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From Wikipedia

How to Solve It

How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.

Four principles

How to Solve It suggests the following steps when solving a mathematical problem:

  1. First, you have to understand the problem.
  2. After understanding, then make a plan.
  3. Carry out the plan.
  4. Look backon your work.How could it be better?

If this technique fails, Pólya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

First principle: Understand the problem

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:

  • What are you asked to find or show?
  • Can you restate the problem in your own words?
  • Can you think of a picture or a diagram that might help you understand the problem?
  • Is there enough information to enable you to find a solution?
  • Do you understand all the words used in stating the problem?
  • Do you need to ask a question to get the answer?

The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.

Second principle: Devise a plan

Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

  • Guess and check
  • Make an orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation

Also suggested:

  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be creative
  • Use your head/noggin

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.

Fourth principle: Review/extend

Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.

The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:

The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.

The book has achieved "classic" status because of its considerable influence (see the next section).

Other books on problem solving are often related to more creative and less concrete techniques. See lateral thinking, mind mapping, brainstorming, and creative problem solving.


  • It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
  • Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems."
  • Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.
  • Russian physicistZhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with Pólya's famous book.
  • Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels Pólya's work.

From Yahoo Answers

Question:Can anyone explain to me how to solve this math problem? Directions: Find the slope of the line through each pair of points. Pairs of points: A(1,1) and B(6,3) I hope this made sense? haha. I would appreciate it if anyone could atleast try to explain it? Thanks :D

Answers:You have to use slope intercept formula to solve this problem Formula: Y2-Y1 over X2-X1 or if that doesn't make sense, its the second y minus the first y over the second x minus the first x. Your equation would now be, 3-1 over 6-1 then that equals 2 over 5 Your slope is 2/5

Question:QUESTION: I walk somewhere. I walk 18 miles North, then get bored and walk 24 miles East. If I want to now get directly home in a straight line, and I walk at 2 miles an hour, how much time will the trip back take? THE CHOICES: 15 3 18 30 6 I completly guessed 3 randomly and got it WRONG. (I THINK the answer is 15)(But I maybe wrong) So how do you work this problem? (This is for practice of the ACT I have coming up in a couple of days)

Answers:First, you need to know if you are searching for the longest side of the triangle or not. Since North and East have a right angle, it's a right triangle and the hypotenuse is your way home. hypotenuse^2 = (18)^2 + (24)^2 = 324 + 576 = 900 => hypotenuse = 30 To find how many hours to get home you need to divide the distance by your speed. .30miles . . . 30miles * 1hours . . .30*1hours --------------- = ------------------------- = ----------------- = 15hours . 2miles. . . . . . . .2miles. . . . . . . . . .2 .------------ . 1hours

Question:Person 1 invested $3000 in an account that has an interest rate of 5.4% and in compounded twice a year. Person 2 invested $2500 in an account that has an interest rate of 7% and in compounded once a year. This is an algebraic question using the formula A=P(1+i/c)^(t)(c) My equations: Person 1: A=3000(1.027)^(2)(x) Person 2: A=2500(1.07)^(x) Please show me how to find x!

Answers:The problem is in the definition of the compounded interest rate. In order for the rate that person 1 invested in to return 5.4% per year, it would have to be 2.6645% every six months because (1.026645)^2 = 1.054. However, back in junior high school when they first taught the summation of a geometric series, they always use compounded interest as an example but since logarithms aren't taught till high school, the teachers always tell the students to divide the annual interest rate by 12 instead of doing the 12th root to get the monthly interest rate. This and with the unfortunate fact that business does not attract the best math students has lead to an entire industry that miscalculates compound interest by deriving the interest per interval from the per annum interest rate as if they were simple interest rates instead of compounded and then projecting that back into the future as if they were compounded interest rates. This would mean that using an 5.4%/2 or 2.7% per six months would be considered the accepted method although it would be mathematically wrong. The mathematically right answer to your question would be: 3000 * (1.02664)^(2 * x) = 2500 * (1.07)^(x) .: 3000 * (1.02664)^2) * (1.02664)^(x) = 2500 * (1.07)^(x) .: ln( 3000 * (1.02664)^2) + ln(1.02664) * x = ln(2500) + ln(1.07) * x .: ln( 3000 * (1.02664)^2) - ln(2500) = ( ln(1.07) - ln(1.02664)) * x .: x = ( ln( 3000 * (1.02664)^2) - ln(2500) )/( ln(1.07) - ln(1.02664)) .: x = 5.6785 years so it would take 6 years for the $2,500 investment to overtake the $3,000 investment However, the more accepted method with bad math would yield: x = ( ln( 3000 * (1.027)^2) - ln(2500) )/( ln(1.07) - ln(1.027)) .: x = 5.7441 years so even the mathematically illiterate would still say it would take 6 years for the $2,500 investment to overtake the $3,000 investment.

Question:If you put a coin on the first square of a checker board and then 2 on the next then 4 and just keep doubling the previous amount on all 64 squares, how many coins would be on the board when you finish? My friend needs help with her logic class and she manually did this on her calculator and got the answer, but we wanted to know how you would generally solve this problem. Is there an equation for this?

Answers:When you add powers of two, you will get as a sum the next highest power of two minus one: 1 +2 +4 =7 =8 -1 1 + 2 +2^2 + ... + 2^9 =2^10 - 1 So for the checkerboard problem, the answer is 2^64 -1.

From Youtube

Applied Linear Equations: Investment Problem :Free Math Help at Brightstorm! www.brightstorm.com How to solve an interest problem when given the total amount deposited.

A-level Maths : Geometric Series (investment problem) :In this video we show you how to answer the following problem. A savings scheme is offering a rate of interest of 3.5% per annum for the lifetime of the plan. Alan wants to save up 20000. He works out that he can afford to save 500 every year, which he will deposit on 1st Janary. If interest is paid on 31st December, how many years will it be before he has saved up his 20000?