I'm sure a lot of people who are interested in this kind of stuff have probably already heard of this, but for those who have not, here is a fun little set of challenges for you.
http://projecteuler.netProject Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.
Ideally you want the program to efficient enough to solve the problem within a minute on the average computer.
The challenges require some programming ability, but ultimately they seem like mathematics challenges, as you have to find out how to solve them before you can implement it into a program.