Table of contents
How to Solve It
This is basically a summary of the book that appears at the very beginning of the book.
You have to understand the problem.
UNDERSTANDING THE PROBLEM
What is the unknown? What are the data? What is the condition?
Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
Draw a figure. Introduce suitable notation.
Separate the various parts of the condition. Can you write them down?
Find the connection between the data and the unknown.
You may be obliged to consider auxiliary problems if an immediate connection cannot be found.
You should obtain eventually a plan of the solution.
DEVISING A PLAN
Have you seen it before? Or have you seen the same problem in a slightly different form?
Do you know a related problem? Do you know a theorem that could be useful?
Look at the unknown! And try to think of a familiar problem having the same or similar unknown.
Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
Could you restate the problem? Could you restate it still differently? Go back to definitions.
If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or the data, or both if necessary, so that the new unknown and the new data are nearer to each other? Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
Carry out your plan.
CARRYING OUT THE PLAN
Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?
Examine the solution obtained.
Can you check the result? Can you check the argument?
Can you derive the result differently? Can you see it at a glance?
Can you use the result, or the method, for some other problem?
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]
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?
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 are new things that you introduce to a problem in the hope that they will further the solution.
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
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.
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.
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, or 'good idea', or 'seeing the light', is a colloquial expression describing a sudden advance toward the solution."
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?
Could you derive something useful from the data?
Could you restate the problem?
Decomposing and recombining
Determination, hope, success
Did you use all the data?
Draw a figure
Examine your guess
Have you seen it before?
If you cannot solve the proposed problem
Induction and mathematical induction
Is it possible to satisfy the condition?
Look at the unknown
Pedantry and mastery
Problems to find, problems to prove
Progress and achievement
Reductio ad absurdum and indirect proof
Rules of discovery
Rules of style
Rules of teaching
Separate the various parts of the condition
Setting up equations
Signs of progress
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?
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.
- The first thing to do is to thoroughly understand the problem and desire a solution to it.
- Who understands ill, answers ill.
- Think on the end before you begin. / Respice Finem.
- A fool looks to the beginning, a wise man regards the end.
- A wise man begins in the end, a fool ends in the beginning.
- Where there is a will, there is a way.
- Devising a plan is the next step.
- We will have to deserve a good idea.
- Diligence is the mother of good luck.
- Perseverence kills the game.
- An oak is not felled at one stroke.
- If at first you don't succeed, try, try again.
- We must try different means, vary our trials.
- Try all the keys in the bunch.
- Arrows are made of all sorts of wood.
- We must adapt to the circumstances.
- As the wind blows you must set your sail.
- Cut your coat according to the cloth.
- We should start carrying out our plan at the right moment.
- Looking back at the completed solution is an important and instructive step.
- He then creates some new proverbs:
- The end suggests the means.
- 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.
- Do not believe anything but doubt only what is worth doubting.
- Look around when you have got your first mushroom or made your first discovery; they grow in clusters.
- TODO: Continue summarizing this.
- "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)
- [Polya's] If you have a 9-quart bucket and a 4-quart bucket, how can you measure out exactly 6 quarts?
- [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?