A tiling is a way of covering a region with pieces ("tiles") so that there are no gaps or overlaps between the tiles. It is not surprising that there can be very many ways of tiling a large finite region with tiles of a specified shape and size. What is surprising is that there are tiling-problems such that, if you choose randomly from the set of tilings, the tiling you get is very likely to look very un-random! I will describe why this is true, and talk about the role undergraduate research assistants have played in this work.