Tracking the location of NFC chips on a board
It sounds like a similar problem that chess computers do. A regular grid, with 64 spaces and 32 pieces to track.

The chess boards that do this range from simple detection: knowing only that a piece is present in a square, to identifying each individual piece. For chess, knowing only “present/not present” is sufficient because the beginning state of the board is fixed, so it’s easy logic to  track the moves and the game progresses. That is, until someone picks up two pieces, I suppose. :)

Some ideas for tracking chess pieces, including reed switches, Hall effect sensors, capacitive sensors, and NFC tags are discussed here: http://chess.fortherapy.co.uk/home/chess-piece-identification-technology/