how about this.
place a fibreoptic and a LED underneath every whole.
the fiberoptics go to a panel, where it is captures by a webcam or similar and sent to computer with processing.
as soon as a stone is placed (no matter if black or white), the fiberoptic will turn black, because the light is covered from the stone.
now the LED turns on and the fiberoptic gets a new color.
if the stone is while, enough light will be reflected and the fiberoptic turns white for a moment.
if the stone is black, not enough light is reflected, and the fiberoptic stays black. the stone must be matt for that matter.
here a quick and dirty sketch 
the LED could have a specific color, i.e. red.
then you'd have three fiberoptic states.
"white"/"bright" is ambient light -> no stone
"black" means no light reflected -> blackstone
"red" means red light reflected from white stone -> white stone
another way, without any fiber optics, would be a table (kinda microsoft surface like) solution.
it would need to have something other than stones, something that looks like a pin, or maybe stones with a pin attached.
the pins are black or white and have a flat bottom.
when the pin is put in the hole, a latch is moved (dont know how to describe it, but drawed it in the picture), and the flat surface of the pin is shown.
the surface of the board should have a different color than black or white (i choose red).
so when no pin is insert, the hole is covered with a red latch.
when the pin is insert, the latch is moved away and the pin color is revealed.
now you need to have enough light in the box/table and a webcam that captures the board from underneath.
since the camera is fixed, the position of the holes is also fixed and can be programmed in processing.
then you can check with processing if the "position" of the hole is red (no pin), white or black.
you can put a little computer (eeepc, or an intel atom board) inside the table and analyse the picture right inside the table.
here's a bit better sketch
you can see the latched that are just flipped down when a pin is insert.
i think brown would be a better color than red since it doesnt reflect that much and would maybe make the white pin look like a red pin.
i think this would be a good solution, since its easy to build, just drill holes and dumps in a wooden plate, and some kind of latch to each hole.
and all you'd need is a webcam and a computer. no need for complicated soldering or multiple arduinos etc.