I suggest unipolar Hall effect switches, such as http://www.digikey.co.uk/product-detail/en/AH337-WG-7/AH337-WGCT-ND/2122985. They are smaller than reed switches, so it will be easier to fit them in the space available, and they don't bounce. You can surface-mount them on a PCB that goes under the board, which will save you a lot of wiring. Make sure you put the magnets the right way round on the bottom of the pieces.To connect them to an Arduino, as they have open drain outputs, I think you could multiplex them. Use an active high output 8-way demultiplexer chip (74HC238) to power all the sensors in a single column. In each row, connect all the outputs together, with a common pullup resistor. Then use an 8-input multiplexer to select which row to read (or 8 input pins).
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