Problem Solving Methods

Cheatsheet Content

1. Polya's Four-Step Method A classic heuristic for problem-solving, especially in mathematics. 1. Understand the Problem: What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory? Draw a figure. Introduce suitable notation. Separate the various parts of the condition. Can you write them down? 2. Devise a Plan: Have you seen it before? Or have you seen the same problem in a slightly different form? Do you know a related problem? Do you know a theorem that could be useful? Look at the unknown! Try to think of a familiar problem having the same or a similar unknown. Here is a problem related to yours and solved before. Can you use it? Can you use its result? Can you use its method? Could you restate the problem? Could you restate it still differently? If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Can you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Change the unknown or the data, or both if necessary, so that the new unknown and the new data are nearer to each other. Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem? 3. Carry out the Plan: Carry out your plan. Check each step. Can you see clearly that the step is correct? Can you prove that it is correct? 4. Look Back: Can you check the result? Can you check the argument? Can you derive the result differently? Can you see it at a glance? Can you use the result, or the method, for some other problem? 2. IDEAL Problem Solving A structured approach often used in educational and corporate settings. I dentify the problem: Define the problem clearly. D efine and represent the problem: Frame the problem, considering goals, knowns, and unknowns. E xplore possible strategies: Brainstorm potential solutions. A ct on the strategies: Implement the chosen solution. L ook back and evaluate: Review the outcome and process. 3. Root Cause Analysis (RCA) A method of problem-solving used for identifying the underlying causes of problems rather than just treating symptoms. 3.1. Common RCA Techniques The 5 Whys: Ask "Why?" five times (or until the root cause is found) to drill down into the problem. Example: Car won't start. 1. Why? - The battery is dead. 2. Why? - The alternator is not functioning. 3. Why? - The alternator belt broke. 4. Why? - The belt was old and worn. 5. Why? - Vehicle maintenance was not performed regularly. (Root Cause) Fishbone Diagram (Ishikawa Diagram): Categorizes potential causes into categories (e.g., Manpower, Methods, Machines, Materials, Measurement, Environment). Helps visualize potential causes and their relationships. Pareto Analysis: The 80/20 rule. Focus on the 20% of causes that lead to 80% of problems. Prioritize problems based on impact. Failure Mode and Effects Analysis (FMEA): Identifies potential failure modes in a system, their causes, and their effects. Used to prevent problems. 4. Creative Problem Solving (CPS) A structured process for generating novel and useful solutions to problems. 1. Clarify: Identify the challenge, gather information, and define the problem. Tools: 5 W's and H (Who, What, Where, When, Why, How), Brainstorming. 2. Ideate: Generate a wide range of potential solutions. Defer judgment. Tools: Brainstorming, Mind Mapping, SCAMPER (Substitute, Combine, Adapt, Modify, Put to another use, Eliminate, Reverse/Rearrange). 3. Develop: Evaluate, refine, and select the most promising ideas. Tools: NUF Test (New, Useful, Feasible), PMI (Plus, Minus, Interesting). 4. Implement: Plan for action and put the solution into practice. Tools: Action Plans, Force Field Analysis. 5. Decision-Making Matrix A tool to evaluate and compare multiple options based on a set of criteria. Criteria Weight Option A (Score) Option B (Score) Option C (Score) Cost 0.4 7 ($7 \times 0.4 = 2.8$) 5 ($5 \times 0.4 = 2.0$) 8 ($8 \times 0.4 = 3.2$) Effectiveness 0.3 6 ($6 \times 0.3 = 1.8$) 8 ($8 \times 0.3 = 2.4$) 5 ($5 \times 0.3 = 1.5$) Feasibility 0.2 9 ($9 \times 0.2 = 1.8$) 7 ($7 \times 0.2 = 1.4$) 6 ($6 \times 0.2 = 1.2$) Time 0.1 5 ($5 \times 0.1 = 0.5$) 9 ($9 \times 0.1 = 0.9$) 7 ($7 \times 0.1 = 0.7$) Total Score 1.0 6.9 6.7 6.6 Note: Scores are typically 1-10 (1=worst, 10=best) for each criterion. The option with the highest total score is preferred. 6. Systems Thinking Understanding how parts of a system interrelate and how problems can arise from these interactions, rather than focusing on isolated events. Key Concepts: Interconnectedness: Everything is connected. Feedback Loops: Outputs of a system can become inputs. Positive Feedback: Reinforces change. Negative Feedback: Stabilizes or counteracts change. Emergence: Whole is greater than the sum of its parts. Leverage Points: Small changes can produce large results. Application: Useful for complex, wicked problems where simple cause-and-effect is insufficient. 7. TRIZ (Theory of Inventive Problem Solving) A methodology that provides an algorithmic approach to inventive solutions, based on patterns of invention in patents. Key Principles: Contradiction Matrix: Identifies inventive principles to resolve technical contradictions. 40 Inventive Principles: General solutions to specific problems (e.g., Segmentation, Extraction, Asymmetry). Ideality: Striving for a system that delivers maximum benefit with minimum cost/harm. Steps: Define the specific problem. Map it to a generic TRIZ problem. Find generic TRIZ solutions. Map generic solutions back to the specific problem. 8. Heuristics & Biases Understanding common mental shortcuts (heuristics) and systematic errors (biases) can improve problem-solving. Common Heuristics: Availability Heuristic: Relying on immediate examples that come to mind. Representativeness Heuristic: Judging probability by similarity to a prototype. Anchoring and Adjustment: Relying too heavily on an initial piece of information (the "anchor"). Common Biases: Confirmation Bias: Seeking out information that confirms existing beliefs. Framing Effect: Decisions influenced by how information is presented. Sunk Cost Fallacy: Continuing an endeavor because of invested resources, even if it's no longer optimal. Mitigation: Be aware of these tendencies, seek diverse perspectives, and use structured decision-making tools.