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5 edge commutators

Joined
Mar 31, 2017
Messages
50
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9
WCA
2016CUEV04
Thread starter #1
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.
 
Joined
Aug 29, 2018
Messages
8
Likes
4
#2
Hello,

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

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. ;)
 
Joined
Aug 29, 2018
Messages
8
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4
#3
100 5-edges commutators with 6 moves :

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
 
Joined
Aug 29, 2018
Messages
8
Likes
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#5
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) :
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) :
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) :
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 :
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. ;)
 
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#7
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.
 
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#8
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.
 
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#9
Hello,

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

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. ;)

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
 
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#13
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.
 
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abunickabhi
#14
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.
 
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#15
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.
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.
 
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#16
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.
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.

Depth

Elements

Reduced(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
 
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abunickabhi
#17
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.

Depth

Elements

Reduced(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
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.
 
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#18
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.

Depth

Elements

Reduced(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
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
 
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