HOW TO SOLVE IT
Alain Fournier (stolen from George Polya)
Computer Science Department University of British Columbia
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)
The Goals
Help
the students Help the teachers Develop problem solving skills in general Practice, practice
How to Solve It (the 4 steps)
Understanding Devising
the problem
a plan Carrying out the plan Looking back
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
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?
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
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?
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)?
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
An Example (ctd)
Draw
a figure
An Example (ctd)
Relax
the conditions
We get more than one solution
An Example (ctd)
How
can the solution vary?
An Example (ctd)
Is
it correct? Is it unique?
Can
we use the method for something else?
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
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
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.
Physical Problems
Data
from experience Looking back to experience Tides Sap rising (Occam’s razor) Spinning book
Computer Science Problems
Some
mathematical Some physical Some neither: solution is a creation, mathematical engineering (actually often problem itself is a creation)
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)
Conclusion
go
solve your own problems