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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
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?