A typical homework problem from my Graph Theory course:

There are 12 people applying to a company which has 15 different jobs available. Each person is qualified for at least 6 of the jobs. No two people are qualified for the same three jobs. Also, every job has at least two people qualified for it. Prove that it is possible to assign all twelve people to twelve different jobs such that each is qualified for the job assigned.

Three months ago I would have no idea how to solve it. In fact, I still don't, but I have to figure it out by Monday. I love problems where I can sit and ponder them at random moments during the day, and in fact, that's when I usually figure them out.