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An algorithm

The algorithm (several instances may run in parallel):

  1. be curious and open minded, do retain concepts, beyond techniques
  2. when you catch a new idea (for instance in a seminar) try it on all your problems
  3. if your imagination fails to produce a promising problem, return to first step
  4. if the problem is already sparsely solved by others, write down an expository synthesis
  5. if not, never hesitate to learn new techniques to solve your problem
  6. work hard, generalize, specialize, follow your intuition, use simulations
  7. if the problem is still too difficult, seek for complementary friends
  8. if the problem is still unsolved, keep it in mind for future use
  9. it is time to take vacations… maybe
  10. return to first step (and never forget to eat and to sleep 😉

For a poor human like me, this algorithm is more a paradigm than a reality… 🙂

Paul Halmos: Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

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