Message #2141

From: Andrew Gould <agould@uwm.edu>
Subject: RE: [MC4D] Faulty Logic Counting States for 4D Center-Edge Cubies
Date: Fri, 11 May 2012 15:43:03 -0500

notation: 3C piece = 3C cubie = 3-color cubie = 3-sticker cubie

Counterexample for the 2nd to last sentence of the 1st paragraph of David
Vanderschel’s "valid argument":

He claimed that an even length cubie permutation cycle (of 3C pieces)
implies 3 even length permutation cycles of stickers. Counterexample: a
2-cycle of the 3C pieces, but a single 6-cycle of their stickers. I believe
you would still be able to arrive at the last sentence of that paragraph
("Thus the parity of the sticker permutation is equal to the parity of the
cubie permutation."), but I believe this approach may be easier (it seems to
be what David Smith was getting at in Roice’s last link):

Note that an element of the generating set (of twists) performs an odd
permutation on the 3C pieces iff (if and only if) it performs an odd
permutation on the 3C stickers. Thus, any composition of generators
performs an odd permutation on the 3C pieces iff it performs an odd
permutation on the 3C stickers.

–

Andy

From: 4D_Cubing@yahoogroups.com [mailto:4D_Cubing@yahoogroups.com] On Behalf
Of Roice Nelson
Sent: Friday, May 11, 2012 13:45
To: 4D_Cubing@yahoogroups.com
Subject: Re: [MC4D] Faulty Logic Counting States for 4D Center-Edge Cubies

On Thu, May 10, 2012 at 9:51 PM, David Vanderschel wrote:

If a more rigorous argument has been presented somewhere,
I would appreciate learning about it.

The following paper has shown up on our list a number of times. They derive
permutation counts for all piece types, so it should be relevant.

http://udel.edu/~tomkeane/RubikTesseract.pdf

Also, David Smith has put a lot of thought into this topic, and I think his
writings are valuable. The links in some of our archive emails appear
broken, but this one works:

http://seti.weebly.com/channel.html

In particular, have a look at the one titled "A Paper Which Derives a
4-Dimensional Rubik’s Cube Permutation Formula", section 4, "The 3^4 Cube".

I hope this is useful information, as far as collecting arguments with the
end goal of more rigor.

Roice