• Welcome to the Speedsolving.com, home of the web's largest puzzle community!
    You are currently viewing our forum as a guest which gives you limited access to join discussions and access our other features.

    Registration is fast, simple and absolutely free so please, join our community of 35,000+ people from around the world today!

    If you are already a member, simply login to hide this message and begin participating in the community!

I'm writing a group theory essay

Jude

Member
Joined
Mar 27, 2008
Messages
292
Location
Sheffield, England
WCA
2008WRIG02
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
 
Last edited:

wontolla

Member
Joined
Oct 21, 2010
Messages
300
WCA
2011GRAJ01
YouTube
Visit Channel
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.
 

Ilkyoo Choi

Member
Joined
Nov 6, 2008
Messages
146
Location
Champaign, IL
WCA
2008CHOI04
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.
 

LNZ

Premium Member
Joined
Jul 13, 2009
Messages
595
Location
Adelaide, SA
YouTube
Visit Channel
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.
 

mrCage

Member
Joined
Jun 17, 2006
Messages
654
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
 

mrCage

Member
Joined
Jun 17, 2006
Messages
654
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
 

macky

Premium Member
Joined
Apr 4, 2006
Messages
402
Location
Stanford, CA
WCA
2003MAKI01
I second Ilkyoo's suggestion, to write down the structure of the cube group.

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.
 

Cielo

Member
Joined
Mar 22, 2009
Messages
35
Location
Beijing, China
WCA
2008LUYU01
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.

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.
 

mrCage

Member
Joined
Jun 17, 2006
Messages
654
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
 

Jude

Member
Joined
Mar 27, 2008
Messages
292
Location
Sheffield, England
WCA
2008WRIG02
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.
 
Top