A list of all 15 btm "pure" one dedge flips
I don't know if this has been done before (probably, but I don't know where it is mentioned on the web), but I am going to list all of the UNIQUE 15 btm "pure" one dedge flip algorithms.
Clément Gallet listed all possible 15 btm solutions 3.5 years ago (it was probably done well before that though) in
this post. I translated all of the algorithms in his list to WCA notation, and I rotated all of them so that they affect the FU dedge and the U center.
Eliminating exact duplicates (which were many), here is the "complete" list (the mirrors and inverses are also possible solutions).
Note:
There are only 4 unique algorithms which flip the FU dedge and affect the top center. All other algorithms are transformations (or mirrors or inverses). Hence, I list them showing how the algorithms are directly related.
Symmetrical Algorithms (16)
r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2|Alg.1(v1)|
r2 B2 U2 l U2 l' B2 r B2 U2 r U2 r' B2 r2|Alg.1(v2)|
r2 F2 D2 r' D2 r F2 l' F2 D2 l' D2 l F2 r2|Alg.1(v3)|
r2 F2 D2 r' D2 l D2 l' D2 B2 l' B2 r F2 r2|Alg.1(v4)|
r2 B2 U2 r B2 r' B2 l U2 F2 r F2 l' B2 r2|Alg.1(v5)|
r2 F2 D2 l' F2 l F2 r' D2 B2 l' B2 r F2 r2|Alg.1(v6)|
r2 B2 U2 r B2 l' D2 r D2 B2 l U2 r' B2 r2|Alg.1(v7)|
r2 F2 D2 l' F2 r U2 l' U2 F2 r' D2 l F2 r2|Alg.1(v8)|
r2 B2 U2 l' U2 r U2 r' U2 B2 r' B2 l B2 r2|Alg.2(v1)|
r2 B2 U2 l' U2 l F2 r' F2 U2 r' U2 r B2 r2|Alg.2(v2)|
r2 F2 D2 r D2 r' B2 l B2 D2 l D2 l' F2 r2|Alg.2(v3)|
r2 F2 D2 r D2 l' D2 l D2 F2 l F2 r' F2 r2|Alg.2(v4)|
r2 B2 U2 r' F2 r F2 l' U2 B2 r' B2 l B2 r2|Alg.2(v5)|
r2 B2 U2 r' F2 l D2 r' D2 F2 l' U2 r B2 r2|Alg.2(v6)|
r2 F2 D2 l B2 r' U2 l U2 B2 r D2 l' F2 r2|Alg.2(v7)|
r2 F2 D2 l B2 l' B2 r D2 F2 l F2 r' F2 r2|Alg.2(v8)|
Non-Symmetrical Algorithms (10)
r' U2 l F2 l' F2 r2 U2 r U2 r' U2 F2 r2 F2 |Frédérick Badie (v1)|
r' U2 r U2 l' U2 l2 F2 r F2 l' U2 F2 r2 F2 |Frédérick Badie (v2)|
r B2 r' U2 r U2 r2 F2 r' D2 l D2 B2 r2 F2 |Frédérick Badie (v3)|
r B2 r' U2 r U2 l2 B2 r' U2 l U2 F2 l2 F2 |Frédérick Badie (v4)|
r B2 l' B2 l U2 r2 F2 l' F2 r U2 F2 l2 F2 |Frédérick Badie (v5)|
r B2 r' U2 l B2 l2 B2 U2 l U2 l' U2 l2 B2 |Alg.2(v1)|
r B2 r' U2 r D2 r2 U2 F2 l F2 r' D2 r2 B2 |Alg.2(v2)|
r B2 r' U2 r D2 l2 D2 B2 l B2 r' U2 l2 B2 |Alg.2(v3)|
r B2 l' B2 l D2 l2 D2 B2 r D2 l' D2 r2 B2 |Alg.2(v4)|
r' U2 r U2 l' D2 r2 D2 B2 l' D2 r D2 l2 B2 |Alg.2(v5)|
Thus there are exactly 26 (52 if you including mirrors and 104 if you also include inverses) one dedge flip 15 btm algorithms which flip the FU dedge and affect the top center.
And actually,
[1] Shifting any of the symmetrical algorithms from the first group and adding back the two setup moves yields algorithms from the second group. (and vice versa).
[2] Shifting any of the non-symmetrical algorithms from the first group yields algorithms from the second group (and vice versa).
*Therefore, in reality, there are only two unique paths for 15 btm move solutions: Frédérick Badie and the (old) standard algorithm. This has been my guess for a while, but I wasn't absolutely sure until I looked at all possible solutions and found relationships between them.
Would it be too much if I list all 26 of these algorithms here? (Two of them are there already).
[wiki]Impure Dedge Flips with Wide Turns/Pure Flips with Inner Layer Turns (SiGN Notation)[/wiki]
