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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.
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4 Steps to Efficiently Solve Problems
Problems—we all have to deal with minor or major problems in our personal or professional lives. Having a consistent problem-solving approach can be very helpful, and demonstrating strong problem-solving skills can help you stand out in your career.
In this blog post, I’m going to cover a simple problem-solving framework. Although much of what I discuss can be applied to any type of problem, I’ll focus on using the framework from a professional standpoint.
“We cannot solve our problems with the same thinking we used when we created them.” – Albert Einstein
Categories of Problems
Work-related problems can generally be categorized by the area they impact most. That’s not to say a problem can’t impact multiple areas, but usually there is an area of primary impact. I find it useful to categorize problems into the following three categories:
- People —These problems center around people, their expectations, and their interactions with other people.
- Product —These problems are related to what you produce at work. The “product” can be tangible or intangible. If you’re a home builder, your product would be houses. If you’re a software developer, the product would be the application you work on. If you’re a sales professional, you produce sales. Problems in this category are often related to the “product” not meeting the expectations of the customer or stakeholder.
- Process —These problems are related to the processes you use at work, generally in the context of producing the work product. The problem could be the process isn’t producing the desired result, the process isn’t being followed, or the process doesn’t account for enough scenarios.
Although the framework described in the sections below works with each of these categories, the specific approaches you take might vary. For example, if you’re dealing with a process-related problem, a group discussion to analyze the problem likely makes sense. If it’s a people problem, group discussions can be counterproductive, particularly in the early stages.
The Steps (and the Pre-Step)
The framework consists of four steps and a very important pre-step. The four steps are as follows:
- Analyze —Understand the root cause.
- Plan —Determine how to resolve the problem.
- Implement —Put the resolution in place.
- Evaluate —Determine if the resolution is producing the desired results.
I’ll discuss these steps further below, but first I want to discuss an important precursor—triage. In emergency medical situations, the triage process is used to prioritize patients: do they need immediate attention to survive, or do they have injuries that aren’t immediately life threatening? Sometimes, we’re faced with more problems than we can immediately solve, so it’s helpful to prioritize them. I find the following questions to be useful in this process:
- Is there an immediate action I need to take to reduce the impact of the problem?
- Is there a reasonable degree of likelihood I can solve this problem?
- If I can solve the problem, can I solve it in a timely manner?
- If I can solve the problem, will it make a significant difference?
The answers to these questions can help you prioritize the order in which you should focus on particular problems. If a problem is causing significant and immediate pain, then you need to stabilize the situation first—often by addressing the symptoms.
For example, if a customer is upset, you need to address their immediate pain before attempting to resolve the root problem. Once you’ve done so, you can move on to prioritization. If a problem is solvable, can be solved quickly, and has a significant impact, you should focus on it first. If you aren’t sure the problem can be solved, or solving it won’t have a positive impact, then it should be lower on the priority list.
Once this prioritization has been completed, you can analyze the problem.
The goal for analyzing the problem is to understand the root cause(s). (Yes, problems can have more than one root cause.) If you can address the root cause, you can prevent the problem from recurring. It’s important during this process to get multiple perspectives on why the problem occurs. If the problem is in the Product or Process categories, I like to use a group of approximately five people to discuss the root causes. If it’s a person problem, a group setting might be counterproductive and individual conversations are better. However, for Person problems, it’s critical to get multiple perspectives.
There are many techniques for getting to the root cause of problems. One popular and effective approach is the “ 5 whys .” With this approach, you iteratively ask “Why?” about the problem and then each answer until you get to a root cause. For example:
- Why did the upgrade fail? -> The prerequisite updates weren’t installed.
- Why weren’t the prerequisites installed? -> The person performing the install didn’t know there were prerequisites.
- Why didn’t the person performing the install know there were prerequisites? -> They didn’t read the release notes.
- Why didn’t they read the release notes? -> The release notes aren’t included or linked to from the installer.
- Why aren’t the release notes included or linked to from the installer? -> Because the release notes aren’t always required reading for an upgrade.
When using the “5 whys” approach, it’s important to look for process failures as the root cause. In many cases, it’s easy to get to a why such as “There wasn’t enough time” or “We didn’t have enough people.” If you want to fix the root cause, you need to get to “Why did the process fail to alert us of the problem?”
Once you have one or more root causes, you can start looking at how to resolve them going forward. This is another great time in the process to involve multiple people. Having multiple perspectives can produce innovative approaches to address the root causes. It’s also important to remember you might need multiple solutions if you have multiple root causes.
Brainstorming is a good way to generate ideas, but it’s helpful to have a method to manage all the ideas that can be produced. Affinity Grouping is an approach that has been around for a long time, and for good reason—it works well. After generating ideas, you group and potentially combine the similar ones. The various ideas in each group can lead to a better, more rounded solution.
An important aspect of the solution(s) you develop is that you can measure the outcomes. I’ve seen many great ideas that simply didn’t result in the desired outcomes for reasons that couldn’t be anticipated. If you’re able to measure successful outcomes (and unsuccessful outcomes), it helps you adjust more quickly and pivot to different solutions if needed.
Now it’s time to put the solution in place. How you do so can vary significantly depending on what the solution is. However, a key consideration should be how the solution will be monitored. This is why it’s important to define what success looks like in the planning stage. Those measurements are what you will monitor.
It’s important to allow some time before moving to the next step. How much time? It depends—it can be helpful to look at how many times the new solution has been used when determining this. For the example above about release notes, imagine you decided to add an “IMPORTANT” note in a new version of an installer to link people to the release notes. If a week has passed, but only one person has downloaded the new version, then you probably don’t have a large enough sample size to evaluate the solution yet. Conversely, if it’s only been 24 hours, but 50 people have downloaded the new version, you have a much better sample to work with.
Evaluating the solution requires looking at the outcomes objectively and determining if they match expectations. Often, you will find the solution did improve things, but perhaps not as much as you would have liked. If that’s the case, you can refine and iterate on the solution. It might take a few iterations to get the outcomes you would like.
What if the outcomes really don’t match expectations? This scenario often indicates the root cause wasn’t fully understood, and you might need to jump back to the Analyze step. Revisiting the problem with the additional insight of what did not work can help you uncover other root causes.
The next time you’re faced with a problem at work, think TAPIE :
Problem solving is a process—and it’s one we need to be able to carry out in a thoughtful and timely manner throughout our careers. Our ability to consistently and efficiently address problems can be what sets us apart.
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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.
- Goal-Oriented Decision Making - The APE Model
- Generative AI and Decision Making
- The OODA Loop
- The RPD Model
- Reducing the Dunning-Kruger Effect
- Using a Premortem
- The Planning Fallacy
- Accelerated Expertise
- Conduct a SWOT Analysis
- 4D's on a To-Do-List
- Mere Exposure Effect
- The Trolley Problem
- Wicked Problems
- Reciprocity Bias
- Motivated Change
- Correlation vs. Causation
- Maslow's Hierarchy and Innovation
- Understanding Psychological Anchors
- IDEA 4-Step Problem Solving
- Using SMART Goals
- How to Gain Insights
- The Eisenhower Matrix
- SMART Goals - 60 Seconds
- Tactical Decision Games
How to Solve a Problem in Four Steps - The I.D.E.A. Model
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Problem Solving Steps
Solving problems is an important part of any math course. Techniques used for solving math problems are also applicable to other real-world situations. When solving problems it is important to know what to look for and to understand possible strategies for solving. As with most things, the more problems you solve, the better you will get at it. As you solve different types of math problems, you will gain a better understanding of the techniques that can be used with each type of problem.
In his book called How to Solve It, George Polya (1887 1985), proposed a four-step process for problem solving. In this module, we will take a look at those four common problem-solving principles as well as examples of techniques to be applied when solving problems.
Step 1: Understand the problem
In order to solve a problem, it is important to understand what you are being asked to find.
Strategies for understanding the problem:
- Review the problem. If you are solving a word problem, read through the entire problem.
- Seek to understand all the words used in stating the problem. Look for key words that will help you determine whether you will need to add, subtract, multiply, divide, or use a combination of these functions.
- Determine what you are being asked to find or show.
- Restate the problem in your own words.
- Try drawing a picture or diagram to better understand the problem.
- Make sure you have all of the information you need to solve the problem.
Step 2: Develop a plan
Determine how you will solve the problem. Some problems are solved by using a formula and others require you to develop an equation. Pictures, tables, or charts may also be used.
Keep in mind, you may be solving problems that require multiple steps. When you encounter these, break them down into smaller steps and solve each piece. The more problems you solve, the easier it will get to develop a plan for solving problems. You will begin to learn what techniques work best for each type of problem you solve.
Here are some of the common problem-solving techniques:
- Guess and check
- Make a table or list
- Eliminate possibilities
- Use symmetry
- Consider special cases
- Solve as an equation
- Look for a pattern
- Draw a sketch or a picture
- Solve a simpler problem
- Use a model
- Start from the end - work backward
- Use a formula
Step 3: Carry out the plan
Once you have determined your plan for solving the problem, the next step is putting that plan to work. Use the approach that makes sense for the problem and solve it. In most cases, this step will be easier than actually determining the plan. Having an understanding of basic math (pre-algebra) skills will help as you perform the necessary steps to solve the problem. Memorizing the simple multiplication and division tables at least to 10 can make solving problems much easier as well.
If you find that the problem-solving approach you chose does not work, you will need to go back to step two and choose a new approach. Having patience while carrying out your plan is important. It is not uncommon for mathematicians to have to try multiple approaches when solving problems.
Step 4: Look back and check
Here is where you check your logic. If you solved an equation, fill your answer into the equation and check to make sure it works. If you solved a word problem, consider whether or not your answer makes sense. If the problem asked for the height of a ball in the air and your answer was -10 feet, that does not make sense. Just because you get a number doesnt mean it is right. It is important to check your answer and see if it logically makes sense. If your answer does not make sense, you should review the approach you chose as well as your math calculations. Many errors are corrected in this final step of the problem-solving process.
Example 1 Difference in temperature
The hottest temperature ever recorded in Death Valley, CA was 134 degrees on July 10, 1913. The coldest temperature ever recorded there, 15 degrees, occurred earlier that year on January 8, 1913. What was the difference between these record temperatures in 1913? (www.nps.gov)
Step 1: Understand the problem: After reading through the problem, you will find that the problem to solve is clearly stated. You will need to determine the difference in temperature between the hottest and coldest recorded days in Death Valley, CA.
Step 2: Develop a plan: As you were reading the problem, you noted the key word difference. To find the difference between two numbers, you will need to use subtraction. You are now ready to set up an equation. You can assign the variable, x, to the unknown. In this case, the unknown is the difference in temperatures.
x = 134 - 15
Step 3: Carry out the plan: Now that you have developed your equation, go ahead and solve it. x = 134 - 15 x = 119 degrees
Step 4: Look back and check: To check your equation, fill the answer back into the equation and make sure it works. Also, consider whether or not your answer makes sense. If you had received an answer that was significantly higher or lower than either of the temperatures in the problem, that would indicate that there may be an error in your calculations. 119 = 134 15
Example 2 Interest Earned
If a savings account balance of $2650 earns 4% interest in one year, how much interest is earned? What will the account balance be after the interest is earned?
Step 1: Understand the problem: After reading through the problem, you recognize that there are actually two problems to solve. You will need to determine the amount of interest on the account balance and you will need to determine what the account balance will be when the interest is earned.
Step 2: Develop a plan: The first problem you need to solve is the amount of interest earned on $2650. To find the amount of interest, you will need to multiply the current account balance by the interest rate. In order to complete the multiplication problem, you will need to change the percent into a decimal. You can assign the variable, x, to the unknown. In this case, the unknown is the amount of interest earned.
Once you determine how much interest will be earned, you can solve the second problem. You will need to add the current balance and the interest earned to determine how much will be in the account once the interest is applied. To establish an equation for the second problem, you can assign a variable, y, to the unknown. You establish this equation to solve the second problem.
y = $2650 + x
Step 3: Carry out the plan: Now that you have developed your equations, go ahead and solve them.
Part 1: x = .04 x $2650 x = $106
Part 2: y = $2650 + x y = $2650 + $106 y = $2756
Step 4: Look back and check: To check your equations, fill the answers back into them. Also, consider whether or not your answers makes sense. If you had received an answer that was lower or significantly higher than the original account balance, that would indicate there may be an error in your calculations.
Part 1: $106 = .04 x $2650
Part 2: $2756 = $2650 + $106
Example 3 - Book Buyers
In a recent sample of book buyers, 70 more shopped at large-chain bookstores than at small-chain/independent bookstores. A total of 442 book buyers shopped at these two types of stores. How many buyers shopped at each type of bookstore?
Step 1: Understand the problem: After reading through the problem, you determine you are asked to find the number of buyers shopping at each type of bookstore.
Step 2: Develop a plan: To solve this problem you will need to assign a variable, x, for one of the unknowns. If x is the number of book buyers shopping at large-chain bookstores, then (x 70) = the number of book buyers shopping at small-chain/independent bookstores.
To solve the problem, you come up with this equation: x + x - 70 = 442
Step 3: Carry out the plan: Now that you have developed your equation, go ahead and solve it.
x + x - 70 = 442 2x 70 + 70 = 442 + 70 2x /2= 512 /2 x = 256
After solving the equation, you determine that 256 people shopped at large-chain bookstores. You can plug 256 into the equation representing those shopping at small-chain bookstores (x - 70).
256 70 = 186
186 people shopped at small-chain bookstores
Step 4: Look back and check: To check your answers, fill them back into the original problem. Also, consider whether or not your answers makes sense. If the number of small-chain store shoppers was greater than the number of large-chain store shoppers or if the two numbers did not equal 442 that would indicate there was an error in your calculations.
The number of large chain shoppers (256) is 70 more than the number of small-chain store shoppers (186), and the total number of these shoppers (256 + 186) is 442.