Techniques in Solving Math Problems.

1. HOW TO SOLVE IT
Alain Fournier
(stolen from George Polya)
Computer Science Department
University of British Columbia

2. 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)

3. The Goals
 Help the students
 Help the teachers
 Develop problem solving skills in general
 Practice, practice

4. How to Solve It (the 4 steps)
 Understanding the problem
 Devising a plan
 Carrying out the plan
 Looking back

5. 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

6. 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?

7. 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

8. 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?

9. 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)?

10. 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

11. An Example (ctd)
 Draw a figure

12. An Example (ctd)
 Relax the conditions
We get more than one solution

13. An Example (ctd)
 How can the solution vary?

14. An Example (ctd)
 Is it correct?
 Is it unique?
 Can we use the method for something else?

15. Some strategies
 Start at the beginning
 Visualize
 Take it apart
 Look for angles
 Don’t dismiss foolish ideas right away
 Restart often
 Sweat the details
 Do not assume
 Try to solve again

16. Key Principles
(among many others)
 Analogy
 Auxiliary problem
 Conditions (redundant, contradictory)
 Figures
 Induction
 Inventor’s 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

17. 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.

18. Physical Problems
 Data from experience
 Looking back to experience
 Tides
 Sap rising (Occam’s razor)
 Spinning book

19. Computer Science Problems
 Some mathematical
 Some physical
 Some neither: solution is a creation, mathematical
engineering (actually often problem itself is a
creation)

20. Problems Found in Past Year
 Area of spherical triangles (-> simpler problem)
 Models of animal patterns
(growth and distance measure, analogy)
 Efficient storage of wavelet coefficients (engineering,
similarity)

21. Conclusion
 go solve your own problems