I have done independent study projects with undergraduates ranging from the mathematics of chess to study of the stock market, but mostly I enjoy working on the Lights Out Puzzle with undergraduates. The standard puzzle is made up of a rectangular board of squares, each of which can be either on or off. All squares start on, and when one is clicked, it switches states but so do the squares directly adjacent (horizontally and vertically). The question is not just how to solve this puzzle by turning off all the lights but, perhaps more interestingly, how many solutions there are. There are lots of unstudied variants, such as changing the number of possible states of squares or the shape of the board that students can enjoy exploring, making conjectures, and proving them to develop a whole mathematical theory of their own.