Far as I can tell, the actual name of this puzzle is "Guimo Yijie Ghost Cube", which translates into the wonderfully redundant "One-Layer Ghost Cube Ghost Cube", and "floppy" isn't part of its official name. (I assume it's a translation convention, like changing "yileng" to "Fisher".)
I'm posting this in the theory section because I don't actually have this puzzle yet and I was a bit curious about its solving process. Essentially, while the shape of the puzzle changes with every move, the shapeshifting aspect is only for "obfuscation" purposes (like mirror blocks or the original ghost cube) and doesn't complicate the solve if you know all the shapes by heart. There also aren't any identical pieces.
Just like on a floppy cube, the five "edges" have orientation (they can be flipped), but they cannot move. Unlike a floppy cube, the five "corners" also have orientation, and any corner piece can move to any of the five corner locations in any of two orientations. (On a floppy cube, it might look like corners can be flipped, but a corner can never be flipped "in place".)
Every move has order 2: if you turn one side 180° twice, it's the same as not turning it at all. Also, each move flips one edge, flips two corners, and does a 2-cycle on adjacent corners, so all 5! = 120 corner permutations are attainable, EO parity is always the same as corner permutation parity, and CO parity is always even. It turns out that there aren't any other restrictions (you can get 3-cycles pretty easily along with a pure edge flip; pure corner flip is a bit trickier but still possible), so the puzzle has 5! 2^(5−1) 2^(5−1) = 30720 states.
30720 is pretty small for a computer to handle, so I did my thing with it. Here's the distance distribution.
0: 1
1: 5
2: 15
3: 40
4: 105
5: 275
6: 670
7: 1500
8: 3140
9: 5825
10: 7752
11: 6415
12: 3395
13: 1270
14: 282
15: 30
So God's number for this puzzle is 15, which seems surprisingly high for such a small puzzle. Anyway, before I wrote the code to enumerate the cases, I also tried to work out (on paper!) bounds for God's number, just for fun, and I managed to get as far as a lower bound of 10 moves and an upper bound of 22 moves, which weren't too far off from the true value! (I'll type out the details when I get the time to—I think there's something interesting here.)
I'm posting this in the theory section because I don't actually have this puzzle yet and I was a bit curious about its solving process. Essentially, while the shape of the puzzle changes with every move, the shapeshifting aspect is only for "obfuscation" purposes (like mirror blocks or the original ghost cube) and doesn't complicate the solve if you know all the shapes by heart. There also aren't any identical pieces.
Just like on a floppy cube, the five "edges" have orientation (they can be flipped), but they cannot move. Unlike a floppy cube, the five "corners" also have orientation, and any corner piece can move to any of the five corner locations in any of two orientations. (On a floppy cube, it might look like corners can be flipped, but a corner can never be flipped "in place".)
Every move has order 2: if you turn one side 180° twice, it's the same as not turning it at all. Also, each move flips one edge, flips two corners, and does a 2-cycle on adjacent corners, so all 5! = 120 corner permutations are attainable, EO parity is always the same as corner permutation parity, and CO parity is always even. It turns out that there aren't any other restrictions (you can get 3-cycles pretty easily along with a pure edge flip; pure corner flip is a bit trickier but still possible), so the puzzle has 5! 2^(5−1) 2^(5−1) = 30720 states.
30720 is pretty small for a computer to handle, so I did my thing with it. Here's the distance distribution.
0: 1
1: 5
2: 15
3: 40
4: 105
5: 275
6: 670
7: 1500
8: 3140
9: 5825
10: 7752
11: 6415
12: 3395
13: 1270
14: 282
15: 30
So God's number for this puzzle is 15, which seems surprisingly high for such a small puzzle. Anyway, before I wrote the code to enumerate the cases, I also tried to work out (on paper!) bounds for God's number, just for fun, and I managed to get as far as a lower bound of 10 moves and an upper bound of 22 moves, which weren't too far off from the true value! (I'll type out the details when I get the time to—I think there's something interesting here.)