How to Solve It

Working note — Part of my personal research notebook. Entries may be incomplete and are revised over time.

“A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiousity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triump of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.”

How to Solve It

• understanding the problem: know the data, unknown, and condition
  • what is the unknown
  • what are the data
  • what is the condition?
• devising a plan: find a connection between the data and the unknown
  • have you seen it before?
    • think of a familiar problem having the same or a similar unknown
  • have you seen the same problem in a slightly different form?
  • do you know a related problem?
    • here a related problem to yours and solve before.
      • could you use it?
      • could you use its results?
      • could you use its method?
      • should you introduce some auxiliary element in order to make its use possible?
  • do you know a theorem that could be useful?
  • could you restate the problem?
    • could you restate it still differently?
  • could you imagine a more accessible related problem?
    • a more general problem?
    • a more special problem?
    • an analogous problem?
    • a part of the problem?
  • keep some condition, how far is the unknown then determined, how can it vary?
  • did you use all the data?
  • did you use the whole condition?
  • have you taken into account all essential notions involved in the problem?
• carrying out the plan
  • check each step
  • can you see clearly that the step is correct?
  • can you prove that it is correct?
• looking back
  • can you check the result?
  • can you check the argument?
  • can you derive the result differently?
  • can you see it at a glance?
  • can you use the result, or the method, for some other problems?