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I am looking for a book which would teach me heuristic principles for problem solving. I am not necessarily looking for a book with lots of new theory, but more principles I could rely on when solving problems. I have recently been working from Arthur Engel's Problem Solving Strategies, but the book is quite dense and the explanation can get too brief. On the other hand, what is covered in the book is exactly what I'm looking for (invariance principle, extremal principle, pigeonhole principle, colouring etc.).

Is there a book that contains more explanation that covers similar material? Any help would be much appreciated.

I have looked into Textbooks for math contests apart from AoPs. I feel many of the books suggested cover the theory of a particular area of math, rather than ways of thinking - I do know of 'the art and craft of problem solving', but I wonder whether there are books more focused on these ways of thinking rather than the theory itself?

math-physicist
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IChoi
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  • Does this answer you question: Textbooks for math contests apart from AoPs ? – math-physicist Jul 08 '22 at 14:34
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    I feel many of the books suggested cover the theory of a particular area of maths, rather than ways of thinking - I do know of 'the art and craft of problem solving', but I wonder whether there are books more focused on these ways of thinking rather than the theory itself? – IChoi Jul 08 '22 at 14:39
  • @IChoi So you think there's some magic craft of problem solving independent of "theory" (= mathematics)? – wasn't me Jul 08 '22 at 14:43
  • I wouldn't call it a 'magic craft', but yes; I do believe that there are certain ways of thinking about problems which are widely applicable. – IChoi Jul 08 '22 at 14:55
  • I think you'll learn those "ways of thinking" with experience. – Wizard0001 Jul 08 '22 at 14:57
  • @Wizard0001 I agree practice is important but, for example, I would not have known how to solve many questions which use the invariance principle had I not read Engel's book. And it is not something so natural that one would come up with it easily. There do seem to be some widely applicable techniques which don't belong to a particular area of maths.. – IChoi Jul 08 '22 at 15:01
  • I am talking about that only, once you are done with Engel, I think you are equipped with most of the standard tools. It is the appropriate time to start solving ISLs, Choose the problems that are appropriate as per your current level, then go beyond. You might come up with new "tricks" yourselves, and if you are not able to solve then reading the solution will give you some new insight i.e. it will teach you a new technique. If you want to know techniques before solving problems, then maybe you can look at evan chen's or yufei zhao's handouts. – Wizard0001 Jul 08 '22 at 19:20
  • Try Problem Solving Tactics by Angelo Di Pasquale and others. Another great one is Solving Mathematical Problems: A Personal Perspective, by Terence Tao. – TheBestMagician Jul 09 '22 at 03:50

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