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Maths › 00. General Mathematics ›

08.5 Counterexamples and Edge Cases

8.5 Counterexamples and Edge Cases A counterexample is a single instance that shows a statement is false. An edge case is a boundary or extreme situation where a statement is most likely to fail or require adjustment. Together, they form one of the most reliable tools in problem solving. A universal statement has the form: $$ \text{for all } x,\ P(x). $$ To disprove it, one example with $\neg P(x)$...
mathematics problem-solving counterexamples edge-cases May 2, 2026
Maths › 00. General Mathematics ›

08.4 Heuristics and Experimentation

8.4 Heuristics and Experimentation A heuristic is a rule of thumb that guides search. It does not guarantee a solution, but it helps decide what to try next. Experimentation is the systematic use of examples, computations, diagrams, and small cases to discover patterns. These methods are useful when the direct route is unclear. A problem may have no obvious theorem to apply. Its structure may be hidden. In that situation,...
mathematics problem-solving heuristics experimentation May 2, 2026
Maths › 00. General Mathematics ›

08.3 Analogy and Transfer

8.3 Analogy and Transfer Analogy is the recognition that one problem has the same shape as another. Transfer is the use of that recognition to move ideas, methods, or results from one setting to another. A useful analogy is not based on surface resemblance. It is based on structure. Two problems may use different objects and notation, but if their relationships match, the same method may apply. For example, the...
mathematics problem-solving analogy transfer May 2, 2026
Maths › 00. General Mathematics ›

08.2 Generalization and Specialization

8.2 Generalization and Specialization Generalization and specialization are complementary strategies. One moves outward to a broader setting. The other moves inward to simpler cases. Effective problem solving uses both directions. Generalization replaces a specific problem with a more general one. The goal is not to make the problem harder, but to expose a pattern that is hidden in the original form. For example, instead of studying a single sum, one...
mathematics problem-solving generalization specialization May 2, 2026
Maths › 00. General Mathematics ›

08. Problem Solving Strategies

Chapter 8. Problem Solving Strategies Mathematics is not only a collection of results, but a practice of solving problems. While proof establishes correctness, problem solving determines how one arrives at a solution. This chapter gives an overview of the main strategies used to approach unfamiliar or complex problems. A problem rarely yields directly. It must be transformed. One of the most common strategies is reduction. A difficult problem is rewritten...
mathematics problem-solving methods strategy May 2, 2026
Maths › 00. General Mathematics ›

08.1 Reduction and Transformation

8.1 Reduction and Transformation Reduction is the act of replacing a problem with another problem that is easier, smaller, or already understood. Transformation is the act of changing the form of a problem so that its structure becomes clearer. These two strategies often work together. A reduction says: if we can solve this simpler problem, then we can solve the original one. A transformation says: this is the same problem...
mathematics problem-solving reduction transformation May 2, 2026