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 |
|---|
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
|
[TD='top']
6
[/TD]
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
