# Message #545

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] Something interesting and strange about permutations

Date: Fri, 08 Aug 2008 19:30:19 -0500

Hi Lucas,

It took me a bit, but I think I’m now mostly following what you are saying

here. If I am interpreting correctly, I think what you have effectively

discovered is how the MC2D rotations are not analogous to any of the other

puzzles (except David Vanderschel’s extended MC3D functionality) because the

MC2D face rotations allow mirroring. The MC4D and MC5D puzzles don’t permit

mirroring twists, which is in more strict analogy with the physical 3D

Rubik’s cube. There is flexibility in how exactly we want to carry over the

analogy of twists, but I like the MC4D

FAQ<http://www.superliminal.com/cube/FAQ.txt>description of what it

means to make a twist, which says "Take the face you

want to twist and remove it from the larger object. Turn it around any way

you like without flipping it over, and then put it back so that it fits

exactly like it did before.". If we were to adhere to this in MC2D, no

scrambling twists would be possible, and hence it would be degenerately easy

to solve <http://www.gravitation3d.com/magiccube5d/2d_rubik’s_cube.jpg> :)

I found your observation about MC4D twists "only affecting 4 faces"

intriguing! All the MC4D twists (except the identity) do in fact affect all

6 adjacent faces, but it sounds like you are making a distinction with the

3D case where there is no possibility to make a twist and have all stickers

on an adjacent face remain the same color. In MC4D, all the adjacent *

cubies* are getting shuffled around, but some twists (not all!) allow the

sticker colors to remain the same on 2 of the 6 adjacent faces. This was a

cool point for you to make, as I have never explicitly focused on that

contrast with the 3D puzzle before. Likewise in MC5D, the cubies on 8

adjacent faces are always affected with every twist, but some twists allow

stickers on up to 4 of the 8 adjacent faces to not change color (in our MC5D

implementation, this is actually the only possibility since the twists are

not fully worked out). I don’t think that we are understanding the higher

dimensional puzzles wrongly, but that this different behavior arises due to

the extra space in the higher dimensions.

Also, it is possible on the 3D, 4D, and 5D cubes to build a 3-color series

based on two 2-color series, although the 2-color series require 4 moves

instead of 2. An example in the 3D case (with the 2-color series in

parenthesis):

```
(R'FRF') B' (FR'F'R) B
```

Anyway, I hope I was on the right track and that these ramblings are

usefully related to your thoughts…

Roice

P.S. As a short aside, it is an interesting fact that the motion of any

rotation can equivalently be described as a set of two reflections, which is

why your U2 example is achievable as two of the "unreal movements". Visual

Complex Analysis <http://www.usfca.edu/vca/> is a fantastic source to learn

much more about this.

On 8/6/08, lucas_awad <lucasawad@gmail.com> wrote:

>

> After solving the MC5D, I have discovered something a bit strange

> about permutations.

>

> As everyone who read the solution for MC4D know, we can permutate the

> 4-color hypercubies by doing the 3-color series two times (one of them

> the reverse).

>

> But, why we cannot permutate the 3-color pieces with doing two times a

> 2-color permutation with 2 moves on MC2D?

>

> Because the face rotation is different.

>

> When rotating a "face" in MC2D, the move is like this:

>

> 1 2 3 –> 3 2 1

>

> In 3D, the same movement should be:

>

> 1 2 3 –> 3 2 1

> 4 5 6 –> 6 5 4

> 7 8 9 –> 9 8 7

>

> But that’s not what we really do with a rubik’s cube, it is this (it

> would we for example U2, if it is "U" face):

>

> 1 2 3 –> 9 8 7

> 4 5 6 –> 6 5 4

> 7 8 9 –> 3 2 1

>

> If you see, this algorythm (2-color permutation in MC2D) doesn’t only

> do 4-6 permutation, also 2-8, which don’t happen in MC4D with 3-color

> series.

>

> By doing the previous movement (the unreal one) we only affect two

> faces which change their stickers (the same as MC2D), but with a

> rubik’s cube (and also MC4D and 5D) we are affecting 4 adjacent faces

> (the other keep still the same stickers). So with the unreal movement

> we would be able 3-color pieces by doing the sequence: ( F - R ) U ( R

> - F ) U

>

> However, in MC4D we do movements that only affect 4 faces, and that

> allows us to easily permutate the 4-color hypercubies by doing the

> 3-color series algorythm. The fact I’m thinking now is if in MC4D and

> MC5D all adjacent faces should be affected to make the rotation real,

> and we are understanding higher dimensional puzzles wrongly.

>

> I hope that you understand what I have said.

>

> Greetings

> Lucas

>

>

>