So, today is Pi Approximation Day, and unfortunately I have nothing very closely related to π to present today.

Anyway. Let’s have a look at the Rubik’s Cube. There are effectively six moves available, which we may take to be {U,D,L,R,F,B}, as usual. (It is, of course, possible to instead use {U,u,R,r,F,f} or {U,R,F,E,M,S} as a “basis” instead, but using the standard moves fixes the centre facelets.)

It is obvious that with a normal Rubik’s Cube, you’re only allowed to turn the faces in multiples of 90°, for the very simple fact that edge cubies and corner cubies are shaped differently and you can’t just interchange them.

But what if we could? What if we turn the faces by only 45°?

It’s probably not possible to make a good real implementation of this, since, projecting the cubies onto a sphere, edge cubies and corner cubies have different latitudes, so the turns will inevitably have some bumpiness. (The path of motion is also most likely forced to be nondifferentiable.) I presume using magnets would be the most straightforward way of creating this idea IRL, and this has probably already been attempted by puzzle-modding enthusiasts.

There is also the Mixup Cube, where faces are still restricted to 90° turns, but slices can turn by 45°; corners remain corners, but edge and centre cubies are now equals. This is conceptually similar to Mozaika and a few other puzzles, except that edge cubies on the Mixup Cube have orientation. (Depending on stickering, the centre cubies might not have visible orientation, in which case the Mixup Cube’s states don’t form a group.)

Getting back to the idea of a 45°-face-turn cube, centre cubies are still fixed and edge/corners can freely mix. It’s not hard to prove that arbitrary permutations (not just even permutations) can be obtained due to the existence of an eight-cycle. As on the usual Rubik’s Cube, there is an orientation invariant; if we choose an orientation reference as pointing towards the direction of increasing latitude, U/D don’t change the orientation, while any of the other four face turns changes the orientation of four pieces by 30° and the other four pieces by 60°, resulting in zero total orientation change.

The non-centre cubies have six possible orientations. On a normal Rubik’s Cube edge orientation corresponds to a cyclic group of order 2 and corner orientation corresponds to a cyclic group of order 3; since edges can corners can mix here, this means they should have at least six orientations. While it’s mentioned above that a non-horizontal face turn would change the orientation of four cubies by a twelfth of a revolution, it turns out that that’s just an artifact of how we decided to label our reference orientation. Any originally E-slice edge cubie in a non-E-slice position necessarily has orientation an odd multiple of 30°, and likewise for any originally non-E-slice cubie in an E slice position.

The centre cubies have eight orientations. Unlike the non-centre cubies, the total orientation is not an invariant in either the 45°-face-turn cube or the Rubik’s Cube. Instead, the *parity* of the total orientation equals the permutation parity of the non-centre cubies for the former. This can be seen to be because every turn changes both the permutation parity and the centre orientation parity. Like the Rubik’s Cube and unlike the Mixup Cube, ignoring the centre orientation still results in a group.

The 45°-face-turn cube would have a total of states if centre orientation is entirely ignored, if the parity of the centre orientation is used, and if all eight orientations are distinguishable. That said, it’s not really the number of states which determines a puzzle’s difficulty—the Megaminx has *many* more states than a Rubik’s Cube but can be solved in pretty much the same way.

For that matter, the availability of a 45° face turn in our described puzzle would probably make it easier to solve than a normal Rubik’s Cube. At least the permutation aspect reduces into an almost-trivial puzzle that can be solved by spamming commutators/conjugates, after which the orientation could be solved with the same technique or the usual Rubik’s Cube algorithms. Using commutators is a particularly easy “first principles” way of solving puzzles that behave sufficiently like groups, but generally interesting puzzles don’t have simple commutators to exploit. Huge parts of my two gigaminx solves were done through conjugated commutator spam, for example, and considering that they both took more than an hour, this also highlights how incredibly inefficient solving with commutators/conjugates can be when they involve more than just a few moves. In this case, taking just two faces of the 45°-face-turn cube and ignoring orientation reduces to a (5,3,5) rotational puzzle, which is easy to solve with simple commutators.

Now, suppose we also add 45° turns to slices, like the Mixup Cube. This monstrosity can mix every cubie together and in all likelihood can never be manufactured. Every piece has 24 possible orientations and every permutation of the 26 cubies is possible. I’m not sure about what the orientation subgroup is like, but I guess there are either three or six orbits since each move changes the total orientation by a multiple of 45°. Assuming that there are three orbits and that the centre cubie orientations are distinguishable, this would have positions.

Update: I borked the HTML which caused half the post to vanish when viewed from the front page. Also expanded on some stuff.