Chess-Board schematic/pcb layout
As per comments, I'm posting an alternative solution that may better suit your needs.

If you make the base of the piece and the square in the chessboard a closed system, you ~might~ be able to identify parts by how much light they reflect back.

For this I believe you need 4 things:

1 Chessboard and piece to be a closed system (outside light may mess with readings depending on what sensor you use)

2 Paint parts bases in different grayscale tones.

3 Add a fixed/known light (might be IR LED, QRE1113 for example, eliminating the LDR)

4 Use a port expander with analog inputs (quick search gives a few alternatives: ADS1015, MCP3424

I've used the QRE1113 under many conditions and it manages to reject ambient light very well. I've only had problems with direct sunlight on it.

I believe you need 6*2 different "grays" (king, queen, rook, bishop, knight, pawn, black and white). I have no idea how hard or easy it is to achieve this precision on print/light sensing. Its very easy to prototype though. You dont even need the i2c port expander.