Random 20 move scramble

IQubic said:
So, hiw many ways to get Superflip? Would there be 43 Quintillion 20 move, because every scramble takes at least 20 moves to solve.
Click to expand...

It's incorrect to say that every scramble takes 20 moves to solve. But none require more than that (in face turn metric). Of course, the number of 20-move scrambles is more than 43 quintillion.

First, we could consider *arbitrary* 20-move sequences, such as what would be generated by IQubic's simple scramble algorithm in the initial post. The number of such sequences is 18^20 = 12748236216396078174437376. This is enough to generate each position about 294743 times, but of course there will be a wide variation in the number for individual positions. But each position will be generated at least once. I think the number of these sequences producing superflip would be about 2*48*2*20*(2^3) = 30720. This is about 1 in 400 quintillion.

Second, we can consider "canonical" sequences only where we don't allow trivial redundancies like R R2 or L R L, and commuting moves are always in a specific order (as in U D but not D U). A recurrence formula gives the number of such 20-move canonical sequences to be 43946585901564160587264. This is enough to generate each position about 1016 times, but again the count for individual positions will vary. I think the number of times superflip would be generated by canonical sequences would be about 2*48*2*(20+3) = 4416. This is about 1 in 10 quintillion.

We would need to determine exactly the number of superflip maneuvers to get exact answers, but someone correct me if you think I've made any gross error in my estimates.

And I would be very surprised if Mike Reid's assertion would turn out to be incorrect about there being only 2 such optimal maneuvers, up to cyclic shifting, inversion, and conjugation by symmetries of the cube (and ordering of commuting moves, I might add). I note he also claims essentially only 4 optimal maneuvers in QTM. (link)
 
Sorry if I am speaking out of ignorance, but what is the simple proof that picking up exactly 20 random moves in HTM can give you any reachable Rubik's Cube configuration?

Of course it's not true in QTM because half of all the configurations is at even distance, and other half is at odd distance. So if you do, say, 26 random QTM moves, you cannot get position solvable in 25 QTM moves.

Edit: And what if only "canonical" sequences are allowed? Can any configuration be reached in this case?

- Bulat
 
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AustinReed said:
Humans are never random. That's not going to be a "random" scramble.

Plus, what if you pick 16, then 18?
Click to expand...

Very true, that was one of the reason the Brits broke the German Enigma code , the key was generated by the operator....
To generate a real random sequence is not that simple as it sounds.... even the basic random generators are not random enough
 
stannic said:
Sorry if I am speaking out of ignorance, but what is the simple proof that picking up exactly 20 random moves in HTM can give you any reachable Rubik's Cube configuration?

Of course it's not true in QTM because half of all the configurations is at even distance, and other half is at odd distance. So if you do, say, 26 random QTM moves, you cannot get position solvable in 25 QTM moves.

Edit: And what if only "canonical" sequences are allowed? Can any configuration be reached in this case?

- Bulat
Click to expand...

If a position is 20f*, it obviously can be generated by a 20f maneuver. If a position is less than 20f*, you can expand it using trivial identies (e.g. U = U' U2) to get to 20 moves. And no position is more than 20f*. I made no claim that the same goes for canonical maneuvers, or even if just consecutive moves of the same layer are disallowed. And, of course, it's definitely not the case for QTM.
 
cuBerBruce said:
If a position is 20f*, it obviously can be generated by a 20f maneuver. If a position is less than 20f*, you can expand it using trivial identies (e.g. U = U' U2) to get to 20 moves. And no position is more than 20f*. I made no claim that the same goes for canonical maneuvers
Click to expand...
I would be quite interested in seeing a 19f* (or less? 19 seems the most plausible) position with no 20f solutions. I wonder if there's any way of trying to search for this other than by generating random scrambles and exhaustively checking all 20-move solutions.

IQubic said:
But, CubeTimer give hard enough scrambles for me to solve.
Click to expand...
Sure they might be "hard enough", but it's still true that the hardest scrambles are less likely, so your times might be better than they should be. It really starts to matter when you can build blocks and plan two steps ahead in inspection, so it makes sense that you wouldn't be at this level yet.
 
cuBerBruce said:
Second, we can consider "canonical" sequences only where we don't allow trivial redundancies like R R2 or L R L, and commuting moves are always in a specific order (as in U D but not D U). A recurrence formula gives the number of such 20-move canonical sequences to be 43946585901564160587264. This is enough to generate each position about 1016 times, but again the count for individual positions will vary. I think the number of times superflip would be generated by canonical sequences would be about 2*48*2*(20+3) = 4416. This is about 1 in 10 quintillion.

We would need to determine exactly the number of superflip maneuvers to get exact answers, but someone correct me if you think I've made any gross error in my estimates.
Click to expand...

I used my Cube Explorer program to generate all essential different maneuvers for superflip. Cube Explorer reduces the search space by the symmetries of the position, but not really strict. So I had to remove some of the generated positions manually. I hope I did nothing wrong here. What remains is a list of 108 maneuvers, lexicographically sorted in the order U,R,F,D,L,B. Each of these maneuvers can be expanded to 48 different maneuvers by applying the 48 symmetries. So there should be 108*48 =5184 different canonical sequences which give superflip, considerably more than the 4416 of your estimation.

U R U2 R F2 L U2 R F' B' R2 D R' L U2 F2 D2 F R2 D (20f*)
U R U2 R F2 L U2 R F' B' R2 D B2 U2 F2 R' L F R2 D (20f*)
U R D2 R B2 L D2 R F' B' R2 U R' L D2 B2 U2 B R2 D (20f*)
U R D2 R B2 L D2 R F' B' R2 U F2 D2 B2 R' L B R2 D (20f*)
U R D2 F B D B2 D R2 U B2 D R' L' D2 F L2 B2 R2 D' (20f*)
U R D2 F' B' D L2 U F2 D L2 D R L D2 F R2 F2 L2 D' (20f*)
U R L U2 F U' D F2 R2 B2 L U2 F' B' U R2 D F2 U R2 (20f*)
U R L U2 F L2 F2 R2 U' D L U2 F' B' U R2 D F2 U R2 (20f*)
U R L U2 B U' D B2 L2 F2 R U2 F' B' U L2 D B2 U L2 (20f*)
U R L U2 B R2 B2 L2 U' D R U2 F' B' U L2 D B2 U L2 (20f*)
U R L' U2 B2 D2 F L2 U' D' L B2 R U2 L B2 L F B L2 (20f*)
U R L' D2 F2 U2 F L2 U D L D2 L F2 R D2 L F' B' L2 (20f*)
U R2 U R L U2 F U' D F2 R2 B2 L U2 F' B' U R2 D F2 (20f*)
U R2 U R L U2 F L2 F2 R2 U' D L U2 F' B' U R2 D F2 (20f*)
U R2 U B2 D R2 U F' B' U2 L U' D R2 B2 L2 B U2 R L (20f*)
U R2 U B2 D R2 U F' B' U2 L F2 R2 B2 U' D B U2 R L (20f*)
U R2 U2 L2 F' B R F2 U' D' F L2 B U2 F L2 F R L F2 (20f*)
U R2 F U2 F2 D2 R' L U R2 F' B' R D2 L F2 R D2 R D (20f*)
U R2 F R' L F2 D2 B2 U R2 F' B' R D2 L F2 R D2 R D (20f*)
U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2 (20f*)
U R2 F B R B2 R U2 L B2 R U' D' R2 F D2 B2 U2 R' L (20f*)
U R2 F2 L2 F D2 R L D R2 D F2 U R2 D F' B' D2 L D' (20f*)
U R2 F2 L2 F' U2 R' L' U' R2 U' F2 D' R2 U' F B U2 L' D' (20f*)
U R2 F2 L2 B D2 R' L' D F2 U R2 D F2 D F B D2 R D' (20f*)
U R2 F2 L2 B' U2 R L U' F2 D' R2 U' F2 U' F' B' U2 R' D' (20f*)
U R2 F' U2 B2 D2 R L' D' R2 F' B' R' B2 R' D2 L' B2 R' D (20f*)
U R2 F' R L' B2 D2 F2 D' R2 F' B' R' B2 R' D2 L' B2 R' D (20f*)
U R2 F' B' R D2 L F2 R D2 R U D R2 F U2 F2 D2 R' L (20f*)
U R2 F' B' R D2 L F2 R D2 R U D R2 F R' L F2 D2 B2 (20f*)
U R2 D F2 U R2 U R L U2 F U' D F2 R2 B2 L U2 F' B' (20f*)
U R2 D F2 U R2 U R L U2 F L2 F2 R2 U' D L U2 F' B' (20f*)
U R2 D F2 U R' L' U2 B U' D F2 R2 B2 R U2 F B U F2 (20f*)
U R2 D F2 U R' L' U2 B L2 F2 R2 U' D R U2 F B U F2 (20f*)
U R2 D F2 D F B D2 R U D' R2 F2 L2 B D2 R' L' D F2 (20f*)
U R2 D F2 D F B D2 R B2 R2 F2 U D' B D2 R' L' D F2 (20f*)
U R2 D F' B' D2 L U D' R2 F2 L2 F D2 R L D R2 D F2 (20f*)
U R2 D F' B' D2 L B2 R2 F2 U D' F D2 R L D R2 D F2 (20f*)
U R2 D2 L2 F B' L B2 U D B D2 B L2 F D2 B R' L' B2 (20f*)
U R2 B R' L B2 U2 F2 D R2 F' B' R U2 L B2 R U2 R D (20f*)
U R2 B D2 B2 U2 R' L D R2 F' B' R U2 L B2 R U2 R D (20f*)
U R2 B' R L' F2 U2 B2 U' R2 F' B' R' F2 R' U2 L' F2 R' D (20f*)
U R2 B' D2 F2 U2 R L' U' R2 F' B' R' F2 R' U2 L' F2 R' D (20f*)
U R' U2 F B U' L2 D' F2 U' L2 U' R' L' U2 F' R2 F2 L2 D' (20f*)
U R' U2 F' B' U' B2 U' R2 D' B2 U' R L U2 F' L2 B2 R2 D' (20f*)
U R' F2 L' D2 R' F2 R' F' B' R2 D' R L' D2 F2 U2 B' R2 D (20f*)
U R' F2 L' D2 R' F2 R' F' B' R2 D' B2 D2 F2 R L' B' R2 D (20f*)
U R' L U2 F2 D2 B R2 U' D' R F2 L U2 R F2 R F B R2 (20f*)
U R' L D2 B2 U2 B R2 U D R D2 R B2 L D2 R F' B' R2 (20f*)
U R' L' U2 F U' D B2 L2 F2 L U2 F B U B2 U L2 D B2 (20f*)
U R' L' U2 F R2 B2 L2 U' D L U2 F B U B2 U L2 D B2 (20f*)
U R' L' U2 B U' D F2 R2 B2 R U2 F B U F2 U R2 D F2 (20f*)
U R' L' U2 B L2 F2 R2 U' D R U2 F B U F2 U R2 D F2 (20f*)
U R' B2 L' U2 R' B2 R' F' B' R2 U' R L' U2 B2 D2 F' R2 D (20f*)
U R' B2 L' U2 R' B2 R' F' B' R2 U' F2 U2 B2 R L' F' R2 D (20f*)
U D R U2 R F2 L U2 R F' B' R2 D R' L U2 F2 D2 F R2 (20f*)
U D R U2 R F2 L U2 R F' B' R2 D B2 U2 F2 R' L F R2 (20f*)
U D R2 F U2 F2 D2 R' L U R2 F' B' R D2 L F2 R D2 R (20f*)
U D R2 F R' L F2 D2 B2 U R2 F' B' R D2 L F2 R D2 R (20f*)
U D R2 F' U2 B2 D2 R L' D' R2 F' B' R' B2 R' D2 L' B2 R' (20f*)
U D R2 F' R L' B2 D2 F2 D' R2 F' B' R' B2 R' D2 L' B2 R' (20f*)
U D R' F2 L' D2 R' F2 R' F' B' R2 D' R L' D2 F2 U2 B' R2 (20f*)
U D R' F2 L' D2 R' F2 R' F' B' R2 D' B2 D2 F2 R L' B' R2 (20f*)
U D R' B2 L' U2 R' B2 R' F' B' R2 U' R L' U2 B2 D2 F' R2 (20f*)
U D R' B2 L' U2 R' B2 R' F' B' R2 U' F2 U2 B2 R L' F' R2 (20f*)
U D' R D2 F B D B2 D R2 U B2 D R' L' D2 F L2 B2 R2 (20f*)
U D' R D2 F' B' D L2 U F2 D L2 D R L D2 F R2 F2 L2 (20f*)
U D' R2 F2 L2 F D2 R L D R2 D F2 U R2 D F' B' D2 L (20f*)
U D' R2 F2 L2 B D2 R' L' D F2 U R2 D F2 D F B D2 R (20f*)
U2 R U D R2 B R' L B2 U2 F2 D R2 F' B' R U2 L B2 R (20f*)
U2 R U D R2 B D2 B2 U2 R' L D R2 F' B' R U2 L B2 R (20f*)
U2 R U' D R2 B2 L2 F U2 R' L' U B2 D R2 U B2 U F B (20f*)
U2 R U' D L2 F2 R2 F U2 R L U L2 U F2 D L2 U F' B' (20f*)
U2 R F2 U D F D2 F R2 B D2 F R' L' F2 U F' B D2 R2 (20f*)
U2 R F2 R F B R2 U R' L U2 F2 D2 B R2 U' D' R F2 L (20f*)
U2 R F2 R F B R2 U B2 U2 F2 R' L B R2 U' D' R F2 L (20f*)
U2 R F2 R2 B2 U' D F U2 R' L' U B2 D R2 U B2 U F B (20f*)
U2 R F2 L U2 R F' B' R2 D R' L U2 F2 D2 F R2 U D R (20f*)
U2 R F2 L U2 R F' B' R2 D B2 U2 F2 R' L F R2 U D R (20f*)
U2 R F2 L U2 L U D L2 F R L' F2 U2 B2 D L2 F' B' L (20f*)
U2 R F2 L U2 L U D L2 F D2 F2 U2 R L' D L2 F' B' L (20f*)
U2 R F2 L U' D' L2 B R L' F2 U2 B2 U L2 F B L F2 L (20f*)
U2 R F2 L U' D' L2 B D2 F2 U2 R L' U L2 F B L F2 L (20f*)
U2 R F' B' R2 D R' L U2 F2 D2 F R2 U D R U2 R F2 L (20f*)
U2 R F' B' R2 D B2 U2 F2 R' L F R2 U D R U2 R F2 L (20f*)
U2 R L U R2 U B2 D R2 U F' B' U2 L U' D R2 B2 L2 B (20f*)
U2 R L U R2 U B2 D R2 U F' B' U2 L F2 R2 B2 U' D B (20f*)
U2 R L U L2 U F2 D L2 U F' B' U2 R U' D L2 F2 R2 F (20f*)
U2 R L U L2 U F2 D L2 U F' B' U2 R B2 L2 F2 U' D F (20f*)
U2 R L U' F2 D' R2 U' F2 U' F' B' U2 R' U D' R2 F2 L2 B' (20f*)
U2 R L U' F2 D' R2 U' F2 U' F' B' U2 R' B2 R2 F2 U D' B' (20f*)
U2 R L U' B2 D' L2 U' B2 U' F' B' U2 L' U D' L2 B2 R2 F' (20f*)
U2 R L U' B2 D' L2 U' B2 U' F' B' U2 L' F2 L2 B2 U D' F' (20f*)
U2 R L' U L2 F B L F2 L U2 R F2 L U' D' L2 B D2 F2 (20f*)
U2 R L' U' R2 F' B' R' F2 R' U2 L' F2 R' U D R2 B' D2 F2 (20f*)
U2 R L' D L2 F' B' L U2 R F2 L U2 L U D L2 F D2 F2 (20f*)
U2 R L' D' R2 F B R' U2 L' F2 R' U2 R' U' D' R2 F' D2 F2 (20f*)
U2 R B2 U' D' B L2 F D2 B L2 B R L B2 D F B' D2 L2 (20f*)
U2 R B2 L2 F2 U' D F U2 R L U L2 U F2 D L2 U F' B' (20f*)
U2 R2 U B2 R L B L2 B U2 F L2 B U' D' B2 R F B' L2 (20f*)
U2 R2 F B' R B2 U D B U2 B R2 F U2 B R' L' B2 D L2 (20f*)
U2 R2 F B' R' F2 U' D' F' U2 F' R2 B' U2 F' R L F2 D' L2 (20f*)
U2 R2 F B' L B2 U' D' B R2 F U2 B R2 B R L B2 U L2 (20f*)
U2 R2 F B' L' F2 U D F' R2 B' U2 F' R2 F' R' L' F2 U' L2 (20f*)
U2 R2 D B2 R' L' B U2 F L2 B U2 B U D B2 L F B' L2 (20f*)
U2 R2 D2 R B2 U D B U2 B R2 F U2 B R' L' B2 D F B' (20f*)
U2 R2 D2 F B' U B2 R' L' B D2 F R2 B D2 B U D B2 R (20f*)
U2 R2 D2 F B' D B2 R L B R2 B D2 F R2 B U' D' B2 L (20f*)
U2 R2 D2 L B2 U' D' B R2 F U2 B R2 B R L B2 U F B' (20f*)
 
Herbert Kociemba said:
So there should be 108*48 =5184 different canonical sequences which give superflip, considerably more than the 4416 of your estimation.
Click to expand...
I think Herbert's value of 5184 is an overcount. I think the sequences starting with U D and U D' may lead to duplicates (after canonicalization) when expanding by symmetries. It's not clear to me that accounts for the difference between my 4416 and Herbert's 5184, though. I will try to analyze this further.
 
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Every sequence Kociemba posted is equivalent to either U2 R2 D2 F B' D B2 L R B R2 B D2 F R2 B D' U' B2 L or U2 R2 D2 F B' U B2 L' R' B D2 F R2 B D2 B D U B2 R by move cycling, rotation, mirroring, inversion, and swapping two adjacent moves on the same axis. (Also, Reid's two sequences are equivalent to those two.)
 
I have generated the 5184 sequences generated by applying the 48 cube symmetries to the 108 sequences posted by Herbert. After canonicalizing these sequences, my code indicates there are 768 duplicates among these. 5184 - 768 = 4416, so I am pretty confident that the 4416 canonical optimal maneuvers I estimated for superflip is in fact the correct number.
 
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cuBerBruce said:
I have generated the 5184 sequences generated by applying the 48 cube symmetries to the 108 sequences posted by Herbert. After canonicalizing these sequences, my code indicates there are 768 duplicates among these. 5184 - 768 = 4416, so I am pretty confident that the 4416 canonical optimal maneuvers I estimated for superflip is in fact the correct number.
Click to expand...
It would be very helpful for me to understand where my reasoning went wrong if you name a few duplicates.

Edit1: Ok, I found at least one error, indeed concerning maneuvers starting with UD.

U D R' F2 L' D2 R' F2 R' F' B' R2 D' R L' D2 F2 U2 B' R2 (20f*)
and
U D R' B2 L' U2 R' B2 R' F' B' R2 U' R L' U2 B2 D2 F' R2 (20f*)
are for example equivalent by CR2 rotation.
Will have to look where the other 15 duplicates are...

Edit2: I really took a closer look, especially at the sequences starting U D.. and U D' .

I found another pair
U D R' F2 L' D2 R' F2 R' F' B' R2 D' B2 D2 F2 R L' B' R2 (20f*)
and
U D R' B2 L' U2 R' B2 R' F' B' R2 U' F2 U2 B2 R L' F' R2 (20f*)
also related by CR2
but still 14 duplicates are missing. Bruce maybe you can give me a hint.
 
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Herbert Kociemba said:
It would be very helpful for me to understand where my reasoning went wrong if you name a few duplicates.
Click to expand...
OK, I get the following as the first duplicate pair:

Line 7: U R L U2 F U' D F2 R2 B2 L U2 F' B' U R2 D F2 U R2 (20f*)
Line 9: U R L U2 B U' D B2 L2 F2 R U2 F' B' U L2 D B2 U L2 (20f*)

The full list of duplicate pairs are the following:
(7, 9)
(8, 10)
(11, 47)
(12, 48)
(49, 51)
(50, 52)
(61, 63)
(62, 64)
(85, 87)
(86, 88)
(89, 91)
(90, 92)
(93, 94)
(95, 96)
(100, 101)
(102, 103)
 
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cuBerBruce said:
OK, I get the following as the first duplicate pair:

Line 7: U R L U2 F U' D F2 R2 B2 L U2 F' B' U R2 D F2 U R2 (20f*)
Line 9: U R L U2 B U' D B2 L2 F2 R U2 F' B' U L2 D B2 U L2 (20f*)
Click to expand...

You are totally right. I forgot lots of situations where two moves like RL commute.
I just thought for example that two different maneuvers which both start with U R cannot be equivalent because U R itself has no symmetry. Which is not true.
Thanks for the correction.
The following list of 92 maneuvers should be ok now and should expand indeed to 92*48 = 4416 different maneuvers. Maybe you can run your program again over this set to doublecheck the result.

U R U2 R F2 L U2 R F' B' R2 D R' L U2 F2 D2 F R2 D (20f*)
U R U2 R F2 L U2 R F' B' R2 D B2 U2 F2 R' L F R2 D (20f*)
U R D2 R B2 L D2 R F' B' R2 U R' L D2 B2 U2 B R2 D (20f*)
U R D2 R B2 L D2 R F' B' R2 U F2 D2 B2 R' L B R2 D (20f*)
U R D2 F B D B2 D R2 U B2 D R' L' D2 F L2 B2 R2 D' (20f*)
U R D2 F' B' D L2 U F2 D L2 D R L D2 F R2 F2 L2 D' (20f*)
U R L U2 F U' D F2 R2 B2 L U2 F' B' U R2 D F2 U R2 (20f*)
U R L U2 F L2 F2 R2 U' D L U2 F' B' U R2 D F2 U R2 (20f*)
U R L' U2 B2 D2 F L2 U' D' L B2 R U2 L B2 L F B L2 (20f*)
U R L' D2 F2 U2 F L2 U D L D2 L F2 R D2 L F' B' L2 (20f*)
U R2 U R L U2 F U' D F2 R2 B2 L U2 F' B' U R2 D F2 (20f*)
U R2 U R L U2 F L2 F2 R2 U' D L U2 F' B' U R2 D F2 (20f*)
U R2 U B2 D R2 U F' B' U2 L U' D R2 B2 L2 B U2 R L (20f*)
U R2 U B2 D R2 U F' B' U2 L F2 R2 B2 U' D B U2 R L (20f*)
U R2 U2 L2 F' B R F2 U' D' F L2 B U2 F L2 F R L F2 (20f*)
U R2 F U2 F2 D2 R' L U R2 F' B' R D2 L F2 R D2 R D (20f*)
U R2 F R' L F2 D2 B2 U R2 F' B' R D2 L F2 R D2 R D (20f*)
U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2 (20f*)
U R2 F B R B2 R U2 L B2 R U' D' R2 F D2 B2 U2 R' L (20f*)
U R2 F2 L2 F D2 R L D R2 D F2 U R2 D F' B' D2 L D' (20f*)
U R2 F2 L2 F' U2 R' L' U' R2 U' F2 D' R2 U' F B U2 L' D' (20f*)
U R2 F2 L2 B D2 R' L' D F2 U R2 D F2 D F B D2 R D' (20f*)
U R2 F2 L2 B' U2 R L U' F2 D' R2 U' F2 U' F' B' U2 R' D' (20f*)
U R2 F' U2 B2 D2 R L' D' R2 F' B' R' B2 R' D2 L' B2 R' D (20f*)
U R2 F' R L' B2 D2 F2 D' R2 F' B' R' B2 R' D2 L' B2 R' D (20f*)
U R2 F' B' R D2 L F2 R D2 R U D R2 F U2 F2 D2 R' L (20f*)
U R2 F' B' R D2 L F2 R D2 R U D R2 F R' L F2 D2 B2 (20f*)
U R2 D F2 U R2 U R L U2 F U' D F2 R2 B2 L U2 F' B' (20f*)
U R2 D F2 U R2 U R L U2 F L2 F2 R2 U' D L U2 F' B' (20f*)
U R2 D F2 U R' L' U2 B U' D F2 R2 B2 R U2 F B U F2 (20f*)
U R2 D F2 U R' L' U2 B L2 F2 R2 U' D R U2 F B U F2 (20f*)
U R2 D F2 D F B D2 R U D' R2 F2 L2 B D2 R' L' D F2 (20f*)
U R2 D F2 D F B D2 R B2 R2 F2 U D' B D2 R' L' D F2 (20f*)
U R2 D F' B' D2 L U D' R2 F2 L2 F D2 R L D R2 D F2 (20f*)
U R2 D F' B' D2 L B2 R2 F2 U D' F D2 R L D R2 D F2 (20f*)
U R2 D2 L2 F B' L B2 U D B D2 B L2 F D2 B R' L' B2 (20f*)
U R2 B R' L B2 U2 F2 D R2 F' B' R U2 L B2 R U2 R D (20f*)
U R2 B D2 B2 U2 R' L D R2 F' B' R U2 L B2 R U2 R D (20f*)
U R2 B' R L' F2 U2 B2 U' R2 F' B' R' F2 R' U2 L' F2 R' D (20f*)
U R2 B' D2 F2 U2 R L' U' R2 F' B' R' F2 R' U2 L' F2 R' D (20f*)
U R' U2 F B U' L2 D' F2 U' L2 U' R' L' U2 F' R2 F2 L2 D' (20f*)
U R' U2 F' B' U' B2 U' R2 D' B2 U' R L U2 F' L2 B2 R2 D' (20f*)
U R' F2 L' D2 R' F2 R' F' B' R2 D' R L' D2 F2 U2 B' R2 D (20f*)
U R' F2 L' D2 R' F2 R' F' B' R2 D' B2 D2 F2 R L' B' R2 D (20f*)
U R' L' U2 F U' D B2 L2 F2 L U2 F B U B2 U L2 D B2 (20f*)
U R' L' U2 F R2 B2 L2 U' D L U2 F B U B2 U L2 D B2 (20f*)
U R' B2 L' U2 R' B2 R' F' B' R2 U' R L' U2 B2 D2 F' R2 D (20f*)
U R' B2 L' U2 R' B2 R' F' B' R2 U' F2 U2 B2 R L' F' R2 D (20f*)
U D R U2 R F2 L U2 R F' B' R2 D R' L U2 F2 D2 F R2 (20f*)
U D R U2 R F2 L U2 R F' B' R2 D B2 U2 F2 R' L F R2 (20f*)
U D R2 F U2 F2 D2 R' L U R2 F' B' R D2 L F2 R D2 R (20f*)
U D R2 F R' L F2 D2 B2 U R2 F' B' R D2 L F2 R D2 R (20f*)
U D R2 F' U2 B2 D2 R L' D' R2 F' B' R' B2 R' D2 L' B2 R' (20f*)
U D R2 F' R L' B2 D2 F2 D' R2 F' B' R' B2 R' D2 L' B2 R' (20f*)
U D R' F2 L' D2 R' F2 R' F' B' R2 D' R L' D2 F2 U2 B' R2 (20f*)
U D R' F2 L' D2 R' F2 R' F' B' R2 D' B2 D2 F2 R L' B' R2 (20f*)
U D' R D2 F B D B2 D R2 U B2 D R' L' D2 F L2 B2 R2 (20f*)
U D' R D2 F' B' D L2 U F2 D L2 D R L D2 F R2 F2 L2 (20f*)
U D' R2 F2 L2 F D2 R L D R2 D F2 U R2 D F' B' D2 L (20f*)
U D' R2 F2 L2 B D2 R' L' D F2 U R2 D F2 D F B D2 R (20f*)
U2 R U D R2 B R' L B2 U2 F2 D R2 F' B' R U2 L B2 R (20f*)
U2 R U D R2 B D2 B2 U2 R' L D R2 F' B' R U2 L B2 R (20f*)
U2 R U' D R2 B2 L2 F U2 R' L' U B2 D R2 U B2 U F B (20f*)
U2 R U' D L2 F2 R2 F U2 R L U L2 U F2 D L2 U F' B' (20f*)
U2 R F2 U D F D2 F R2 B D2 F R' L' F2 U F' B D2 R2 (20f*)
U2 R F2 R F B R2 U R' L U2 F2 D2 B R2 U' D' R F2 L (20f*)
U2 R F2 R F B R2 U B2 U2 F2 R' L B R2 U' D' R F2 L (20f*)
U2 R F2 R2 B2 U' D F U2 R' L' U B2 D R2 U B2 U F B (20f*)
U2 R F2 L U2 R F' B' R2 D R' L U2 F2 D2 F R2 U D R (20f*)
U2 R F2 L U2 R F' B' R2 D B2 U2 F2 R' L F R2 U D R (20f*)
U2 R F2 L U2 L U D L2 F R L' F2 U2 B2 D L2 F' B' L (20f*)
U2 R F2 L U2 L U D L2 F D2 F2 U2 R L' D L2 F' B' L (20f*)
U2 R F2 L U' D' L2 B R L' F2 U2 B2 U L2 F B L F2 L (20f*)
U2 R F2 L U' D' L2 B D2 F2 U2 R L' U L2 F B L F2 L (20f*)
U2 R F' B' R2 D R' L U2 F2 D2 F R2 U D R U2 R F2 L (20f*)
U2 R F' B' R2 D B2 U2 F2 R' L F R2 U D R U2 R F2 L (20f*)
U2 R L U R2 U B2 D R2 U F' B' U2 L U' D R2 B2 L2 B (20f*)
U2 R L U R2 U B2 D R2 U F' B' U2 L F2 R2 B2 U' D B (20f*)
U2 R L U' F2 D' R2 U' F2 U' F' B' U2 R' U D' R2 F2 L2 B' (20f*)
U2 R L U' F2 D' R2 U' F2 U' F' B' U2 R' B2 R2 F2 U D' B' (20f*)
U2 R L' U L2 F B L F2 L U2 R F2 L U' D' L2 B D2 F2 (20f*)
U2 R L' D L2 F' B' L U2 R F2 L U2 L U D L2 F D2 F2 (20f*)
U2 R B2 U' D' B L2 F D2 B L2 B R L B2 D F B' D2 L2 (20f*)
U2 R B2 L2 F2 U' D F U2 R L U L2 U F2 D L2 U F' B' (20f*)
U2 R2 U B2 R L B L2 B U2 F L2 B U' D' B2 R F B' L2 (20f*)
U2 R2 F B' R B2 U D B U2 B R2 F U2 B R' L' B2 D L2 (20f*)
U2 R2 F B' L B2 U' D' B R2 F U2 B R2 B R L B2 U L2 (20f*)
U2 R2 D B2 R' L' B U2 F L2 B U2 B U D B2 L F B' L2 (20f*)
U2 R2 D2 R B2 U D B U2 B R2 F U2 B R' L' B2 D F B' (20f*)
U2 R2 D2 F B' U B2 R' L' B D2 F R2 B D2 B U D B2 R (20f*)
U2 R2 D2 F B' D B2 R L B R2 B D2 F R2 B U' D' B2 L (20f*)
U2 R2 D2 L B2 U' D' B R2 F U2 B R2 B R L B2 U F B' (20f*)
 
Herbert Kociemba said:
Maybe you can run your program again over this set to doublecheck the result.
Click to expand...

Done. With the new list, I do not find any duplicates, 4416 total after expanding by symmetries.

I think many scramble programs may generate adjacent commuting moves in either order, as in U D2 or D2 U, but never the same layer twice before switching axis. This would result with yet a different ratio for superflip maneuvers vs. total 20-moves scrambles.
 
Good point. Random-move scramblers usually don't regularize their output, they just choose possible moves one at a time, so U D2 and D2 U are both possible.

It's probably reasonable to conclude that the superflip is by far the most likely 20f* position to pop up if you do 20 random moves on the cube.
 
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