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I realized the FB+1 stage in Roux, although parallel to XCross in difficulty, hasn't yet been studied carefully, so I programmed a solver to investigate its move count, along with some other interesting stats including FB square, LSE subsets and more! I have summarized my findings in this sheet.

Here's a preview of my FB+1 findings :

Code:

FB+1 fixed = ~7.64 moves
FB+1 on y = ~6.93 moves
FB+1 on x2y = ~6.59 moves
FB+1 on CN = ~6.14 moves.
For comparison, XCross on fixed cross is 7.35 moves, opposite is 6.99 and CN is 6.54.
(XCross data from: https://www.speedsolving.com/threads/2x2x2-x-cross-and-3x2x2-move-count.12403/#post-185213 )

Some numbers seem suspect. For example, 5.14 moves for first block (optimal), then 6.14 for FB+1. That might actually be the correct numbers, but you would get a different number if you started with a million scrambles that have first block already solved, and see how many moves it takes to solve the extra edge. I think the computer in this case is preferentially selecting FB solves where the next edge is already solved by luck, which is not possible for a human. So if someone wanted to know the average number of moves to solve the extra edge (after FB), the number is not 1 (even intuitively it can't be an average of 1 move).

Some numbers seem suspect. For example, 5.14 moves for first block (optimal), then 6.14 for FB+1. That might actually be the correct numbers, but you would get a different number if you started with a million scrambles that have first block already solved, and see how many moves it takes to solve the extra edge. I think the computer in this case is preferentially selecting FB solves where the next edge is already solved by luck, which is not possible for a human. So if someone wanted to know the average number of moves to solve the extra edge (after FB), the number is not 1 (even intuitively it can't be an average of 1 move).

Some numbers seem suspect. For example, 5.14 moves for first block (optimal), then 6.14 for FB+1. That might actually be the correct numbers, but you would get a different number if you started with a million scrambles that have first block already solved, and see how many moves it takes to solve the extra edge. I think the computer in this case is preferentially selecting FB solves where the next edge is already solved by luck, which is not possible for a human. So if someone wanted to know the average number of moves to solve the extra edge (after FB), the number is not 1 (even intuitively it can't be an average of 1 move).

Influencing DR is very important EDIT: Like the same could be said about the xcross. Solving cross than first pair is almost always going to be worse than solving it directly.

Yup you both have a point here. Starting from nothing to FB+DR might involve DR influencing that's beyond human comprehension. But solving DR with no influencing at all doesn't reflect reality either.
Solution? I think another stats from FS to FB+DR might help here, it also closely mimics a Roux speedcuber's thought process (tracks where DR is after FS is a common strategy). I might add that later.

Great data, but I think there is something amiss in your program, or perhaps just in the way I'm reading it.

For FB+DR edge (fixed orientation), you give an upper bound of 9 moves. HARCS gives an upper bound of ~10 HTM - here is the distribution:

Here is an example scramble where it takes 10 moves to solve the Left Block and the DR edge (fixed orientation):

Code:

B2 D U' R2 U' B F D L' U2

Both HARCS and CubeExplorer give a 10-move optimal solution:

Spoiler: HARCS

first 3 solutions:

Code:

UF - fb:
1. U2 L D' F' B' U R2 U D' B2 (10)
2. U2 F' U D B D2 B2 D R D2 (10)
3. U2 F' U B' D R U2 B2 D2 B (10)
3 solution(s) generated in 1221 ms

Spoiler: CubeExplorer

all solutions:

Code:

U2 F' U D B D2 B2 D R D2 (10f*)
U2 F' U B' D R U2 B2 D2 B (10f*)
U2 L D' F' B' U R2 U D' B2 (10f*)
U2 B2 L D2 F' U2 R B' D' B' (10f*)
U' F2 R' B2 R B' U' D' L' F' (10f*)
U' F2 B U D L' B' R2 U' F2 (10f*)
U' L2 B' D' L2 D2 F R B' D2 (10f*)
U' B' U F D R' B' R2 D2 F2 (10f*)
U' B' U D R' B' R2 U D2 F2 (10f*)
R D L2 B' U' B2 R U' L2 B (10f*)
R B R L2 F' U' L2 B R2 F (10f*)
R B2 R D2 F U R' U2 B D (10f*)
R2 F B U' B R' D2 R2 F D' (10f*)
R2 F2 U F' U' B2 R' B' L2 D (10f*)
R2 F2 U2 R' B2 D2 F' U B D (10f*)
R2 F2 B2 U F' U' R' B' L2 D (10f*)
R2 F2 B2 R' F' B' L' U L' D (10f*)
R2 D2 L F2 U2 D2 F' U B D (10f*)
R2 B U' F B R' D2 R2 F D' (10f*)
R2 B2 U F R' D2 R2 L F D' (10f*)
R2 B2 D2 L F U R' U2 B D (10f*)
R' U2 F' U2 R B2 D2 B D' B' (10f*)
R' F R' B2 U' D2 L2 F D B (10f*)
D R2 F R' L U2 L2 B' D F' (10f*)
D L2 F2 B2 D2 R' F R B D' (10f*)
D L2 F2 B2 D2 R' B U F D' (10f*)
D L2 F2 B2 D2 B R' F R D' (10f*)
D B D2 L2 F' R' F R B2 D' (10f*)
D B D2 L2 F' B2 R' F R D' (10f*)
D B' U2 R F2 D2 B' L' B D (10f*)
D2 B U' D' R' F' L2 B' R D2 (10f*)
D2 B U' D' F' R' L2 B' R D2 (10f*)
D' R2 F B2 R' D2 R2 L2 F D' (10f*)
D' F' R2 F B2 R' D2 R2 F D' (10f*)
D' L B R L2 D2 B2 D2 B D (10f*)
D' L B R' U2 F2 U2 L2 B D (10f*)
D' L2 F B U L B' R2 F' L (10f*)
D' L2 F2 B R' U2 F2 L2 B D (10f*)
D' L2 F2 B' R2 F2 R' D2 B' D (10f*)
D' L2 B R2 L2 B2 R' D2 B' D (10f*)
D' L2 B2 D2 R' F R B D' B2 (10f*)
D' L2 B2 D2 R' B U F D' B2 (10f*)
D' L' B' R' U2 L' U2 F L2 D' (10f*)
D' B D2 F' U' F' L2 B' R D2 (10f*)
D' B L2 F B2 R F' D2 B' D (10f*)
D' B L2 F B2 R B' L2 F' D (10f*)
D' B L2 B2 R U' D2 B' R2 D (10f*)
D' B L2 B2 R D2 B' U' R2 D (10f*)
D' B2 D2 R' F R B U2 D' B2 (10f*)
D' B2 D2 R' B U F U2 D' B2 (10f*)
D' B2 D2 B R' F R U2 D' B2 (10f*)
L D' B R L2 D2 B2 D2 B D (10f*)
L D' B R' U2 F2 U2 L2 B D (10f*)
L B D' R L2 D2 B2 D2 B D (10f*)
L B D' R' U2 F2 U2 L2 B D (10f*)
L B2 U D2 F B D R' D2 B' (10f*)
L B2 D2 R' U F R B D' B' (10f*)
L B2 D2 R' U B U F D' B' (10f*)
L B2 D2 B' R' F R B2 D' B' (10f*)
L2 U2 B L D2 F' R B2 D' B' (10f*)
L2 F2 U2 L D F' B' U' R2 D' (10f*)
L2 F2 D B2 D2 R' F R B D' (10f*)
L2 F2 D B2 D2 R' B U F D' (10f*)
L2 F2 D B2 D2 B R' F R D' (10f*)
L2 F2 D' B L2 B2 R D2 B' D (10f*)
L2 F2 B U' D F' L2 B' R D2 (10f*)
L2 F2 B D' L2 B2 R D2 B' D (10f*)
L2 D' B L2 B2 R U D2 B' D (10f*)
L2 D' B L2 B2 R D2 B' R D (10f*)
L2 B U' F' L2 D B' D' R D2 (10f*)
L2 B U' D R2 F' L2 B' R D2 (10f*)
L2 B2 D2 R' U' F R B D' B' (10f*)
L2 B2 D2 R' U' B U F D' B' (10f*)
L2 B2 D2 R' F U' R B D' B' (10f*)
L' D B L2 B2 R D2 B' D B (10f*)
L' D B' R2 F2 R' D2 B' D B (10f*)
L' D' F B L' B2 R' F L' D' (10f*)
L' D' B L2 B2 R D2 B' R' D (10f*)
L' D' B L' B2 R' L F L2 D' (10f*)
L' B D' L2 B2 R U' D2 B' D (10f*)
L' B2 D2 R' F R U2 B D' B' (10f*)
L' B2 D' B2 D' B' R' F D' B' (10f*)
B U D' L2 D2 B' R2 L' F' L (10f*)
B U2 L' U' F' B L' D B' R2 (10f*)
B U2 L' D2 B R F2 D' B' D2 (10f*)
B U' F D F2 R L2 B' R D2 (10f*)
B U' D F' L2 B' R2 U R' D2 (10f*)
B U' D B' U2 F' D R D2 R (10f*)
B U' D' R' D2 F' L2 B' R D2 (10f*)
B U' D' F R2 U2 R2 L F' L (10f*)
B U' D' F R' L2 D2 L B L' (10f*)
B U' D' F L F2 R' L2 B L' (10f*)
B U' B' U2 B2 D R2 L' F' B' (10f*)
B R2 U' B R U' F D2 F D (10f*)
B R' U2 F' D' R' U F' D L2 (10f*)
B R' U2 D' F2 R L2 B D B' (10f*)
B R' U2 D' F' R' U F' D L2 (10f*)
B D R' U2 R2 F2 L2 B' R2 D (10f*)
B D R' U2 F2 L2 B U' D B2 (10f*)
B D F' U' F' R L2 B' R D2 (10f*)
B D F' R' U2 F2 L2 B D B2 (10f*)
B D2 R U' D2 F2 L2 D B' D (10f*)
B D2 R' U2 D' R2 F2 L2 B' D (10f*)
B D2 F' U' D' F' L2 B' R D2 (10f*)
B D2 F' D L2 B2 R D2 B' D (10f*)
B D2 L2 U D' B R2 L' F' L (10f*)
B D2 L2 F' R' B U F B D' (10f*)
B D2 L' F' B2 R2 L2 D' F L (10f*)
B D' R' U2 F2 U2 L2 B R' D (10f*)
B D' R' U2 D F2 L2 B D B' (10f*)
B D' R' F L' B2 L' U R' D' (10f*)
B D' R' L2 B2 R2 D2 B' R' D (10f*)
B D' F2 R' U2 F2 L2 B R' D (10f*)
B D' L2 B2 R U2 D2 B' R2 D (10f*)
B D' L' B2 L' D2 R B' R2 D (10f*)
B D' B2 R' U2 L' U2 F L2 D' (10f*)
B L U R2 F2 L2 U' F L B (10f*)
B L D B2 R2 F2 D' F L B (10f*)
B L2 D F2 U F' R2 L F' B' (10f*)
B L2 D' F R D' B2 L2 F' D' (10f*)
B L2 B2 R U' D2 B' R' D B' (10f*)
B L2 B2 R D2 B' U' R' D B' (10f*)
B L' B2 L' U' D R B' R D2 (10f*)
B2 R L2 F' B' R2 L' U L' D (10f*)
B2 D2 R' F R B D' B U2 B2 (10f*)
B2 D2 R' B U F D' B U2 B2 (10f*)
B2 D2 B R' F R D' B U2 B2 (10f*)
B' U2 R2 L' U2 F' L2 F' D' B' (10f*)
B' U2 L D2 F2 D' R2 F2 B' D (10f*)
B' R2 F2 R' U' D2 B' R' D B' (10f*)
B' R2 F2 R' D2 B' U' R' D B' (10f*)

Can you clarify? What does your code give for the above state/scramble?

Unless maybe you shave a move by picking a certain orientation beforehand - which I don't think Johannes' results (other thread) would do, because that wasn't how his JARCS software worked - that's very unusual. (Edit: I just ran that scramble in every orientation in CE, and I still get 10 moves.) It's probably because you are not solving with respect to centers, whereas the other solvers are. I wonder how many professional Roux users can predict the optimal orientation - or do they just choose an ergonomic orientation and come up with an optimal solution. It's an interesting difference.

So if someone wanted to know the average number of moves to solve the extra edge (after FB), the number is not 1 (even intuitively it can't be an average of 1 move).

Here is an example scramble where it takes 10 moves to solve the Left Block and the DR edge (fixed orientation):

Code:

B2 D U' R2 U' B F D L' U2

Both HARCS and CubeExplorer give a 10-move optimal solution:

Spoiler: HARCS

first 3 solutions:

Code:

UF - fb:
1. U2 L D' F' B' U R2 U D' B2 (10)
2. U2 F' U D B D2 B2 D R D2 (10)
3. U2 F' U B' D R U2 B2 D2 B (10)
3 solution(s) generated in 1221 ms

Spoiler: CubeExplorer

all solutions:

Code:

U2 F' U D B D2 B2 D R D2 (10f*)
U2 F' U B' D R U2 B2 D2 B (10f*)
U2 L D' F' B' U R2 U D' B2 (10f*)
U2 B2 L D2 F' U2 R B' D' B' (10f*)
U' F2 R' B2 R B' U' D' L' F' (10f*)
U' F2 B U D L' B' R2 U' F2 (10f*)
U' L2 B' D' L2 D2 F R B' D2 (10f*)
U' B' U F D R' B' R2 D2 F2 (10f*)
U' B' U D R' B' R2 U D2 F2 (10f*)
R D L2 B' U' B2 R U' L2 B (10f*)
R B R L2 F' U' L2 B R2 F (10f*)
R B2 R D2 F U R' U2 B D (10f*)
R2 F B U' B R' D2 R2 F D' (10f*)
R2 F2 U F' U' B2 R' B' L2 D (10f*)
R2 F2 U2 R' B2 D2 F' U B D (10f*)
R2 F2 B2 U F' U' R' B' L2 D (10f*)
R2 F2 B2 R' F' B' L' U L' D (10f*)
R2 D2 L F2 U2 D2 F' U B D (10f*)
R2 B U' F B R' D2 R2 F D' (10f*)
R2 B2 U F R' D2 R2 L F D' (10f*)
R2 B2 D2 L F U R' U2 B D (10f*)
R' U2 F' U2 R B2 D2 B D' B' (10f*)
R' F R' B2 U' D2 L2 F D B (10f*)
D R2 F R' L U2 L2 B' D F' (10f*)
D L2 F2 B2 D2 R' F R B D' (10f*)
D L2 F2 B2 D2 R' B U F D' (10f*)
D L2 F2 B2 D2 B R' F R D' (10f*)
D B D2 L2 F' R' F R B2 D' (10f*)
D B D2 L2 F' B2 R' F R D' (10f*)
D B' U2 R F2 D2 B' L' B D (10f*)
D2 B U' D' R' F' L2 B' R D2 (10f*)
D2 B U' D' F' R' L2 B' R D2 (10f*)
D' R2 F B2 R' D2 R2 L2 F D' (10f*)
D' F' R2 F B2 R' D2 R2 F D' (10f*)
D' L B R L2 D2 B2 D2 B D (10f*)
D' L B R' U2 F2 U2 L2 B D (10f*)
D' L2 F B U L B' R2 F' L (10f*)
D' L2 F2 B R' U2 F2 L2 B D (10f*)
D' L2 F2 B' R2 F2 R' D2 B' D (10f*)
D' L2 B R2 L2 B2 R' D2 B' D (10f*)
D' L2 B2 D2 R' F R B D' B2 (10f*)
D' L2 B2 D2 R' B U F D' B2 (10f*)
D' L' B' R' U2 L' U2 F L2 D' (10f*)
D' B D2 F' U' F' L2 B' R D2 (10f*)
D' B L2 F B2 R F' D2 B' D (10f*)
D' B L2 F B2 R B' L2 F' D (10f*)
D' B L2 B2 R U' D2 B' R2 D (10f*)
D' B L2 B2 R D2 B' U' R2 D (10f*)
D' B2 D2 R' F R B U2 D' B2 (10f*)
D' B2 D2 R' B U F U2 D' B2 (10f*)
D' B2 D2 B R' F R U2 D' B2 (10f*)
L D' B R L2 D2 B2 D2 B D (10f*)
L D' B R' U2 F2 U2 L2 B D (10f*)
L B D' R L2 D2 B2 D2 B D (10f*)
L B D' R' U2 F2 U2 L2 B D (10f*)
L B2 U D2 F B D R' D2 B' (10f*)
L B2 D2 R' U F R B D' B' (10f*)
L B2 D2 R' U B U F D' B' (10f*)
L B2 D2 B' R' F R B2 D' B' (10f*)
L2 U2 B L D2 F' R B2 D' B' (10f*)
L2 F2 U2 L D F' B' U' R2 D' (10f*)
L2 F2 D B2 D2 R' F R B D' (10f*)
L2 F2 D B2 D2 R' B U F D' (10f*)
L2 F2 D B2 D2 B R' F R D' (10f*)
L2 F2 D' B L2 B2 R D2 B' D (10f*)
L2 F2 B U' D F' L2 B' R D2 (10f*)
L2 F2 B D' L2 B2 R D2 B' D (10f*)
L2 D' B L2 B2 R U D2 B' D (10f*)
L2 D' B L2 B2 R D2 B' R D (10f*)
L2 B U' F' L2 D B' D' R D2 (10f*)
L2 B U' D R2 F' L2 B' R D2 (10f*)
L2 B2 D2 R' U' F R B D' B' (10f*)
L2 B2 D2 R' U' B U F D' B' (10f*)
L2 B2 D2 R' F U' R B D' B' (10f*)
L' D B L2 B2 R D2 B' D B (10f*)
L' D B' R2 F2 R' D2 B' D B (10f*)
L' D' F B L' B2 R' F L' D' (10f*)
L' D' B L2 B2 R D2 B' R' D (10f*)
L' D' B L' B2 R' L F L2 D' (10f*)
L' B D' L2 B2 R U' D2 B' D (10f*)
L' B2 D2 R' F R U2 B D' B' (10f*)
L' B2 D' B2 D' B' R' F D' B' (10f*)
B U D' L2 D2 B' R2 L' F' L (10f*)
B U2 L' U' F' B L' D B' R2 (10f*)
B U2 L' D2 B R F2 D' B' D2 (10f*)
B U' F D F2 R L2 B' R D2 (10f*)
B U' D F' L2 B' R2 U R' D2 (10f*)
B U' D B' U2 F' D R D2 R (10f*)
B U' D' R' D2 F' L2 B' R D2 (10f*)
B U' D' F R2 U2 R2 L F' L (10f*)
B U' D' F R' L2 D2 L B L' (10f*)
B U' D' F L F2 R' L2 B L' (10f*)
B U' B' U2 B2 D R2 L' F' B' (10f*)
B R2 U' B R U' F D2 F D (10f*)
B R' U2 F' D' R' U F' D L2 (10f*)
B R' U2 D' F2 R L2 B D B' (10f*)
B R' U2 D' F' R' U F' D L2 (10f*)
B D R' U2 R2 F2 L2 B' R2 D (10f*)
B D R' U2 F2 L2 B U' D B2 (10f*)
B D F' U' F' R L2 B' R D2 (10f*)
B D F' R' U2 F2 L2 B D B2 (10f*)
B D2 R U' D2 F2 L2 D B' D (10f*)
B D2 R' U2 D' R2 F2 L2 B' D (10f*)
B D2 F' U' D' F' L2 B' R D2 (10f*)
B D2 F' D L2 B2 R D2 B' D (10f*)
B D2 L2 U D' B R2 L' F' L (10f*)
B D2 L2 F' R' B U F B D' (10f*)
B D2 L' F' B2 R2 L2 D' F L (10f*)
B D' R' U2 F2 U2 L2 B R' D (10f*)
B D' R' U2 D F2 L2 B D B' (10f*)
B D' R' F L' B2 L' U R' D' (10f*)
B D' R' L2 B2 R2 D2 B' R' D (10f*)
B D' F2 R' U2 F2 L2 B R' D (10f*)
B D' L2 B2 R U2 D2 B' R2 D (10f*)
B D' L' B2 L' D2 R B' R2 D (10f*)
B D' B2 R' U2 L' U2 F L2 D' (10f*)
B L U R2 F2 L2 U' F L B (10f*)
B L D B2 R2 F2 D' F L B (10f*)
B L2 D F2 U F' R2 L F' B' (10f*)
B L2 D' F R D' B2 L2 F' D' (10f*)
B L2 B2 R U' D2 B' R' D B' (10f*)
B L2 B2 R D2 B' U' R' D B' (10f*)
B L' B2 L' U' D R B' R D2 (10f*)
B2 R L2 F' B' R2 L' U L' D (10f*)
B2 D2 R' F R B D' B U2 B2 (10f*)
B2 D2 R' B U F D' B U2 B2 (10f*)
B2 D2 B R' F R D' B U2 B2 (10f*)
B' U2 R2 L' U2 F' L2 F' D' B' (10f*)
B' U2 L D2 F2 D' R2 F2 B' D (10f*)
B' R2 F2 R' U' D2 B' R' D B' (10f*)
B' R2 F2 R' D2 B' U' R' D B' (10f*)

Can you clarify? What does your code give for the above state/scramble?

Unless maybe you shave a move by picking a certain orientation beforehand - which I don't think Johannes' results (other thread) would do, because that wasn't how his JARCS software worked - that's very unusual. (Edit: I just ran that scramble in every orientation in CE, and I still get 10 moves.) It's probably because you are not solving with respect to centers, whereas the other solvers are. I wonder how many professional Roux users can predict the optimal orientation - or do they just choose an ergonomic orientation and come up with an optimal solution. It's an interesting difference.

18 possible positions
(0[moves]*1[cases]+1*3+2*3+3*6+4*2+5*3)/18 = 2.78 moves

Hey thanks for pointing this out!
There is indeed something I was wrong about: I truncated the cumulative distribution at the nearest thousandths, so any small distribution < 0.1% would be rounded to 0. That is, you are right in that there exist 10 movers, and it is also true that my program found them correctly in simulation, but I messed up on the presentation part, thus giving a false impression that the first number where cum. percentage becomes 1.0 is the upper bound.

I updated the accuracy and here is the new result on 100k simulations:

Code:

Avg fixed FB+DR over 100000 sample is 7.644710
(0 0.00000)(1 0.00000)(2 0.00001)(3 0.00007)(4 0.00085)(5 0.00701)(6 0.06230)(7 0.36582)(8 0.91927)(9 0.99996)(10 1.00000)

Also, your solver gives an average of ~8.06 moves for fixed FBDR because it doesn't take misaligned M-slice centers into consideration. My definition of the solved states is essentially yours plus the three states that are (L' R), (L2R2) and (LR') away from it.
If I limit my solver to aligned centers only, then it also gives similar results:

Code:

Avg fixed FB+DR no misaligned over 100000 sample is 8.069930
(0 0.00000)(1 0.00000)(2 0.00001)(3 0.00006)(4 0.00035)(5 0.00284)(6 0.02562)(7 0.18689)(8 0.71710)(9 0.99720)(10 1.00000)

As for choosing the optimal orientation, I think fast roux solver would almost always start with ready-made pairs first. Maybe a stats on how often that would lead to the optimal orientation would be very helpful in understanding how effective this strategy is.