### Why cubie orientations are preserved

A while ago in “Rotating only one cubie”, I said there was “extra structure” to the Rubik’s Cube’s moves that preserves the sum-total of all corner cubie orientations, and of all edge cubie orientations, but didn’t preserve the sum-total of center cubie orientations. I didn’t get into that then, because it’s hard to describe, and *why* they’re preserved wasn’t the point of that post. But I did make a note to get to it later, as I wasn’t aware of any good explanations of it that I could link to, and it’s high time for that “later” to become “now”.

The dynamics of the corner and edge cubies are pretty different in this regard, which is probably expected by everybody who’s played around with the Rubik’s Cube that much — edge and corner cubies have very distinct characters in general. As is usual with the Rubik’s Cube, the corner cubies are easier to think about. (I think this is mostly because there are only 8 of them, while there are 12 edge cubies.) So, let’s do them first.

As is often the case, the first thing to do is to simplify the problem. You’ll notice that for any pair of opposite-face colors, every corner cubie will have exactly one face with a member of that pair. We can exploit that fact by “re-coloring” our Cube so that two opposing faces are black and the other four are white. (Two colors are *much* easier to reason about than six!) This results in each corner cubie having one black face and two white ones, which makes for an obvious way to define orientations numerically: A black cubie face on a black Cube face is rotated by 0°, otherwise it’s considered rotated by 120° or −120° as seen around the axis running through the cubie’s corner. This simplification is “valid” because any time a corner cubie is in its home position on the six-colored Cube, it’s oriented exactly the same way it is on the two-colored Cube.

If we rotate one of the Cube’s black faces, we observe that the orientations of all four affected corner cubies have the same values after the rotation as they did before it. If we rotate one of the Cube’s white faces, we see that two of the corner cubies have been rotated by 120° by this definition of orientation, and the other two by −120°. In either case, the sum-total of the cubie orientations is unchanged.

The edge cubies are trickier. We want to define them as either “flipped” or “unflipped”, but our two-color Cube has four edge cubies whose faces are both white, so we can’t stick with the simplicity of having everything black and white anymore. We need to add a third color, gray, which we assign to opposing faces so our Cube has a pair of opposite faces in each color. The 12 edge cubies of this Cube consist of four black/white cubies, four black/gray cubies, and four gray/white cubies.

How do we define “(un)flipped”? Most temping (at least to me) was to have the cubie be unflipped if one (or both) of its faces matched the Cube face it was on. This produces a “valid” convention: flipping a flipped cubie makes it unflipped and vice-versa, and in the home position it’s flipped or not exactly as it is on the six-color Cube. But when we rotate a white face, we have a problem. Consider the case where the face’s two gray/white edge positions are filled by unflipped gray/white cubies and one of the black/white positions is filled by an unflipped black/white cubie, but the other black/white position is occupied by a black/gray cubie with gray facing the black side and black facing the white side. That fourth cubie will be considered “flipped” per the proposed convention, but after rotating the white face its status changes to unflipped, as now its gray side is facing one of the Cube’s gray faces. But the other three edge cubies on the face still have their white sides facing the white face, so all three remain unflipped, so total orientation was not preserved. So while this convention is *valid*, it’s sadly not *useful* for our purpose, and we need to choose a different convention.

What if we say an edge cubie is unflipped if its darker side faces the darker of the two Cube faces it’s between? That passes the validity checks: flipping reverses the “(un)flipped” status and when it’s in the home position it matches the six-color Cube’s orientation. If we rotate a black face, then no edge cubies change their (un)flipped status: each one that had its darker side face the black face was and remains unflipped, and each one that had its lighter side face it remains flipped. Similarly for rotating white faces: edge cubies that do (or don’t) have their lighter sides facing the white face before the move, each still will (or won’t) after the rotation. Well now, for the “flipped” status to not be vacuous, we need something to change it — have we been analyzing a re-coloring that’s valid only because it’s trivial? No, because rotating a gray face “flips” all four of its edge cubies! Each cubie that was between gray and black (so gray was the lighter side) is now between gray and white (so gray is the darker side), and vice-versa.

Flipping groups of four changes the number of flipped cubies by 0, ±2, or ±4 only, so under the darker-faces-darker^{*} convention, the number of flipped (equivalently, 180°-rotated) edge cubies is always even. Hence the sum-total of the edge cubies’ orientations is always zero (under the typical convention that orientations are equivalent mod 360°). And since the sum-total of all corner cubies’ orientations is also always zero, the intuition in “Rotating only one cubie” is indeed justified. (Though it was many years after I developed that intuition before I realized that by re-coloring the Cube I could justify it logically.)

^{*} This “darker” business is why I chose the black/gray/white color scheme. Upon first working this stuff out, I used the “colors” primary/secondary/tertiary, with secondary being “effectively primary” when it was paired with tertiary. Once I knew what I needed to know, thinking about it as darker and lighter shades is easier and clearer than as primary/effectively-primary, but it’s the destination that’s important, the journey can be relegated to footnotes.

[…] Since each turn rotates the edge cubies by 360° in total and every sequence is a composition of face turns, the “total rotation” of all the edge cubies together is a multiple of 360° and so equivalent to 0, meaning there can only be an even number of 180° edge cubie flips. The corner cubie rotations sum the same way, hence the corner cubie rotations that don’t cancel themselves out must sum to a multiple of 360°. The conclusions are correct: no sequence can do an odd number of edge flips or corner rotations that aren’t equivalent to zero. However, the argument is unsound: adding rotations around different axes together willy-nilly like this can (in other contexts) produce complete nonsense — there’s some extra structure in the Rubik’s Cube that makes this OK [for different reasons explained here]. […]

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