I may be carbon-dating myself here, but one of the first things I implemented on my Apple IIe in the early 80s was John Conway's Game of Life. If you're not familiar with it, it's a mathematical game that simulates cellular automata with fixed rules. There's an infinite square grid containing spaces for cells. Every generation, cells can be born or they can die, according to very simple rules. Despite the simple rules, very complex structures can be created. See the wiki page for more information.
All the code here can be found on github: https://github.com/pauljchang/sandbox/blob/master/life/life.sql
