# I'm writing a group theory essay



## Jude (Dec 7, 2011)

At my uni there's a compulsory essay that all 2nd year maths students have to do, on whatever we like. The only requirements is that the maths is advanced enough that it would take a 2nd year a few times reading through before they fully get it.

So, my choice of topic is group theory and it's applications in puzzles - e.g. rubik's cube. My question is, can anyone recommend any particularly interesting theorems/papers/books etc I could research? I need some sort of goal to work towards, some sort of objective. For example, I could work towards proving the maximum minimum amount of moves needed to solve the rubik's cube but I spoke to my tutor and he said that proof was probably a bit too advanced for this essay. Are there any other significant results in the mathematics behind cubing (or not neccessarily cubing but a puzzle of some sort) that I could try and prove?

Thanks 


EDIT: Finished essay can be downloaded here: http://www28.zippyshare.com/v/97472928/file.html


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## wontolla (Dec 7, 2011)

Although I don't know what level of math corresponds to "2nd year math". I suggest to write about the calculation of all possible solved states of a 7x7, for example.

Regarding the maximum number of moves to solve a 3x3, I think that no one ever has proved God's number to be 20 mathematically, so your tutor is probably right. But you can still write about the runs made by Google to obtain that number.


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## Ilkyoo Choi (Dec 7, 2011)

God's number and the Devil's number are good topics, but there is no pure math proof as of now. Describing the group theoretical structure of the "Rubik's Cube group" might be a topic that "would take a 2nd year a fe times reading through before they fully get it."

Look into the following book: "Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys", by David Joyner.


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## Lucas Garron (Dec 7, 2011)

I could try to link to them, but just search for "group theory" here to find a bunch of threads with decent discussion and references.


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## LNZ (Dec 7, 2011)

The devil's algorithm has been proved for the 1x3x3 Floppy Cube. So the idea that a devils algorithm is real. But computing this for other puzzles is much harder than doing Gods's algorithm for the same puzzle.


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## mrCage (Dec 7, 2011)

wontolla said:


> Although I don't know what level of math corresponds to "2nd year math". I suggest to write about the calculation of all possible solved states of a 7x7, for example.


 
Great! I get my name in the essay 

Per


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## mrCage (Dec 9, 2011)

Another great topic for a short essay. *Show that the 2x2x2 cube can always be solved with turning 3 layers only.* Those 3 layers should be all-adjacent, like U-F-R or D-B-L etc. Show also that non adjacent 3 layers like U-F-D won't work (in general).

Per


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## Krag (Dec 9, 2011)

http://geometer.org/rubik/group.pdf
This might be helpfull even though it might be too simple for 2nd. math student but you might get some ideas from it.


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## macky (Dec 9, 2011)

I second Ilkyoo's suggestion, to write down the structure of the cube group.



mrCage said:


> Another great topic for a short essay. *Show that the 2x2x2 cube can always be solved with turning 3 layers only.*


That's obvious.



> Those 3 layers should be all-adjacent, like U-F-R or D-B-L etc. Show also that non adjacent 3 layers like U-F-D won't work (in general).


That's false.


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## ASH (Dec 9, 2011)

Group Theory is soooo boring.

Do a essay on proper math, like nonlinear analysis!? 
There is a lot of fun stuff!

(Hairy ball, ham-sandwich ...)


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## chris410 (Dec 9, 2011)

Here is a paper you may find helpful: 

http://www.google.com/url?sa=t&rct=...sg=AFQjCNGLZnP0ZQAeMA7QiURQr-b0DA34OQ&cad=rja


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## Cielo (Dec 9, 2011)

I've read _Inside the Rubik's cube and beyond_(by Christoph Bandelow) and _Handbook of cubik math_(by A.H.Frey & D.Singmaster). These might help.



macky said:


> That's false.


I tried and solved, turning only L-U-R. 
At first, I only thought of the alg I used: R' U L' U2 R U' R' U2 R2. Then I realized that L' R, then F turns into U.


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## Lucas Garron (Dec 9, 2011)

Anyone up for starting a wiki with references and links for (group) theory resources?


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## macky (Dec 9, 2011)

[wiki]List of Puzzle Theory References[/wiki]
Go crazy.


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## cmhardw (Dec 10, 2011)

Jaap's articles on puzzles are very interesting. I found this one particularly interesting. In particular I like his discussion about the center of the cube group. I've used the trick he describes here on kids, and they find it very neat (plus the math supporting why it works is very interesting).


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## Ilkyoo Choi (Dec 12, 2011)

ASH said:


> Group Theory is soooo boring.
> 
> Do a essay on proper math, like nonlinear analysis!?


 
Oh, such a delicate topic.


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## blah (Dec 12, 2011)

nonstandard analysis


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## mrCage (Dec 13, 2011)

ASH said:


> Group Theory is soooo boring.


 
I found GT boring also at uni, mainly because there was no good real work examples to demonstrate the theory. The cube and other puzzles are great visual aids!!

Per


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## Jude (Dec 22, 2011)

lots of great ideas here, thanks guys! i'll let you know how it turns out


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## Jude (Mar 6, 2012)

Hm, so I have a conjecture which I'm not keen to focus on because it's not very interesting, but just out of curiosity is it true that the group of all the states reachable by peeling off the stickers and sticking back on them anywhere is isomorphic to S56/S9? I think the cardinality of that group is right and there's what appears to me to be an intuitive isomorphism but I haven't tried proving it at all. 

FWIW by S56 I mean the Symmetric Group on 56 letters.


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## cuBerBruce (Mar 8, 2012)

Jude said:


> Hm, so I have a conjecture which I'm not keen to focus on because it's not very interesting, but just out of curiosity is it true that the group of all the states reachable by peeling off the stickers and sticking back on them anywhere is isomorphic to S56/S9? I think the cardinality of that group is right and there's what appears to me to be an intuitive isomorphism but I haven't tried proving it at all.
> 
> FWIW by S56 I mean the Symmetric Group on 56 letters.


If you consider the unstickered cubies and their orientations to be distinguishable, I believe there are \( 54!/(9!^6) \) possible stickerings. If you consider the orientation of the cube not to matter, it would be approximately 24 times fewer (although I believe the exact number to be rather tricky to calculate). As with the 4x4x4 cube with its indistinguishable pieces, these arrangements of indistinguishable stickers do not form a group. Also S56/S9 (or S54/S9, if that's what you meant) is not a group, since S9 is not a normal subgroup of S56 (or S54). Even though those coset spaces are not groups, we may still be able to talk about an isomorphism, provided the the number of elements matched (although that is not the case here either).

It might also be a somewhat interesting question to determine the total number of possible stickerings if we allow stickerings to be considered equivalent if one can be converted to the other by some element of <U,x,y>. One might also allow stickerings to be considered equivalent if one can be converted to the other by disassembling and reassembling the cube. (I am assuming the stickers to have square symmetry, are a solid color with no markings, and are divided into 6 sets of 9 indistinguishable stickers.)


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## Jude (Mar 8, 2012)

cuBerBruce said:


> If you consider the unstickered cubies and their orientations to be distinguishable, I believe there are \( 54!/(9!^6) \) possible stickerings. If you consider the orientation of the cube not to matter, it would be approximately 24 times fewer (although I believe the exact number to be rather tricky to calculate). As with the 4x4x4 cube with its indistinguishable pieces, these arrangements of indistinguishable stickers do not form a group. Also S56/S9 (or S54/S9, if that's what you meant) is not a group, since S9 is not a normal subgroup of S56 (or S54). Even though those coset spaces are not groups, we may still be able to talk about an isomorphism, provided the the number of elements matched (although that is not the case here either).
> 
> It might also be a somewhat interesting question to determine the total number of possible stickerings if we allow stickerings to be considered equivalent if one can be converted to the other by some element of <U,x,y>. One might also allow stickerings to be considered equivalent if one can be converted to the other by disassembling and reassembling the cube. (I am assuming the stickers to have square symmetry, are a solid color with no markings, and are divided into 6 sets of 9 indistinguishable stickers.)


 
Excellent post, everything there makes perfect sense and sounds spot on to me! Cheers!


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## AbstractAlg (Mar 9, 2012)

chris410, thanks for the link. 

also, my contribution: http://kociemba.org/cube.htm
there are some group theory stuff that might be useful.


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## TMOY (Mar 11, 2012)

cuBerBruce said:


> If you consider the unstickered cubies and their orientations to be distinguishable, I believe there are \( 54!/(9!^6) \) possible stickerings. If you consider the orientation of the cube not to matter, it would be approximately 24 times fewer (although I believe the exact number to be rather tricky to calculate).



Actually it's not that hard. Since there are 9 undistinguishable sticers of each color and 24 possible orientations, the order of the group of symmetries of any given position must divide gcd(9,24)=3, hence the only possible symmetries are invariances by 120-degree rotation around an axis going through two opposite corners. A position satisfying that condition is entirely determined by the choice of a pair of opposite corners and the way some given pair of opposite faces is stickered; moreover, such a pair of faces must contain exactly 3 stickers of each color, hence \( 18!/(3!^2) \) possiblities. Now if you consider positions which are undistinguishable by rotation, the pair of corners doesn't matter and any position is undistinguishable only from its image by the action consistiong of switching opposite faces, which is always reachable by some rotation (for example x2 y if the pair of corners is UBR-DFL) and always different from the starting position. The number of such positions is then \( 18!/(2*3!^2) \), and we must add twice that number to \( 54!/(24*9!^6) \) to gat the total number of positions undistinguishable by rotation on the cube.


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## bselena (Mar 12, 2012)

I have also thought of writing my own take on how speed cubing could be and rather than talk about technique, they practically base things on carefully plotted patterns that they just have to follow.

And while a lot of people find that interesting and great, there is actually a side to it that considers it as a plain and straight up puzzle without that much of its sleeves.


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## Jude (May 3, 2012)

Essay finished and handed in! Thanks for the suggestions everyone  If you're interested in reading it you can find it here http://www.sendspace.com/file/84ulri


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## blah (May 3, 2012)

protip: use backticks (`) for open single quote, like

```
`this'
```

Edit: Good read. But don't you find it awkward to describe angles of rotation in radians?

Also, protip2: \( n \equiv 0 \pmod 3 \)

```
n \equiv 0 \pmod 3     % stands for "parentheses mod"
```


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## macky (May 9, 2012)

I'm amused that you cited Demazure and Gabriel for the definition of semi-direct product.
For references in math you should use MathSciNet to get BibTeX entries!

Some more tips. If you \usepackage{enumerate} in the preamble, you can use

\begin{enumerate}[(a)]
\item ...
\item ...
\end{enumerate}

to generate (a), (b), etc.
align and align* environments are also useful.
DeclareMathOperator{\sgn}{sgn} and same for \im and \ker to make those straight.
Write out "for all" using mbox instead of using the symbol.



blah said:


> Also, protip2: \( n \equiv 0 \pmod 3 \)
> 
> ```
> n \equiv 0 \pmod 3     % stands for "parentheses mod"
> ```


Lots of real pros seem to like \pod for congruences in Z and \bmod for more general congruences modulo an ideal.


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## Jude (Oct 20, 2012)

does anyone have a copy of this by any chance? lost my version, would be great if someone could re-up!

oh and thanks for the feedback guys


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## Christopher Mowla (Oct 26, 2012)

Jude said:


> does anyone have a copy of this by any chance? lost my version, would be great if someone could re-up!
> 
> oh and thanks for the feedback guys


I'm sorry for not responding sooner. I thought someone would have done it by now. Maybe someone already emailed you a copy, but your post asking for a copy was still here.

Anyway, I didn't think I had a copy, but I just realized that I did. I have uploaded it on the web. You can download it from here. As soon as you confirm that you've downloaded it, I will remove it from my archive (because it's your work).

I renamed it "Jude's essay" when I saved it to my computer (if that wasn't its name already) because I have several group theory documents on twisty puzzles.

Just in case you uploaded more than one version, I downloaded this version on 5/3/2012.

EDIT:
Link removed. There were a lot of views, so please provide a new download link, Jude, because people seem to be interested.


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## Jude (Oct 26, 2012)

Great! Downloaded it now, thank you very much!!


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## Jude (Nov 23, 2012)

cmowla said:


> EDIT:
> Link removed. There were a lot of views, so please provide a new download link, Jude, because people seem to be interested.



Done, but by all means keep it on your website if you want and distribute as you wish!

http://www.sendspace.com/file/8d2866


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## Jude (Feb 13, 2014)

Got a request for a re-upload of this so here it is

http://www28.zippyshare.com/v/97472928/file.html


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