🧭 Topic Routing Heuristics Playbook
Develop the ability to quickly identify problem types and route to the most effective solution strategies. This systematic approach will help you start problems efficiently.
🎯 Problem Identification Framework
First 10 Seconds:
- Read the problem: Get the gist quickly
- Identify key words: Look for topic signals
- Check answer choices: What type of answer is expected?
- Route to strategy: Choose your approach
- Start solving: Begin with confidence
Topic Signal Words
- Algebra: “solve”, “equation”, “function”, “graph”, “slope”
- Number Theory: “divisible”, “remainder”, “prime”, “factor”, “digit”
- Geometry: “triangle”, “circle”, “angle”, “area”, “perimeter”, “coordinate”
- Counting/Probability: “how many”, “ways”, “probability”, “arrange”, “choose”
🔢 Algebra Routing
Problem Signals
- Equations: Linear, quadratic, polynomial, rational
- Functions: Graphs, transformations, compositions
- Inequalities: Linear, quadratic, absolute value
- Systems: Multiple equations, multiple variables
Quick Recognition
- Variables: $x, y, z$ or other letters
- Operations: $+, -, \times, \div, \sqrt{}, \text{^}$
- Relations: $=, <, >, \leq, \geq$
- Functions: $f(x), g(x), h(x)$
Solution Strategies
- Substitution: Replace variables with values
- Elimination: Remove variables systematically
- Factoring: Break down expressions
- Graphing: Visual representation
- Backsolving: Use answer choices
Algebra Example
Problem: Solve $2x + 3 = 11$
Signals: “solve”, equation with variable $x$ Route: Direct algebraic manipulation Solution: $2x = 8$, $x = 4$
🔢 Number Theory Routing
Problem Signals
- Divisibility: “divisible by”, “multiple of”, “factor of”
- Remainders: “remainder when divided by”, “mod”
- Primes: “prime number”, “prime factor”
- Digits: “digit sum”, “last digit”, “digit properties”
Quick Recognition
- Modular arithmetic: $\bmod$, remainders
- Prime numbers: 2, 3, 5, 7, 11, 13, …
- Divisibility rules: 2, 3, 5, 9, 11
- Digit operations: Sum, product, last digit
Solution Strategies
- Modular arithmetic: Use small moduli
- Prime factorization: Break down numbers
- Divisibility rules: Apply standard rules
- Pattern recognition: Look for cycles
- Extreme cases: Test boundary values
Number Theory Example
Problem: Find the remainder when $2^{100}$ is divided by 7
Signals: “remainder when divided by”, modular arithmetic Route: Find pattern in powers of 2 mod 7 Solution: $2^3 \equiv 1 \pmod{7}$, so $2^{100} = 2^{99} \cdot 2 \equiv 2 \pmod{7}$
📐 Geometry Routing
Problem Signals
- Shapes: “triangle”, “circle”, “rectangle”, “square”
- Measurements: “area”, “perimeter”, “volume”, “angle”
- Coordinates: “point”, “line”, “slope”, “distance”
- Transformations: “rotate”, “reflect”, “translate”
Quick Recognition
- Geometric terms: Triangle, circle, angle, area
- Coordinate terms: Point, line, slope, distance
- Measurement terms: Length, area, volume, angle
- Relationship terms: Similar, congruent, parallel, perpendicular
Solution Strategies
- Coordinate geometry: Use coordinates and formulas
- Similar triangles: Proportional relationships
- Power of a point: Circle properties
- Area formulas: Standard area calculations
- Angle properties: Sum of angles, parallel lines
Geometry Example
Problem: Find the area of a triangle with vertices at $(0,0)$, $(3,0)$, and $(0,4)$
Signals: “area”, “triangle”, coordinates Route: Use coordinate geometry or basic area formula Solution: $A = \frac{1}{2} \cdot 3 \cdot 4 = 6$
🎲 Counting/Probability Routing
Problem Signals
- Counting: “how many”, “ways”, “arrange”, “choose”
- Probability: “probability”, “chance”, “likelihood”
- Combinations: “combinations”, “permutations”, “factorial”
- Events: “at least”, “at most”, “exactly”, “or”, “and”
Quick Recognition
- Counting words: “how many”, “ways”, “arrange”
- Probability words: “probability”, “chance”, “likelihood”
- Combinatorial words: “choose”, “select”, “pick”
- Event words: “at least”, “at most”, “exactly”
Solution Strategies
- Complementary counting: Count what you don’t want
- Casework: Break into cases
- Symmetry: Use symmetric properties
- Indicators: Use indicator variables
- Recursion: Build up from smaller cases
Counting Example
Problem: How many ways can 3 people sit in 5 chairs?
Signals: “how many ways”, “arrange”, “sit” Route: Permutation problem Solution: $P(5,3) = 5 \cdot 4 \cdot 3 = 60$
⚡ Quick Routing Decision Tree
🎯 Topic-Specific Quick Starts
Algebra Quick Start
- Identify the equation type: Linear, quadratic, polynomial?
- Choose solution method: Direct solving, substitution, elimination?
- Check for special cases: Factoring, completing the square?
- Verify your answer: Plug back into original equation
Number Theory Quick Start
- Identify the number property: Divisibility, remainder, prime?
- Choose the right tool: Modular arithmetic, prime factorization?
- Look for patterns: Cycles, repetitions, symmetries?
- Test your answer: Verify with small examples
Geometry Quick Start
- Draw a diagram: Visualize the problem
- Identify the shape: Triangle, circle, polygon?
- Choose the approach: Coordinate, synthetic, or analytical?
- Apply the right formula: Area, perimeter, angle, distance?
Counting/Probability Quick Start
- Identify the counting type: Permutation, combination, arrangement?
- Choose the method: Direct counting, complementary, casework?
- Look for symmetry: Can you use symmetric properties?
- Check your answer: Does it make sense?
🚨 Common Routing Mistakes
Avoid These Errors:
- Misidentifying the topic: Don’t force the wrong approach
- Overcomplicating: Don’t use complex methods for simple problems
- Undercomplicating: Don’t miss the complexity in hard problems
- Skipping the diagram: Always draw for geometry problems
Red Flags:
- Can’t identify the topic: Re-read the problem carefully
- Wrong approach: Try a different method
- Getting stuck: Switch to a different strategy
- Answer doesn’t make sense: Check your work
📊 Quick Reference
Topic Identification:
- Algebra: Variables, equations, functions
- Number Theory: Divisibility, remainders, primes
- Geometry: Shapes, angles, coordinates
- Counting/Probability: Arrangements, combinations, likelihood
Solution Strategies:
- Algebra: Solve, substitute, eliminate, factor
- Number Theory: Modular arithmetic, prime factorization
- Geometry: Coordinate geometry, similar triangles
- Counting: Direct counting, complementary, casework
Time Allocation:
- Easy problems: 2-3 minutes
- Medium problems: 3-5 minutes
- Hard problems: 5-10 minutes
- Very hard problems: Skip or guess
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