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It appears I messed up the KQ superflip picture (it has some unintended
transparency). A nicer version without that problem is here:
https://photos.app.goo.gl/Z0PgeEZAJBx8wtS62
Best,
Roice
On Fri, Jan 19, 2018 at 10:09 PM, Roice Nelson roice3@gmail.com [4D_Cubing]
<4D_Cubing@yahoogroups.com> wrote:
> [Attachment(s) <#m_-3120951100068111505_TopText> from Roice Nelson
> included below]
>
> Hi all,
>
> I looked into the 3^4 group center
>
> 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
>
> load these files with the GAP Read command. Once loaded, grab the size o=
f
> the group in GAP with:
>
> Size( puzzle );
>
> Calculate and display the center of the group with:
>
> z :=3D Centre( puzzle );
> 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 reorien=
t
> 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 i=
f
> you made a macro to execute the superflip S, then for any other sequence =
of
> moves X, SXS' =3D X. (Since these superflips are order 2, S is also its =
own
> inverse, S =3D S'.) If a center is the entire group, the group is abelia=
n
> 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 t=
o
> 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. Al=
so
> 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
>
>
>=20
>
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me unintended transparency).=C2=A0 A nicer version without that problem is =
here: https://photo=
s.app.goo.gl/Z0PgeEZAJBx8wtS62
i, Jan 19, 2018 at 10:09 PM, Roice Nelson m">roice3@gmail.com [4D_Cubing] <:4D_Cubing@yahoogroups.com" target=3D"_blank">4D_Cubing@yahoogroups.com=
> wrote:
0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
=20=20=20=20=20=20=20=20
[n:none" href=3D"#m_-3120951100068111505_TopText">Attachment(s) from Roi=
ce Nelson included below]
k">group center=C2=A0question using GAP.=C2=A0 I wrote a short program =
to spit out the generators as a permutation group that GAP can read.=C2=A0 =
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=
=C2=A0=3D"_blank">this Rubik's cube example=C2=A0useful).=C2=A0 You can l=
oad these files with the GAP Read command.=C2=A0 Once loaded, grab the size=
of the group in GAP with:
zle );
group with:
puzzle );
bik's cube.=C2=A0 They have only one non-trivial element that flips all=
2-colored pieces.=C2=A0 In the case of the 3^4, I was hoping the "sup=
erflip" move would also reorient the corners in place with double swap=
s.=C2=A0 That seemed pretty but alas, it only reorients the 24 2C pieces.=
=C2=A0=C2=A0
enter of a group, it is a subgroup that contains all elements that commute =
with every element of the group.=C2=A0 So if you made a macro to execute th=
e superflip S, then for any other sequence of moves X, SXS' =3D X.=C2=
=A0 (Since these superflips are order=C2=A02, S is also its own inverse, S =
=3D S'.)=C2=A0 If a center is the entire group, the group is abelian an=
d probably not an interesting permutation puzzle.=C2=A0 You might say (roug=
hly) that groups with small centers are less abelian.
div>It is interesting to me that GAP can quickly find the center in a group=
with so many elements.=C2=A0 Also worthy of note: it took just a few secon=
ds to find it for the 3^4, but a few minutes for KQ.=C2=A0 I don't know=
why.
The KQ and 3^4 superflips are as far away from pristine as the diameter of=
the state space.=C2=A0 That is, it=C2=A0requires at least God's number=
of moves to get to these positions.=C2=A0 Also in analogy to the 3^3, I be=
t it will be much easier to verify the minimum number of moves to reach a s=
uperflip than to verify that this count is God's number.
=
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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 -----=20
From: Roice Nelson roice3@gmail.com [4D_Cubing]=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Saturday, January 20, 2018 5:09 AM
Subject: [MC4D] Puzzle Group Centers and Superflips [3 Attachments]
=20=20=20=20
[Attachment(s) from Roice Nelson included below]
=20=20=20
Hi all,
I looked into the 3^4 group center question using GAP. I wrote a short p=
rogram 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 g=
roup, in case you want to investigate other properties yourself (I found th=
is Rubik's cube example useful). You can load these files with the GAP Rea=
d command. Once loaded, grab the size of the group in GAP with:
Size( puzzle );
Calculate and display the center of the group with:
z :=3D Centre( puzzle );
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 o=
nly reorients the 24 2C pieces.=20=20
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' =3D X. (Since these superflips are order 2, S is also its o=
wn inverse, S =3D S'.) If a center is the entire group, the group is abeli=
an and probably not an interesting permutation puzzle. You might say (roug=
hly) that groups with small centers are less abelian.
It is interesting to me that GAP can quickly find the center in a group w=
ith so many elements. Also worthy of note: it took just a few seconds to f=
ind 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 aw=
ay 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 mo=
ves to reach a superflip than to verify that this count is God's number.
Cheers,
Roice
=20=20
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