Problem-solving techniques

• How do we start the process of analyzing a given problem?
• How do we start solving a problem?
• What abilities are required?
• What strategies are followed?

Rules:

• Read carefully and clearly identify the purpose and goal of the problem.
• Understand the problem; restate the problem to eliminate ambiguities and clarify its objectives.
• Identify the known information and look for hidden assumptions and consider extreme cases to gain insight into a situation.
• Identify the unknown- and wanted information.
• Restate and simplify the problem in terms of known and unknown information and state any additional assumptions and approximations. Human communication tends to be imprecise and by restating the problem, more than one interpretation may emerge.
• Break the problem, if possible, into smaller, simpler, easy to manage problems.
• Select appropriate notation to identify the known and unknown information, and if beneficial, define intermediate variables.
• Make a graph, figure, or drawing to help visualize the abstract elements of a problem (include a flowchart when appropriate).
• Construct a table. In some cases, a table may indicate a pattern that may lead to a solution or a better understanding of the problem.
• Replace the variables defined in the mathematical relations by their units and check for consistency.
• Construct a physical model when possible.
• Determine which principles, equations, or models best describe the relation that transforms the known information (called inputs) into the unknown (wanted) variables (called outputs).
• Guess a solution and check if indeed the guessed solution makes sense. Use trial and error method.
• State general solution and systematically list other approaches, exhausting all possibilities, eliminating the impossible but not the improbable.
• Select from all possible solutions the best one. The term 'best' should be defined by the problem solver. Best could mean different things to different people- it could mean the shortest, clearest, easiest, simplest, or cheapest solution.
• Once a solution is known, it is appropriate to work backwards. Verify if it is valid and correct. Analyze and test the solution with simple data to see if indeed the solution satisfies the requirements of the problem. Estimate the results and analyze the implications, such as does the solution make mathematical, logical, or physical sense?
• Test the solution using extreme and special cases and search for patterns or symmetries. 
• Find alternate solutions and compare them. 

From Misza Kalechman, 2007 or 2009?. Practical MATLAB for Engineers