Template for Problem Solutions

Template for Problem Solutions (With Guiding Questions)

1. Problem Statement
   Q: What exactly is being asked?

   • Clearly restate the problem.
     Example: “Prove √2 cannot be written as a fraction of two integers.”

2. Initial Intuition
   Q: What’s your first instinct or assumption?

   • Describe patterns, guesses, or starting assumptions.
     Example: “Assume √2 is rational. If this leads to a contradiction, it must be irrational.”

3. Step-by-Step Solution
   Q: How can you break this down logically?

   • Write the solution flow.
     Example:
   • Let √2 = a/b (simplified fraction).
   • Square: 2b² = a² ⇒ a is even.
   • Substitute a = 2k ⇒ b² = 2k² ⇒ b is even. Contradiction. Note that this step is quite involved, so there will be another post on this.

4. Key Insights
   Q: What made the solution work?

   • Highlight pivotal ideas or techniques.
     Example:
   • Proof by contradiction.
   • Divisibility/parity arguments.

5. Fragile Assumptions
   Q: What could break this solution?

   • Identify implicit assumptions or overlooked edge cases.
     Example:
   • Not ensuring a/b is in lowest terms.
   • Missing the parity chain (a even ⇒ b even).

6. Alternative Paths
   Q: Could a different method work?

   • Suggest other approaches.
     Example:
   • Geometric infinite descent (e.g., shrinking right triangles).

7. Generalizations
   Q: Can this apply to similar problems?

   • Extend the core idea to broader cases.
     Example:
   • Replace 2 with any prime p to prove √p is irrational.

8. Connections
   Q: Where else have you seen this pattern?

   • Link to analogous problems or domains.
     Example:
   • Similar contradiction logic applies to proving log₂3 is irrational.

9. Reflection
   Q: What would you do differently next time?

   • Critique your process and identify improvements.
     Example:
   • “Next time, I’d check if the Rational Root Theorem could shortcut the proof.”

Example Flow with Questions (√2 Proof):

• Key Insights: Q: What unlocked the solution?
  • Contradiction + parity analysis.
• Fragile Assumptions: Q: What nearly broke the proof?
  • Forgetting to reduce a/b initially.
• Connections: Q: Similar problems?
  • Irrationality of √3, e, or sums like √2 + √3.
• Reflection: Q: What’s your takeaway?
  • “Contradiction is powerful for impossibility proofs—look for structural mismatches.”

Why This Works: Questions force active engagement, surface hidden assumptions, and link ideas across domains. The template becomes a dialogue with your reasoning.

Final template (concise version):

1. Problem Statement
2. Initial Intuition
3. Potential Paths
4. Step-by-Step Solution
5. Key Insights
6. Fragile Assumptions
7. Alternative Paths
8. Generalizations
9. Connections
10. Reflection