We all like problems. But it's sort of sad that as our careers develop, we get to spend less and less time on problems that don't offer opportunities for resume enhancement. I don't think twice about watching a 3-hour movie, but feel guilty if I spend more than an hour on a problem that has nothing to do with research...
A fellow grad student was TAing an algorithms class and posed a question about the change-making problem. In the change-making problem, you're given a set of denominations c1 > c2 > ... > cN and a value M and asked to find the minimum number of coins that add up to M. The greedy strategy is to choose as many c1's as possible, then as many c2's as possible etc. Is there a polynomial-time algorithm that, given a set of denominations as input, tells if the greedy strategy is always optimal for the set (i.e., optimal for every M)?
We pondered awhile, but then of course I had to ruin things by Googling the very elegant solution published by David Pearson a decade back.
Fresh-faced undergrads and first-year grads out there, resist the urge. Enjoy your freedom and treat each problem the same while you still can, before your research becomes too compelling (= compulsory)