Message #3974

From: Roice Nelson <roice3@gmail.com>
Subject: Puzzle Group Centers and Superflips
Date: Fri, 19 Jan 2018 22:09:40 -0600

Hi all,

I looked into the 3^4 group center
<https://en.wikipedia.org/wiki/Center_(group_theory)> 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
<https://www.gap-system.org/Doc/Examples/rubik.html> 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