🌳 When Stuck Decision Tree
Follow this systematic decision tree when you encounter difficult problems. Don’t waste time spinning your wheels - use this structured approach to find a solution or move on efficiently.
🎯 Initial Assessment (30 seconds)
First, Ask Yourself:
- How long have I been on this problem? (If > 5 minutes, consider moving on)
- Am I making progress? (If no progress, try different approach)
- Do I understand what’s being asked? (If not, re-read carefully)
- Is this problem worth my time? (Consider expected value)
- What’s my confidence level? (Low confidence = try different approach)
Quick Time Check:
- Problems 1-10: Don’t spend more than 3 minutes
- Problems 11-15: Don’t spend more than 5 minutes
- Problems 16-20: Don’t spend more than 7 minutes
- Problems 21-25: Don’t spend more than 10 minutes
🔄 Tactic Switching (1-2 minutes)
If Current Approach Isn’t Working:
- Switch from algebra to geometry: Try coordinate geometry or visual approach
- Switch from geometry to algebra: Try setting up equations
- Switch from direct to indirect: Try complement counting or working backwards
- Switch from general to specific: Try plugging in specific values
- Switch from complex to simple: Try a simpler approach
Tactic Switching Examples:
- Algebra problem: Try backsolving with answer choices
- Geometry problem: Try coordinate geometry
- Counting problem: Try complement counting
- Number theory problem: Try modular arithmetic
- Function problem: Try graphing or specific values
🎯 Complement Approaches (1-2 minutes)
When Direct Approach is Hard:
- Count what you don’t want: Use complement counting
- Work backwards: Start from the answer and work back
- Use “at least” logic: Count “none” and subtract
- Use “at most” logic: Count “more than” and subtract
- Use probability complements: $P(A) = 1 - P(A^c)$
Complement Examples:
- “At least one”: Count “none” and subtract from total
- “At most one”: Count “more than one” and subtract from total
- “Not all”: Count “all” and subtract from total
- “Some but not all”: Count “all” and “none”, then subtract
🔍 Symmetry and Patterns (1-2 minutes)
Look for Symmetry:
- Geometric symmetry: Use symmetric properties
- Algebraic symmetry: Use symmetric substitutions
- Combinatorial symmetry: Use symmetric counting
- Numerical symmetry: Use symmetric patterns
- Functional symmetry: Use even/odd properties
Pattern Recognition:
- Look for cycles: Find repeating patterns
- Look for sequences: Find arithmetic or geometric patterns
- Look for formulas: Find known formulas that apply
- Look for identities: Find trigonometric or algebraic identities
- Look for special cases: Find cases that are easier to handle
🔄 Backsolving and Answer Choice Exploitation (1-2 minutes)
When Direct Solving is Hard:
- Try answer choices: Plug in each choice
- Start with middle choice: Often C or D
- Use binary search: Go higher or lower based on result
- Eliminate obviously wrong: Cross out impossible answers
- Use answer choice patterns: Look for patterns in choices
Backsolving Process:
- Start with middle choice: Try C or D
- Test the value: Substitute into problem
- Compare result: Does it match what’s asked?
- Adjust direction: Go higher or lower based on result
- Narrow down: Use binary search approach
🎯 Coordinate and Visual Approaches (1-2 minutes)
When Abstract Approach is Hard:
- Draw a diagram: Visualize the problem
- Use coordinate geometry: Place in coordinate system
- Use graphing: Plot functions or relationships
- Use visualization: Create mental or physical models
- Use symmetry: Apply symmetric properties
Visual Approach Examples:
- Geometry problem: Draw accurate diagram
- Function problem: Graph the function
- Counting problem: Use Venn diagrams or tree diagrams
- Algebra problem: Graph both sides of equation
- Number theory problem: Use number line or grid
⚡ Quick Approximation and Estimation (1 minute)
When Exact Solution is Hard:
- Use estimation: Approximate the answer
- Use bounds: Find upper and lower bounds
- Use order of magnitude: Estimate the scale
- Use special values: Try $x = 0, 1, -1$
- Use extreme cases: Test boundary values
Approximation Examples:
- Large numbers: Use scientific notation
- Complex expressions: Use approximation formulas
- Geometric problems: Use approximate values
- Counting problems: Use approximation techniques
- Probability problems: Use approximation methods
🚨 Emergency Protocols (30 seconds)
When All Else Fails:
- Make an educated guess: Use elimination and intuition
- Use expected value: Apply guessing strategy
- Move on: Don’t waste more time
- Mark for later: Come back if time permits
- Stay calm: Don’t let one problem ruin your contest
Guessing Strategy:
- Eliminate obviously wrong: Cross out impossible answers
- Use partial information: Apply what you know
- Use symmetry: Choose symmetric answer if applicable
- Use intuition: Go with your gut feeling
- Use expected value: Apply mathematical guessing strategy
📊 Decision Tree Summary
Time-Based Decisions:
- 0-2 minutes: Try different approach
- 2-5 minutes: Try complement or symmetry
- 5-10 minutes: Try backsolving or approximation
- 10+ minutes: Make educated guess or move on
Problem-Based Decisions:
- Easy problems (1-10): Don’t get stuck, move on quickly
- Medium problems (11-15): Try 2-3 different approaches
- Hard problems (16-20): Try 1-2 approaches, then guess
- Very hard problems (21-25): Try 1 approach, then guess
Confidence-Based Decisions:
- High confidence: Continue with current approach
- Medium confidence: Try different approach
- Low confidence: Try complement or backsolving
- No confidence: Make educated guess or move on
⚡ Quick Reference
When Stuck Checklist:
- How long? (If > time limit, move on)
- Making progress? (If no, try different approach)
- Understand problem? (If not, re-read)
- Worth my time? (Consider expected value)
- Confidence level? (Low = try different approach)
Recovery Actions:
- Switch tactics: Try different approach
- Use complement: Count what you don’t want
- Look for symmetry: Use symmetric properties
- Try backsolving: Use answer choices
- Use approximation: Estimate the answer
- Make educated guess: Use elimination and intuition
Prev: During Test Quick Checks | Next: Final Review Pass | Back to: Strategy Guide