# 5 edge commutators



## Theo Leinad (Aug 7, 2018)

Hello everyone
I'm in the seek of all the 5 edge commutators (EO preserve preferably) like this one: 

R2 F2 R2 U' x 2

(all the cycle shifts are already taken into account) 

So if you have any commutator (8 mover or fewer even better) is really well appreaciated.


----------



## Alsamoshelan (Aug 29, 2018)

Hello,

Here are 59 5-edge commutators finded by a program that I've created :



Spoiler



U' L D' U B' D
F' U' B' F R B
L' L2 D' U B' D U'
R' U U2 D B U D'
L' U' D F D' U
L' B L R' U' R
F U B F' L' B'
L B F' D' B' F
B' D U' L D' U
U' D B' D' U R
D' R L' F R' L
F' B U2 U' B' F R'
L' B' L R' U R
F R' L D' R L'
L' F' B D' B' F L2
R' L D L2 L R F'
R' B' R' L U L' R2
U R L L2 B' L R'
R' B' F2 F' D B F'
U D' B D U' L'
F' D' B' F L B' B2
U L R' F' R L'
D' R L' F R' L
R' F' B U F B'
L' U' D F D2 D U
R' L B L' R D'
B F' U F B' R'
D' U F' D U' R
D2 D' L R' B' L' R
B' F L' B F' D
U' U2 L U' D F' D'
U' R U2 D' U' F' D
R B' L R' U L'
F' U' D R D' U
U D' B' U' D L
U R D U' B' D'
U B F' L' F B'
U B' D U U2 L D'
D R U D' F' U'
B R L' D' R2 R L
B L' F B' U F'
U F' B L' B' F
R' U' R L' B L
F B2 B D' B F' R
L' R B' R' L U
L D' L' R F R'
U L R' F' L L2 R
L U' L' R B R'
L' F R' L D' R
D B' F L' F2 B F
L' R B R' L U'
U' D F2 F U D' L
D R' D U D2 F U'
R2 B F' U B' F R
B' F R' B F' U
B' D U' L D' U' U2
R2 R' L' F' L R' D
U' D L' D' U B
U' R' D D2 U F D


I don't know what they do exactly but these are 5-edges commutators.
This program searched between 1 and 7 moves. It would be easy for him to look for algorithms with longer moves if you want. Also it took about 2 minutes to find these 59 algorithms, if you want I could try to find much more 5-edges commutators like these ones.


----------



## Alsamoshelan (Aug 29, 2018)

100 5-edges commutators with 6 moves :



Spoiler



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


----------



## Theo Leinad (Aug 29, 2018)

This is awesome @Alsamoshelan 
Could you help
Me get 5 cycles for up to 8 moves?
Is it possible to check if they preserve EO or not?
Thanks!!!


----------



## Alsamoshelan (Aug 29, 2018)

I will find a way to test the preservation of EO. 

Before that, some other algorithms :

0 algorithm with a height < 6 founded.

Equal to 7 moves (but take care : some of these ones are "fake" 7-moves as #3) :


Spoiler



1 : U D2 L D U' F' D
2 : L R F' L R' D L2
3 : D R' U D' F U U2
4 : R2 U' L' R B L R
5 : U F' B L' B B2 F
6 : B' F D' B2 F' B' R
7 : F' B U B' F' F2 R'
8 : D L R' B B2 L' R
9 : D2 L' R F R' L D
10 : F D' U' U2 L' U' D
11 : D U' L D D2 U B'
12 : R B' F' B2 U' F B'
13 : D' F' B' B2 R F B'
14 : R' F' R' R2 L' U L
15 : R L B' L R' U L2
16 : L2 R' L' B L' R D'
17 : L' B' L R' U R' R2
18 : L' D R L' F' L2 R'
19 : U' R L' B' B2 L R'
20 : B F' R B2 B F D'
21 : D L' R F' L R2 R
22 : B2 L B' F U' F' B'
23 : B U D' R' U' D2 D'
24 : F' U D D2 L D U'
25 : B' R' F B F2 U F
26 : R' U D' F D U U2
27 : D2 D B F' R F B'
28 : F' R' L D' D2 R L'
29 : L2 B' R' L U L R
30 : R' B' L' R' R2 D L
31 : U' B B2 U D' R D
32 : D' F B' L B' F' B2
33 : U' B F' L F B2 B
34 : L' U' D F D' U' U2
35 : B2 F' U' B' F R B'
36 : U L R' F' L' R' R2
37 : B L F' B D' F B2
38 : F B2 B U F' B L'
39 : B L B F' D' B2 F
40 : F D' U L' D' U' D2
41 : R F B' D' B2 B' F'
42 : U2 D U L U D' B'
43 : B F' R B2 F B D'
44 : B' F' R B F' U' F2
45 : R' U' L2 L R B L
46 : B F2 L B' F U' F
47 : U D' R U D U2 B'
48 : L R' B L' R2 R' D'
49 : F' B U F B2 B R'
50 : B R L' D' R2 R L
51 : L' D L R' B' R' R2
52 : L L2 R D' L R' B
53 : F L B F' D' B2 B
54 : B D U D2 R' D U'
55 : B F D B F' R' B2
56 : U' U2 F D U' R' D'
57 : B F' R B B2 F D'
58 : U2 U' R' L F' L' R
59 : U' D' D2 L' U D' B
60 : B2 F' B' D' B' F L
61 : L U U2 D F' D' U
62 : D U' L2 L' D' U B'
63 : D U' B' U2 D' U' R
64 : D' L D' U B' U' D2
65 : B' F D B F F2 R'
66 : L B F' D B' F L2
67 : B' U' D L U D D2
68 : R D' U F' U' D' D2
69 : B' F2 U' B F' L F'
70 : R' D U' B U D D2
71 : L' F2 B F D B' F
72 : U D' L2 L U' D F
73 : B' F D F' B R R2
74 : U D' L' L2 U' D F'
75 : R' D' U F' F2 D U'
76 : D' U F F2 U' D R
77 : U R U D U2 B' D'
78 : R L U L R' F' L2
79 : L2 L B F' D B' F
80 : U' D R' D' U' U2 F
81 : R' U' D B' B2 D' U
82 : R2 L U L' R B' R
83 : U' R L' B L2 L' R'
84 : L' B' F U' U2 B F'
85 : D' U R' R2 D U' B'
86 : R2 U R L' B' L R
87 : L' D' U B D2 U' D'
88 : F' B' F2 L B F' D'
89 : F F2 R' B' F D B
90 : L2 F' B D B' F L
91 : B' F D2 D B F' R
92 : L L2 R F' R' L D
93 : R L' F F2 R' L D
94 : U' D L2 L' D' U B'
95 : R D2 D' L R' B' L'
96 : D' D2 U' L' D' U B
97 : L L2 B F' D F B'
98 : U' F U' D' U2 L' D
99 : B D' F B2 B L F'
100 : F B2 L B F' D' B


Equal to 8 moves (but apparently there are a lot of fake 8-moves) :


Spoiler



1 : F' U U' R' L D L' R
2 : R' B F' U F B' U U'
3 : L' F R2 R2 L R' D' R
4 : F R' L' L2 D' R L L2
5 : R L' L2 L2 B' L R' U
6 : L' L B' L' F' B D F
7 : R' L D L2 R L2 L' F'
8 : U' D2 D' R D' U2 U' F'
9 : D' L R' U L' R B' D
10 : L U U2 L2 R L B R'
11 : F' R2 R2 D F B' L' B
12 : F U' B F' F2 F2 L B'
13 : R2 D2 U' D' B D' U R
14 : F2 B D' F2 B' F' L F
15 : L' R D R' F F' L B'
16 : L F B' D' F' B L' R
17 : F B F2 R' F2 B' F' D
18 : L R' B' U' U R L' D
19 : B F' L' F2 F2 B' F U
20 : D2 U L U' D2 D' F' D
21 : B' L' D2 D2 B F' D F
22 : U L D2 D' U U2 F' D'
23 : D' D2 B' F' F2 L' F' B
24 : R2 U' D F D' U R2 L'
25 : L2 L L2 B' F U' F' B
26 : F B R' B' F D' D2 F2
27 : L' F' B D F' F' B' F'
28 : B' F L B F' U2 U2 D'
29 : F2 B2 R' F B' D B' F
30 : D' U F' U' D R2 R2 R
31 : U2 U' B U U2 D L' D'
32 : F2 U2 F2 R' F2 U2 F2 R'
33 : R D U' B' U2 U' D2 D
34 : R' L' F' R' L D R R
35 : R' L B L2 R L2 L' D'
36 : R2 L' R U' R L' B L2
37 : U' F U2 U' D' L L2 D
38 : L' R F R L' R2 L2 D'
39 : D L' D U' D2 U2 B U'
40 : U2 U' D' R U' D B B2
41 : R' R2 L R2 F L' R U'
42 : R' R R' F R L' U' L
43 : D2 D U' F' D U' R U2
44 : U R2 L' R L2 F' L' R
45 : B' U D' R L2 L2 D U'
46 : B2 B F R' F' B' B2 U
47 : B U D' B B' R' U' D
48 : D2 D' F' B R F B' D2
49 : B F F' L' R D' R' L
50 : D' F' B L2 L2 R B' F
51 : R2 R2 B D' U R' D U'
52 : D' R U U2 D2 D' B' U
53 : L' D U' F D U' D2 U2
54 : R U2 U R' L F L2 L
55 : R2 R F' R L' U L' L2
56 : R' D' D2 R2 L' R' F' L
57 : U D' D2 F' D' U L U2
58 : D R D D2 U F F2 U'
59 : U' R2 R2 F' B L B' F
60 : L' D U2 U F D' U' U2
61 : L' U2 U D F' D' U L2
62 : D2 U' D' B' D' U R2 R'
63 : R B B' F' R' L D L'
64 : R' R2 U2 U R' L F L'
65 : F' D' B' F L B D D'
66 : F' F F' B L B' F U'
67 : L U' D U U' F' U D'
68 : U2 B' U' U2 D' R U D
69 : F R' R2 L' U' L' R' L2
70 : L' B R' B' F D F' L
71 : D F B' D2 D2 L' B F'
72 : U D' F F2 U2 D U R
73 : R2 R' L' D' L R R2 B
74 : U' D B' F2 F2 U D' R
75 : L' R F R' L F' F D'
76 : F' L' B' F L L' U B
77 : F2 B F R' B' F D' D2
78 : R2 F R L' U' L R' R2
79 : B U B' F L' R' L F'
80 : D F2 B F R' B F B2
81 : L R R2 D' D2 R L' F'
82 : B F' R F B' U D' U'
83 : F' F2 D' U2 U' L' U' D
84 : L' D U' R' D' U F L
85 : D U' R D' D' U D F'
86 : F D2 D2 R B F' U' B'
87 : D' U F' D' U L D2 U2
88 : R B D' U R' D U' R'
89 : R B R' L U' B2 B2 L'
90 : R' L B L' F' F R D'
91 : L2 L2 R' U L' R B' L
92 : R D U U2 B B2 D' U
93 : U B U D' U2 D2 L' D'
94 : B' U D' R2 R' U2 D U
95 : F' D U' R' D2 D U F2
96 : U' D' U2 R' U' D2 D' B
97 : F' R L' U R2 R' R2 L
98 : B D2 D U R' U U2 D
99 : L2 B' F U F' B L2 L'
100 : U' B2 B F R B' F' B2


Equal to 9 moves (same remark) :


Spoiler



1 : U2 F2 F2 U F' U D' L D
2 : D2 B' D U D2 R U D' U2
3 : F L' R U' L' L2 U U' R'
4 : L D D2 U2 U U2 B' U' D
5 : F2 F' D U' R' D2 D' D2 U
6 : D U' B' F2 F' B U D' L'
7 : U' B' D' U L L2 R L D
8 : R L' U2 U' R' R2 L R2 F'
9 : F2 F2 L' B' F U F' B2 B'
10 : R' L2 R2 L U' L R' F2 F'
11 : R2 R' L' B R' U U' L U'
12 : D D D U F' U' D' D2 R
13 : B2 B2 D' F' B R B' F2 F'
14 : L B' L2 L R D R2 R' R2
15 : U' D F B B2 L' B F' U
16 : U' U2 D' U' F' D U' R U
17 : D' U D R L R2 F' L' R
18 : L' R2 F B' U2 U' B F' R2
19 : D L2 L2 U L U' D F' D2
20 : R B L L2 R' L L U' L'
21 : R' L' U2 U' L' R' R2 B' L2
22 : U' F B F2 R' B' F U D
23 : U2 R' D2 U' D U2 F U D
24 : F B' L L2 R R' B F' D
25 : U F B' R' B2 F' B' B' B
26 : R2 R' F' L2 L2 B U' F B'
27 : U' F F' B B2 U D' R D
28 : L' R' L F' B U F2 B' F'
29 : R D R L' L2 R2 B' L2 L
30 : F F2 B L' F L L' B' U
31 : F B B2 F2 L' F' B D F2
32 : D' R R2 L D L' R F' D
33 : L D D' B B2 F U' F' B
34 : U L D U' F' R R2 R D'
35 : R B L' L2 R' U' L' F' F
36 : R L' U U2 L B2 B2 R' F
37 : U F U' L L' D R' D D2
38 : D2 U D L' U U2 D2 D' F
39 : R2 R R2 B F' U' B' F2 F'
40 : F2 F L' R' L2 D R' R2 L'
41 : U D' R U2 U' D' D2 U2 B'
42 : U' L U2 U D F' U U D'
43 : F' D' B' F2 B2 F R F2 B'
44 : D' R U' D F F' B B2 U
45 : U B' U2 U D2 D' L D D2
46 : D' D2 B' F L' D' D F' B
47 : D' B F' L2 R L L F B'
48 : R' L' L L B' B2 L' R D'
49 : U' D' R' U' D F2 F2 B U2
50 : D' D2 F' F U' B U D' R'


Eventually, algorithms with a random height between 6 and 12 :


Spoiler



1 : D R' L B' L' R
2 : R B R' U U' L U U2 L'
3 : F B' L B F' D'
4 : B' F R D' D B F' U'
5 : D' D L L' U B' F R' B F'
6 : F' R' U' D B U D' F
7 : B' U D' R D U'
8 : D U' L D' U B'
9 : U' R R' F2 F' U' U B' R' F' B U2
10 : U2 U D B' U D' R
11 : U' L' R' R B L R' U' R U
12 : R' U U' B2 B' F B2 D F' B
13 : B U D' R' D U'
14 : U L' D' U B U' U D U2
15 : F2 B B2 R F B' D' B2 F
16 : U' L' U D' B D2 D'
17 : D2 D' L2 L' D' U B' U'
18 : B F' U' B' F R
19 : D' B F' R B' F
20 : R L' D R' L B'
21 : R L B R L' D' R2
22 : R' D R' L B' L' R2
23 : B' L B' F U' F' B2
24 : F' L' R U' R' L F2
25 : R D2 U D F' U' D
26 : F' B' R' B' R R' F D B B L2 L2
27 : D' L R' B L' R
28 : L' F' R R R2 R L' U R' L2
29 : D' R L' F R' R2 L R2
30 : L' R' B2 B L R' U R2
31 : B' U' B F' L F' F2
32 : B' L' B F' D F
33 : L R' F' L' R U
34 : R' U' D B F' F U D'
35 : D' L' R2 R' F R' L' L2
36 : D' B' F' F2 L F' U2 U2 B' B2
37 : F' B R' F' F R2 F B B2 D'
38 : D B B2 B2 D' U R' U'
39 : F F' U' L L2 R B R' L
40 : B' L' F' B D F
41 : B B2 F2 U U' F' R' B F' U
42 : R U D' F2 F D U'
43 : B' D F' B R R2 F
44 : R2 R L D' L' R F
45 : U' B' D' U R D
46 : R D' L' R F R2 L
47 : B L R' U2 U L' D' D' D2 R
48 : L B' R L' D R'
49 : B' R2 R' B F' U' F2 F'
50 : B D D2 B' F L B F2 F B'
51 : U' L' D' U B D
52 : D' U2 D2 U F' D U D2 L
53 : B' R F2 B' F' D' F' B2
54 : F R2 R L D' R L' U' B' B U
55 : L' R' D R L' F' L L
56 : R2 R2 R' D2 U D F' U2 U D R2
57 : D U D2 R' D2 U' D' B
58 : L2 F' L R' D' D2 L R2 R'
59 : L' D' L' L2 R' B R
60 : R L' F' R' U' U L D
61 : D' U B B2 U' D L
62 : R L' B' L R' U' U2
63 : F F B' R2 R' B F' U' F'
64 : D' U2 U U2 B' D U' L
65 : D' D2 B' U D2 D' D' D D2 R U'
66 : L' U F' F D' B D U'
67 : B B' R2 L R D' R L' F
68 : B' L' R D R L' R L2 R
69 : R' R2 D2 D' U' B' D' U
70 : R R' F' B R' F B' D
71 : D U' F2 B' F2 D' U R
72 : D2 D2 B' F' F2 U B F' L'
73 : L' L D L U D' B' U'
74 : R' R U2 D' U' F' U' D R
75 : F R' B' B' F' B' U B'
76 : U' R L2 L' R2 F R L'
77 : D' B D2 U' D' L' U
78 : U B2 B U' D L D'
79 : D R B R' L U2 U L' D'
80 : F F2 B D' B2 B F L
81 : U' B' U' U2 D' R D
82 : L' B' F U B' B2 B' F F2 B
83 : F F2 D F B' L' B
84 : R' B' F D B' F' B2
85 : R' D' L' R F L' L2
86 : D' R L' D R' L' L2 B' D
87 : D' R L' F L' R' L2
88 : B' F U B F' L'
89 : D U' F' D' U L
90 : U D' F U' D R'
91 : L' R' F L R' D' R2
92 : U2 R' U D' F U D
93 : F' B L' F' F2 B' U U' U
94 : D' U2 B2 B' B' U D' D' B' U D' R
95 : D' L' R F' F2 R' L2 L'
96 : D U' L' D' U B
97 : U' D B2 B U D' R
98 : U D' R D U' B'
99 : U L R' F' R L'
100 : R F B' D L' L D2 L2 L2 F F2 B


My program is very primitive yet, I will improve it in order to avoid fake n-moves and to check EO preservation.


----------



## rokicki (Aug 29, 2018)

What does it mean here to "preserve edge orientation"? What's your edge orientation convention?


----------



## Alsamoshelan (Aug 29, 2018)

I thought that there is a sort of "universal" conventional definition : an edge is oriented if and only if we can put it into its original slot without F, F', B and B' moves.
But after all, it's just a convention and maybe the author of this threat uses another one.


----------



## Bruce MacKenzie (Mar 1, 2019)

Alsamoshelan said:


> I thought that there is a sort of "universal" conventional definition : an edge is oriented if and only if we can put it into its original slot without F, F', B and B' moves.
> But after all, it's just a convention and maybe the author of this threat uses another one.



Another definition of edge flip counts the number of quarter turns required to return the edge piece to the solved position and orientation. Even flip requires an even number of q-turns. Odd flip requires and odd number of q-turns.


----------



## abunickabhi (Mar 2, 2019)

Alsamoshelan said:


> Hello,
> 
> Here are 59 5-edge commutators finded by a program that I've created :
> 
> ...




All the algs are pretty bad here.
Here is the one that I have hand-crafted:
https://github.com/abunickabhi/5style/blob/master/5-style-edge.pdf

https://algsets.jonatanklosko.com/alg-sets/5c712ae91a3b515e307eea47


----------



## abunickabhi (Mar 2, 2019)

I have been developing 5 cycles for both edges and corners for over 3 years now.
With the Giiker cube, I can practise in this fun way now:





Documentation of this method: https://docs.google.com/document/d/1b3tT8Wv18WdzFyY7FujyptwzYeWFH8K-UXTddTvwUWw/edit?usp=sharing


----------



## abunickabhi (Mar 2, 2019)

I also have a YouTube channel where I try to make fingertricks for all these 126,720 edge algs. (I plan to do corner algs later since UFR is so gooood).

https://www.youtube.com/channel/UCa7dTclUqnR9VwSeOCpRwAQ

Also, not all algs are like, R2 F2 R2 U' x 2, there are also many hard 5-cycles, but the hardest ones are about 14 moves long for edges.


----------



## abunickabhi (Mar 16, 2019)

The upper bound for 5 edges commutator is 14 moves STM. I just found it out manually. I have to verify it computationally.


----------



## Bruce MacKenzie (Mar 18, 2019)

I did some back of the envelope calculations:

The number of sets of 5 edge cubies:

12! / 7! = 95,040

Each of these edge sets may be permuted:

5! / 2 x 2^4 = 60 x 16 = 960

giving:

95040 x 960 = 91,238,400 edge permutations of five or fewer cubies.

similarly

12! / 8! x 4!/2 x 2^3 = 1,140,480 edge permutations of four or fewer cubies.

Thus

91,238,400 - 1,140,480 = 90,097,920 cube positions with five unsolved edge cubies.

This is not that big a state space that modern computers could not completely explore it.

Woops, the above is in error. The order one picks the 5 cubies to scramble doesn't matter:

The number of sets of 5 edge cubies:

12! /(7! x 5!) = 792

Each of these edge sets may be permuted:

5! / 2 x 2^4 = 60 x 16 = 960

giving:

792 x 960 = 760,320 edge permutations of five or fewer cubies.

similarly

12! /( 8! * 4!) x 4!/2 x 2^3 = 47,520 edge permutations of four or fewer cubies.

Thus

760,320 - 47,520 = 712,800 cube positions with five unsolved edge cubies.


----------



## abunickabhi (Mar 18, 2019)

You have left out some cases.

Working with the sticker logic,
there are a total of 24 edge stickers.
Assuming buffer to buffer, we have 22 targets for the first edge to move to,
20 targets for the second edge to move to,
18 targets for the third edge to move to,
16 targets for the fourth edge to move to,
giving us 22x20x18x16= 126,720!

As simple as that.

In your second calculation, there is no need of, 5! /2 as 11C4 (=330) takes care of all the cases.
All it is 11C4 since I am assuming buffer(DF) to be the reference piece.

126,720 comes up with your method as: 11C4 x 4! x 2^4 = 126, 720

(Keep in mind, I am not accounting for flipped edge cases, and dedges cases in big cubes, the number would be much higher there, but still not as high as your 712,800 which is just astronomical.) With mirrors and inverses, the number of cases in 5-style becomes like ~40k which is manageable with 2 years of dedicated training. Since I have to generate optimal fingertrickable algs for myself, it will take me 4 years. And I started out in 2016 yo.

Looks like your intention was to just bump up the numbers and make 5-style sound harder. :|
The number 712,800 is close to 5 times 126,720, so there are 5 times more cases if we try to go for full floating buffer 5-style.
Even I would find complete floating buffer 5-style ridiculously hard and never speculate it or make a SS thread on it lol.


----------



## Bruce MacKenzie (Mar 18, 2019)

abunickabhi said:


> You have left out some cases.
> 
> Working with the sticker logic,
> there are a total of 24 edge stickers.
> ...



You are correct. My calculation includes cases which are not 5 cycles. There are three cycles + two cubies flipped in place, etc. The requirement to preserve edge orientation is ambiguous since that depends on how one defines edge orientation.


----------



## Bruce MacKenzie (Mar 19, 2019)

Bruce MacKenzie said:


> I did some back of the envelope calculations:
> 
> The number of sets of 5 edge cubies:
> 
> ...



I knocked together some code today and enumerated the set of cube states with five unsolved edge cubies. As it turns out the above calculation is not valid. There are 462,528 cube states with 5 unsolved edges not 712,800. The latter number includes a lot of duplicates of states with less than 5 unsolved cubies.

The 462,528 states reduce to 19,272 symmetry equivalence classes by the 24 cubic rotation symmetries. I ran representative elements of these classes through an optimal solver giving the following results. There are none less than 6 turns from solved and all may be solved in 15 or fewer moves. The 5-Cycles column are those instances where all five cubies are moved from their home position. That is, none of the five are simply flipped in place.

DepthElementsReduced(O)5-Cycles

6​


192​


8​


8​

7​
480​
20​
20​
8​
2,112​
88​
88​
9​
5,472​
228​
228​
10​
25,632​
1,068​
958​
11​
58,320​
2,430​
2,256​
12​
138,384​
5,766​
4,700​
13​
143,496​
5,979​
3,914​
14​
81,384​
3,391​
496​
15​
7,056​
294​
4​


Sum​
462,528​
19,272​
12,672​
 


Here are representative members of the four depth 15 5-Cycles. None of these has any symmetry so they each represent a 24 element symmetry equivalence class whose elements differ only in the orientation of the cube to which the maneuver is applied.

F' R F R' F2 R' U F L F L' F2 U' R F2 15f
D R F R2 D2 R2 D2 R' B R' B' D2 R' F' D' 15f
U R U L' B2 R2 B R L' U B R' B' U' L2 15f
D R' D L' D2 R2 D2 R2 B R B' L R2 B2 D2 15f


----------



## abunickabhi (Mar 19, 2019)

Bruce MacKenzie said:


> I knocked together some code today and enumerated the set of cube states with five unsolved edge cubies. As it turns out the above calculation is not valid. There are 462,528 cube states with 5 unsolved edges not 712,800. The latter number includes a lot of duplicates of states with less than 5 unsolved cubies.
> 
> The 462,528 states reduce to 19,272 symmetry equivalence classes by the 24 cubic rotation symmetries. I ran representative elements of these classes through an optimal solver giving the following results. There are none less than 6 turns from solved and all may be solved in 15 or fewer moves. The 5-Cycles column are those instances where all five cubies are moved from their home position. That is, none of the five are simply flipped in place.
> 
> ...



Nice work, in BLD however, using cube symmetry and reducing it just 12,672 cases won't work. Since each alg needs to be fingertricky, rotationless and from [R U D F M S E] set, the number of cases is 126,720/2(Inverses) = 63,360. (Some Mirror Algs are slow to execute)
Also the target shooting should be consistent which only one visit to each piece.

In the 15 mover, F' R F R' F2 R' U F L F L' F2 U' R F2, by BLD tracing, there will always be a cycle break from a buffer.


----------



## Bruce MacKenzie (Mar 19, 2019)

Bruce MacKenzie said:


> I knocked together some code today and enumerated the set of cube states with five unsolved edge cubies. As it turns out the above calculation is not valid. There are 462,528 cube states with 5 unsolved edges not 712,800. The latter number includes a lot of duplicates of states with less than 5 unsolved cubies.
> 
> The 462,528 states reduce to 19,272 symmetry equivalence classes by the 24 cubic rotation symmetries. I ran representative elements of these classes through an optimal solver giving the following results. There are none less than 6 turns from solved and all may be solved in 15 or fewer moves. The 5-Cycles column are those instances where all five cubies are moved from their home position. That is, none of the five are simply flipped in place.
> 
> ...



I extended the above analysis to the Quarter Turn Metric and the Slice Turn Metric.

(I'm done trying to use the table editor here. You'll have to copy and paste into a spreadsheet if you're interested. The columns are the same as above.)

*QTM*

6 192 8 8
8 1,536 64 64
10 13,248 552 552
12 79,728 3,322 3,248
14 211,296 8,804 7,590
16 147,744 6,156 1,210
18 8,784 366 0
Sum 462,528 19,272 12,672


*STM*

4 96 4 4
5 288 12 12
6 1,632 68 68
7 5,424 226 208
8 20,064 836 758
9 53,568 2,232 1,976
10 136,464 5,686 4,586
11 158,496 6,604 4,204
12 78,432 3,268 852
13 8,064 336 4
Sum 462,528 19,272 12,672

And here are the depth 13s 5-Cycles:

R L F MU2 F' U F' MU2 F U MR U2 L2 • CR' 13s
R' D R' D' MR' D MR D' MR' D L B' R • CR 13s
R' F L' F' MR F2 R MU L' U' L D' F • CR' CF' 13s
R U F' U' B MU2 F' L F L' MU2 MF' U' • CF 13s


----------



## iheartgeo (Apr 27, 2019)

Alsamoshelan said:


> Hello,
> 
> Here are 59 5-edge commutators finded by a program that I've created :
> 
> ...



Thanks for producing these algs! I would like to understand the structure of 5-cycles. I know how to construct 3-cycles on the fly because I have internalized their structure, and would like to be able to do the same with 5-cycles. Is anyone else interested in developing the theory with me?


----------



## iheartgeo (Apr 27, 2019)

Alsamoshelan said:


> 100 5-edges commutators with 6 moves :
> 
> 
> 
> ...



The first alg is equivalent to [E’ , F].


----------



## iheartgeo (Apr 27, 2019)

Alsamoshelan said:


> 100 5-edges commutators with 6 moves :
> 
> 
> 
> ...



The 2nd alg is equivalent to D: [R’ , E].


----------



## mark49152 (Apr 27, 2019)

iheartgeo said:


> The 2nd alg is equivalent to D: [R’ , E].


I haven't checked them all, but there are no slice moves used, and it looks like all these are equivalent to 4-move comms with a slice, some with a single setup parallel to the slice.


----------



## iheartgeo (Apr 27, 2019)

mark49152 said:


> I haven't checked them all, but there are no slice moves used, and it looks like all these are equivalent to 4-move comms with a slice, some with a single setup parallel to the slice.



Great observation! This is really helpful. I want to understand how to recognize all 4-move edge 5-cycles, as well as cases that setup nicely into these.

Here are the first 10 algs written as comms:

1. [E' , F]
2. D: [R' , E]
3. [D , M']
4. U: [B' , E']
5. [L , E']
6. [L' , S]
7. U': [R , E]
8. [M' , D]
9. [R' , E']
10. [M , B']


----------



## iheartgeo (Apr 27, 2019)

mark49152 said:


> I haven't checked them all, but there are no slice moves used, and it looks like all these are equivalent to 4-move comms with a slice, some with a single setup parallel to the slice.



I think all the 4-move comms can be reduced to four basic cases. All others can be setup into these with cube rotations:

[M' , U]
[M' , U']
[U , M']
[U' , M']


----------



## iheartgeo (Apr 27, 2019)

Alsamoshelan said:


> 100 5-edges commutators with 6 moves :
> 
> 
> 
> ...



All 100 cases above can be solved using the same basic procedure.

Case 1: 4 edges are in one layer, the 5th edge is in the opposite layer. In this case, rotate the cube so that the 4 edges are in the U layer and the 5 edge is in the DF position.

Case 2: 3 edges in one layer, one in the middle layer, one in the opposite layer. In this case, just rotate the edge from the middle layer to the top layer to put the cube into the same state as Case 1.

To solve Case 1, you just do two things: 1) place the UB edge, then 2) place the
DF edge. Here are the sub-cases:

A. UB cubie is in the UF spot: fix it with M'. Now fix the DF cubie with UM or U'M. Do an AUF to finish the commutator.

B. UB cubie is in the UL or UR spot: fix it with U or U'. Now fix the DF cubie with M'U'M or M'UM.

So it all comes down to one of four cases:
1. M' U M U'
2. M' U' M U
3. U M' U' M
4. U' M' U M

All of which are subsumed under the rule: 1) fix UB then 2) fix DF.

To solve Case 2, you just set it up to Case 1, solve it, then undo the setup move.


----------



## iheartgeo (Apr 27, 2019)

mark49152 said:


> I haven't checked them all, but there are no slice moves used, and it looks like all these are equivalent to 4-move comms with a slice, some with a single setup parallel to the slice.



You are 100% right! I checked all 100 cases and posted an analysis just now. They all reduce to the same solution: 1) place the UB cubie, then 2) place the DF cubie.


----------



## mark49152 (Apr 28, 2019)

iheartgeo said:


> You are 100% right! I checked all 100 cases and posted an analysis just now. They all reduce to the same solution: 1) place the UB cubie, then 2) place the DF cubie.


Yes, the way they work is quite simple to understand. Using DF buffer and [M', U], I think of it as follows: the first two stickers solved (by M') are FU and UB, after which the U moves the next two stickers into those same locations to be solved in reverse order (by M). So, UL and RU. 

That particular type of 5-cycle is potentially attractive because it's so move efficient. The challenge of course is making it practical in a real (timed) solve. Intuitively setting up to one of these cases is not trivial, and the thinking time involved will usually more than cancel out whatever move advantage it offers relative to two 3-cycles.


----------



## iheartgeo (Apr 28, 2019)

mark49152 said:


> Yes, the way they work is quite simple to understand. Using DF buffer and [M', U], I think of it as follows: the first two stickers solved (by M') are FU and UB, after which the U moves the next two stickers into those same locations to be solved in reverse order (by M). So, UL and RU.
> 
> That particular type of 5-cycle is potentially attractive because it's so move efficient. The challenge of course is making it practical in a real (timed) solve. Intuitively setting up to one of these cases is not trivial, and the thinking time involved will usually more than cancel out whatever move advantage it offers relative to two 3-cycles.



I agree that it's hard to see how this method could have a practical advantage of 3-style. I'm still interested in 5-style for theoretical reasons - I want to understand the cube! If I can develop a procedure on the fly for an arbitrary 5-cycle the way I can for 3-cycles, I will feel like I have power over the cube!


----------



## iheartgeo (Apr 28, 2019)

Alsamoshelan said:


> I will find a way to test the preservation of EO.
> 
> Before that, some other algorithms :
> 
> ...



The "7-move" alg set can all be solved with a version of [M', U*] or its inverse (plus one possible setup move). The only differences between this set and the algs posted in the "6-move" set are two things: 1) sometimes the setup move is a double turn (e.g. R2) as opposed to a quarter turn, and 2) sometimes the setup move is on the U face to achieve the necessary position. Examples of this include algs #9, 66, and 90.


----------



## iheartgeo (Apr 28, 2019)

mark49152 said:


> I haven't checked them all, but there are no slice moves used, and it looks like all these are equivalent to 4-move comms with a slice, some with a single setup parallel to the slice.



I'm stuck on this one:
29 : F2 B2 R' F B' D B' F 

Can this be solved by a variant of [M' , U] with setup moves?


----------



## iheartgeo (Apr 28, 2019)

iheartgeo said:


> I'm stuck on this one:
> 29 : F2 B2 R' F B' D B' F
> 
> Can this be solved by a variant of [M' , U] with setup moves?



I got it! It needs an M2 setup move (after cube rotation).


----------



## mark49152 (Apr 28, 2019)

iheartgeo said:


> I got it! It needs an M2 setup move (after cube rotation).


Yep - (F2 B2) R' (F B') D (B' F)


----------



## iheartgeo (Apr 28, 2019)

mark49152 said:


> Yep - (F2 B2) R' (F B') D (B' F)



My solution was to rotate the cube so that the 3 edges in the same layer are on UL-UB-UR, then set it up with M2, then do [M' , U'].


----------



## iheartgeo (Apr 28, 2019)

Alsamoshelan said:


> I will find a way to test the preservation of EO.
> 
> Before that, some other algorithms :
> 
> ...



In the "8-move" alg set, 99/100 cases can be solved using a variation of [M' , U'] or [M' , U] after cube rotation. The one exception is this:

32 : F2 U2 F2 R' F2 U2 F2 R'

This case is a variant of the alg that started this thread:
(R2 F2 R2 U') 2

It is notable that the five edges involved are all oriented with respect to the L/R faces. This distinguishes the case from the other 99/100 cases, which all have four bad edges that get fixed with M' U M or M' U' M.

The following cases require interesting setup moves because of the relative position of the misplaced edges:

70 : L' B R' B' F D F' L
87 : D' U F' D' U L D2 U2
88 : R B D' U R' D U' R'


----------



## iheartgeo (Apr 28, 2019)

Alsamoshelan said:


> I will find a way to test the preservation of EO.
> 
> Before that, some other algorithms :
> 
> ...



All 50 "9-movers" are variations on [M' , U'] or [M' , U].


----------



## abunickabhi (Apr 30, 2019)

iheartgeo said:


> Thanks for producing these algs! I would like to understand the structure of 5-cycles. I know how to construct 3-cycles on the fly because I have internalized their structure, and would like to be able to do the same with 5-cycles. Is anyone else interested in developing the theory with me?



Ahem ahem, I have already started developing commutator theory for it since the last 2 years. Some 5-Style cases are not as intuitive as 3-Style algs, so there is more work in understanding it.

Also, I am now just genning optimal algs, and not observing the types as of yet. I think the type of commutators that can be there can be like 50+ as opposed to just 8 types of commutators in BH/3-Style corners.

For an alg like this U M D M D' U' S U M2 U' S, you can somewhat construct a commutator out of it,

but for alg like this E2 L F L D M D' R' D' L2 r E2*, *it is tougher to come up with commutator notation for it and takes a lot of understanding to derive it.


----------

