Thread: "Cube puzzles and math"

From: e.kennedy.a@gmail.com
Date: Sun, 26 Jun 2016 16:41:22 +0200
Subject: Cube puzzles and math



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Hi!

A bit of a quick (and short) answer. I believe that the two areas of
mathematics that are most closely related to Rubik's Cube and similar
puzzles (aka twisty puzzles) are group theory and combinatorics.
Combinatorics, obviously, for calculating the number of configurations of
the puzzles (although some group theory might be required as well).

Group theory is the branch of mathematics that really describe the dynamics
of twisty puzzles. In general, group theory is a branch of mathematics that
studies groups (which are a kind of abstract algebraic structure). Twisty
puzzles can be seen as a special kind of group, namely a permutation group,
where each sticker is a member of the group. It's through group theory that
twisty puzzles are analyzed and we can prove what restrictions a twisty
puzzle has by using group theory. Moreover, the concept of commutators and
conjugates comes from group theory so it also helpful when trying to solve
twisty puzzles. If you would like to get an introduction to group theory
and how it can be applied on Rubik's Cube I highly recommend that you
read *Group
Theory via Rubik's Cube *by Tom Davis (
http://www.geometer.org/rubik/group.pdf)

Best regards,
Joel

2016-06-26 15:22 GMT+02:00 e.kennedy.a@gmail.com [4D_Cubing] <
4D_Cubing@yahoogroups.com>:

>
>
> Hello everyone!
>
> There has been a question on my mind for some time and after solving the
> 3x3x3x3 recently I getting to know that this group existed I thought that=
I
> could manage to find the answer here.
>
> I would like to know what's the connection between maths and puzzles like
> the rubik's cube. What specific areas of mathematics explain how rubik's
> cube dinamics work?
>
> What do you think?
>
>
> P.S.: I was also wondering why does parity also exist in a 4x4x4 which
> orientable center pieces like the 4x4x4 axis cube.
>
>
> Thank you and best wishes!
>
>=20
>

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Hi!

A bit of a quick (and short) answer. I believe =
that the two areas of mathematics that are most closely related to Rubik=
9;s Cube and similar puzzles (aka twisty puzzles) are group theory and comb=
inatorics. Combinatorics, obviously, for calculating the number of configur=
ations of the puzzles (although some group theory might be required as well=
).

Group theory is the branch of mathematics that really describe th=
e dynamics of twisty puzzles. In general, group theory is a branch of mathe=
matics that studies groups (which are a kind of abstract algebraic structur=
e). Twisty puzzles can be seen as a special kind of group, namely a permuta=
tion group, where each sticker is a member of the group. It's through g=
roup theory that twisty puzzles are analyzed and we can prove what restrict=
ions a twisty puzzle has by using group theory. Moreover, the concept of co=
mmutators and conjugates comes from group theory so it also helpful=C2=A0wh=
en trying to solve twisty puzzles. If you would like to get an introduction=
to group theory and how it can be applied on Rubik's Cube I highly rec=
ommend that you read=C2=A0Group Theory via Rubik's Cube by Tom D=
avis (http://www.geomet=
er.org/rubik/group.pdf)

Best regards,
Joel
"gmail_extra">
2016-06-26 15:22 GMT+02:00 href=3D"mailto:e.kennedy.a@gmail.com">e.kennedy.a@gmail.com [4D_Cubing]=
<=3D"_blank">4D_Cubing@yahoogroups.com>:
=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padd=
ing-left:1ex">












=20

=C2=A0







=20=20=20=20=20=20
=20=20=20=20=20=20

Hello everyone!=C2=A0

There has be=
en a question on my mind for some time and after solving the 3x3x3x3 recent=
ly I getting to know that this group existed I thought that I could manage =
to find the answer here.=C2=A0

I would like to know what=
's the connection between maths and puzzles like the rubik's cube. =
What specific areas of mathematics explain how rubik's cube dinamics wo=
rk?=C2=A0

What do you think?=C2=A0

r>

P.S.: I was also wondering why does parity also exist in a =
4x4x4 which orientable center pieces like the 4x4x4 axis cube.


p>

Thank you and best wishes!




=20=20=20=20=20

=20=20=20=20







=20=20









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From: "Eduard Baumann" <ed.baumann@bluewin.ch>
Date: Sun, 26 Jun 2016 16:46:23 +0200
Subject: Re: [MC4D] Cube puzzles and math



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Rubik and Math, see:
Christoph Bandelow, "Einf=C3=BChrung in Cubologie", Verlag Vieweg.

Best regards
Ed Baumann

----- Original Message -----=20
From: e.kennedy.a@gmail.com [4D_Cubing]=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Sunday, June 26, 2016 3:22 PM
Subject: [MC4D] Cube puzzles and math


=20=20=20=20

Hello everyone!=20

There has been a question on my mind for some time and after solving the =
3x3x3x3 recently I getting to know that this group existed I thought that I=
could manage to find the answer here.=20

I would like to know what's the connection between maths and puzzles like=
the rubik's cube. What specific areas of mathematics explain how rubik's c=
ube dinamics work?=20

What do you think?=20




P.S.: I was also wondering why does parity also exist in a 4x4x4 which or=
ientable center pieces like the 4x4x4 axis cube.




Thank you and best wishes!


=20=20
------=_NextPart_000_0054_01D1CFCA.4D6CE2A0
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charset="UTF-8"
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=EF=BB=BF




Rubik and Math, see:

Christoph Bandelow, "Einf=C3=BChrung in Cu=
bologie",=20
Verlag Vieweg.

 

Best regards

Ed Baumann

 

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:e.kennedy.a@gmail.com [4D_Cubing]">e.kennedy.a@gmail.com=20
[4D_Cubing]
To: ps.com=20
href=3D"mailto:4D_Cubing@yahoogroups.com">4D_Cubing@yahoogroups.com <=
/DIV>
Sent: Sunday, June 26, 2016 3:22 P=
M

Subject: [MC4D] Cube puzzles and=20
math


 =20


Hello everyone! 


There has been a question on my mind for some time and after sol=
ving=20
the 3x3x3x3 recently I getting to know that this group existed I thought =
that=20
I could manage to find the answer here. 


I would like to know what's the connection between maths and puz=
zles=20
like the rubik's cube. What specific areas of mathematics explain how rub=
ik's=20
cube dinamics work? 


What do you think? 




P.S.: I was also wondering why does parity also exist in a 4x4x4 which=
=20
orientable center pieces like the 4x4x4 axis cube.




Thank you and best wishes!




------=_NextPart_000_0054_01D1CFCA.4D6CE2A0--



From: Sid Brown <synjanbrown@gmail.com>
Date: Sun, 26 Jun 2016 15:48:56 +0100
Subject: Re: [MC4D] Cube puzzles and math



From: Sid Brown <synjanbrown@gmail.com>
Date: Sun, 26 Jun 2016 15:56:52 +0100
Subject: Re: [MC4D] Cube puzzles and math



From: Sid Brown <synjanbrown@gmail.com>
Date: 26 Jun 2016 12:57:43 -0700
Subject: Re: [MC4D] Cube puzzles and math



From: apturner@mit.edu
Date: Mon, 27 Jun 2016 15:49:49 +0100
Subject: Re: Cube puzzles and math



From: Sid Brown <synjanbrown@gmail.com>
Date: 27 Jun 2016 13:07:21 -0700
Subject: Re: [MC4D] Re: Cube puzzles and math



From: apturner@mit.edu
Date: 27 Jun 2016 16:51:22 -0700
Subject: Re: Cube puzzles and math



From: apturner@mit.edu
Date: 27 Jun 2016 23:14:09 -0700
Subject: Re: Cube puzzles and math



From: apturner@mit.edu
Date: Tue, 28 Jun 2016 08:09:59 +0100
Subject: Re: Cube puzzles and math



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Andrew,

I thought about defining equivalence within my group but it is not
intuitive as as you have to convert both configurations into Joel's group.
Converting from my group into Joel's one is pretty straight forward but
converting back again is difficult as you must follow a solution algorithm
and invert it. Modding into the smallest member of the group is an
extremely difficult task and I would not know where to start, maybe using
equivalence rules for sub words in the language? I'm not sure. But if this
is possible then modding out to the smallest representation (which I'm
coining as the prime numbers of this group) will give you the shortest
possible solution for the given Rubik's cube configuration. Although
verifying that is the smallest is also a very difficult task; the only way
I have thought of doing this so far is a kind of brute force algorithm
which would take an age to complete. What may be easier is verifying that
the equivalence rules for reduction is fully comprehensive, but again I'm
not sure how to prove this.

Any thoughts on this?

Regards,
Sid

On 28 June 2016 at 07:14, apturner@mit.edu [4D_Cubing] <
4D_Cubing@yahoogroups.com> wrote:

>
>
> Dear Sid,
>
> Yes, the object you defined is simply a nonabelian (non-commutative) grou=
p.
>
> The comment you made about mapping to Joel's group and back is
> interesting, and accurate, since you never actually mentioned taking the
> quotient. The set R =3D S* (with * the Kleene star) does have an infinite
> number of elements that correspond to each physical state of the Rubik's
> cube. With the group operation you described and no other relations in th=
e
> presentation of the group, you have the free group over the set S. To
> actually get the Rubik's Cube group (the group that Joel describes), you
> have to take the quotient by the equivalence relation that relates all
> sequences of moves resulting in the same physical state, R/~. An equivale=
nt
> approach would be to add relations to the presentation of the group (such
> as L^4 =3D e, because rotating the left face four quarter turns is equiva=
lent
> to the identity). Really, it's R/~ that is isomorphic to Joel's group, no=
t
> the free group as you described it. In all my previous posts I had assume=
d
> you were modding out the equivalent move sequences.
>
> Thanks for the discussion!
>
> Cheers,
> Andrew
>
>=20
>

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Andrew,

I thought about defining equiva=
lence within my group but it is not intuitive as as you have to convert bot=
h configurations into Joel's group. Converting from my group into Joel&=
#39;s one is pretty straight forward but converting back again is difficult=
as you must follow a solution algorithm and invert it. Modding into the sm=
allest member of the group is an extremely difficult task and I would not k=
now where to start, maybe using equivalence rules for sub words in the lang=
uage? I'm not sure. But if this is possible then modding out to the sma=
llest representation (which I'm coining as the prime numbers of this gr=
oup) will give you the shortest possible solution for the given Rubik's=
cube configuration. Although verifying that is the smallest is also a very=
difficult task; the only way I have thought of doing this so far is a kind=
of brute force algorithm which would take an age to complete. What may be =
easier is verifying that the equivalence rules for reduction is fully compr=
ehensive, but again I'm not sure how to prove this.

>
Any thoughts on this?

Regards,
Sid=
=C2=A0

>On 28 June 2016 at 07:14, apturner@mit=
.edu [4D_Cubing] <ogroups.com" target=3D"_blank">4D_Cubing@yahoogroups.com> wro=
te:
left:1px #ccc solid;padding-left:1ex">












=20

=C2=A0







=20=20=20=20=20=20
=20=20=20=20=20=20

Dear Sid,


Yes, the object you defined is si=
mply a nonabelian (non-commutative) group.

The com=
ment you made about mapping to Joel's group and back is interesting, an=
d accurate, since you never actually mentioned taking the quotient. The set=
R =3D S* (with * the Kleene star) does have an infinite number of elements=
that correspond to each physical state of the Rubik's cube. With the g=
roup operation you described and no other relations in the presentation of =
the group, you have the free group over the set S. To actually get the Rubi=
k's Cube group (the group that Joel describes), you have to take the qu=
otient by the equivalence relation that relates all sequences of moves resu=
lting in the same physical state, R/~. An equivalent approach would be to a=
dd relations to the presentation of the group (such as L^4 =3D e, because r=
otating the left face four quarter turns is equivalent to the identity). Re=
ally, it's R/~ that is isomorphic to Joel's group, not the free gro=
up as you described it. In all my previous posts I had assumed you were mod=
ding out the equivalent move sequences.

Thanks for=
the discussion!

Cheers,
Andrew

=




=20=20=20=20=20

=20=20=20=20







=20=20









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From: apturner@mit.edu
Date: 28 Jun 2016 00:36:20 -0700
Subject: Re: [MC4D] Re: Cube puzzles and math



From: apturner@mit.edu
Date: Tue, 28 Jun 2016 08:42:26 +0100
Subject: Re: [MC4D] Re: Cube puzzles and math



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Andrew,

Yeah I understand that you are talking about it quite abstractly. But what
I aim to do is program an algorithm that can always determine the shortest
possible solution. I can implement my brute force one for a 3x3x3 as that
won't take too long, but for 3^4 it would take too long. I will use the
brute force of 3^3 to verify an algorithm of reduction of 3^3 and then
attempt to expand this to 3^4, but what I'm trying to do might prove near
impossible and I'm not sure whether I will actually achieve it. I will have
a decent stab at it though.

Sid

On 28 June 2016 at 08:36, apturner@mit.edu [4D_Cubing] <
4D_Cubing@yahoogroups.com> wrote:

>
>
> Sid,
>
> You're talking about actually finding an explicit presentation of the
> Rubik's Cube group. Just from some quick Googling, it looks like there ha=
s
> been a fair amount of discussion of this online, but no definitive result=
s
> as far as I can see.
>
> But in terms of just talking about the group, it isn't necessary to
> actually determine the presentation. We still understand what it means to
> say that we quotient out the free group by the equivalence relation that
> relates physically equivalent states.
>
> Another way to think about it, which sort of bridges the gap between your
> formulation and Joel's, is given on Wikipedia. They just consider the 48
> movable stickers on the cube, and the permutations of these stickers that
> correspond to the moves L, R, U, D, F, B, and then say that the Rubik's
> Cube group is the subset of the symmetric group S_{48} generated by those
> permutations: < L, R, U, D, F, B>. This phrases the setup in terms of
> moves of the cube, but automatically deals with the physically equivalent
> states, because it inherits the relations from the presentation of S_{48}=
.
> In fact, this approach gives you a presentation of the Rubik's Cube group=
,
> in the form < L, R, U, D, F, B| all relations of S_{48}>. This
> presentation is almost certainly not the smallest, however!
>
> Cheers,
> Andrew
>
>
>=20
>

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Andrew,

Yeah I understand that you are =
talking about it quite abstractly. But what I aim to do is program an algor=
ithm that can always determine the shortest possible solution. I can implem=
ent my brute force one for a 3x3x3 as that won't take too long, but for=
3^4 it would take too long. I will use the brute force of 3^3 to verify an=
algorithm of reduction of 3^3 and then attempt to expand this to 3^4, but =
what I'm trying to do might prove near impossible and I'm not sure =
whether I will actually achieve it. I will have a decent stab at it though.=

Sid

v class=3D"gmail_quote">On 28 June 2016 at 08:36, r@mit.edu">apturner@mit.edu [4D_Cubing] <=3D"mailto:4D_Cubing@yahoogroups.com" target=3D"_blank">4D_Cubing@yahoogrou=
ps.com> wrote:
margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">












=20

=C2=A0







=20=20=20=20=20=20
=20=20=20=20=20=20

Sid,


You're talking about actually find=
ing an explicit presentation of the Rubik's Cube group. Just from some =
quick Googling, it looks like there has been a fair amount of discussion of=
this online, but no definitive results as far as I can see.

=
But in terms of just talking about the group, it isn't neces=
sary to actually determine the presentation. We still understand what it me=
ans to say that we quotient out the free group by the equivalence relation =
that relates physically equivalent states.

Another=
way to think about it, which sort of bridges the gap between your formulat=
ion and Joel's, is given on Wikipedia. They just consider the 48 movabl=
e stickers on the cube, and the permutations of these stickers that corresp=
ond to the moves L, R, U, D, F, B, and then say that the Rubik's Cube g=
roup is the subset of the symmetric group S_{48} generated by those permuta=
tions: <=C2=A0word-spacing:normal">L, R, U, D, F, B>. This phrases the setup in terms =
of moves of the cube, but automatically deals with the physically equivalen=
t states, because it inherits the relations from the presentation of S_{48}=
. In fact, this approach gives you a presentation of the Rubik's Cube g=
roup, in the form=C2=A0<n>=C2=A0ing:normal">L, R, U, D, F, B| all relations of S_{48}word-spacing:normal">>. This presentation is almost certainly not the sm=
allest, however!

<=
/span>
Cheers,
iv>Andrew
=C2=A0
=




=20=20=20=20=20

=20=20=20=20







=20=20









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From: phamthihoa4444@gmail.com
Date: 29 Jun 2016 11:12:28 -0700
Subject: Re: [MC4D] Re: Cube puzzles and math




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