I was thinking the same thing. I love the idea of using something like kick starter to publicly fund prizes for math / science, not necessarily related to 4-color rectangles.
You'd have to be careful about it, and have clear rules about who holds it and when the reward is given out. You could conceivably fund research that way as well, such as things that big business or the government shy away from, but has popular demand or value. It sounds approachable.