So, I'll admit to not having read the entirety of this thread, but on the subject of triangles, and more specifically, deltas, there is apparently a type of robot referred to as a "Delta Bot" that is often used to move materials in industry (
http://en.m.wikipedia.org/wiki/Delta_robot I may have misinterpreted the entry, but it's still a better lead than 'water game!' ;-) ) Maybe instead of game pieces going directly from the human players to the robot, there will be a delta bot transferring them?