Ishkur:
Chagrin: not really sure if I understood what kind of contacts you mean. The different resistance per piece idea is interesting, but maybe since its something that people are supposed to handle that will have an interference on it?
Using the example of a round playing piece like a chess piece, each piece would have taped to its bottom a unique value resistor with one lead of the resistor terminating in the center of the piece and one lead terminating at the periphery. When placed on a pad of the board (see image of an example 2x2 board) the four analog pins could be checked for resistance to determine where the piece is.
Using just one resistance value per piece you could only reliably identify ~6 different resistance values. More practically you'd need to use a few resistance values per piece with multiple concentric rings per pad.
...but really the more I think about this suggestion the less I like it; it would only scale well to 3 contact points (~36 unique pieces) unless you used spring loaded pins or such to make contact with the board.
Can you provide more description as to what kind of game this is (how many unique pieces / how big is the board)? Don't be shy; you are in a forum with a bunch of nerds and I'm sure many people here have played D&D or such at one time or another 
