problem solving definition wiki

What is Problem Solving? (Steps, Techniques, Examples)

By Status.net Editorial Team on May 7, 2023 — 5 minutes to read

What Is Problem Solving?

Definition and importance.

Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a crucial skill that allows you to tackle complex situations, adapt to changes, and overcome difficulties with ease. Mastering this ability will contribute to both your personal and professional growth, leading to more successful outcomes and better decision-making.

Problem-Solving Steps

The problem-solving process typically includes the following steps:

• Identify the issue : Recognize the problem that needs to be solved.
• Analyze the situation : Examine the issue in depth, gather all relevant information, and consider any limitations or constraints that may be present.
• Generate potential solutions : Brainstorm a list of possible solutions to the issue, without immediately judging or evaluating them.
• Evaluate options : Weigh the pros and cons of each potential solution, considering factors such as feasibility, effectiveness, and potential risks.
• Select the best solution : Choose the option that best addresses the problem and aligns with your objectives.
• Implement the solution : Put the selected solution into action and monitor the results to ensure it resolves the issue.
• Review and learn : Reflect on the problem-solving process, identify any improvements or adjustments that can be made, and apply these learnings to future situations.

Defining the Problem

To start tackling a problem, first, identify and understand it. Analyzing the issue thoroughly helps to clarify its scope and nature. Ask questions to gather information and consider the problem from various angles. Some strategies to define the problem include:

• Brainstorming with others
• Asking the 5 Ws and 1 H (Who, What, When, Where, Why, and How)
• Analyzing cause and effect
• Creating a problem statement

Generating Solutions

Once the problem is clearly understood, brainstorm possible solutions. Think creatively and keep an open mind, as well as considering lessons from past experiences. Consider:

• Creating a list of potential ideas to solve the problem
• Grouping and categorizing similar solutions
• Prioritizing potential solutions based on feasibility, cost, and resources required
• Involving others to share diverse opinions and inputs

Evaluating and Selecting Solutions

Evaluate each potential solution, weighing its pros and cons. To facilitate decision-making, use techniques such as:

• SWOT analysis (Strengths, Weaknesses, Opportunities, Threats)
• Decision-making matrices
• Pros and cons lists
• Risk assessments

After evaluating, choose the most suitable solution based on effectiveness, cost, and time constraints.

Implementing and Monitoring the Solution

Implement the chosen solution and monitor its progress. Key actions include:

• Communicating the solution to relevant parties
• Setting timelines and milestones
• Assigning tasks and responsibilities
• Monitoring the solution and making adjustments as necessary
• Evaluating the effectiveness of the solution after implementation

Utilize feedback from stakeholders and consider potential improvements. Remember that problem-solving is an ongoing process that can always be refined and enhanced.

Problem-Solving Techniques

During each step, you may find it helpful to utilize various problem-solving techniques, such as:

• Brainstorming : A free-flowing, open-minded session where ideas are generated and listed without judgment, to encourage creativity and innovative thinking.
• Root cause analysis : A method that explores the underlying causes of a problem to find the most effective solution rather than addressing superficial symptoms.
• SWOT analysis : A tool used to evaluate the strengths, weaknesses, opportunities, and threats related to a problem or decision, providing a comprehensive view of the situation.
• Mind mapping : A visual technique that uses diagrams to organize and connect ideas, helping to identify patterns, relationships, and possible solutions.

Brainstorming

When facing a problem, start by conducting a brainstorming session. Gather your team and encourage an open discussion where everyone contributes ideas, no matter how outlandish they may seem. This helps you:

• Generate a diverse range of solutions
• Encourage all team members to participate
• Foster creative thinking

When brainstorming, remember to:

• Reserve judgment until the session is over
• Encourage wild ideas
• Combine and improve upon ideas

Root Cause Analysis

For effective problem-solving, identifying the root cause of the issue at hand is crucial. Try these methods:

• 5 Whys : Ask “why” five times to get to the underlying cause.
• Fishbone Diagram : Create a diagram representing the problem and break it down into categories of potential causes.
• Pareto Analysis : Determine the few most significant causes underlying the majority of problems.

SWOT Analysis

SWOT analysis helps you examine the Strengths, Weaknesses, Opportunities, and Threats related to your problem. To perform a SWOT analysis:

• List your problem’s strengths, such as relevant resources or strong partnerships.
• Identify its weaknesses, such as knowledge gaps or limited resources.
• Explore opportunities, like trends or new technologies, that could help solve the problem.
• Recognize potential threats, like competition or regulatory barriers.

SWOT analysis aids in understanding the internal and external factors affecting the problem, which can help guide your solution.

Mind Mapping

A mind map is a visual representation of your problem and potential solutions. It enables you to organize information in a structured and intuitive manner. To create a mind map:

• Write the problem in the center of a blank page.
• Draw branches from the central problem to related sub-problems or contributing factors.
• Add more branches to represent potential solutions or further ideas.

Mind mapping allows you to visually see connections between ideas and promotes creativity in problem-solving.

Examples of Problem Solving in Various Contexts

In the business world, you might encounter problems related to finances, operations, or communication. Applying problem-solving skills in these situations could look like:

• Identifying areas of improvement in your company’s financial performance and implementing cost-saving measures
• Resolving internal conflicts among team members by listening and understanding different perspectives, then proposing and negotiating solutions
• Streamlining a process for better productivity by removing redundancies, automating tasks, or re-allocating resources

In educational contexts, problem-solving can be seen in various aspects, such as:

• Addressing a gap in students’ understanding by employing diverse teaching methods to cater to different learning styles
• Developing a strategy for successful time management to balance academic responsibilities and extracurricular activities
• Seeking resources and support to provide equal opportunities for learners with special needs or disabilities

Everyday life is full of challenges that require problem-solving skills. Some examples include:

• Overcoming a personal obstacle, such as improving your fitness level, by establishing achievable goals, measuring progress, and adjusting your approach accordingly
• Navigating a new environment or city by researching your surroundings, asking for directions, or using technology like GPS to guide you
• Dealing with a sudden change, like a change in your work schedule, by assessing the situation, identifying potential impacts, and adapting your plans to accommodate the change.
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problem-solving

Definition of problem-solving

Examples of problem-solving in a sentence.

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'problem-solving.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Dictionary Entries Near problem-solving

Cite this entry.

“Problem-solving.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/problem-solving. Accessed 8 May. 2024.

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The Problem Definition Process

The Problem Definition Process is a critical step in problem-solving and decision-making, where the goal is to identify and articulate the problem at hand accurately. This process is essential because it sets the foundation for the subsequent stages of problem-solving, such as analysis, generating alternatives, and selecting the best solution. A well-defined problem statement helps ensure that the solution will effectively address the core issue and meet the desired objectives.

Purpose: The main purpose of the Problem Definition Process is to clarify the nature, scope, and boundaries of the problem, ensuring that all stakeholders have a shared understanding of the issue to be addressed. A clear problem definition allows for more efficient and focused problem-solving efforts.

Role: The Problem Definition Process is crucial in problem-solving and decision-making, as it serves as the starting point for any subsequent analysis, alternative generation, and solution selection. By accurately defining the problem, it helps ensure that the problem-solving process remains focused on addressing the core issue.

Components: The Problem Definition Process can be broken down into several key steps:

• Identify the problem: Recognize that a problem exists and determine the symptoms or indicators of the issue.
• Gather information: Collect relevant data and information better to understand the problem's context and root causes.
• Define the problem statement: Clearly articulate the problem, its scope, and its boundaries concisely.
• Identify constraints and criteria: Determine any limitations or constraints that may impact potential solutions, as well as the criteria that will be used to evaluate and select the best solution.
• Validate the problem definition: Ensure that the problem definition accurately captures the core issue and is agreed upon by all stakeholders.

Importance: The Problem Definition Process is important because it lays the groundwork for effective problem-solving and decision-making. A well-defined problem helps ensure that subsequent efforts are focused on addressing the core issue, ultimately leading to better solutions and more efficient use of resources.

History: The concept of problem definition has been recognized as an essential component of problem-solving and decision-making processes for many years. Various methodologies and frameworks, such as the scientific method, design thinking, and systems thinking, emphasize the importance of accurately defining the problem before attempting to find a solution.

Benefits: Some benefits of the Problem Definition Process include the following:

• Improved clarity and focus: A well-defined problem helps maintain focus on the core issue and prevents distractions or tangents.
• Better decision-making: By understanding the problem's context and root causes, decision-makers can make more informed choices.
• Increased efficiency: A clear problem definition allows for more targeted problem-solving efforts, saving time and resources.
• Enhanced communication: A shared understanding of the problem among stakeholders facilitates collaboration and effective communication throughout the problem-solving process.

Pros and cons:

• Enhances clarity and focus on the core issue.
• Facilitates better decision-making and problem-solving.
• Improves communication among stakeholders.
• Can be time-consuming, especially in complex or ambiguous situations.
• Requires effort to gather relevant information and validate the problem definition.

Examples to illustrate key concepts:

• A manufacturing company faces declining productivity in one of its production lines. Instead of jumping to solutions like investing in new equipment or changing work schedules, the company first goes through the Problem Definition Process. By gathering data and analyzing the root causes, they determine that the primary issue is a bottleneck in one part of the line. With a well-defined problem, they can explore targeted solutions to address the bottleneck and improve productivity.
• A software development team receives customer complaints about the slow performance of their application. Instead of immediately attempting to optimize the code, the team first defines the problem by analyzing the performance metrics, identifying specific areas where performance is lagging, and determining the impact on user experience. With a clear problem definition, the team can prioritize their efforts and address the most critical performance issues, ultimately improving the application's overall performance and user satisfaction.

In both examples, the Problem Definition Process helps ensure that the problem-solving efforts focus on addressing the core issue rather than treating symptoms or making assumptions. By gathering relevant information, understanding the context, and clearly articulating the problem, decision-makers can explore targeted solutions and make more informed choices.

In conclusion, the Problem Definition Process is a critical step in problem-solving and decision-making that helps clarify the nature, scope, and boundaries of the problem. By accurately defining the problem, gathering relevant information, and ensuring a shared understanding among stakeholders, the process lays the foundation for more effective problem-solving efforts, ultimately leading to better solutions and more efficient use of resources.

• Problem Solving

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Invariant Principle

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• Suyeon Khim

Generally speaking, an invariant is a quantity that remains constant during the execution of a given algorithm . In other words, none of the allowed operations changes the value of the invariant. The invariant principle is extremely useful in analyzing the end result (or possible end results) of an algorithm, because we can discard any potential result that has a different value for the invariant as impossible to reach.

Formal Definition

Basic examples, monovariants, problem-solving techniques.

Consider a set of states \(S = (s_1, s_2, \dots, s_n)\), and a set of transitions \(T \subseteq S \times S\), defined as follows: \((s_i, s_j) \in T\) if and only if we can transition from state \(s_i\) to state \(s_j\). An invariant with respect to \(T\) is a function \(f: S \rightarrow \mathbb{R}\) such that \((s_i,s_j) \in T \implies f(s_i)=f(s_j)\).

In particular, given a starting state \(s_1\) and a rule for transitions, invariants allow us to determine which states we can reach from \(s_1\).

Many problems give the starting state and the transition rule, then ask whether a certain result is achievable (or equivalently, ask which results are achievable). For example,

Alice writes the numbers 1, 2, 3, 4, 5, and 6 on a blackboard. Bob selects two of these numbers, erases both of them, and writes down their sum on the blackboard. For example, if Bob chose the numbers 3 and 4, the blackboard would contain the numbers 1, 2, 5, 6, and 7. Bob continues until there is only one number left on the board. What are the possible values of that number? In this problem, the invariant is the sum of the numbers on the blackboard, \(n\). If Bob chooses to erase the numbers \(a\) and \(b\), he will write \(a+b\) on the blackboard, making the new sum \(n-a-b+(a+b)=n\), so \(n\) is indeed an invariant. This means that at any time during the process, the sum of the numbers on the blackboard will be \(n=1+2+3+4+5+6=21\), which means that the final number must be 21. In a more formal sense, the states of this problem are possible sets of numbers on the blackboard, and the starting state is \(s_1=\{1, 2, 3, 4, 5, 6\}.\) The transition in this problem is erasing two numbers and writing their sum. The invariant function, \(f(S)\), is the sum of the numbers in \(S,\) and the invariant rule is verified as above. Therefore, since \(f(s_1)=21,\) the end state \(S_{\text{final}}\) must also satisfy \(f(S_{\text{final}})=21,\) and since \(S_{\text{final}}\) has only one number, it must be 21. \(_\square\)

Although the invariant was able to determine precisely what would happen in the previous problem, they are usually only able to determine what cannot happen. For example, if the problem was slightly changed as follows:

Alice writes the numbers 1, 2, 3, 4, 5, and 6 on a blackboard. Bob selects two of these numbers, erases both of them, and writes down their positive difference on the blackboard. For example, if Bob chose the numbers 3 and 4, the blackboard would contain the numbers 1, 1, 2, 5, and 6. Bob continues until there is only one number left on the board. What are the possible values of that number?

Then many different results are possible. However, invariants are still useful in excluding possibilities:

If Bob chooses the numbers \(a\) and \(b\), where \(a \geq b,\) the sum changes from \(n\) to \(n-a-b+(a-b)=n-2b\). Therefore, the sum always changes by an even number, meaning \(n \pmod{2}\) is an invariant. Originally, the sum of the numbers on the board is \(1+2+3+4+5+6=21\), so at any point of the process, the sum of the numbers must be odd. Therefore, the final number cannot be even. \(_\square\)

Note that the invariant says nothing about whether all odd final results are possible; it merely says that no even results are possible.

A monovariant is very similar to an invariant, but instead of remaining unchanged under transitions, monovariants either increase or decrease under transitions. They are very useful in showing algorithms terminate; for example, if a particular monovariant is a positive integer that decreases at every step of an algorithm, the algorithm must eventually terminate since decreasing integers cannot stay positive forever.

In more formal language, a monovariant with respect to \(T\) is a function \(f: S \rightarrow \mathbb{R}\) such that \((s_i,s_j) \in T \implies f(s_i)>f(s_j)\), or a function \(f: S \rightarrow \mathbb{R}\) such that \((s_i,s_j) \in T \implies f(s_i)<f(s_j)\).

Note that the definition uses \(>\) and \(<\) rather than \(\geq\) and \(\leq\). This is because if an algorithm can leave the monovariant equal to its previous value, it becomes much less useful at demonstrating the algorithm terminates. However, if it can be shown that the algorithm must change the value of the monovariant in a finite number of steps, then the non-strict inequality would be sufficient.

As a general rule, invariants are useful whenever several different actions are possible, and especially when a problem asks whether a specific result is possible. In particular, invariants are especially helpful in the analysis of combinatorial games , where the potential transitions are given by legal moves, and the result asked about is the winner of the game.

Alice and Bob have a large chocolate bar, in the shape of a \(10 \times 10\) grid. Each turn, a player may either eat an entire bar of chocolate, or break any chocolate bar into two smaller rectangular chocolate bars along a grid line. The player who moves last loses. Who wins this game? (adapted from NIMO) Every turn, the number of chocolate bars either increases by one (if the player breaks a chocolate bar into two chocolate bars), or decreases by one (if the player eats a chocolate bar). Therefore, the number of chocolate bars Alice will have to choose from is invariant modulo 2. At the beginning of the game, Alice has only one chocolate bar to choose from. Since the players cannot break the chocolate bar forever (since they must break the chocolate along grid lines), eventually Alice will have to eat the final piece of chocolate, so Bob wins regardless of how the players choose to play the game. \(_\square\)

In more advanced problems, the use of invariants will not be set up in such an obvious manner. In those cases, it is usually necessary to transform the problem into a transitional one in some way. Many coloring problems fall under this category. For example, a problem asking whether is it possible to tile some shape with dominoes can be considered a transitional problem, where the starting state is the whole shape, the set of transitions is removing two adjacent squares, and the end state is an empty board.

Two opposite corners are removed from a standard chessboard. Is it possible to tile the resulting board with dominoes? Suppose that the removed corners were both white squares. In every transition, we remove one white square and one black square, so \(S=(\text{black squares})-(\text{white squares})\) is invariant. Originally, there are 32 black squares and 30 white squares, so \(S=2\) at any point. The empty board has \(S=0\), so it is impossible to reach it from the original board; in other words, the board cannot be tiled by dominoes. \(_\square\)

Many problems will also use monovariants in a non-obvious way, usually by asking to prove that some arrangement is possible. These problems are often solved by beginning with a random arrangement, describing an algorithm that improves the situation at each step, and showing the algorithm terminates with the help of a monovariant.

In the parliament of Sikinia, each member has at most three enemies . Prove that the house can be separated into two houses, so that each member has at most one enemy in his own house. (Engel, Problem Solving Strategies, p.2) Initially, separate the members into two houses in any manner. Let \(H\) be the total sum of all enemies each member has in his own house. Now suppose there is a person \(A\) who has at least two enemies in his own house. Then he has at most one enemy in the other house, so if \(A\) switches houses, \(H\) will decrease. Since \(H\) is necessarily a non-negative integer, it cannot decrease forever, so at some point this process ends. This means we cannot any longer find a person who has at least two enemies in their own house, which is precisely the assignment requested. \(_\square\)

Another big clue that a problem involves invariants is being asked to prove something for all possible inputs, such as points in a plane. A common strategy to try in these problems is picking a random point, and examining what changes when the point is moved around.

As the first two problems showed, the sum of all the numbers (possibly modulo some constant) is a common invariant to try. More generally, a weighted sum is often useful, such as \(a+2b+4c+8d+16e\) or \(a-b+c-d+e-f\). When dealing with transforming lists of numbers, another common invariant is the number of inversions , or pairs \((i,j)\) such that \(i<j\), but the \(i^\text{th}\) element is bigger than the \(j^\text{th}\) element. For example, the list \((5, 1, 2, 3, 4)\) has four inversions.

A common monovariant to look out for, when dealing with tuples of numbers, is distance from the origin. In many cases, the set of transitions will either cause the distance from the origin to always decrease, often demonstrating that the end result must be \((0,0,\dots, 0)\); in some cases, this is even an invariant.

Start with the set \(\{2, 3, 4\}\). In each step, choose two numbers \(a\) and \(b\), and replace them with \(0.6a-0.8b\) and \(0.8a+0.6b\).

(a) Is it possible to reach the set \(\{1, 3, 5\}\)? (b) Is it possible to reach the set \(\{0, 2, 5\}\)?

(adapted from Engel, Problem Solving Strategies, p.9)

On the island of Camelot live 13 gray, 15 brown and 17 crimson chameleons. If two chameleons of different colors meet, they both simultaneously change color to the third color (e.g. if a gray and a brown chameleon meet each other, they both change to crimson).

(a) Is it possible that they will eventually all be the same color? (b) Is it possible that there will eventually be the same numbers of gray, brown, and crimson chameleons?

(1984 Tournament of Towns, Problem 1)

There are 12 boys seated around a big round table. They play a game with 12 cards. Initially a boy \(A_1\) has all the 12 cards with him.

Every minute, if any boy has 2 or more cards with him, he passes a card to the boy on the left, and a card to the boy on the right. The game ends when each and every boy has 1 and only 1 card with him.

How many minutes does it take for this game to end?

In a chess tournament there are 12 players participating. Every player has to play against every other player in the first round. Winners get 3 points , losers get -1 point and the players who tie get 1 point each.

After first round what is the sum of points of all players?

A dragon has 100 heads. A knight can cut off 15, 17, 20, or 5 heads, respectively, with one blow of his sword. In each of these cases 24, 2, 14, or 17 new heads grow on its shoulders, respectively. If all heads are blown off, the dragon dies. Can the dragon ever die?

There is a mob of 3000 people, 1 of whom is a zombie while the other 2999 are not.

Every minute, all 3000 people form 1000 groups of three. If there are any zombies in a group of three, all three end up zombies.

What is the probability that after 5 minutes there will be exactly 100 zombies in the mob?

Please provide up to 5 decimal places as necessary.

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