Remember Oskar's Geared Mixup
? It's a cross between the Gear Cube and the Mixup Cube. On the Gear Cube, you can only make 180° (antislice) turns; if you stop at 90°, the cube will be blocked on the other axes.
The Geared Mixup removes that restriction. This means that centers and edges are interchangeable.
I bought a Geared Mixup from Oskar last weekend, mostly solved it on the plane on the way back from Gathering for Gardner, and then got the last couple of edge rotations later that night. Oskar says I'm the first person to solve it, so I thought I'd post about my solution.
The challenge here is that it's difficult to build any useful kind of commutators -- every turn affects most of the pieces. You can't turn just one face, which is how you usually build commutators. Also building conjugate sequences to wrap around transforms is challenging. The same is true of the Gear Cube, but then the moves are so restrictive that you can't mess that up a whole lot anyway. Not so the Geared Mixup!
Most of my solution is built around one basic commutator: R4 U' R4 U. Yeah, R4! That actually does something on the Geared Mixup -- effectively it's a slice-squared move.
Call this sequence A. Now, A results in two 3 cycles and two swaps. Do A3, and you get just the two swaps. These are of a center slot and the opposite center, and an adjacent edge and its opposite edge. Those pieces are also all rotated 180°. Also two other pieces are rotated 90° in opposite directions. This is normally invisible, because those are center slots, but if you have an edge in that slot it makes a difference.
So, the solution. First, solve the corners like a 2x2x2. Then, using A, get the center pieces in center slots. Not necessarily matching colors, but on the right "axes": blue & green centers in the blue & green slots (either way), red & orange in those slots, etc. Leaving the flexibility of swapping them back and forth later is useful. This step can normally be done with simple one- or two-move conjugate sequences to position the centers, without messing up the other centers.
Similarly, use A to get all the edges in their correct slots, letting the centers swap back and forth freely. Here you have to be a little careful in your conjugate sequences positioning the edges for A not to mess up existing edges (by getting them in the center slots to be swapper), but there are only a handful of cases to care about, and they can all be handled.
Now is where it gets tricky -- edge rotation. What I did for the first solve is build a sequence that counterrotates two edge pairs. This is a bit problematic, because (1) the sequence is very long, and (2) sometimes finding the right conjugate sequence to make this useful is tricky.
Here's how you build that sequence. First, you can get a 3-cycle like this (or many similar ways): A3 B' A3 B. This also twists some edges. Do that three times, and you are left with just the edge twists. But that adds up to 114 moves! (Counting each R4 as two moves, because physically it is.)
Finally, get the centers in the right slots. If they are not already, then two pairs will be swapped. Perform A, swapping one pair, and the edge pair between them, then do A again on the same edge pair and the other center pair. Solved.
Now I have a better way to handle edge rotation. Do B' A B, and you have swapped two centers and two edges, and counterrotated two other edges 90°. Only problem is, the two edges you swapped are not opposite, so you have to put them back without undoing the edge rotations you just did. To do that, do R2 B A B' R'2. The result is two pairs of centers have swapped (which you don't care about at this stage), the two edges you swapped are rotated 180° (which you also don't care about, because here we're solving the 90°-rotated edges), and two edges are counterrotated 90°. Do this until you get rid of all 90° rotations. If there's an odd number, you'll have to first get a center into the slot of one of the rotated edges for this sequence.
Dealing with the 180°-rotated edges is easy -- B' A2 B rotates two edges 180°. There's also a simple way to flip 4 edges.
I still have a little more to learn, I think, to optimize the edge rotators -- also I'm still investigating other basic commutators besides the A sequence. But that's so simple, I don't really expect to find anything better.
I'm having a blast with this puzzle! It's not impossible, like many new jumbling puzzles; neither is it just another basic cube variant you can solve by finding commutators in the obvious way. You have to think about it and do a fair amount of visualization. If I put it down for a while, it will be a challenging new puzzle again. Also, the one I bought from Oskar turns amazingly smoothly. I highly recommend the Geared Mixup!