Okay I have some basic notes about how this puzzle behaves and some general thoughts on solving it. First some piece terminology:
celtic_cube_piece_labels.png [ 1.24 MiB | Viewed 235 times ]
- 1 Centers
- 2 Edges
- 3 Batman pieces
- 4 Corners
- 5 Edge squares / Center squares
- 6 Center triangles / Wing triangles
Notice even though the edges look chiral, they're labeled the same. They're in the same orbit so an edge can go anywhere. They can also be put in backwards via jumbling.
The Edge squares and Center squares are also in the same orbit and behave as the same piece type.
The Center triangles / Wing triangles also look like three different pieces but they're all in the same orbit too. It is easy to see if you've put a right-handed Wing triangle into a left-handed spot or vice-versa because they'll stick out of the spot in an odd way.
I classify the piece into three categories. The first is the centers and they're the guide to unjumbling the puzzle correctly. The second is the "big pieces" which are the edges, batman pieces, and corners. The third category is the "small pieces" which are the center / edge squares and the center / wing triangles.Working with the big pieces:
The big pieces jumble into all sorts of horrible configurations. All of the blockage that prevents some axes from turning when the puzzle is jumbled is due to big pieces. No small piece can cause any blockage so to unjumble the puzzle you only focus on the big pieces.
The corners, edges, and batman pieces are all interchangeable with jumbling. When they're interchanged with each other they usually cause blocking and this can make finding available turns hard. Here is an example of jumbled batman pieces:
celtic_cube_batman_commutator_jumbling.png [ 1.02 MiB | Viewed 235 times ]
And here a corner+edge has been cycled with some batman pieces:
celtic_cube_edge_backwards.png [ 1.16 MiB | Viewed 235 times ]
Your goal should be to resolve the jumbling of the big pieces without worrying about the sticker colors. Once the big pieces are unjumbled and match up with the centers in their solved twist state the puzzle should be much easier to work with. You'll probably find that fixing all of the flipped edge+corner pairs is both much harder and much more useful that fixing flipped batman pieces.
Once you've unjumbled all of the big pieces you can start cycling them into place to match up the colors. I first solved all of the batman pieces because they were really easy to do. Here are the setups I used to put the puzzle into a simple state for a [1,1] commutator to 3-cycle the batman pieces without leaving them jumbled:
celtic_cube_batman_commutator_setups.png [ 927.45 KiB | Viewed 235 times ]
Then I worked on the edges (completely ignoring the corners). Here are the setups to 3-cycle edges with a [1,1] commutator without leaving them jumbled:
celtic_cube_edge_commutator_setups.png [ 1.17 MiB | Viewed 235 times ]
Note that both of the simple [1,1] commutators shown above end up treating the edges and the batman pieces like they're in orbits. You'll have to use jumbling to get pieces not nicely in the orbit your commutator is restricted to. It can be really exceptionally hard to find the right jumbling setups needed to cycle the edges into place. You can also run into a parity with the edges. To resolve it you will need to involve batman pieces.
There is also a parity in the corners but I'm not sure if it is automatically resolved when the parity in the edges is fixed. It was for me but I think I see a way of separating corners from edges in a way that would decouple the parities.
I solved the corners by wrapping my edge commutator in an addition few setups and turning it into a large commutator that just affects edges.Working with the small pieces:
Once the big pieces are solved the small pieces are pretty easy. There isn't anything inherently hard about them except that there are a lot of them and it can be really hard to move them around the puzzle the way you want because of the jumbling. I solved the center / edge squares with simple [1,1] commutators and then the center / wing triangles with [7,1] commutators. The [7,1] commutator might sound intimidating but it's easy to find. It's really just a simple [5,1] commutator with an extra conjugate nested in the "5 part" to avoid some jumbling problems.
Because there are 72 center / wing triangles and the setups are awful, I suggest using [1,1] commutator edge / center square cycles to move the center / wing triangles around. I didn't start doing this until later in the solve after I'd already wasted a lot of time. This is what the "setup" commutators look like:
celtic_cube_small_pieces.png [ 1.12 MiB | Viewed 235 times ]
If I had to guess, I'd say re-solving this puzzle would take me about 6 hours based on what I know about it now.