Lately I have been puttering around with 2011 Putnam problems during lunch. I think I know the answer to one of the questions. By this I mean I can work through a demonstration in my head. I call this “making the moves.” But I can’t show it, that is, prove what I think I know. In mathematics we really don’t know something until it has been proven and checked by other people. So I can’t claim I really know the solution. But I think my difficulty in not in my insight but in my limited proving skills – in the case of the Putnam problem, my lack of skill in manipulating infinite series notation. I guess I could say that privately, very privately in the confines of my skull I know the solution, but in public my lips are sealed.
This situation reminds me of a problem I worked on in a graduate class in large scale linear systems. We were to develop algorithms to find an initial feasible solution to a system with a special structure by using the methods that we had learned in class. I thought I had come up with a new method using dynamic programming. For a month I would wake up early, go down to a small dorm library and work on a proof. I was fairly sure I had one but never totally satisfied. I handed in the paper and nothing came back for a month. Finally I got the graded paper back with the notation “I’m not sure I believe this!” That wasn’t a very mathematical thing to say, but having graded very many papers I know that graders have only so much time to devote to arcane student-generated proofs. Still I was disappointed because it was one of the few times I experienced what F. Scott Fitzgerald called “the fine quiet of the scholar which is nearest of all things to heavenly peace”.