Table of contents

Related pages

• Mathematics

PDF: Polya_HowToSolveIt.pdf

Contents

How to Solve It

This is basically a summary of the book that appears at the very beginning of the book.

Part I. In the Classroom

Part II. How to Solve It

This is an example of a dialogue between teacher and student in which the teacher leads the student through the solving of a problem using the heuristics discussed in the book.

Part III. Short Dictionary of Heuristic

[this makes up the majority of the book's length]

Analogy

Can you think of a simpler situation/problem that shares many of the characteristics of the larger problem you are trying to solve? How would you go about solving that simpler problem?

Examples:
1. [his example; pp38-42] If you are asked to find the center of gravity of a homogeneous tetrahedron (a uniformly-dense 3D block w/ 4 triangular sides), you can first ask yourself the simpler question how to find the center of gravity of a homogeneous triangle. And to answer that question you can simplify things by imagining the triangle is made up of lots of tiny rectangles piled on one another (getting narrower at the top).
2. [his example; p43] Scientists try to devise cures to many human ills by first devising cures to the same or similar ills in simpler creatures.
3. [my example; mentioned elsewhere] If you're trying to decide what kind of career you should pursue among the vast number open to you, consider asking yourself how you might make such a decision in a small village in which your range of options was greatly diminished (and assuming you had the freedom to do whatever you wanted).
p.42 - discussion of the difference between using the method vs the result of an analogous problem

Auxiliary elements

Auxiliary elements are new things that you introduce to a problem in the hope that they will further the solution.

Examples:
1. [his example; pp48-50] I'm not 100% sure I understand his example, but I think it's this: i give you the height of a triangle, the angle of the top corner, and the perimeter, and i want you to find the equations for the other sides and angles (i'm not sure about that last part after the comma b/c he never says it explicitly). My impression is that this is considered a hard question initially b/c the student won't have any math formulas memorized for how to tackle problems like this; but you can make use of your existing math knowledge by drawing out a bigger triangle that allows you to make some deductions.
2. [my example] Using something to weigh down a string in the two-string puzzle:
http://www.shadowboxdesign.com/portfoli ... 09305.html

Auxiliary problem

An auxiliary problem is a different problem you try to solve when you get stuck on your main problem; the goal is to use the solution to the different problem to help you solve the main problem.

Examples:
1. [my example; mentioned elsewhere] If you're trying to decide what kind of career you should pursue among the vast number open to you, consider asking yourself how you might make such a decision in a small village in which your range of options was greatly diminished (and assuming you had the freedom to do whatever you wanted).
2. [his] If i say, "Solve for 'x' in the following equation: x^4 + 13x^2 + 36 = 0", most math students wouldn't have a memorized tool for solving it without an intermediate step. In this case, the intermediate step is realizing that we can say, "Let's say y = x^2, and then rewrite the problem as y^2 + 13y + 36 = 0". At this point a math student would be able to use their memorized method of solving quadratic equations to figure out what y is. The "y... = 0" bit is the auxiliary problem.

Bolzano

Bernard Bolzano is a guy who wrote about using heuristics to solve problems. Check him out. [that's pretty much all polya says]

Bright idea

"Bright idea, or 'good idea', or 'seeing the light', is a colloquial expression describing a sudden advance toward the solution."

Examples:
1. [Aristotle's example] "...if you see a person talking with a rich man in a certain way, you may instantly guess that that person is trying to borrow money."
2. [Aristotle's example] "...observing that the bright side of the moon is always toward the sun, you may suddenly perceive why this is; namely, because the moon shines by the light of the sun."

Can you check the result?

Can you derive the result differently?

Can you use the result?

Carrying out

Condition

Contradictory

Corollary

Could you derive something useful from the data?

Could you restate the problem?

Decomposing and recombining

Definition

Descartes

Determination, hope, success

Diagnosis

Did you use all the data?

Do you know a related problem?

Draw a figure

Examine your guess

Figures

Generalization

Have you seen it before?

Here is a problem related to yours and solved before

Heuristic

Heuristic reasoning

If you cannot solve the proposed problem

Induction and mathematical induction

Inventor's paradox

Is it possible to satisfy the condition?

Leibnitz

Lemma

Look at the unknown

Modern heuristic

Notation

Pappus

Pedantry and mastery

Practical problems

Problems to find, problems to prove

Progress and achievement

Puzzles

Reductio ad absurdum and indirect proof

Reductant

Routine problem

Rules of discovery

Rules of style

Rules of teaching

Separate the various parts of the condition

Setting up equations

Signs of progress

Specialization

Subconscious work

Symmetry

Terms, old and new

Test by dimension

The future mathematician

The intelligent problem-solver

The intelligent reader

The traditional mathematics professor

Variation of the problem

What is the unknown?

Why proofs?

Wisdom of proverbs

• Summary: Solving problems is a fundamental human activity; we spend much of our time trying to solve problems of one kind or another. Certain proverbs seem to contain knowledge that has been passed down about the best way to solve problems.
  
  1. The first thing to do is to thoroughly understand the problem and desire a solution to it.
     1. Who understands ill, answers ill.
     2. Think on the end before you begin. / Respice Finem.
     3. A fool looks to the beginning, a wise man regards the end.
     4. A wise man begins in the end, a fool ends in the beginning.
     5. Where there is a will, there is a way.
  2. Devising a plan is the next step.
     1. We will have to deserve a good idea.
        1. Diligence is the mother of good luck.
        2. Perseverence kills the game.
        3. An oak is not felled at one stroke.
        4. If at first you don't succeed, try, try again.
     2. We must try different means, vary our trials.
        1. Try all the keys in the bunch.
        2. Arrows are made of all sorts of wood.
     3. We must adapt to the circumstances.
        1. As the wind blows you must set your sail.
        2. Cut your coat according to the cloth.
  3. We should start carrying out our plan at the right moment.
     1. ...
  4. Looking back at the completed solution is an important and instructive step.
     1. ...
  5. He then creates some new proverbs:
     1. The end suggests the means.
     2. Your five best friends are What, Why, Where, When, and How. You ask What, you ask Why, you ask Where, When, and How–and ask nobody else when you need advice.
     3. Do not believe anything but doubt only what is worth doubting.
     4. Look around when you have got your first mushroom or made your first discovery; they grow in clusters.
• TODO: Continue summarizing this.

Working backwards

• "Exceptionally able people, or people who had the chance to learn in their mathematics classes something more than mere routine operations, do not spend too much time in [guessing a path forward if they hit a situation with many possible paths forward] but turn around, and start working backwards. There is some sort of psychological repugnance to this reverse order which may prevent a quite able student from understanding the method if it is not presented carefully."
• Apparently Plato called this the method of analysis(?). (it's mentioned by Polya in this section)
• Examples:
  1. [Polya's] If you have a 9-quart bucket and a 4-quart bucket, how can you measure out exactly 6 quarts?
  2. [Polya's] If you place a fence on 3 sides of an animal, and put a tasty treat on the opposite side of the middle of that fence, can the animal figure out that it should initially move away from the food so it can go around the fence?