HOW TO SOLVE IT

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HOW TO SOLVE IT
Alain Fournier (stolen from George Polya)
Computer Science Department
University of British Columbia
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Relevant Books by Polya

• Induction and Analogy in Mathematics
• Patterns of Plausible Inference
• This one
• How to Solve it
• A New aspect of Mathematical Method
• Princeton University Press
• 1957 (Second Edition)

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The Goals

• Help the students
• Help the teachers
• Develop problem solving skills in general
• Practice, practice

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How to Solve It (the 4 steps)

• Understanding the problem
• Devising a plan
• Carrying out the plan
• Looking back

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Understanding the problem

• What is the unknown?
• What are the data?
• What are the conditions?
• Are the conditions sufficient to determine the
  unknown, unsufficient, redundant, contradictory?
• Draw a figure
• Devise suitable notation
• Separate the various parts of the conditions
• Write down the conditions

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Devising a Plan I

• Have you seen that before?
• Is the problem already solved?
• Do you know a related problem?
• Look at the unknown
• is there another problem with the same unknown?
• Is there a related problem solved?
• can you use its result?
• can you use its method?
• can you establish a new link?
• Can you restate the problem? Can you re-restate
  it?

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Devising a Plan II

• Find an easier related problem
• More general
• More restricted
• Solve part of the problem
• Simplify the conditions
• Change the data (do you need more, less?)
• Change the unknown
• Any notion missing in the statement?
• Change the problem

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Carrying out the Plan

• Go step by step
• Check each step
• are you sure it is correct?
• can you convince others it is correct?
• can you prove it is correct?

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Looking Back

• Can you check the result?
• Is the result unique?
• Can you check the arguments
• Can you derive the result differently
• Can you use the result, or the method, for some
  other problem (or the original one if you changed
  it)?

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An Example

• Inscribe a square in a given triangle. Two
  vertices of the square should be on the base of
  the triangle, the two other vertices of the
  square on the two other sides of the triangle,
  one on each.
• Unknown a square
• Data a triangle
• Conditions positions of 4 corners of square

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An Example (ctd)

• Draw a figure

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An Example (ctd)

• Relax the conditions

We get more than one solution
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An Example (ctd)

• How can the solution vary?

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An Example (ctd)

• Is it correct?
• Is it unique?
• Can we use the method for something else?

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Some strategies

• Start at the beginning
• Visualize
• Take it apart
• Look for angles
• Dont dismiss foolish ideas right away
• Restart often
• Sweat the details
• Do not assume
• Try to solve again

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Key Principles (among many others)

• Analogy
• Auxiliary problem
• Conditions (redundant, contradictory)
• Figures
• Induction
• Inventors paradox (a more ambitious problem
  might be easier to solve)
• Notation
• Reductio at absurdum
• write numbers using each of the ten digits
  exactly once so that the sum of the numbers is
  exactly 100
• Working backwards

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Working Backwards

• Get from the river exactly 6 quarts of water
  when you have only a four quart pail and a nine
  quart pail to measure with.

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Physical Problems

• Data from experience
• Looking back to experience
• Tides
• Sap rising (Occams razor)
• Spinning book

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Computer Science Problems

• Some mathematical
• Some physical
• Some neither solution is a creation,
  mathematical engineering (actually often problem
  itself is a creation)

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Problems Found in Past Year

• Area of spherical triangles (-gt simpler problem)
• Models of animal patterns
• (growth and distance measure, analogy)
• Efficient storage of wavelet coefficients
  (engineering, similarity)

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Conclusion

• go solve your own problems