How To Solve Problems Creatively?

How do you make the leap from proving things with standard techniques (like proof by contradiction, induction, etc.) to coming up with lemmas to solve problems?

• I understand that the answer to all of these questions is practice.However, it has been my observation that simply practicing for a large amount of time is of no great consequence if you don't do it the right way. In particular, I have 'seen' through a problem before and just knew that I had to prove a lemma, and from toying around with the problem arrived at a statement which I had to prove that would make the previous problem almost trivial. My question is therefore in a sense misworded but really means to ask how can I do this systematically, or in other words how can I become adept at just 'seeing' through problems when, really, just seeing through problems doesn't cut it as something I can rely on. The type of problems which I want to solve are usually the ones not chosen for Olympiad contests because they require particular insights that are too much to ask fairly in a given time frame but are still well posed and all of that stuff.Alternatively, some Russian Olympiad problems fit into this category as well.

• Answer:

  The short answer is: Read lots and lots of papers. Read lots and lots of proofs. Solve lots and lots of exercises. When you've done this enough, you'll begin to see non-obvious patterns more easily. You'll begin to come up with original ideas for proofs. Another exercise I've found rewarding is to try to formulate mini research problems and work on them for weeks or months. These problems should be easy enough for you to tackle with your current tools but they should also go a little beyond your comfort zone. They should be hard enough to keep you thinking for days or weeks. You can start with reading a published paper and trying to reprove every lemma in the paper differently. Then pick another paper and do the same. And another. And another... You can also try creating a conjecture and proving or disproving it. It's also useful to keep a mathematical diary when doing all these. I hope this advice is useful.