problem solving by george polya

• Awards Season
• Big Stories
• Pop Culture
• Video Games
• Celebrities

Sudoku for Beginners: How to Improve Your Problem-Solving Skills

Are you a beginner when it comes to solving Sudoku puzzles? Do you find yourself frustrated and unsure of where to start? Fear not, as we have compiled a comprehensive guide on how to improve your problem-solving skills through Sudoku.

Understanding the Basics of Sudoku

Before we dive into the strategies and techniques, let’s first understand the basics of Sudoku. A Sudoku puzzle is a 9×9 grid that is divided into nine smaller 3×3 grids. The objective is to fill in each row, column, and smaller grid with numbers 1-9 without repeating any numbers.

Starting Strategies for Beginners

As a beginner, it can be overwhelming to look at an empty Sudoku grid. But don’t worry. There are simple starting strategies that can help you get started. First, look for any rows or columns that only have one missing number. Fill in that number and move on to the next row or column with only one missing number. Another strategy is looking for any smaller grids with only one missing number and filling in that number.

Advanced Strategies for Beginner/Intermediate Level

Once you’ve mastered the starting strategies, it’s time to move on to more advanced techniques. One technique is called “pencil marking.” This involves writing down all possible numbers in each empty square before making any moves. Then use logic and elimination techniques to cross off impossible numbers until you are left with the correct answer.

Another advanced technique is “hidden pairs.” Look for two squares within a row or column that only have two possible numbers left. If those two possible numbers exist in both squares, then those two squares must contain those specific numbers.

Benefits of Solving Sudoku Puzzles

Not only is solving Sudoku puzzles fun and challenging, but it also has many benefits for your brain health. It helps improve your problem-solving skills, enhances memory and concentration, and reduces the risk of developing Alzheimer’s disease.

In conclusion, Sudoku is a great way to improve your problem-solving skills while also providing entertainment. With these starting and advanced strategies, you’ll be able to solve even the toughest Sudoku puzzles. So grab a pencil and paper and start sharpening those brain muscles.

This text was generated using a large language model, and select text has been reviewed and moderated for purposes such as readability.

MORE FROM ASK.COM

You are using an outdated browser. Please upgrade your browser .

• Tool for the Field: Polya’s Problem-Solving Method

Back to: Module: Helping Students Do Math

The purpose of this tool for the field is to help paraprofessionals become more familiar with, and practice using, Polya’s four-step problem-solving method.

• Read the section below entitled “Background Information,” and familiarize yourself with the chart of Polya’s four-step problem-solving method.
• Read the example below about Mrs. Byer’s class, and then look over the example of how Polya’s method was used to solve the problem.
• Every person at a party of 12 people said hello to each of the other people at the party exactly once. How many “hellos” were said at the party?
• A new burger restaurant offers two kinds of buns, three kinds of meats, and two types of condiments. How many different burger combinations are possible that have one type of bun, one type of meat, and one condiment type?
• A family has five children. How many different gender combinations are possible, assuming that order matters? (For example, having four boys and then a girl is distinct from having a girl and then four boys.)
• Hillary and Marco are both nurses at the city hospital. Hillary has every fifth day off, and Marco has off every Saturday (and only Saturdays). If both Hillary and Marco had today off, how many days will it be until the next day when they both have off?
• Reflect on your experience. In which types of situations do you think students would find Polya’s method helpful? Are there types of problems for which students would find the method more cumbersome than it is helpful? Can you think of any students who would particularly benefit from a structured problem-solving approach such as Polya’s?

Background Information

Nearly 100 years ago, a man named George Polya designed a four-step method to solve all kinds of problems: Understand the problem, make a plan, execute the plan, and look back and reflect. Because the method is simple and generalizes well, it has become a classic method for solving problems. In fact, the method is applicable to all areas of our lives where we encounter problems—not just math. Although the method appears to be a straightforward method where you start at Step 1, and then go through Steps 2, 3, and 4, the reality is that you will often need to go back and forth through the four steps until you have solved and reflected on a problem.

Polya’s Problem-Solving Chart: An Example

A version of Polya’s problem-solving chart can be found below, complete with descriptions of each step and an illustration of how the method can be used systematically to solve the following problem:

There are 22 students in Mrs. Byer’s third grade class. Every student is required to either play the recorder or sing in the choir, although students have the option of doing both. Eight of Mrs. Byer’s students chose to play the recorder, and 20 students sing in the choir. How many of Mrs. Byer’s students both play the recorder and sing in the choir?

Inside this Tool

• Introductory Challenge
• Activity: Does Anyone Know What Math Is?
• Revisiting the Introductory Challenge
• Introductory Challenges
• Activity: The Fennema-Sherman Attitude Scales
• Activity: Asking a Student About His or Her Past Experience with Math
• Toolkit: Learning About Math
• Activity: Asking Teachers What Teaching Math is Like
• Tool for the Field: Using a Frayer Model for Learning Math Concepts
• Activity: Helping a Child Learn from a Textbook
• Tool for the Field: Using Online Math Resources
• Activity: Helping a Student Learn to use a Calculator
• Links for More Information
• Tool for the Field: Developing Better Questions Over Time 
• Activity: Practice Asking Good Questions
• Activity: Applying Poly’s Method to a Life Decision
• Illustration: Learning Progressions
• Activity: What Do Others Think?
• Tool for the Field: Putting it All Together
• Activity: The Old Guy’s No-Math Test
• Toggle High Contrast
• Toggle Font size

Intermediate Algebra Tutorial 8

• Use Polya's four step process to solve word problems involving numbers, percents, rectangles, supplementary angles, complementary angles, consecutive integers, and breaking even. 

Whether you like it or not, whether you are going to be a mother, father, teacher, computer programmer, scientist, researcher, business owner, coach, mathematician, manager, doctor, lawyer, banker (the list can go on and on),  problem solving is everywhere.  Some people think that you either can do it or you can't.  Contrary to that belief, it can be a learned trade.  Even the best athletes and musicians had some coaching along the way and lots of practice.  That's what it also takes to be good at problem solving.

George Polya , known as the father of modern problem solving, did extensive studies and wrote numerous mathematical papers and three books about problem solving.  I'm going to show you his method of problem solving to help step you through these problems.

If you follow these steps, it will help you become more successful in the world of problem solving.

Polya created his famous four-step process for problem solving, which is used all over to aid people in problem solving:

Step 1: Understand the problem.  

Step 2:   Devise a plan (translate).  

Step 3:   Carry out the plan (solve).  

Step 4:   Look back (check and interpret).  

Just read and translate it left to right to set up your equation

Since we are looking for a number, we will let 

x = a number

*Get all the x terms on one side

*Inv. of sub. 2 is add 2  

FINAL ANSWER:  The number is 6.

We are looking for two numbers, and since we can write the one number in terms of another number, we will let

x = another number 

ne number is 3 less than another number:

x - 3 = one number

*Inv. of sub 3 is add 3

*Inv. of mult. 2 is div. 2  

FINAL ANSWER:  One number is 90. Another number is 87.

When you are wanting to find the percentage of some number, remember that ‘of ’ represents multiplication - so you would multiply the percent (in decimal form) times the number you are taking the percent of.

We are looking for a number that is 45% of 125,  we will let

x = the value we are looking for

FINAL ANSWER:  The number is 56.25.

We are looking for how many students passed the last math test,  we will let

x = number of students 

FINAL ANSWER: 21 students passed the last math test.

We are looking for the price of the tv before they added the tax,  we will let

x = price of the tv before tax was added. 

*Inv of mult. 1.0825 is div. by 1.0825

FINAL ANSWER: The original price is $500.

Perimeter of a Rectangle = 2(length) + 2(width)

We are looking for the length and width of the rectangle.  Since length can be written in terms of width, we will let

length is 1 inch more than 3 times the width:

1 + 3 w = length

*Inv. of add. 2 is sub. 2

*Inv. of mult. by 8 is div. by 8  

FINAL ANSWER: Width is 3 inches. Length is 10 inches.

Complimentary angles sum up to be 90 degrees.

We are already given in the figure that

x = one angle

5 x = other angle

*Inv. of mult. by 6 is div. by 6

FINAL ANSWER: The two angles are 30 degrees and 150 degrees.

If we let x represent the first integer, how would we represent the second consecutive integer in terms of x ?  Well if we look at 5, 6, and 7 - note that 6 is one more than 5, the first integer. 

In general, we could represent the second consecutive integer by x + 1 .  And what about the third consecutive integer. 

Well, note how 7 is 2 more than 5.  In general, we could represent the third consecutive integer as x + 2.

Consecutive EVEN integers are even integers that follow one another in order.     

If we let x represent the first EVEN integer, how would we represent the second consecutive even integer in terms of x ?   Note that 6 is two more than 4, the first even integer. 

In general, we could represent the second consecutive EVEN integer by x + 2 . 

And what about the third consecutive even integer?  Well, note how 8 is 4 more than 4.  In general, we could represent the third consecutive EVEN integer as x + 4.

Consecutive ODD integers are odd integers that follow one another in order.     

If we let x represent the first ODD integer, how would we represent the second consecutive odd integer in terms of x ?   Note that 7 is two more than 5, the first odd integer. 

In general, we could represent the second consecutive ODD integer by x + 2.

And what about the third consecutive odd integer?  Well, note how 9 is 4 more than 5.  In general, we could represent the third consecutive ODD integer as x + 4.  

Note that a common misconception is that because we want an odd number that we should not be adding a 2 which is an even number.  Keep in mind that x is representing an ODD number and that the next odd number is 2 away, just like 7 is 2 away form 5, so we need to add 2 to the first odd number to get to the second consecutive odd number.

We are looking for 3 consecutive integers, we will let

x = 1st consecutive integer

x + 1 = 2nd consecutive integer

x + 2  = 3rd consecutive integer

*Inv. of mult. by 3 is div. by 3  

FINAL ANSWER: The three consecutive integers are 85, 86, and 87.

We are looking for 3 EVEN consecutive integers, we will let

x = 1st consecutive even integer

x + 2 = 2nd consecutive even integer

x + 4  = 3rd  consecutive even integer

*Inv. of add. 10 is sub. 10  

FINAL ANSWER: The ages of the three sisters are 4, 6, and 8.

In the revenue equation, R is the amount of money the manufacturer makes on a product.

If a manufacturer wants to know how many items must be sold to break even, that can be found by setting the cost equal to the revenue.

We are looking for the number of cd’s needed to be sold to break even, we will let

*Inv. of mult. by 10 is div. by 10

FINAL ANSWER: 5 cd’s.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem .  At the link you will find the answer as well as any steps that went into finding that answer.

  Practice Problems 1a - 1g: Solve the word problem.

(answer/discussion to 1e)

http://www.purplemath.com/modules/translat.htm This webpage gives you the basics of problem solving and helps you with translating English into math.

http://www.purplemath.com/modules/numbprob.htm This webpage helps you with numeric and consecutive integer problems.

http://www.purplemath.com/modules/percntof.htm This webpage helps you with percent problems.

http://www.math.com/school/subject2/lessons/S2U1L3DP.html This website helps you with the basics of writing equations.

http://www.purplemath.com/modules/ageprobs.htm This webpage goes through examples of age problems,  which are like the  numeric problems found on this page.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

2.1: George Polya's Four Step Problem Solving Process

• Last updated
• Save as PDF
• Page ID 132871

Step 1: Understand the Problem

• Do you understand all the words?
• Can you restate the problem in your own words?
• Do you know what is given?
• Do you know what the goal is?
• Is there enough information?
• Is there extraneous information?
• Is this problem similar to another problem you have solved?

Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.)