Recently, I read an article by M. Chaves of Universidad de Costa Rica titled CHESS PURE STRATEGIES ARE PROBABLY CHAOTIC. http://arxiv.org/pdf/cs.CC/9808001
From the article:
To describe a particular state in chess we use a 64 dimensional vector space, so that to each square of the board we associate a coordinate that takes a different value for each piece occupying it. A possible convention could be, for instance, the following: A value of zero for the coordinate of the dimension corresponding to a square means that there is no piece there. For the White pieces the convention would be: a value of 1 for the coordinate means the piece is a pawn, of 2 means it is a pawn without the right to the en passant move, of 3 that it is a knight, of 4 a bishop, of 5 a rook, of 6 a queen, of 7 a king, and of 8 a king without the right to castle. The values for the Black pieces would be the same but negative.
The question I have is:
Is he right? Shouldn’t vector spaces obey the laws of vector addition and scalar multiplication? Sticking to his description, adding two chess board configurations do not lead to another board configuration… Multiplying a board configuration matrix with a scalar does not lead to another board configuration.