2L' BL BR — Case 72
BL' 2L' BL BR' BL' 2L BL BR — Case 73
BL' BR BL 2L BL' BR' BL 2L' — Case 74
BL' BR BL 2L' BL' BR' BL 2L — Case 75
BL' BR' BL 2L' BL' BR BL 2L — Case 76
BL' F BL 2F BL' F' BL 2F' — Case 77
BL' F BL 2F' BL' F' BL 2F — Case 78
BL' F' BL 2F BL' F BL 2F' — Case 79
BL' F' BL 2F' BL' F BL 2F — Case 80
BL' U BL 2D BL' U' BL 2D' — Case 81
BL' U BL 2D' BL' U' BL 2D — Case 82
BL' U' BL 2D BL' U BL 2D' — Case 83
BL' U' BL 2D' BL' U BL 2D — Case 84
BR 2BL BR' BL BR 2BL' BR' BL' — Case 85
BR 2BL BR' BL' BR 2BL' BR' BL — Case 86
BR 2BL' BR' BL BR 2BL BR' BL' — Case 87
BR 2D BR' U BR 2D' BR' U' — Case 88
BR 2D' BR' U BR 2D BR' U' — Case 89
BR 2D' BR' U' BR 2D BR' U — Case 90
BR 2F BR' F BR 2F' BR' F' — Case 91
BR 2F BR' F' BR 2F' BR' F — Case 92
BR 2F' BR' F BR 2F BR' F' — Case 93
BR 2F' BR' F' BR 2F BR' F — Case 94
BR BL BR' 2BL' BR BL' BR' 2BL — Case 95
BR BL' BR' 2BL BR BL BR' 2BL' — Case 96