Message #3463

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Calculate number of permutation of restricted cube
Date: Sun, 17 Jul 2016 21:36:49 -0500

Very cool investigation, and I’ve been thinking about this. To start, I
looked up the factors of 1344, and was struck by some of them (24, 42, 56,
84, 168, … check out this article by John Baez
<https://johncarlosbaez.wordpress.com/2013/05/25/42/> for some
justification of why these numbers are so compelling). I was also
surprised 5 was not there.

When applying half-turns to the 2^4, it is quickly clear that there are two
independent orbits for cubies like Hao said, and that orientation need not
be considered - you can’t return a piece to its original position in a
different orientation.

GAP is one way to verify questions like this (see this Rubik’s Cube example
<http://www.gap-system.org/Doc/Examples/rubik.html>), so I used that to
check Hao’s brute force code. I got the same size for the group (1344),
additional evidence he is correct. One nice thing about GAP is you can ask
further questions about the group structure, so I could see that the
restricted 2^3 has the structure of the group S4
<http://groupprops.subwiki.org/wiki/Symmetric_group:S4>. The restricted
2^4 has the structure of the group (C2 x C2 x C2) : PSL(2,7), meaning
a semidirect
product <https://en.wikipedia.org/wiki/Semidirect_product> of (a direct
product of) cyclic groups and the group PSL(2,7). PSL(2,7) is the group of
symmetries of the Klein Quartic and so many other things, and explains why
the factors contain all those interesting numbers.

So I think there is lots more to understand here. I still don’t have a
nice intuition about why 5 is not a factor of the order of the group, or
why PSL(2,7) shows up. And while this might not be the most difficult
puzzle, it is a special one with a special group structure!

Best,
Roice

A few post scripts…

Searching for "restricted pocket cube" didn’t find much, but found this
page: http://anttila.ca/michael/devilsalgorithm/ Melinda has a mention
there :) It is helpful to think about the 3D pocket cube as a warm up to
the 4D case.

I can (relatively easily) produce graph definition files from groups in
GAP, so if there is interest I will do that. I may go ahead and do it
anyway. I’ll attach my current GAP script for those who want to dig
further, and I can describe how I generated it if desired.

On Fri, Jul 1, 2016 at 1:04 AM, phamthihoa4444@gmail.com [4D_Cubing] <
4D_Cubing@yahoogroups.com> wrote:

>
>
> Why should I visualize it? What I need to prove now is why only 1344
> positions are reachable. But is 1344 too large to visualize?
>
> Just like how permutation of 2^4 = (15!/2) * (12^14 * 4) is calculated,
> hold one corner piece in its place and rotate the rest.
>
> I tried to solve the puzzle by MPUlt (the structure of puzzle file is
> really difficult to understand, and I have not understand many parts yet)
> and can solve it intuitively, then I think number of permutation is correct.
>
> Computer brute force given:
> + Any 7 cubies in same parity cannot be moved without affecting other
> pieces. Thus 3, 4, 5, 6 and 7-cycle are all impossible.
> + Any 4 cubies in same parity cannot be moved if 4 other cubies in that
> parity is fixed and the other parity can be moved.
> (let a piece be even if all edge-adjacent pieces of it are odd and vice
> versa)
>
>
>
>