problem solving wiki

• Art of Problem Solving

Art of Problem Solving ( AoPS ) is an educational resources company founded by Richard Rusczyk in 2003.

• 1.1 Mission
• 1.2 Tradition of Excellence
• 1.3 The Staff
• 2.1 AoPS Wiki
• 2.2.1 Subject textbooks
• 2.2.2 Competition preparation books
• 2.2.3 Other math books
• 2.3 AoPS Online School
• 2.4 AoPS Forums
• 2.5 AoPS: For The Win
• 2.6 TeX/LaTeX Resources
• 2.7 Math contest problem database
• 2.8 Articles
• 3 Philanthropy

The Company

The main goal of AoPS is to create interactive educational opportunities for avid students of mathematics . As time goes on, AoPS is reaching out to students of other problem solving disciplines as well, including informatics , physics , programming, and others.

Tradition of Excellence

The accomplishments of every student are unique, and there is no way to measure that success. However, we try to record and celebrate achievements of AoPS students, faculty, and community members .

For a list of all the current staff at AoPS and some previous staff go to the AoPS Administrators page.

Resources on AoPS

You're in the AoPS Wiki now!

• AoPSWiki:Table of Contents -- a basic guide to AoPSWiki content
• Academic competitions including a huge List of mathematics competitions
• Academic scholarships including a large list of Mathematics scholarships

Subject textbooks

AoPS subject texts provide instruction to excellent students of mathematics. This modern curriculum is both comprehensive and challenging enough for brilliant young mathematical minds.

• Introduction to Algebra
• Introduction to Counting & Probability
• Introduction to Geometry
• Introduction to Number Theory
• Intermediate Algebra
• Intermediate Counting & Probability
• Precalculus

Competition preparation books

The Art of Problem Solving competition preparation books cover a variety of topics of interest to students of mathematics interested in competitive math.

• Art of Problem Solving Volume 1: the Basics
• Art of Problem Solving Volume 2: and Beyond
• Competition Math for Middle School

Other math books

AoPS sells numerous other math books at the AoPS Bookstore . Many of these books focus on competitions such as the AMC , the Mandelbrot Competition , Mathcounts , or MOEMS .

AoPS Online School

The AoPS Online School hosts math classes primarily for bright middle and high school students. Students of the online math school include winners of nearly every major mathematics competition in the U.S.

AoPS Forums

The AoPS Forums are a place where students and problem solvers of all ages can discuss mathematics and problem solving. Visit AoPSWiki:AoPS forums for more details.

AoPS: For The Win

For The Win is an online interactive game for students to participate in activities similar to the MATHCOUNTS Countdown Round.

TeX/LaTeX Resources

The AoPS Forums are LaTeX -enabled, allowing users to post nice mathematics.

Math contest problem database

AoPS members help organize a large and well-organized [aops.com/resources.php list of problems and solutions] from mathematics competitions all over the world.

AoPS resources include numerous very good articles for students, parents, and teachers.

Philanthropy

Many of the features of the Art of Problem Solving website are made available for free so that every student interested in mathematics can use them.

Additionally, AoPS instructors provide hundreds of hours of teaching and support time for the San Diego Math Circle .

The Art of Problem Solving Foundation is a nonprofit corporation founded by AoPS CEO Richard Rusczyk with largely the same fundamental mission as AoPS itself, but with greater focus on providing educational resources where they are lacking. AoPS donates a portion of its book and class sales to the foundation. A link can be found here . Make sure it is .org if you want to see the foundation website, otherwise, you'll see the homepage of the current website you're on.

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Problem solving

• 1 Introduction
• 2 Learning support, assessment and certification
• 3.1 Define and analyse a problem
• 3.2 Evaluate solutions
• 3.3 Implement the solution
• 4 Assignment

Introduction [ edit | edit source ]

How do you react when you think of "problems"? Do they mean something’s wrong? Do you see hours of your time disappearing in damage control and the search for a solution? Or are you an enduring optimist: problems are early warning signs to issues that need to be addressed . Perhaps your view is more pragmatic: problems simply signify differences between what is actually happening and what needs to happen .

Whichever way you look at problems, you would know from experience that problem solving requires good management. Excellent communication skills, the ability to gather relevant information and properly analyse it, the ability to discern what’s important and what’s not, and ability to actually make decisions, in a timely and appropriate fashion. How does your thinking and practice measure up?

An alternative, more complete course, on solving problems is also available on Wikiversity.

This unit is used in:

• First Line Management
• International Tourism

Learning support, assessment and certification [ edit | edit source ]

The following learning institutions offer learning support, assessment services, and accreditation for courses using this unit:

• Otago Polytechnic in Dunedin, New Zealand offers both face to face and online support in this programme in the School of Business Travel and Tourism , through the (New Zealand) National Certificate in First Line Management .

Content [ edit | edit source ]

Define and analyse a problem [ edit | edit source ], evaluate solutions [ edit | edit source ], implement the solution [ edit | edit source ].

we can define a problem & provide possible solutions

Assignment [ edit | edit source ]

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• Suyeon Khim
• Silas Hundt

For recursion in computer science, see recursive functions .

Recursion formalizes the process of recognizing how solutions to smaller cases of a problem can, layer by layer, be built up to solve any case of a problem, no matter how enormous. Needless to say, it can be tricky to figure out how to solve infinitely many problems simultaneously. That's what this wiki page will explain, so brace yourself for some problem solving that feels a bit loopy and oddly cyclic at times.

When to Use Recursion

Recursive problem solving, patterns with a longer memory, infinite recursion, further topics in recursion, unsolved recursion-related combinatorics problems.

Some signs that recursion might be a good method for attacking a problem are as follows:

The problem asks for the value of a large case, and you can determine small values or they are given. There is a clear connection between small cases and large cases.

This example problem can be solved straight, without recursion. But the recursive solution is a beautiful and efficient approach.

How many red triangle tiles will be needed for the next \((\)the \(4^\text{th})\) layer of this star pattern? There's no shame in solving this puzzle by counting (although doing so accurately might be trickier than you think). You might even come up with a clever way to count – say, by recognizing that each arm of the star will need the same number of triangles. But neither of these methods is as practical or as elegant as recursion.

If you're not convinced yet that a recursive method is worth the investment, consider counting all of the triangles in the \(1000^\text{th}\) level of this pattern. There are 23988 of them.

Why use recursion? Because no one would ever want to count that high! Using recursion, you'll be able to solve this problem easily by the end of the next section.

Recursion should be applied as a technique when the problem you're solving is like an onion. Using recursion requires realizing that a big, complex problem that's likely to make you cry is really just a slightly smaller problem, plus one super-thin layer. And that problem, in turn, is just a slightly smaller problem, plus one super-thin layer...

Steps of Recursive Problem Solving Create and analyze smaller cases of the problem. Try to construct larger cases using smaller cases. Make a conjecture (a guess) about how small cases are generally related to larger cases. Prove your conjecture and translate it into a general formula that uses the answers to smaller/simpler cases of the problem to find the answer to a larger/more difficult case. Use the formula from step 4 to build up to the solution to the specific layer you care about.

Critically, in step 4, the relationship between smaller and larger cases needs to be a general one for constructing any case of the problem from earlier/smaller cases, not just a single example of making a specific complex case from specific earlier cases.

These steps are the secret to efficiently and beautifully solving many problems that would make the average Joe weep.

How many red triangle tiles will be needed for the next (the 4th) layer of this star pattern? We will solve this problem using recursion. Thinking recursively solves this problem beautifully and efficiently. Step 1 Create and analyze smaller cases of the problem. The natural cases in this problem are the sequential layers of the star: The first layer has 12 triangles. The second layer has 36 triangles. The third layer has 60 triangles. Step 2 Try to construct larger cases using smaller cases. Compare the \(1^\text{st}\) layer to the \(2^\text{nd}\) layer, and the \(2^\text{nd}\) layer to the \(3^\text{rd}\) layer: Step 3 Make a conjecture (a guess) about how small cases are generally related to larger cases. Notice that by moving groups of the triangles in one layer away from the center and then adding 24 extra triangles in diamond-shaped pairs, we can make the next layer of the pattern from each previous layer. This geometric relationship between two sequential layers is a general one that works for any two sequential layers. Step 4 Prove your conjecture and translate it into a general formula that uses the answers to smaller/simpler cases of the problem to find the answer to a larger/more difficult case. Each layer will require 24 more triangles than the layer just before it! If \(C_n\) is the answer to case \(n\) of the problem, then \(C_n = C_{n-1} + 24\) and \(C_1\) = 12 since with one layer, there are 12 triangles in the star. Step 5 Use the formula from step 4 to build up to the solution to the specific layer you care about. Finding the number of triangles in the \(4^\text{th}\) layer: Case 1: There are 12 triangles in the initial star. Case 2: There are \(12 + 24 = 36\) triangles in the \(2^\text{nd}\) layer. Case 3: There are \(36 + 24 = 60\) triangles in the \(3^\text{rd}\) layer. Case 4: There are \(60 + 24 = \boxed{84}\) triangles in the \(4^\text{th}\) layer. \(_\square\)

How many white triangles would you need to make the \(5^\text{th}\) layer of this tile pattern?

Too big for counting? Definitely. Too big for recursion? Never!

How many white triangles would be needed to make the \(1001^\text{st}\) layer of this tiling pattern?

Finding a recurrence relation requires identifying some regular relationship between each case of a problem and the earlier cases. But many times it won't be solely the case just before that's relevant. Sometimes, later cases will evolve from combining two, three, or any number of previous cases together.

You're creating a catalog of tile designs for a friend. In this series of designs, the area to be tiled is 2'' x 16'' and the tiles are each 1'' x 2''. How many different ways are there to cover the area with tiles? A few examples:

Step 1 Create and analyze smaller cases of the problem.

The length of the area can be decreased in order to make this problem simpler. It is definitely easier to count how many ways a 2'' x 5'' rectangle could be tiled for example. Additionally, our wishful thinking at this stage is that somehow the solution to the 2'' x 16'' case would be easy to figure out if only we knew the solution to the 2'' x 15'' case already... The sequence of problems this reasoning suggests is to consider tiling a 2'' x 1'' area, then a 2'' x 2'' area, then a 2'' x 3'' area, etc. Therefore, the general case is 2'' x n''. Solving the Smallest Cases \(C_1=1\) There is 1 way to tile a 1'' x 2'' area. \(C_2=2\) There are 2 ways to tile a 2'' x 2'' area. \(C_3=3\) There are 3 ways to tile a 3'' x 2'' area. \(C_4=5\) There are 5 ways to tile a 4'' x 2'' area.

Step 2 Try to construct larger cases using smaller cases.

We could keep doing out cases one at a time or, at any point, we can start looking for a pattern. Try using the sets above as a tool to identify all of the tiling patterns for \(C_5\). How can you make a larger tile pattern out of a smaller one?

Step 3 Make a conjecture (a guess) about how small cases are generally related to larger cases.

Say, for example, that we're trying to figure out the number of tiling patterns for \(n = 5\), aka, the number of ways to tile a 5'' x 2'' area with dominoes. To start, we know that there are at least 5 ways to do it: take each of the five \(n = 4\) solutions and add one new vertical tile on the left: But are there more solutions? Yes. Above are all of the solutions that start with one vertical tile, but what about all of the solutions that start with two horizontal tiles on the far left? If there are 3 inches left to fill after these two tiles on the left, how many ways are there to fill these 3 inches? \(C_3 = 3.\)

Step 4 Prove your conjecture and translate it into a general formula that uses the answers to smaller/simpler cases of the problem to find the answer to a larger/more difficult case.

Generalizing this observation, for any \(n\)'' x 2'' area, there are \(C_{n-1}\) tilings with one vertical domino on the far left, and \(C_{n-2}\) tilings with two horizontal dominoes on the far left. So \(C_n = C_{n-1} + C_{n-2}\). In words, the solution to each case is the sum of the solution to the two previous cases. That's the Fibonacci sequence ! Apart for the base case terms which are, in this case, \(C_1 = 1\) and \(C_2 = 2\).

Step 5 Use the formula from step 4 to build up to the solution to the specific layer you care about.

Using 1'' x 2'' dominoes, there's one way to tile a 1'' x 2'' area, 2 ways to tile a 2'' x 2'' area, 3 ways to tile a 3'' x 2'' area, and 5 ways to tile a 4'' x 2'' area.

How many ways are there to tile a 5'' x 2'' area?

You're creating a catalog of tile designs for a friend. In this series of designs, the area to be tiled is 2'' x 16'' and the tiles are each 1'' x 2''.

How many different ways are there to cover the area with tiles?

Note: The image shows a few examples that describe such scenario.

At this point, stopping at any specific point in the recursion process might feel kind of arbitrary--why get off at the \(1001^\text{st}\) floor of a building when you can take the elevator up to the moon... and beyond?

Whenever you define a recursive process, you can imagine taking the infinite limit of that process, asking, "What are we approaching as we repeat the process over and over and over again?" Sometimes, infinity does amazing things to recursion.

Fractals are a beautiful application of the infinite extension of recurrence relations. One particular kind of fractal called an iterated function system (IFS) is pretty much just visual recursion on an infinite kick. These fractals are made using recursive processes. Here are a few examples of IFS fractals:

Sierpinski's Triangle 1) Start with a right isosceles triangle of side length 1. 2) Draw lines connecting the centers of each edge and remove the inverted triangle that these edges form. 3) Draw lines connecting the centers of each edge of the remaining solid triangles. Remove the inverted triangles that these edges form. ... repeat step 3 process over and over again, each time removing smaller, inverted triangles from the center of each remaining triangle.

Infinitely many. For example, imagine the left corner of the triangle at the origin of a graph and the left edge of the triangle as a vertical line from 0 to 1. In the first iteration of the process, the center point \(\frac12\) is removed, then \(\frac14\) and \(\frac34,\) then \(\frac18, \frac38, \frac58,\) and \(\frac78,...\) etc. But \(\frac13\) will never be removed, nor will \(\frac19, \frac1{27},...\) etc. In fact, on the vertical edge of the figure, even though infinitely many points are removed, more points remain in the fractal.

There will be no line segments left in the image. The vertical left edge of the final fractal looks like a line, but so many points have been removed from it that there are no continuous sections of line left. In other words, eventually, every line segment in the image has at least one point removed from it, breaking it. So the final 'length' of every segment will be 0.

None. For any patch of solid area that initially exists, eventually, some portion of it is removed by the recursive process. Therefore, no solid patches of area remain after executing the recursive process infinitely many times.

Conclusion: Fractals are weird !

• Linear Recurrence Relations
• Finding Irregular Closed Form Solutions via the Substitution Method (Converting a recursive pattern into a formula that immediately gives out an answer for any specific case, without needing to sequentially figure out every smaller solution.)
• Finding the Closed Form of Recurrence Relations via the Master Theorem
• Even More about Recurrence Relations
• Generating Functions - Solving Recurrence Relations
• The Y-Combinator (Lambda Calculus)

Main article: Unsolved Problems in Combinatorics

• TODO: Example 1
• TODO: Example 2

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Problem Solving

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Problem Solving

What is problem-solving.

Problem-solving is the act of defining a problem; determining the cause of the problem; identifying, prioritizing, and selecting alternatives for a solution; and implementing a solution. [1] It is a process of finding solutions to difficult or complex issues. Problem-solving is important because it allows people to take on challenges and solve problems. American workers rank "dead last" out of 18 industrial nations when it comes to problem-solving skills using technology, which highlights the importance of having these skills.

Problem-solving is the process of taking action to identify and resolve problems. It involves applying knowledge, skills, tools, and techniques to explore alternative solutions, and choosing the best possible option.

There are many different types of problem-solving - from finding the root cause of an issue to developing creative new ideas for products or services. The key is to select the right type of problem-solving for the situation you're facing.

Some problem-solving techniques include brainstorming, trial and error, and breaking the task down into smaller parts.

What are the skills needed for problem-solving?

1. critical thinking.

Critical thinking is defined as the ability to think clearly and rationally, understanding the logical connection between ideas. Critical thinkers engage in reflective and independent thinking. They do not accept arguments or conclusions at face value, but instead question the claims, evaluating the evidence and testing the reasoning. They are able to look at a problem from multiple angles and identify the most efficient solution.

Problem-solving is a critical thinking process that involves finding the best possible solution to a problem. It is a process that requires you to use your analytical and creative skills to find the most efficient and effective solution.

Critical thinking is important for problem-solving because it allows people to explore different alternatives and consider the possible consequences of each. Creative thinking skills help people develop the ability to think critically and logically.

2. Analytical thinking

Analytical thinking is a process of breaking down a problem or situation into smaller, more manageable parts in order to better understand it. This process often involves looking at the problem from all angles and can help lead to finding a true solution. Those who use analytical thinking skills effectively often find decision-making to be easier.

3. Organizing skills

Problem-solving is essential for efficiency and effectiveness in any workplace. Good problem solvers are proactive and collaborative; they understand the root of a problem and work with others to come up with solutions. Problem-solving is important in the workplace because it helps employees learn and grow. Employees can use problem-solving skills to research patterns, experiment, and brainstorm ideas. Organizing skills help employees to understand how to solve problems effectively.

4. Systematic problem-solving

Systematic problem-solving is an important tool for organizations to improve their performance. It involves taking a step-by-step approach to solving problems, which can help identify the root causes of problems and find solutions that meet all the requirements.

5. Critical monitoring and feedback

Critical monitoring and feedback are important for problem-solving because they establish metrics that are aligned with company goals. This allows for an accurate and timely assessment of the effectiveness of the chosen solution and its ability to meet needs. Feedback channels help to solidify future success by providing essential information about how well the problem-solving process is working.

6. Effective communication

Effective communication is essential for problem-solving, as it helps identify and clarify the problem. Without effective communication, teams would be unable to work together to solve problems, which would lead to delays and potential issues.

7. Effective teamworking

Good problem-solvers need to be proactive and collaborative in order to identify and resolve problems effectively. teamwork skills help individuals become self-starters and collaborative teammates, which aids in the identification and resolution of problems.

What is the process of problem-solving?

The problem-solving process can be divided into four steps:

• Identify the problem
• Generate possible solutions
• Evaluate the possible solutions
• Choose the most efficient and effective solution

The first step in the problem-solving process is to identify the problem. This can be done by brainstorming with a group or by yourself. Once the problem has been identified, you need to generate possible solutions. This can be done by brainstorming with a group or by yourself. Once you have generated a list of possible solutions, you need to evaluate each one. You need to consider the pros and cons of each solution. After you have evaluated the possible solutions, you need to choose the most efficient and effective solution. This is the final step in the problem-solving process.

Step 1: Precisely identify the problem

When solving a problem, it is important to precisely identify the problem. This is because incorrect identification can lead to inefficient solutions. Strategies for identifying the problem include asking questions, breaking the problem down into smaller pieces, looking at the problem from different perspectives, and conducting research.

Step 2: Collect information and plan

In this step, you collect information about the problem and things related to it such as its environment. Information is gathered from available sources such as systems and stakeholders such as users, and customers. This step helps you understand the problem, separate it from its symptoms, and define plausible hypotheses about its root cause.

Step 3: Brainstorm solutions

Brainstorming refers to the spontaneous free generation of ideas, in which members of a group suggest potential solutions to a problem without critical judgment. The goal is to generate as many ideas as possible in order to identify a variety of potential solutions that can be further explored. When you're trying to solve a problem, it can be helpful to brainstorm with a group. Brainstorming allows you to generate a lot of ideas quickly, without judging them. This can help you come up with potential solutions that you can explore further. Brainstorming is important to the problem-solving process because it allows people to identify potential solutions to a problem. When brainstorming, it is important to focus on the problem statement and not get overwhelmed by unnecessary information. Brainstorming should include planning and setting a timeline.

Step 4: Decide on a solution

Using the information gathered different hypotheses are analyzed to identify the root cause of a problem. Then, through further analysis, the best solution for a situation can be selected from those identified in the previous step. The process of choosing a solution involves eliminating less desirable options, evaluating them based on their advantages and disadvantages, and then selecting the best one. It is important to practice due diligence throughout the process in order to ensure successful implementation from the start.

Step 5: Implement the solution

Implementation is a complex undertaking with its own challenges. It is important to understand that is it an indispensable part of the problem-solving process because without this step no problem can really be solved - it is important to implement the solution in order to achieve success.

Step 6: Evaluate the outcome

Evaluation is important to ensure that the best possible solution is reached. Evaluation can be immediate or delayed, and it can focus on different aspects of the solution. Evaluating solutions in terms of their advantages and disadvantages helps to determine which one is best.

Step 7: Monitoring progress

Monitoring progress is important in problem-solving in order to ensure that a problem is actually solved and no new problems arise. Governance and its frameworks, processes, and tools are used to monitor and control the actual outcomes of implementing a solution. Metrics play a central role in this step keeping in mind the adage: what cannot be measured cannot be managed.

Step 8: Evaluating the results

It is important to evaluate the results of a problem-solving process in order to determine the best course of action. This evaluation can be immediate or delayed, depending on the situation.

Step 9: Modify the solution

This process usually reveals problems, overlaps, gaps, and improvement opportunities in the problem-solving process or the solution. These must be addressed to make sure we have an ever-improving solution based on real-world results.

How can you improve your problem-solving skills?

1. practice problem-solving regularly.

Problem-solving is an important skill for a variety of reasons. It can help boost creativity, decision-making and project management skills. Additionally, problem-solving can help you find solutions to complex problems. Practicing problem-solving regularly is thus important in order to maintain these skills.

2. Use effective problem-solving tools

Problem-solving involves using a variety of tools to identify the problem and choose the best solution. Heuristic methods are general guides used to find solutions to problems. Heuristic methods include IDEAL, which stands for "Identify, Evaluate, and Attack." Problem-solving with heuristic methods can be improved by building a toolbox of problem-solving strategies. Problem solving also involves recognizing and using multiple strategies to solve complex problems. Watching a chimpanzee solve a peanut problem can help you learn how to solve similar problems.

3. Take the time to analyze the problem

The reference explains that taking the time to analyze the problem is an important part of improving your problem-solving skills. This is because it allows you to look at the issue from a variety of perspectives and identify the true source of the problem. By reflecting on your process and evaluating results, you can also learn from your mistakes and become a better problem solver.

4. Use intuition and creativity to come up with solutions

When brainstorming solutions, focus on generating creative ideas. Evaluate the ideas that come up in your mind to see if they are effective and manageable. Make sure the solution is possible and easy to implement without causing any problems. Use intuition and creativity to generate new ideas that may be more effective in solving the problem. When brainstorming solutions, pay attention to the problem statements and goals, and try not to rule any ideas out. Solutions are more like a roadmap with sets of actions. Lean success in mind means minimizing waste and improving business ecosystems.

5. Use a structured approach to problem-solving

There are four steps to good problem solving: define the problem, generate several possible solutions, choose the best one, and put it into action. This is a structured approach to problem-solving that can help improve your skills.

6. Be patient and keep an open mind

When trying to improve problem-solving skills, it is important to be patient and keep an open mind. This allows for constructive controversy and inductive reasoning, which can help clarify thinking and kick-start the process of implementing solutions.

What is the difference between a problem and a goal?

A problem is something that you are experiencing, while a goal is what you want to achieve.

The difference between a problem and a goal is that a goal is something that you want to achieve, while a problem is something that you have to overcome in order for you to achieve your objective. The difference between the two is that a problem is something that you need to fix, while a goal is something that you want to achieve. A problem can be frustrating, but goals are always satisfying because they give you hope for the future.

A problem is a collection of goals that need to be completed in order for you to achieve an objective.

What is the difference between a problem and a symptom?

A problem is something that actually exists, whereas a symptom is just a sign that there is a problem. It's important to be careful not to treat problems lightly because often they are actually underlying causes of other problems. You need to be able to research and identify underlying causes in order to properly address a problem.

What is the difference between a problem and a root cause?

A problem is something that you don't like or is causing you trouble. A root cause is an actual reason why a problem exists. In other words, a problem is something that is causing difficulty or distress, while a root cause is the underlying cause of the problem. It's important to know the real cause and the root of the problem in order to solve small problems easily.

What is the difference between problem-solving and decision-making?

Problem-solving is a process of identifying and resolving problems while decision-making is the process of picking the best course of action based on the information available.

Problem-solving occurs automatically for many of the small decisions that need to be made on a daily basis. When making a decision about whether to get up now or sleep in for an extra 10 minutes, the possible choices and the relative risks and benefits of obeying the alarm clock or sleeping later come automatically to mind. Larger problems are addressed in a similar way - for example: “I have tasks that need to be done by the end of the week. How am I going to get them all done on time?” After considering the possible strategies, 1 is chosen and implemented.

What are the different types of problem-solving?

There are four main types of problem-solving: analytical, creative, technical, and operational. Each type has its own strengths and weaknesses. It is important to know which type of problem-solving is appropriate for the task at hand in order to be successful.

Analytical Problem Solving

Analytical problem-solving is a logical and methodical approach to problem-solving. It involves breaking down a problem into smaller pieces, analyzing each piece, and then putting the pieces back together to find a solution. This type of problem-solving is often used in math and science.

The strengths of analytical problem-solving are:

• It is a systematic approach that can be followed step-by-step.
• All the information needed to solve the problem is usually available.
• The solution is often definitive and final.

The weaknesses of analytical problem-solving are:

• It can be time-consuming.
• It can be difficult to find the right information.
• The solution may not be obvious

Creative problem-solving

Creative problem-solving is a more open-ended approach to problem-solving. It involves coming up with new and innovative solutions to problems. This type of problem-solving is often used in arts and design.

The strengths of creative problem-solving are:

• It can be used to solve problems that are difficult or impossible to solve using analytical methods.
• It can lead to new and innovative solutions.

The weaknesses of creative problem-solving are:

• The solution may not be obvious.

Technical problem-solving

Technical problem-solving is a more hands-on approach to problem-solving. It involves using a specific set of tools and techniques to solve problems. This type of problem-solving is often used in engineering and computer science.

The strengths of technical problem-solving are:

• The solutions are often more efficient and less time-consuming than those found using other methods.

The weaknesses of technical problem-solving are:

• It can be difficult to find the right tool for the job.
• The solutions may not be obvious.

Operational problem-solving

Operational problem-solving refers to solving problems that occur in everyday life. This type of problem-solving often requires the use of common sense and logic.

The strengths of operational problem-solving are:

• It is often easy to find the right tool for the job.
• The solutions are usually obvious.

The weaknesses of operational problem-solving are:

• It can be difficult to solve problems that are not routine.
• The solutions may not be optimal.

All these different types of problem-solving are important and have their own advantages and disadvantages. It is important to know which type of problem-solving is appropriate for the situation. Analytical problem-solving is most appropriate for problems that have a clear and concise answer. Creative problem-solving is most appropriate for problems that do not have a clear answer or when the best solution is not obvious. Technical problem-solving is most appropriate for problems that require a deep understanding of the subject matter. While operational problem solving is most appropriate for problems that require a detailed understanding of the process.

What are some best practices in problem-solving?

When it comes to problem-solving, there is no one-size-fits-all approach. The best way to solve a problem depends on the situation and the people involved. However, there are some things to keep in mind when trying to solve a problem:

• breaking the problem down into smaller parts because it is easier to find a solution when the problem is smaller
• writing down what you know about the problem so you can organize your thoughts and have a reference
• trying different approaches because one approach might not work for every problem
• enlisting the help of a friend or colleague because two heads are better than one!
• Brainstorming
• Rätselvermehrung
• Problem Tree Analysis
• Metaplan Method
• Mind Mapping
• Bisociation
• Six Thinking Hats
• SHARP (Structured, Holistic Approach for a Research Proposal)
• 8D Problem Solving
• ↑ What is problem-solving?

• Educational psychology
• Neuropsychological assessment
• Pages using ISBN magic links

Problem solving

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Problem solving forms part of thinking . Considered the most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of more routine or fundamental skills ( Goldstein & Levin, 1987 ). It occurs if an organism or an artificial intelligence system does not know how to proceed from a given state to a desired goal state. It is part of the larger problem process that includes problem finding and problem shaping .

• 3 USA and Canada
• 4 Characteristics of difficult problems
• 5 Problem-solving techniques
• 6 Problem-solving methodologies
• 9 References
• 10 External links

Overview [ ]

The nature of human problem solving methods has been studied by psychologists over the past hundred years. There are several methods of studying problem solving, including; introspection , behaviorism , simulation and computer modeling , and experiment .

Beginning with the early experimental work of the Gestaltists in Germany (e.g. Duncker, 1935 ), and continuing through the 1960s and early 1970s, research on problem solving typically conducted relatively simple, laboratory tasks (e.g. Duncker's "X-ray" problem; Ewert & Lambert's 1932 "disk" problem, later known as Tower of Hanoi ) that appeared novel to participants (e.g. Mayer, 1992 ). Various reasons account for the choice of simple novel tasks: they had clearly defined optimal solutions, they were solvable within a relatively short time frame, researchers could trace participants' problem-solving steps, and so on. The researchers made the underlying assumption, of course, that simple tasks such as the Tower of Hanoi captured the main properties of "real world" problems, and that the cognitive processes underlying participants' attempts to solve simple problems were representative of the processes engaged in when solving "real world" problems. Thus researchers used simple problems for reasons of convenience, and thought generalizations to more complex problems would become possible. Perhaps the best-known and most impressive example of this line of research remains the work by Newell and Simon (1972).

In Europe, two main approaches have surfaced, one initiated by Donald Broadbent (1977; see Berry & Broadbent, 1995) in the United Kingdom and the other one by Dietrich Dörner (1975, 1985; see Dörner & Wearing, 1995) in Germany. The two approaches have in common an emphasis on relatively complex, semantically rich, computerized laboratory tasks, constructed to resemble real-life problems. The approaches differ somewhat in their theoretical goals and methodology, however. The tradition initiated by Broadbent emphasizes the distinction between cognitive problem-solving processes that operate under awareness versus outside of awareness, and typically employs mathematically well-defined computerized systems. The tradition initiated by Dörner, on the other hand, has an interest in the interplay of the cognitive, motivational, and social components of problem solving, and utilizes very complex computerized scenarios that contain up to 2,000 highly interconnected variables (e.g., Dörner, Kreuzig, Reither & Stäudel's 1983 LOHHAUSEN project; Ringelband, Misiak & Kluwe, 1990). Buchner (1995) describes the two traditions in detail.

To sum up, researchers' realization that problem-solving processes differ across knowledge domains and across levels of expertise (e.g. Sternberg, 1995) and that, consequently, findings obtained in the laboratory cannot necessarily generalize to problem-solving situations outside the laboratory, has during the past two decades led to an emphasis on real-world problem solving. This emphasis has been expressed quite differently in North America and Europe, however. Whereas North American research has typically concentrated on studying problem solving in separate, natural knowledge domains, much of the European research has focused on novel, complex problems, and has been performed with computerized scenarios (see Funke, 1991, for an overview).

USA and Canada [ ]

In North America, initiated by the work of Herbert Simon on learning by doing in semantically rich domains (e.g. Anzai & Simon, 1979 ; Bhaskar & Simon, 1977 ), researchers began to investigate problem solving separately in different natural knowledge domains - such as physics, writing, or chess playing - thus relinquishing their attempts to extract a global theory of problem solving (e.g. Sternberg & Frensch, 1991). Instead, these researchers have frequently focused on the development of problem solving within a certain domain, that is on the development of expertise (e.g. Anderson, Boyle & Reiser, 1985 ; Chase & Simon, 1973 ; Chi, Feltovich & Glaser, 1981 ).

Areas that have attracted rather intensive attention in North America include such diverse fields as:

• Reading ( Stanovich & Cunningham, 1991 )
• Writing ( Bryson, Bereiter, Scardamalia & Joram, 1991 )
• Calculation ( Sokol & McCloskey, 1991 )
• Political decision making ( Voss, Wolfe, Lawrence & Engle, 1991 )
• Managerial problem solving ( Wagner, 1991 )
• Lawyers' reasoning ( Amsel, Langer & Loutzenhiser, 1991 )
• Mechanical problem solving ( Hegarty, 1991 )
• Problem solving in electronics ( Lesgold & Lajoie, 1991 )
• Computer skills ( Kay, 1991 )
• Game playing ( Frensch & Sternberg, 1991 )
• Personal problem solving ( Heppner & Krauskopf, 1987 )
• Mathematical problem solving ( Polya , 1945; Schoenfeld, 1985 )
• Social problem solving (D'Zurilla & Goldfreid, 1971; D'Zurilla & Nezu, 1982)
• Problem solving for innovations and inventions: TRIZ (Altshuller, 1973, 1984, 1994)

Characteristics of difficult problems [ ]

As elucidated by Dietrich Dörner and later expanded upon by Joachim Funke , difficult problems have some typical characteristics that can be summarized as follows:

• commencement opacity
• continuation opacity
• inexpressiveness
• enumerability
• connectivity (hierarchy relation, communication relation, allocation relation)
• heterogeneity
• temporal constraints
• temporal sensitivity
• phase effects
• dynamic unpredictability

The resolution of difficult problems requires a direct attack on each of these characteristics that are encountered.

In reform mathematics , greater emphasis is placed on problem solving relative to basic skills, where basic operations can be done with calculators. However some "problems" may actually have standard solutions taught in higher grades. For example, kindergarteners could be asked how many fingers are there on all the gloves of 3 children, which can be solved with multiplication. [1]

Problem-solving techniques [ ]

• Abstraction : solving the problem in a model of the system before applying it to the real system
• Analogy : using a solution that solved an analogous problem
• Brainstorming : (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum is found
• Divide and conquer : breaking down a large, complex problem into smaller, solvable problems
• Hypothesis testing : assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption
• Lateral thinking : approaching solutions indirectly and creatively
• Means-ends analysis : choosing an action at each step to move closer to the goal
• Method of focal objects : synthesizing seemingly non-matching characteristics of different objects into something new
• Morphological analysis : assessing the output and interactions of an entire system
• Reduction : transforming the problem into another problem for which solutions exist
• Research : employing existing ideas or adapting existing solutions to similar problems
• Root cause analysis : eliminating the cause of the problem
• Trial-and-error : testing possible solutions until the right one is found
• Working Backwards (Halpern,2002)
• Forward-Looking Strategy (Halpern, 2002)
• Simplification (Halpern, 2002)
• Generalization (Halpern, 2002)
• Specialization (Halpern, 2002)
• Random Search (Halpern, 2002)
• Split-Half Method (Halpern,2002)

Problem-solving methodologies [ ]

• Eight Disciplines Problem Solving
• How to solve it
• Kepner-Tregoe
• Southbeach Notation
• RPR Problem Diagnosis
• TRIZ (Teoriya Resheniya Izobretatelskikh Zadatch, "theory of solving inventor's problems")

See also [ ]

• Abductive reasoning
• Anagram problem solving
• Artificial intelligence
• Brainstorming
• Common sense
• Common sense reasoning
• Cognitive hypothesis testing
• Cognitive processes
• Creative problem solving
• Decision making
• Declarative knowledge
• Deductive reasoning
• Divergent thinking
• Executive function
• Expert systems
• Facilitation
• Gagné's hierarchy of learning
• General problem solver
• Group problem solving
• Inductive reasoning
• Intelligence amplification
• Morphological Analysis
• Newell, Allen
• Problem-solving in psychotherapy
• Problem Solving Therapy
• Problem Statement
• Simon, Herbert
• Soar (cognitive architecture)
• Transdisciplinary Studies
• Troubleshooting
• Wicked problem
• ↑ 2007 Draft, Washington State Revised Mathematics Standard

References [ ]

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• Anderson, J. R., Boyle, C. B., & Reiser, B. J. (1985). Intelligent tutoring systems. Science 228 : 456–462.
• Anzai, K., & Simon, H. A. (1979) (1979). The theory of learning by doing . Psychological Review 86 : 124–140.
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• Berry, D. C., & Broadbent, D. E. (1995). Implicit learning in the control of complex systems: A reconsideration of some of the earlier claims. In P.A. Frensch & J. Funke (Eds.), Complex problem solving: The European Perspective (pp. 131-150). Hillsdale, NJ: Lawrence Erlbaum Associates.
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• Halpern, Diane F. (2002).Thought & Knowledge. Lawrence Erlbaum Associates. Worldcat Library Catalog
• Hayes, J. (1980). The complete problem solver . Philadelphia: The Franklin Institute Press.
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• Hübner, R. (1989). Methoden zur Analyse und Konstruktion von Aufgaben zur kognitiven Steuerung dynamischer Systeme [Methods for the analysis and construction of dynamic system control tasks]. Zeitschrift für Experimentelle und Angewandte Psychologie , 36, 221-238.
• Hunt, E. (1991). Some comments on the study of complexity. In R. J. Sternberg, & P. A. Frensch (Eds.), Complex problem solving: Principles and mechanisms (pp. 383-395). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Hussy, W. (1985). Komplexes Problemlösen - Eine Sackgasse? [Complex problem solving - a dead end?]. Zeitschrift für Experimentelle und Angewandte Psychologie , 32, 55-77.
• Kay, D. S. (1991). Computer interaction: Debugging the problems. In R. J. Sternberg & P. A. Frensch (Eds.), Complex problem solving: Principles and mechanisms (pp. 317-340). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Kluwe, R. H. (1993). Knowledge and performance in complex problem solving. In G. Strube & K.-F. Wender (Eds.), The cognitive psychology of knowledge (pp. 401-423). Amsterdam: Elsevier Science Publishers.
• Kluwe, R. H. (1995). Single case studies and models of complex problem solving. In P. A. Frensch & J. Funke (Eds.), Complex problem solving: The European Perspective (pp. 269-291). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Kolb, S., Petzing, F., & Stumpf, S. (1992). Komplexes Problemlösen: Bestimmung der Problemlösegüte von Probanden mittels Verfahren des Operations Research ? ein interdisziplinärer Ansatz [Complex problem solving: determining the quality of human problem solving by operations research tools - an interdisciplinary approach]. Sprache & Kognition , 11, 115-128.
• Krems, J. F. (1995). Cognitive flexibility and complex problem solving. In P. A. Frensch & J. Funke (Eds.), Complex problem solving: The European Perspective (pp. 201-218). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Lesgold, A., & Lajoie, S. (1991). Complex problem solving in electronics. In R. J. Sternberg & P. A. Frensch (Eds.), Complex problem solving: Principles and mechanisms (pp. 287-316). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Mayer, R. E. (1992). Thinking, problem solving, cognition . Second edition. New York: W. H. Freeman and Company.
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• Newell, A., & Simon, H. A. (1972). Human problem solving . Englewood Cliffs, NJ: Prentice-Hall.
• Paradies, M.W., & Unger, L. W. (2000). TapRooT - The System for Root Cause Analysis, Problem Investigation, and Proactive Improvement . Knoxville, TN: System Improvements.
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• Riefer, D.M., & Batchelder, W.H. (1988). Multinomial modeling and the measurement of cognitive processes. Psychological Review , 95, 318-339.
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• Stanovich, K. E., & Cunningham, A. E. (1991). Reading as constrained reasoning. In R. J. Sternberg & P. A. Frensch (Eds.), Complex problem solving: Principles and mechanisms (pp. 3-60). Hillsdale, NJ: Lawrence Erlbaum Associates.
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• Wagner, R. K. (1991). Managerial problem solving. In R. J. Sternberg & P. A. Frensch (Eds.), Complex problem solving: Principles and mechanisms (pp. 159-183). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Wisconsin Educational Media Association. (1993). "Information literacy: A position paper on information problem-solving." Madison, WI: WEMA Publications. (ED 376 817). (Portions adapted from Michigan State Board of Education's Position Paper on Information Processing Skills, 1992).
• Altshuller, Genrich (1973). Innovation Algorithm , Worcester, MA: Technical Innovation Center.
• Altshuller, Genrich (1984). Creativity as an Exact Science , New York, NY: Gordon & Breach.
• Altshuller, Genrich (1994). And Suddenly the Inventor Appeared , translated by Lev Shulyak, Worcester, MA: Technical Innovation Center.

External links [ ]

• Computer Skills for Information Problem-Solving: Learning and Teaching Technology in Context
• Problem solving-Elementary level
• CROP (Communities Resolving Our Problems)
• Teach Kids Math With Model Method
• The Altshuller Institute for TRIZ Studies, Worcester, MA
• Problem Solving Skills
• 1 Pregnancy fetishism
• 2 Race and intelligence (test data)
• 3 Prostitution

• Problem solving

Problem solving consists in using generic or ad hoc methods, in an orderly manner, for finding solutions to specific problems

Quotes [ edit ]

A - f [ edit ].

• Russell L. Ackoff (1956) The development of operations research as a science
• Julian Assange , "Julian Assange, monk of the online age who thrives on intellectual battle" . The Guardian . 2010-08-01 . Retrieved on 2010-08-01 .  
• Barker (2003, p. 410), cited in Charles Zastrow 'The practice of social work. (1995) p. 2
• Georg Cantor (1867) Doctoral thesis
• C. West Churchman (1957) Introduction to Operations Research , p. 8
• C. West Churchman (1982) Thought and Wisdom .

G - L [ edit ]

• Richard W. Hamming, Richard Wesley Hamming (1997) The Art of Doing Science and Engineering: Learning to Learn . p. 200
• Michael Hardt and Antonio Negri (2004), Multitude , pp. 91-92
• Michael Hardt and Antonio Negri (2004), Multitude , pp. 111-112
• John F. Kennedy , address at the University of Washington's 100th anniversary program, Seattle, Washington (November 16, 1961), reported in Public Papers of the Presidents of the United States: John F. Kennedy (1961), p. 726.

M - R [ edit ]

• H. L. Mencken , "The Divine Afflatus", originally published in the New York Evening Mail (November 16, 1917); reprinted in Prejudices: Second Series (1920), and A Mencken Chrestomathy (1949), chapter 25, p. 443.
• Marvin Minsky (2006) The Emotion Machine
• Swamp Thing : Always guns . Are they...your only... solution ? You can shoot...the animals...in the forest...but you cannot shoot the forest.
• Alan Moore Saga of the Swamp Thing #53
• Allen Newell and Herbert Simon (1975) Computer Science as Empirical Inquiry: Symbols and Search . Turing Award Lecture. p. 120-121
• Nyasha in Nervous Conditions (1988) by Tsitsi Dangarembga , p. 93 . Zimbabwe Publishing House. ISBN 978-0-949225-79-5 .  
• Lars Paul (2010) A method for developing Churchmanian Knowledge Management Systems. p. 2 Citing: C. West Churchman (1967) Guest editorial: Wicked problems .

S - Z [ edit ]

• Herbert Simon (1957) Models of Man . p. 198
• Herbert A. Simon, "Bounded rationality and organizational learning." Organization science 2.1 (1991): 125-134.
• Henry J. Tillman , as quoted in Scientific English: A Guide for Scientists and Other Professionals, Third Edition (2011) by Robert A. Day, Nancy Sakaduski, ABC-CLIO
• Colin Wilson in Access to Inner Worlds , p. 13 (1990)

See also [ edit ]

• Artificial intelligence
• Methodology

External links [ edit ]

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