Message #3976

From: Eduard Baumann <ed.baumann@bluewin.ch>
Subject: Re: [MC4D] Puzzle Group Centers and Superflips [3 Attachments]
Date: Sat, 20 Jan 2018 12:25:08 +0100

Wow !!
Marveleous world of GAP.
I had to find out what GAP means!

Groups, Algorithms and Programming

Not everybody is familiar with that or knows that GAP exists.

Best regards
Ed

—– Original Message —–
From: Roice Nelson roice3@gmail.com [4D_Cubing]
To: 4D_Cubing@yahoogroups.com
Sent: Saturday, January 20, 2018 5:09 AM
Subject: [MC4D] Puzzle Group Centers and Superflips [3 Attachments]

[Attachment(s) from Roice Nelson included below]

Hi all,

I looked into the 3^4 group center question using GAP. I wrote a short program to spit out the generators as a permutation group that GAP can read. I’ve attached files for both the 3^4 group and the Klein Quartic puzzle group, in case you want to investigate other properties yourself (I found this Rubik’s cube example useful). You can load these files with the GAP Read command. Once loaded, grab the size of the group in GAP with:

Size( puzzle );

Calculate and display the center of the group with:

  z &#58;= Centre( puzzle );<br>
  GeneratorsOfGroup( z );

For both puzzles, the group centers are like the original Rubik’s cube. They have only one non-trivial element that flips all 2-colored pieces. In the case of the 3^4, I was hoping the "superflip" move would also reorient the corners in place with double swaps. That seemed pretty but alas, it only reorients the 24 2C pieces.

If you aren’t familiar with the center of a group, it is a subgroup that contains all elements that commute with every element of the group. So if you made a macro to execute the superflip S, then for any other sequence of moves X, SXS’ = X. (Since these superflips are order 2, S is also its own inverse, S = S’.) If a center is the entire group, the group is abelian and probably not an interesting permutation puzzle. You might say (roughly) that groups with small centers are less abelian.

It is interesting to me that GAP can quickly find the center in a group with so many elements. Also worthy of note: it took just a few seconds to find it for the 3^4, but a few minutes for KQ. I don’t know why.

Conjecture in analogy to the 3^3: The KQ and 3^4 superflips are as far away from pristine as the diameter of the state space. That is, it requires at least God’s number of moves to get to these positions. Also in analogy to the 3^3, I bet it will be much easier to verify the minimum number of moves to reach a superflip than to verify that this count is God’s number.

Cheers,
Roice