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 the problem Devising 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