While technically calling it a game may be an overstatement, I think this would be the most appropriate place to tell you about Project Euler. This website is a collection of math problems which are meant to be solved by devising efficient algorithms. It does require a little bit of programming and math knowledge, as you may guess. The answers you find are generally very large numbers, which then work as the password to "beat" the problem. There are also achievements for clearing specific sets of problems, and for every 25 problems solved you advance a level.

Here is an automatically-updating picture with my own progress. Feel free to post yours too!

Here is the URL:

http://projecteuler.net/Have fun!