Thread: "Puzzle Group Centers and Superflips [3 Attachments]"

From: Roice Nelson <roice3@gmail.com>
Date: Fri, 19 Jan 2018 22:37:12 -0600
Subject: Re: [MC4D] Puzzle Group Centers and Superflips [3 Attachments]



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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
> 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 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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It appears I messed up the KQ superflip picture (it has so=
me unintended transparency).=C2=A0 A nicer version without that problem is =
here: https://photo=
s.app.goo.gl/Z0PgeEZAJBx8wtS62

Best,
Roice=

On Fr=
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]








Hi all,

I looked into the 3^4=C2=A0ref=3D"https://en.wikipedia.org/wiki/Center_(group_theory)" target=3D"_blan=
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:

=C2=A0 =C2=A0 Size( puz=
zle );

Calculate and display the center of the=
group with:

=C2=A0 =C2=A0 z :=3D Centre(=
puzzle );
=C2=A0 =C2=A0 GeneratorsOfGroup( z );
=

For both puzzles, the group centers are like the original Ru=
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

If you aren't familiar with the c=
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.

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.=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.

=
Cheers,
Roice
























--001a114a5d722e57b505632dc089--




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



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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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charset="UTF-8"
Content-Transfer-Encoding: quoted-printable

=EF=BB=BF




Wow !!

Marveleous world of GAP.

I had to find out what GAP means!DIV>
=
 

style=3D"TEXT-ALIGN: justify; WIDOWS: 1; TEXT-TRANSFORM: none; BACKGROUND-C=
OLOR: rgb(255,255,255); TEXT-INDENT: 0px; DISPLAY: inline !important; FONT:=
medium 'Times New Roman'; WHITE-SPACE: normal; FLOAT: none; LETTER-SPACING=
: normal; COLOR: rgb(0,0,0); WORD-SPACING: 0px; -webkit-text-stroke-width: =
0px">Groups,=20
Algorithms and Programming

 

Not everybody is familiar with that or kno=
ws that=20
GAP exists.

 

Best regards

Ed

 

 

 

style=3D"BORDER-LEFT: #000000 2px solid; PADDING-LEFT: 5px; PADDING-RIGHT: =
0px; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px">
----- Original Message -----

style=3D"FONT: 10pt arial; BACKGROUND: #e4e4e4; font-color: black">Fro=
m:
=20
href=3D"mailto:roice3@gmail.com [4D_Cubing]">Roice Nelson roice3@gmail.co=
m=20
[4D_Cubing]

To: ps.com=20
href=3D"mailto:4D_Cubing@yahoogroups.com">4D_Cubing@yahoogroups.com
<=
/DIV>
Sent: Saturday, January 20, 2018 5=
:09=20
AM

Subject: [MC4D] Puzzle Group Cente=
rs and=20
Superflips [3 Attachments]


 =20


Hi all,


I looked into the 3^4  href=3D"https://en.wikipedia.org/wiki/Center_(group_theory)" target=3D_bl=
ank>group=20
center question using GAP.  I wrote a short program to spit=
out=20
the generators as a permutation group that GAP can read.  I've attac=
hed=20
files for both the 3^4 group and the Klein Quartic puzzle group, in case =
you=20
want to investigate other properties yourself (I found  href=3D"https://www.gap-system.org/Doc/Examples/rubik.html" target=3D_bla=
nk>this=20
Rubik's cube example useful).  You can load these files wit=
h the=20
GAP Read command.  Once loaded, grab the size of the group in GAP=20
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=20
cube.  They have only one non-trivial element that flips all 2-color=
ed=20
pieces.  In the case of the 3^4, I was hoping the "superflip" move w=
ould=20
also reorient the corners in place with double swaps.  That seemed p=
retty=20
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=20
contains all elements that commute with every element of the group. =
So=20
if you made a macro to execute the superflip S, then for any other sequen=
ce of=20
moves X, SXS' =3D X.  (Since these superflips are order 2, S is=
also=20
its own inverse, S =3D S'.)  If a center is the entire group, the gr=
oup is=20
abelian and probably not an interesting permutation puzzle.  You mig=
ht=20
say (roughly) that groups with small centers are less abelian.



It is interesting to me that GAP can quickly find the center in a gr=
oup=20
with so many elements.  Also worthy of note: it took just a few seco=
nds=20
to find it for the 3^4, but a few minutes for KQ.  I don't know=20
why.



Conjecture in analogy to the 3^3: The KQ and 3^4 superflips a=
re as=20
far away from pristine as the diameter of the state space.  That is,=
=20
it requires at least God's number of moves to get to these=20
positions.  Also in analogy to the 3^3, I bet it will be much easier=
to=20
verify the minimum number of moves to reach a superflip than to verify th=
at=20
this count is God's number.



Cheers,

Roice



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