Arduino Project Idea: RFID or Bluetooth or What
Bluetooth location ranging is not sensitive enough under 10 inch or so. RFID readers are at least a magnitude more expensive than tags. 

My recommendations is to use magnets for snap together action. Create a 4x4 matrix of magnets/meeting metal plates but do populate all locations on the matrix with magnets to encode a position. Use Hall sensors to check what positions are populated.