As an example I will share one that I'm thinking about: it's a jigsaw puzzle, but one which is made in a very clever way. Lots of pieces look like they could fit, but the fit is only slightly, maddeningly off. I'm interested in discovering how likely any particular configuration is, given that all the pieces must fit together.
Your project must have some math in it, of course; recursion is coming up, and that's a likely target. You could also improve on applets I've seen for the Logic Game of Lewis Carrol, of course. You could implement the Tautology test. You could also do something out of sections we've skipped (e.g. prolog).
The description should include at least the equivalent of five typed, double-spaced pages about the game you're examining. It will be well-written, as if you were submitting it for publication somewhere.