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 Post subject: braid group puzzles
PostPosted: Fri Nov 30, 2001 6:36 am 
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Joined: Sun May 27, 2001 7:03 pm
i know this site is mainly about physically making puzzles, but i have a theoretical idea for an interesting puzzle. it would be like no other puzzle before it. almost all the puzzles we talk about here have pieces which can be swapped in some form or another - eg the corner pieces on a rubiks cube can be swapped in various ways, and so too can the edges. this gives rise to the idea of a permutation. Jaap's page has an excellent explanation of this idea:

http://www.org2.com/jaap/puzzles/theory.htm

now a permutation can always be undone simply by repeating it enough times. eg, if you twist the top faces of a rubiks cube and then the right face, you can repeat this over and over (and over...) until the cube fixes itself up. an interesting way to generalise this idea is to introduce "braids". on Jaap's page (above) he draws diagrams of permutations with lines crossing over each other. however, it is not important to know which line crosses OVER and which crosses UNDER at an intersection point. however, a braid is a diagram like Jaap's in which it IS important. so there are two types of crossings - one in which the left line (string) crosses over the right string, and one in which the right string crosses over the left string. imagine this like you are actually twisting up strings. so you could see that you could keep twisting two strings around themselves for as long as you want without ever untwisting it. this is very different to the permutation situation in which you will have achieved nothing if two lines cross pver themselves twice in a row.

i was thinking it would be very interesting to create a puzzle with the difference that instead of pieces being swapped via permutations, they be swapped via braids. before you get too worried, here is a simple example:

think of a pyramorphix which is in its solved state, and imagine that it can only be twisted in such a way that after every twist it is still in the right shape. also imagine that the only pieces which matter are the corner pieces. so there are four pieces on this puzzle (i will call the puzzle "P" for short). if you do a single twist of P two of the corners will be swapped. now on a real pyramorphix, if you repeated this twist, you would have solved the puzzle. however, i would like to have a P in which repeating the twist would not solve it but rather would just make it worse - the pieces are in their correct position, but are in some way twisted up. the only way to fix a twist would be to go back in the exact oposite direction.
perhaps this could be achieved by assigning each piece not only its POSITION, but also an integer (positive or negative or 0) which tells us by how much the piece is "twisted up" with each of the other pieces.

(FOR PEOPLE WHO KNOW SOME MATHS: this P puzzle has the symmetric group on four letters (S_4) as its group of symmetries. however I would like to modify S so it has the braid group on four strings (B_4) as its group of "symmetries".)

now before you get too worried, it wouldnt become a matter of mixing up the puzzle and then the only way to fix it is to go backwards exactly the same way as you got there. but i think it would make it extremely interesting as there would actually be an infinite number of positions the puzzle could be put into. this obviously means it would be better to have on a computer which has a reset button...

so what do you people think? id like to hear some ideas. has anyone thought of something like this before?


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 Post subject: there was a confusing typo in that last post...
PostPosted: Sat Dec 01, 2001 6:36 am 
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Joined: Sun May 27, 2001 7:03 pm
part of my message should have read:

(FOR PEOPLE WHO KNOW SOME MATHS: the (simple) P puzzle has the symmetric group on four letters (S_4) as its group of symmetries. however I would like to modify P so it has the braid group on four strings (B_4) as its group of "symmetries".)


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 Post subject: Interesting Idea
PostPosted: Sun Dec 02, 2001 6:47 am 
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Joined: Sat Dec 02, 2000 8:27 pm
Location: Wilmington, NC, USA
If you implement this puzzle with a computer program, make the program display somehow whether the braids are becoming more tangled or less tangled with each move. Then the puzzle would provide people with a way to learn how to solve the cube! I want to be your first customer.


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