I received my Okamoto & Haseda Quarter Cube today. It operates slightly differently from what I imagined, but in a good way. Simply put, every face can be in one of two positions, 90° apart and relative to the adjacent middle layer. From the solved state, after I do an F turn then no matter what other faces I turn, the only option available to me on the front face is F'. I can use F to put the FL edge onto the top layer, but I will have to find another way to get the FR edge there (an R would do it).

M, E and S layers are also constrained to 90° turns and can not be turned independently of a parallel adjacent face, however it can be either adjacent face ! You can use a "wing" turn to get a 180° turn of the adjacent outer face. For example, from the solved state Fw F turns the front face 180° but from the solved state you can also do Bw B to turn the back face 180°. Moving the S layer in a wing turn moves the little indicator line on the stationary face. As you would expect, turning a centre into a face causes that face to inherit the centre's properties. For example, U Fw U moves what's left of the U layer by another 90°.

The colour scheme is funky, but the colours are nice and I'm perfectly happy with grey on the bottom. Who needs orange anyway? It doesn't even rhyme with anything (in English, at least) .

A full rhyme for orange is the very rare word sporange. A half rhyme (pararhyme) is lozenge. Oh, and I have ordered this cube from hknowstore, it will be an interesting solve.

The results from last night's experiment: solving the quarter cube with one stationary face. This is fairly easy using the same sequences as with the normal puzzle.

Start by solving the 4 edges of the stationary face, rolling them into position with CCW moves of adjacent faces, then finish the edges as usual.

When permuting the corners, if your algo uses U, R, and D, keep the stationary face at L and place the corners correctly onto the stationary face, one by one. If a corner is incorrectly placed on that face, pull it off, move it around the cube while keeping the stationary face at L, and then push it onto the correct position. If a corner belonging on the stationary face needs moving around the opposite face first, have the stationary face at B for that maneuver. Permuting the last 4 corners around the face opposite the stationary face is easily done without using the stationary face.

Finally, orienting the corners can be done quite easily without using the stationary face. I find that the easiest way is to first restore twist parity for the stationary face and the opposite face, which needs to be done 2/3 of the time. Then it's easy to complete the solve by twisting pairs of corners in either face, without using the stationary face.

So your instincts are very good, Brandon! It looks as though any constrained cube with at least 5 moving faces can be solved fairly simply using a standard set of algos. Soon I'll try the "Earth" and "Venus" varieties: two stationary faces adjacent, and two stationary faces opposite.

Quarter Cube variation -- "Earth", where two adjacent faces are locked/stationary

This can be solved using the same basic method and algos as a regular Quarter Cube, but with a couple of twists.

We need to start solving the edges in a slightly different order than layer-by-layer or the solve can get very difficult. If the D and B faces are fixed, I start by solving the BR and BL edges, then DR, then DL (solved into position with CCW moves of the R and D faces respectively), then FD, then FR and FL. The last two edges listed need a bit of creativity to flip and permute compared to the regular Quarter Cube. Then solve the edges of the U face as usual, apart from avoiding using the B face, which isn't very difficult.

We can permute the corners with the same [3,1] commutators as before, but with a severe restriction on the possible sets of corners that can be permuted. With D and B fixed, I think that only a single group of adjacent corners on each of the F, U, B, and D faces can be cycled. But that's enough to permute the corners from any situation, with some patience and planning.

We finish by orienting the corners in pairs with [8,1] commutators, and this is where things get quite tricky, because it is usually only possible to do this by sometimes permuting corners into a position where an adjacent pair can be twisted without needing to use the fixed faces, then permuting them back again.

This variation makes an interesting challenge, similar in feel to some of the planet cubes.

Yesterday, I recieved my Quarter Cube.
Today I scrambled and solved it (Once so far). I think it provides a nice challenge. Not too easy not too hard.
I can certainly recommend it.