Are three problems better than one?
One wouldn’t think so in `real life’. On the other hand, when doing mathematical research, there is something to be said for having several saucepans on the go (as it were). While looking for some technical maths I chanced upon the following remarks from Danny Calegari to potential graduate students:
Another approach, apparently practiced by Shelah, is to always have three problems that you’re working on – an easy one, usually an exercise to master a recently learned concept, a moderately difficult one, maybe a generalization of some recent result of someone else, and a very hard one, perhaps a well-known conjecture as above. When you’re stuck on one problem, move to the next.
Of course, you may have to be Shelah for this to be fruitful. (It’s also worth emphasising that what Shelah thinks of as moderately difficult may be somewhat harder for mere mortals!)
This reminds me of some old advice, attributed to Feynman, on how to be a genius:
You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, “How did he do it? He must be a genius!”
(I first saw this maxim when reading Rota’s classic sermon Ten Lessons I Wish I Had Been Taught, which despite being ten years old still seems very relevant in my view.)
Some of Calegari’s other remarks seem very apt to me: for instance,
Also very important is to build up a library of examples of geometric phenomena which you understand very well, from many points of view. Whenever you learn a new abstract concept, try to see how your examples fit into this idea, and try to generate new examples which illustrate the main point.
This is a point I hope to return to in future postings.