I'm writing a group theory essay

Discussion in 'Puzzle Theory' started by Jude, Dec 7, 2011.

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  1. Jude

    Jude Member

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    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: Feb 13, 2014
  2. wontolla

    wontolla Member

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    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.
     
  3. Ilkyoo Choi

    Ilkyoo Choi Member

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    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.
     
  4. Lucas Garron

    Lucas Garron Super-Duper Moderator Staff Member

    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.
     
  5. LNZ

    LNZ Premium Member

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    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.
     
  6. mrCage

    mrCage Member

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    Great! I get my name in the essay :)

    Per
     
  7. mrCage

    mrCage Member

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    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
     
  8. Krag

    Krag Premium Member

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  9. macky

    macky Premium Member

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

    That's obvious.

    That's false.
     
  10. ASH

    ASH Member

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    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 ...)
     
  11. Cielo

    Cielo Member

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    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.

    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.
     
  12. Lucas Garron

    Lucas Garron Super-Duper Moderator Staff Member

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

    macky Premium Member

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

    cmhardw Premium Member

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    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).
     
    Last edited: Dec 10, 2011
  15. Ilkyoo Choi

    Ilkyoo Choi Member

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    Oh, such a delicate topic.
     
  16. blah

    blah brah

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    nonstandard analysis ;)
     
  17. mrCage

    mrCage Member

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    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
     
  18. Jude

    Jude Member

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    lots of great ideas here, thanks guys! i'll let you know how it turns out :)
     
  19. Jude

    Jude Member

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    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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