# Message #151

From: Guy Padfield <guy@guypadfield.com>

Subject: RE: [MC4D] Cubes, beer and Beethoven

Date: Mon, 09 May 2005 19:54:47 +0200

Thanks Roice. Remigiusz is supposed to be producing a solution to the 4^4 at

this very moment, though he keeps getting sidetracked by his mission to

unscramble a 2^4 in under 20 moves. I hope he will report back on any parity

problems he encounters.

I havent had time yet to study Davids and Erics articles, though I did

read Erics when I first found the site, before I had attempted any of the

cubes. I first thought about parity issues, at a much simpler level, at

about the same time David was writing that article in 1980 (when I was a

schoolboy). Once I had solved the Rubik cube (there were no published

solutions then) I realised the box was wrong to boast there was only one

solution, since the face centres could be orientated any way without anyone

noticing. So I marked up my cube to yield only one correct solution and

rapidly made an interesting discovery. If you leave the orientation of the

centres to the end, there are two possibilities. 50% of the time their net

rotation is 0º (mod 360), in which case they are easily solved in pairs. 50%

of the time the net rotation is 180º, in which case you are forced to break

and remake the edges and corners of a face to align the remaining centre

correctly. By coincidence I chose the word parity to describe my

discovery, calling the first situation even and the second odd.

Few people seem to have attempted the 5^4. The 5^3 has just one parity issue

(providing you dont mark it up) once you have aligned the edges correctly

the corners always fall into place. It would be interesting to know if the

same is true for the 5^4.

What would happen if you worked outside in? The corners have only one

possible correct alignment with respect to each other so completing the

edge pieces inside them would presumably always work without rebreaking the

corners. I strongly suspect, without proof, it would then be possible to

align the faces without rebreaking the edges and the centres without

rebreaking the faces (because of the independence of the parity issues when

working in the reverse direction). However, I also suspect the required

sequences would be much longer, undoing the benefit gained.

Roice indirectly revealed his 4^4 solution strategy in his last post

responding to my query. Would it breach protocol for me to publish my

slightly different strategy here, if that would give future solvers a hint

or two? It is far less economical in terms of moves but also, I think, far

easier to execute. Or would that upset people trying to solve it for

themselves?

Guy

_____

From: 4D_Cubing@yahoogroups.com [mailto:4D_Cubing@yahoogroups.com] On Behalf

Of Roice Nelson

Sent: 09 May 2005 06:07

To: 4D_Cubing@yahoogroups.com

Subject: RE: [MC4D] Cubes, beer and Beethoven

Hi Guy,

I had a few comments…

> I will check the sites you refer me to and

> let you know if they answer my question.

> I suspect I was not very clear in my original

> posting, but by 3D version I meant the 4^3,

> not the 3^3 and it was from this that I made

> guesses about the 4^4. Each of us when solving

> probably creates his own lingo but I should be

> more careful when communicating: by centres

> I meant the 8 one-colour pieces on the interior

> of each cubic face of the 4^4. Since you cannot

> distinguish orientations and positions of these

> I wondered if their arrangement influenced the

> subsequent placing of the two-colour pieces,

> such that if it was wrong these last could

> not be aligned.

My memory had failed me as usual, but I dug up some emails from when I

had worked on the 4^4. I did have exactly what you are describing

here happen to me (a parity issue such that I could not line up all

the 2 color pieces after placing the centers, forcing me to undo work

on the centers).

I had the parity issue you had as well, forcing me to break and

rearrange 2 color pieces to fully align the 3 color ones. Very

thankfully, I did not have the issues with the corners! After working

through the other 2 problems, I distinctly remember hoping hard for

that :) Maybe some of other 4^4 solvers have had a corner parity

issue, allowing us to empirical verify the existence of all of these

using the whole group.

I think there could be something to your question about the solution

method affecting parity problems arising in the 4^4 (centers-out vs.

layer by layer). Say for instance there are multiple situations that

can lead to parity problems for 4 color corners, and that these depend

not just on the 3 color edges (if an even/odd number of twists was

used to solve them or whatever), but on both the 3 color and 2 color

pieces. In the centers-out approach, certain situations leading to 4

color problems might never show up since some work to get 2 color and

3 color pieces "in parity" has already been done, and so maybe such

issues could only arise in a layer-by-layer approach. This is

hypothetical, but my point is that the solution algorithm itself

places additional constraints on the state of the puzzle and may limit

what is possible.

In fact, I wonder if the fully unconstrained 4^4 has more than 3

contributing situations that can lead to parity problems (maybe 3! or

something). I am not familiar enough with group theory to back this

up mathematically, but others might be able to. As David suggested,

maybe the information could be gleaned from Eric’s page on the number

of permutations. It would be cool to have a specific explanation of

the number of these problems and their occurrence percentage.

Take Care,

Roice

P.S. I had put marks in my 4^4 solution at the time, so you can check

out the parity problems I encountered on it if you like. Here are

descriptions for my marks…

a Centers Placed

b First parity problem

c Done Combining Face Pieces

d Done Combining Edge Pieces

e Second parity problem

f Fixed Parity Problem

g Faces Placed

h Edges Placed

i Finished

_____

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