If you used LEDs or Lamps, and had a red, green, and blue element for each square, you could easily set it up to generate random colors, by making lights light up at different intensities depending on the color.
All Off = Black
Red + Green = Yellow
Green + Blue = Cyan
Red + Blue = Magenta
Red + Green + Blue = White
Red at Full, plus Green at half = Orange
Red at Full + Green + Blue at 3/4 brightness = Pink
Green at full + Blue at half = Bluish Green
Red+Blue+Green at half = Grey
and so on!