One of the exercises I assigned last week proved interesting:
Consider n points on a circle, labeled clockwise from 0 to n-1. Initially a wolf begins at 0 and there is a sheep at each of the remaining n-1 points. The wolf takes a random walk on the circle; at each step, it moves with probability 1/2 to one neighbor and with probability 1/2 to the other neighbor. (0 and n-1 are neighbors.) The first time the wolf visits any point it eats the sheep that is there. (The wolf can return to points with no sheep.) Which sheep is most likely to be the last eaten?
If you haven't seen it before, you might try it; don't put the answer in the comments, though, since I'll use the problem again.
While grading the assignment, I found a number of students had simulated the process, figured out the answer from the simulations, and then used that knowledge to prove the desired result. The problem didn't ask for them to do it, but they did it themselves.
That was great (and I told them so). That's how solving research problems often works for me. I have to understand what's going on, and in many cases, that understanding comes about by simulating a process to figure out how things behave. Then I go back and try to prove what I think I'm seeing in the simulations.
My worry, though, is that the students that did it this way were primarily the "non-theorists" in the class, who did it because they knew they didn't know the answer, and thought it was easier to code to figure it out. And that the "theorists" in the class correspondingly thought they knew the answer (rightly or wrongly) and went ahead with the calculations without doing a simulation. That's not necessarily a bad thing, certainly not for this problem (which is easy enough), but I'd also like for the theorists to also get into a mindset of doing simulations in this sort of setting, both as a tool to gain insight before trying to prove things and as a check on their proofs.
I think they're probably getting the lesson from other, harder exercises I give. Still, it was nice that a number of people in the class went that direction (and thought to write it down in their assignment).