How To Solve It
- 1. How to Solve It Completely based on the seminal work of G Polya.
- 2. Trying to find / understand not only the solution of some problem, but also the motives and procedures of the solution.
- 3. Four Phases Understanding the Problem To make a plan towards the Solution Carry out the Plan Look back and analyze the Solution
- 4. Understanding the Problem what is the unknown what are the data what are the conditions
- 5. consider all parts from different angles draw figures introduce suitable notation whenever necessary
- 6. Example : Find length of the diagonal of the Parallelepiped when the length of the sides are known.
- 7. Devising the Plan To have (at least) an outline of the procedure to obtain the solution or the computation / calculation required to do so
- 8. “ In fact, the main achievement in the solution of a problem is to conceive the idea of a plan.”
- 9. “ ... it may be hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge.”
- 10. Good ideas are based on past experience and formerly acquired knowledge.
- 11. Devising a Plan Do you know a related problem ? Do you know a problem with a similar unknown ? Can you restate the problem ? Can you transform the problem ?
- 12. back to example what similar problem do you know of ? right angled triangle ? can you introduce an auxiliary element ?
- 13. umm...
- 14. y^2 = a^2 + b^2 (Pythagorus theorem) x^2 = c^2 + y^2 x^2 = c^2 + a^2 + b^2 x = sqrt (c^2 + a^2 + b^2)
- 15. Can you check the result ? Can you check the argument ? Can you arrive at the result differently ? Have you used all required data ? Can you reduce the problem ? Look Back
- 16. Summarizing the procedure ...
- 17. Getting Acquainted Where should I start ? Problem Statement !!
- 18. Getting Acquainted What can I do ?? Visualize the problem as a whole and as vividly and clearly as you can (ignoring the details).
- 19. Getting Acquainted What can I gain by doing so ?? Familiarity with the problem and its purpose. All relevant points will become more accessible and stimulate your thinking.
- 20. Towards Better Understanding Where should I start ?? Digest the problem statement !
- 21. Towards Better Understanding What can I do ?? Isolate the principle parts of your problem. The ‘hypothesis’ and the ‘conclusion’ are the principle parts of a ‘problem to prove’. The ‘unknown’ and ‘data’ are principle parts of a ‘problem to solve’
- 22. Towards Better Understanding What can I gain by doing so ?? Clarify details which are likely to play a role in the ‘plan’ towards the solution.
- 23. Ideation ! Where should I start ?? your now complete understanding of the problem statement.
- 24. Ideation ! What can I do ?? Consider the problem from various angles seeking contact with your formerly acquired knowledge. Scrutinize all aspects from this context.
- 25. Ideation ! What can I gain by doing so ?? More and more ideas will make your conception of the problem more coherent and better balanced. Path towards the end solution will become clearer with ideas giving the intermediate steps.
- 26. Carrying out the Plan What can I do ?? Deal with all the details. Convince yourself of the correctness of each step.
- 27. Carrying out the Plan What should be the result ?? A presentation / implementation of the solution each step of which is correct beyond doubt.
- 28. Looking Back What can I do ?? Consider the completed solution and try to simplify. Try to fit parts into patterns from your previously acquired knowledge. Scrutinize the result and see if you could arrive at it differently.
- 29. Looking Back What can I gain by doing so ?? Maybe a better solution. New and interesting sub-results or facts. Scrutiny of the method of solution will help develop the ability to solve problems.
- 30. Dictionary of Heuristics. A short
- 31. Analogy Can you find a problem analogous to your problem and solve it ? Map
- 32. Auxiliary Elements Can you add some new elements to your problem to get closer to a solution ? Extension
- 33. Decomposing and Recombining Can you decompose the problem and “recombine its elements in a new manner” ? Divide and Conquer
- 34. Definition Is your definition of the concepts involved precise ? Exactness
- 35. Figures and Diagrams Can you draw a picture of the problem ? Diagrammatic reasoning
- 36. Working Backwards Can you start with the goal and work backwards to something you already know ? Backward chaining
- 37. Generalization Can you find a problem more general than your problem ?
- 38. Specialization Can you find a problem more specialized ?
- 39. Notation Can you introduce suitable notation to abstract and simplify the problem space ?
- 40. thank you for the patience !