Author: Daniel Pineo
Date: 14:19:23 05/20/05
Go up one level in this thread
On May 20, 2005 at 05:44:50, Werner Kraft wrote: >Basics of Group Theory for Chess Players ( ca 800 words) > >I try to give here a basic introduction to Group Theory. I will use plain , >jargon free English to the best of my abilities. The whole text is based on an >article in the McGraw-Hill Encyclopedia of Science and Technology, page 240ff >and hopefully inspired by the spirit of "advanced hippie-ology " > >A set of chess moves ( the operation of chess figures ) is called a group. >Lets call the group of all legal chess moves in the universe G , or , more >graphic : GOB ( Gods Opening Book) . >So , each chess move ( m(1), m(2) , m(3) ¡K must belong to GOB.... > >There are four requirements , that have to be fulfilled , if chess moves really >should form a group: Closure , Associative Law, Identity element and Inverse. > >1. Closure > >The "product " , or better the sum of two chess moves , for instance >m(1) = e2-e3 and m(2) = e3-e4 gives another element m(3) in the group > >m(1) + m(2) = e2-e3 + e3-e4 = m(3) = e2-e4 What about m(1) = e2-e4 and m(2) = e4-e5. Are you trying to say that m(1)+m(2) = e2-e5 is a legal move? >2. Associative law > >I have struggled with that one - intuitively it should be what is called a " >Zugumstellung " - if you change the sequence of moves e.g in the opening , but >come to the same result at the end. > >Simple example : 1.e2-e4 e7-e5 > 2.d2-d4 d7-d5 > >gives the same arrangement as 1. d2-d4 d7-d5 2.e2-e4 e7-e5 > >More mathematically: The sum of the chess moves gives the same result , just >the order / sequence is different. That's not associativity, that's commutivity. Associativity means (m1 m2) m3 = m1 (m2 m3) >3. Existence of an identity element > >"There is in the group an element e ( called the identity) which satisfies for >every element g of the group : eg = ge = g " > >Wow ...now lets define e = <NothingHappens> > > h2-h4 * <NothingHappens>= h2-h4 . That¡¦s incredible ¡K incredibly simple. However, e is not in the group of moves, since there is no "Nothing Happens" move, it can't be the identity. >4. Existence of the Inverse > >¡§ For each element g of the group there is an element g(-1) (called the inverse >of g ) which satisfies g(-1)*g = e ¡§ . e is the identity in mathematics. >On the chessboard it is simply the situation : ¡§ nothing has happened ¡§ . There is no move that is the inverse of e2-e4, so there is no inverse.
This page took 0 seconds to execute
Last modified: Thu, 15 Apr 21 08:11:13 -0700
Current Computer Chess Club Forums at Talkchess. This site by Sean Mintz.