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How to solve stuff, theory

Phase 1: Understanding the Problem (Define Goal & Scope)

• Deconstruct the Query: ◦ Identify the explicit "unknown" (what needs to be found, proven, or achieved). ◦ Identify the "data" (all given information, conditions, initial states, or parameters). ◦ Identify the "conditions" (rules, constraints, definitions, or allowable operations).

• Clarify Ambiguities: If any part of the query is unclear or can be interpreted in multiple ways, seek clarification.

• Visualize and Represent: ◦ Draw a figure or diagram if the problem involves geometry, physical setups, or abstract relationships. ◦ Introduce suitable notation for variables, sets, or quantities. ◦ Algebraize by translating relationships into mathematical expressions or equations where appropriate.

• Define Success (Expected Outcome): ◦ Clearly articulate what a complete and correct solution looks like. Is it a specific numerical value, a proof, a set of solutions, or a general formula? ◦ Understand the properties the desired outcome should possess (e.g., must be an integer, continuous, unique, etc.).

• Categorize the Problem: Determine the broad area(s) of mathematics or logic to which the problem belongs (e.g., Number Theory, Geometry, Combinatorics, Algebra, Functional Equations, Optimization, Proof). This helps narrow down potential strategies.

Phase 2: Devising a Plan (Gather Knowledge & Select Strategies)

• Relate to Knowns: ◦ Recall similar problems you have encountered. How were they approached or solved? What underlying concepts did they involve? ◦ Consider related theorems, formulas, or principles that might apply to the current problem

• Simplify or Generalize: ◦ Explore simpler cases: If the problem is complex, try solving a reduced or simpler version of it (e.g., smaller numbers, fewer elements, specific values) ◦ Look for analogous problems in different contexts that share a similar structure ◦ Consider generalizing the problem: Sometimes, a broader perspective can reveal patterns or underlying principles that simplify the specific instance . However, be mindful of when extending (e.g., to the "left" in sequences, per Jacobi's dictum) is more fruitful

• Choose Problem-Solving Strategies: ◦ Invariance Principle: If the problem involves repeated operations, transformations, or algorithms, search for quantities or properties that remain unchanged (invariants) or that change predictably (semi-invariants). This is particularly useful in games or transformations ◦ Extremal Principle (Variational Method): When proving existence or seeking an optimal value (maximum or minimum), identify and analyze the extremal elements (e.g., the largest, smallest, closest, farthest) within the problem's domain. This often relies on the well-ordering principle for integers. ◦ Working Backwards: If the final state or desired outcome is clear, reverse the process step-by-step to deduce the initial conditions or preceding states. This method is effective when the number of possibilities at each backward step is limited. This is closely related to Jacobi's dictum: "You must always invert!" ◦ Divide and Conquer: Break the problem down into smaller, independent sub-problems that are easier to solve. Once solved, combine the partial solutions to construct the solution to the whole problem ◦ Pattern Recognition & Recursion: For sequences, iterative processes, or enumerative problems, look for repeating patterns, sequences, or relationships between terms. Formulate a recurrence relation to describe the pattern ◦ Transformation/Substitution: Rephrase the problem by changing the representation of objects or variables. Examples include: ▪ Introducing new variables or functions ▪ Using coordinate geometry to translate geometric problems into algebraic ones ▪ Applying trigonometric substitutions to simplify algebraic expressions ▪ Using complex numbers to represent points or transformations in geometry ◦ Proof Techniques (General): ▪ Argument by Contradiction (Reductio ad Absurdum): To prove a statement, assume its negation is true and then demonstrate that this assumption leads to a logical inconsistency or absurdity ▪ Mathematical Induction: To prove a statement for all natural numbers, prove a base case and an inductive step (assuming true for k, prove for k+1) ▪ Pigeonhole Principle: If you have more "pigeons" than "pigeonholes," at least one pigeonhole must contain more than one pigeon ▪ Counting by Bijection: To determine the size of a set A, establish a one-to-one correspondence (bijection) with a set B whose size is already known ▪ Count in Two Ways: Derive combinatorial identities by calculating the size of a set or outcome in two different, but equivalent, manners ▪ Inclusion-Exclusion Principle: For counting elements in unions of sets, systematically add the sizes of individual sets, subtract the sizes of pairwise intersections, add triple intersections, and so on ◦ Domain-Specific Techniques: Apply specialized methods relevant to the problem's category (e.g., modular arithmetic and number theory theorems , specific factoring or Viète's relations in algebra, derivatives and integrals in analysis, geometric congruence/similarity, probability formulas like desired/total outcomes, casework, expected value).

• Formulate Hypotheses: If guessing a solution or a pattern, articulate it as a clear hypothesis that can then be tested and rigorously proven.

Phase 3: Carrying Out the Plan (Execution & Verification)

• Execute Steps Rigorously: Follow the chosen plan meticulously, performing all necessary calculations, derivations, or logical deductions with precision

• Self-Check Continuously: At each critical step, verify the correctness of your intermediate results and ensure the logical flow is sound. Do not make unproven assumptions

• Document Progress: Maintain a clear and organized record of your steps, assumptions, and results. This aids in debugging, reviewing, and communicating your solution.

Phase 4: Looking Back (Review & Refinement)

• Validate the Solution: ◦ Does the answer directly address the original query? ◦ Does it satisfy all stated conditions and constraints? ◦ Is the answer consistent with common sense, known facts, or simpler cases? ◦ If a proof, is it rigorous and free from circular reasoning or unproven patterns?

• Consider Alternative Solutions: Can the problem be solved using a different method? Exploring alternative approaches deepens understanding and confirms correctness

• Generalize and Abstract: ◦ Can the specific method used, or the solution itself, be generalized to a broader class of problems? ◦ What underlying principles or concepts were most crucial in finding the solution?

• Reflect on Difficulties: ◦ Identify where the problem-solving process was most challenging. ◦ What mistakes were made, and how can similar pitfalls be avoided in future problems?

• Learn and Incorporate: Integrate new insights, methods, and strategies into your problem-solving repertoire. Problem-solving is an iterative process of continuous learning and refinement KEY SUCCESS FACTORS FOR THE SOLVER

• Emphasize Understanding "Why": Focus on comprehending the rationale and derivations behind methods and formulas, rather than just memorizing them

• Cultivate Creativity and Intuition: Develop a "creative understanding" and "number sense" that goes beyond rote application of rules. Problem-solving is an art that benefits from imaginative approaches

• Practice and Experience: Problem-solving skills are honed through consistent practice. Engage with a wide variety of problems to build experience and internalize strategies

• Persistence and Resilience: Do not be discouraged by initial failures or difficulties. View challenges as opportunities for learning and growth, and maintain enthusiasm throughout the process

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