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First we have to define what a twist is. I propose that a twist

is a twisting of face or faces about an axis such that there is

only one plane on which sliding friction happens. This means that

slice turns are not single turns, but rather multiple turns. This

is consistent with the 3x3x3 definition, and it is generalizable

to nxnxn. Also, I think it is wisest to consider quarter twists

as the only single twist "move" (don't count half twists as one!!)

One thing to note: Even order cubes should have a STANDARD

coloring. This is nice to have since there is now visible axis to

align the corners on. I know it isn't necessary, but if I own a

cube and you own a cube, and they are colored differently, and we

swap for a day, both of us will probably have a hard time trying

to get used to a different color arrangement than what each is

used to. May I suggest

Front Right Up Back Left Down

RED WHITE BLUE ORANGE YELLOW GREEN

The general PONS can be described as follows (for cubes, not

necessarily dodecahedra):

Pick an axis. Make 180 twists along all twistable planes.

Pick another axis. Do the same.

Do the same on the last axis.

This will produce a checkerboard pattern on all faces using

complementary (from the opposite side of the cube) colors. On the

4x4x4 (and in general on even order cubes), to corner cubies do

not move (but then again it is hard to tell without a fixed

reference).

I have done more thinking about the 5x5x5 than the 4x4x4 since

the 5x5x5 offers several immediate advantages. First of all, from

solved we can do things like: do a transform as we would on a

3x3x3 and consider the 3x3 set of cubies in the center of the

faces as the center of our favorite 3x3x3 Hungarian. Another way

to view this is to do normal Hungarian rotations using only one

level deep of cubies. Then run the algorithm backwards doing

twists using two levels deep (considering theface center as a

center and 2x2 at the corners as corners, and 1x2 around the

edges as edges. (Bad explanation, but it is 1:30 in the morning).

I believe doing the traditional SIX-DOTS pattern forward then

backward using this method will produce something that looks

like:

+++++ +++++

+@@@+ +@@@+

+@+@+ +@x@+

+@@@+ +@@@+

+++++ +++++

The one on the right is what happens if you do it again in the

same direction. I think the Plummer's Cross will look like:

+@+@+

@@+@@

+++++

@@+@@

+@+@+

And if I am not mistaken, the approriate EXTENDED-SIX-DOTS

mentioned above performed on this will produce a 3-cycle

checkerboard. And the PONS on top of that will produce a 6-cycle

checkerboard. Of course, I could be wrong, but we wont know until

(a) somebody carfully works it out on paper (b) somebody writes a

computer program to simulate the crazy thing, or (c) somebody

actually builds one.

That's it for tonight folks, aren't you glad!!