I mentioned

*Enigma* - this was certainly enigmatic for its time, without Turing's insights into computable numbers and programming a machine to write a code, things might have been quite different for history.

But that aside, Rubik's original puzzle quickly became enigmatic, it was 'modded' almost immediately and the pocket rocket appeared. There is a k-complete table for this reduced ("edge-less") version, by which I mean the lengths of all algorithms have been computed. The formulations used to do this are not the above, which are specific to N=3 and d=3. I expect the set of formulas are also reduced, so that the group's symmetry is preserved between N=2 and N=3.

(the convention here is that normal text "N" is the number of sections, in the edge basis, the italicised form is recursive, it acts to enumerate all possible positions given n).

Note the symmetry here: there are twice as many slices when N goes from 2 to 3, and the number of sections per face increases geometrically from 4 to 9. The 'deck' goes from two to three

*layers*, and the middle layer has no elements belonging to the N=2 group.

In the 2-cube there are no elements of rotation, but reflection instead, so the colors permute over reflections. A way to model this is to color only the middle slice layer of an original, and keep it oriented so the elements

*stay in-plane*. This is in fact what the computer model of the 2-cube does, pivoting rows or columns of elements one at a time.

Each of the formulas above, given a value for n, will return the

*same* value which is the number of positions n can 'step to' via recursion. If n is 1, that means only one member of the FTM is selected (X), if n is 2, two members are selected (XY), and fully permuted, and so on.

Here, colors can be minimized: the initial color-symmetry has 1 position in the full topograph of G = all faces with equal but separate color. Separate is the operative term, since a pair of opposite faces can be the same color (interference is impossible, opposing pairs commute), this demonstrates that color is also a separating (selective) operator, colors are fixed values, like integers.

The formulas return 'black + white', not color;

*you* need to color the numbers yourself or get a computer to do it, or maybe buy a few billion plastic ones and start really early every day, for a few billion years, see how it goes. Then move up to the 4-dimensional version.

It looks like a cube (or like an open box if you leave one or more sides uncolored), but it's really a device for encoding stuff. The contents are still under analysis.