30th: "How to solve the problem"

30th: "How to solve the problem"

"How to solve the problem" this book is in reading "Master Programmer Interview" (the Chinese version of the 12th page) This book is not to be discovered, a programmer recommended this book to guide programming design, Google to this book after a cursory look over, found to be a teaching students how to solve mathematical problems of the very year book. With careful reading, it is found that this book is written more to teachers, not just to students, but students can learn a lot of thinking skills and ways of thinking in the process of solving problems. Look at this book to see how to help your own programming, and then one is to let the child master the problem of thinking.

Polya completed the first edition of the book in 1945, the book quickly sold, was translated into 17 languages, 1957 out of the second edition, I read the Chinese version seems to be Sushu in the 90 's translation, there are some words translation is not very accurate, i contrast the English version, Many places have been found to be translating the should of English into wrong.

This book is mainly through the 2-page "How to solve the table" to unfold, the author of the process of solving the problem in order to identify problems, plan, plan and review the 4 steps, each step raised a number of universal issues, these questions are very suitable for teachers to guide students step by step to find the answer, as the saying "to the fish, Rather than give it to the fishing ", Polya to a problem, not to throw the answer to the students, nor to the problem-solving process to the students, but put forward a lot of problems, these problems are very skilled, not too broad, not too specific, through the students to answer questions, come and go to find the solution of the problem. In the "How to solve the table" is a series of questions, and the book in front of some examples to illustrate the purpose of these questions, the following gives some terminology or thinking skills, of course, there are also interspersed with some examples, many problems do not require high school knowledge, but still have considerable difficulty, but follow the book's problems, You can also think of the solution to the problem.

The first part of the book describes the purpose of the table, the main problems of the four stages.

The second part of the book describes what problems can be considered in four stages.

The third part of the book is called the "Exploration of the Small dictionary", the entries are arranged alphabetically in English, after the translation of the author is difficult to find the corresponding relationship between these items, need to find the 67 entries back and forth.

The fourth part of the book gives some exercises and answers, different from other books is to give the question of understanding the problem.

How to solve a table

I. Figuring out the problem [1] You have to figure it out .	What is the unknown [2]? What is known quantity? What is the condition [3]? Is it possible to meet all conditions [4]? To determine this unknown condition is not sufficient? Or not enough? Or is it superfluous? Or is it contradictory? Draw a picture [5], introducing the appropriate symbol [6]notation. Consider all the conditions [7], can you write them down?
Ii. plan   Find out the connection between known quantity and unknowns. If you can't find a direct connection, you might want to consider a secondary issue. eventually you should get a solution to the plan.	Have you ever seen this question before? Or a slightly different form of the same problem? [8] Do you know any questions about this? Do you know a theorem that might be used? look at the unknown! [9] Imagine a familiar problem with the same unknown or similar unknowns. You have a problem that has already been solved and is related to your present problem, can you use it? Can you take advantage of its results [10]? Can you use it for a way? Is it possible to introduce an auxiliary element [11] so that it can be exploited? Can you re-describe the problem? Can you just re-describe it in a different way? goes back to the definition [12]. If you cannot solve the problem you are proposing [13], you can first resolve a problem related to this. Can you come up with a question that is easier to get started with? A more general question? A more special question? A question of analogy? Can you solve a part of the problem? What is the extent to which the unknowns can be determined by simply keeping the rest of the condition? How will the unknowns change? Can you export something useful from a known data? Can you come up with other data that is appropriate for determining the unknowns? If necessary, can you change the unknown or known data, or both, so that the new unknowns and new data are closer to each other? Did you take advantage of all the known data? Did you take advantage of all the conditions? Have you considered all the major concepts in the problem?
Iii. implementation of the plan Implement your plan.	Execute your solution plan and test each step. Can you clearly see that this step is correct? Can you prove that this is the right step?
Iv. Review The solution obtained by checking the	Can you verify the result? [14] Can you export this result in a different way [15]? Can you see the result in a sudden? Can you use this result or method for other problems?

[1] This table of problems is mainly for solving the problem to illustrate, another class of questions is the question (or proof), you can see "42".

[2] Unknown English original is unknown, for a problem, this unknown is easy to understand, for a drawing problem, this unknown refers to the figure to make, for a riddle, this unknown refers to the answer. The problem to be solved is to prove that conclusion. The author points out three elements of the solution problem (translation into the main part of the book Principal Parts): unknown unknown, known data and conditional condition.

[3] In the new and old term "57" describes the conditions, conditions used to connect the unknown and known.

[4] in "10" indicates that the conditions can be divided into multiple parts, these conditions may just be sufficient, there may be some conditions are redundant redundant "46", or the paradox of "11". For example: Three unknowns of the ternary one-time equation, if there are 3 equations, then it may be sufficient; If 4 equations are given, it is either redundant or contradictory.

[5] The author emphasizes the importance of the figure "figures" in solving the problem. For geometrical problems, it is necessary to draw a picture, and it is a waste of time to paint a precise picture, but the sketch cannot be too grass, or it may cause misunderstanding. A normal triangle can be drawn in three corners, about 45 degrees, 60 degrees, and 75 degrees, so the triangle is neither isosceles triangle nor right triangle. You can use a solid line, a thick line, or a different color line to represent different features, such as blue to distinguish two similar triangles. Even if it is not a geometrical problem, it is often possible to draw graphs to help solve problems.

[6] The author points out in "38" that the use of good mathematical symbols is sometimes helpful for solving problems.

[7] in "15" refers to the problem can be decomposed and re-combined, in "51" explained that the known data can be considered first with a small number of conditions, see what can be obtained, divided into consideration after the combination, and sometimes immediately out of the results. An example is given in the book, which is known as the top, high, and bottom of a triangle, and draws the triangle. First consider fixing the bottom edge, what shape will the vertices of the same high triangle form? Then consider fixing the bottom edge, what is the vertex of the triangle with the same vertex? Consider all the conditions together and the problem will be solved!

[8] in "26" talked about to actively search for previously learned knowledge, previously done similar problems.

[9] For solving the problem is to look at the unknown, for the question, is to look at the conclusion. Do not forget your goal "36" in the process of doing the problem.

[10] The use of universal "generalization", specialized "specialization", analogy "1" or decomposition combination "15" and other methods, it is easy to come up with a new problem "8".

[11] Do the geometry of the Auxiliary element "2" is usually referred to the auxiliary line, leading to new unknowns, will produce auxiliary questions "3", here is a not simple to do diagram examples, known as the triangle of the apex, high and perimeter, make this triangle.

[12] thoroughly understand definition "16" is very important for solving problems, there is a parabolic example to illustrate the precise definition of the parabola is critical to the problem solving, known as the focus of a parabola, the alignment, there is another line (86th), to make a straight line and the intersection of the parabola. If you do not know the definition of the distance between all points on the parabola and the distance to the alignment, then the problem will not be the same.

[13] When the problem-solving process is blocked, here are some ways to think about the previous related issues, simplifying the problem, the more common problem, analogy, only consider some conditions, change already number or unknown, draw a diagram, the column equation, rethink the definition, derive from known quantity forward, think about symmetry "56", or backwards dry "67 And so on

[14] The test process to avoid simple repetition of "6", in the place where the fall, if the environment and the same as the last time, you will make the same mistake, so in the process of testing to deliberately change the order, re-grouping, or a method, etc., substituting some special values for testing is usually a relatively fast way, In an arbitrary triangular formula, the equilateral triangle of these special triangles can be used to quickly verify the correctness of the results, as well as the dimensional test "58", symmetry test methods such as rough test.

[15] After the problem, if you can find a more convenient solution, will deepen your impression of these types of questions. On the 61st page, another way of thinking about the formula of the side area of the circular table is given, which can find out the side area of the circular table more quickly, and analogy with the trapezoid area formula, so it doesn't need rote memorization.

The following is the third part of the terms of the Chinese and English comparison table

"1" analogy analogy

"2" auxiliary element auxiliary Elements

"3" Auxiliary question auxiliary problem

"4" Bolzano Bolzano

"5" good ideas bright idea

"6" Can you test the result? Can you check the result?

"7" Can you deduce this result in different ways? Can you derive the result differently?

"8" Can you take advantage of this result? Can you use the result?

"9" Implementation plan carrying out (original translation to achieve)

"10" condition condition

"11" Contradiction contradictory

"12" Inference corollary

"13" Can you deduce something useful from the known data? Could derive something useful from the data?

"14" Can you recount this question? Could you restate the problem?

"15" decomposition and re-grouping decomposing and recombining

"16" defines definition

"17" Descartes Descartes

"18" determination, hope, success determination, Hope, Success

"19" Diagnostic diagnosis

"20" Did you take advantage of all the known quantity? Did you use all the data?

"21" Do you know a question about this? Do you know a related problem?

"22" draw a picture draw a figure

"23" Test your guess examine your guess

"24" Graphics figures

"25" universalization of generalization

"26" Have you ever seen it before? Have you seen it before?

The question "27" is related to a problem that you've already solved before. Problem related to yours and sloved before

"28" Exploration method heuristic (I like to translate as heuristics)

"29" heuristic Inference heuristic reasoning (original translation as exploratory argumentation)

"30" If you can't solve the problem raised if you cannot solve the proposed problem

"31" Inductive and mathematical inductive method induction and mathematical induction

"32" Inventor's paradox inventor ' s paradox

Can "33" Meet the requirements? Is it possible to satisfy the condition?

"34" Leibniz Leibnitz

"35" lemma lemma

"36" looks at the unknown look at the unknown

"37" Modern exploration method modern heuristic

"38" Symbol notation

"39" Pappus

"40" and proficient pedantry and mastery

"41" Practical Question practical problems

"42" Solving problem, proving question problems to find, problems to prove

"43" Progress and achievements progress and achievement

"44" Riddle Puzzles

"45" absurdity with indirect proof reductio ad absurdum and indirect proof

"46" redundant redundant

"47" General problem routine problem (translation of the original into the formula problem)

Rules of the "48" innovation Rules of Discovery

"49" style rules rules of style

Rules of the "50" teaching rules of teaching

"51" separates the various parts of the condition separate the various parts of the condition

"52" equation setting up equations

"53" Signs of progress signs of progress

"54" specialized specialization

"55" Subconscious job subconscious work

"56" symmetrical Sysmmetry

"57" New term and old term terms, oid and new

"58" dimensional test test by dimension

"59" Mathematicians of the future mathematician

"60" smart solver the intelligent problem-solver

"61" Smart reader The Intelligent reader

"62" Traditional mathematics professor the traditional mathematics professor

"63" problem changes variation of the problem

What is the "64" unknown? What's the Unknown?

Why do you want to prove "65"? Why proofs?

"66" wisdom of Proverbs Wisdom of Proverbs

"67" inverted dry working backwards

30th: "How to solve the problem"