✅ Sanity Checks & Verification Playbook
Build a systematic approach to catching errors before they cost you points. These verification techniques will help you maintain high accuracy throughout the contest.
🔍 The 20-Second Verification Rule
Before Bubbling Any Answer:
- Plug back: Substitute answer into original problem
- Check units: Do the units make sense?
- Verify constraints: Does answer satisfy all conditions?
- Test extremes: Does answer work at boundaries?
- Check reasonableness: Is the answer in the right ballpark?
🔄 Plug-Back Verification
When to Plug Back
- Every answer: Make it a habit
- Complex calculations: Especially important
- Time pressure: Quick verification saves time
- Uncertainty: When you’re not sure
Plug-Back Process
- Substitute answer: Replace variable with your answer
- Simplify: Work through the calculation
- Check equality: Does left side equal right side?
- Verify all conditions: Check all given constraints
- Confirm: Answer satisfies the problem
Plug-Back Example
Problem: Solve $2x + 3 = 11$
Your answer: $x = 4$
Plug-back: $2(4) + 3 = 8 + 3 = 11$ ✓ Correct!
📏 Units and Dimensional Analysis
Unit Checking
- Length: Should be positive, reasonable scale
- Area: Should be positive, units squared
- Volume: Should be positive, units cubed
- Probability: Must be between 0 and 1
- Time: Should be positive, reasonable duration
Dimensional Analysis
- Check dimensions: Do the units match?
- Verify formulas: Are you using the right formula?
- Test extreme cases: What happens at limits?
- Apply physical intuition: Does it make sense?
Units Example
Problem: Find the area of a circle with radius 3 cm
Your answer: $9\pi$ cm²
Unit check: Area should be length squared ✓ Calculation: $A = \pi r^2 = \pi \cdot 3^2 = 9\pi$ ✓ Units: cm² ✓ Correct!
🔢 Parity and Modulo Checks
Parity (Even/Odd) Checks
- Even numbers: Divisible by 2
- Odd numbers: Not divisible by 2
- Sums: Even + Even = Even, Odd + Odd = Even
- Products: Even × Anything = Even
Modulo Checks
- Mod 3: Sum of digits divisible by 3
- Mod 9: Sum of digits divisible by 9
- Mod 11: Alternating sum of digits
- Mod 4: Last two digits divisible by 4
Parity Example
Problem: Find the sum of first 10 odd numbers
Your answer: 100
Parity check: Sum of odd numbers should be even ✓ Calculation: $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100$ ✓ Verification: 100 is even ✓ Correct!
🎯 Extreme Value Testing
Boundary Cases
- Minimum values: Test smallest possible inputs
- Maximum values: Test largest possible inputs
- Zero cases: Test when variables equal zero
- Negative cases: Test negative inputs when applicable
Extreme Value Examples
- $x = 0$: Often simplifies calculations
- $x = 1$: Good for testing formulas
- $x = -1$: Useful for alternating series
- Large $x$: Test asymptotic behavior
Extreme Value Example
Problem: Find the minimum value of $f(x) = x^2 - 4x + 3$
Your answer: $f(2) = -1$
Extreme value check:
- At $x = 0$: $f(0) = 3$
- At $x = 2$: $f(2) = 4 - 8 + 3 = -1$
- At $x = 4$: $f(4) = 16 - 16 + 3 = 3$
- Minimum occurs at $x = 2$ ✓ Correct!
🧮 Order of Magnitude Checks
Size Verification
- Is answer too big?: Check against reasonable bounds
- Is answer too small?: Verify it’s not zero or negative
- Does scale make sense?: Compare to known quantities
- Are units appropriate?: Check the scale of units
Magnitude Examples
- Distances: Should be positive, reasonable scale
- Areas: Should be positive, appropriate size
- Probabilities: Must be between 0 and 1
- Counts: Should be non-negative integers
🔍 Sign and Direction Checks
Sign Verification
- Positive answers: When answer should be positive
- Negative answers: When answer should be negative
- Zero answers: When answer should be zero
- Sign changes: Check for sign errors
Direction Checks
- Increasing functions: Larger input → larger output
- Decreasing functions: Larger input → smaller output
- Monotonicity: Check if function behavior matches expectation
- Symmetry: Verify symmetric properties
⚡ Quick Verification Checklist
Before Bubbling:
- Plug back: Does answer work in original problem?
- Check units: Do units and dimensions make sense?
- Verify constraints: Does answer satisfy all conditions?
- Test extremes: Does answer work at boundaries?
- Check reasonableness: Is answer in the right ballpark?
- Verify signs: Is the sign correct?
- Check parity: Is even/odd status correct?
Common Error Types:
- Sign errors: Wrong positive/negative
- Unit errors: Wrong units or dimensions
- Calculation errors: Arithmetic mistakes
- Constraint violations: Answer doesn’t satisfy conditions
- Order of magnitude: Answer too big or too small
🚨 Red Flag Warnings
Immediate Red Flags:
- Negative area: Areas can’t be negative
- Probability > 1: Probabilities must be ≤ 1
- Imaginary length: Lengths must be real
- Zero denominator: Division by zero is undefined
- Negative count: Counts must be non-negative
Warning Signs:
- Answer seems too big: Check your calculations
- Answer seems too small: Verify your approach
- Units don’t match: Check your formula
- Sign seems wrong: Check your work
- Answer doesn’t make sense: Re-examine the problem
🎯 Verification by Problem Type
Algebra Problems
- Substitute back: Plug answer into equation
- Check all solutions: Verify each root
- Test constraints: Check domain restrictions
- Verify signs: Check positive/negative requirements
Geometry Problems
- Check dimensions: Verify units and scale
- Test angles: Are angles reasonable?
- Verify formulas: Are you using the right formula?
- Check coordinates: Do coordinates make sense?
Number Theory Problems
- Check divisibility: Verify divisibility rules
- Test modular arithmetic: Check mod calculations
- Verify prime factors: Check prime factorizations
- Test digit properties: Check digit sum rules
Counting/Probability Problems
- Check range: Probability between 0 and 1
- Verify counts: Are counts non-negative integers?
- Test symmetry: Check symmetric cases
- Verify formulas: Are you using the right formula?
📊 Quick Reference
20-Second Verification:
- Plug back (5 seconds)
- Check units (3 seconds)
- Verify constraints (5 seconds)
- Test extremes (4 seconds)
- Check reasonableness (3 seconds)
Error Prevention:
- Make it a habit: Verify every answer
- Use checklists: Follow systematic approach
- Trust your instincts: If something seems wrong, check it
- Don’t rush: Take time to verify
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