🎲 Elimination & Guessing Playbook

Master the art of strategic elimination and expected value-based guessing to maximize your score on difficult problems.

🎯 Expected Value Framework

The Math Behind Guessing

With $k$ total choices and $e$ eliminations:

• Probability of correct guess: $\frac{1}{k-e}$
• Probability of wrong guess: $\frac{k-e-1}{k-e}$

Expected Value of Guessing: $$\text{EV}_\text{guess} = \frac{1}{k-e} \cdot S_C + \frac{k-e-1}{k-e} \cdot S_W$$

Expected Value of Blank: $$\text{EV}_\text{blank} = S_B$$

Decision Rule

Guess if and only if: $\text{EV}\text{guess} > \text{EV}\text{blank}$

🚫 Quick Elimination Techniques

Parity Checks

• Even/odd: Check if answer should be even or odd
• Last digit: Look for patterns in final digits
• Modular arithmetic: Use small moduli (2, 3, 4, 5)

Example: If problem asks for $2n+1$ where $n$ is integer, answer must be odd.

Units & Dimensions

• 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

Example: If asking for area, eliminate negative answers immediately.

Order of Magnitude

• Size bounds: Is answer too big or too small?
• Reasonable range: Does answer make sense?
• Extreme values: Test boundary cases

Example: If problem involves small numbers, eliminate very large answers.

Monotonicity

• Increasing functions: Larger input → larger output
• Decreasing functions: Larger input → smaller output
• Symmetry: Look for symmetric properties

Example: If $f(x) = x^2$ and $a < b$, then $f(a) < f(b)$.

🎯 Strategic Guessing Thresholds

Typical AMC Scoring ($S_C = 6$, $S_W = 0$, $S_B = 1.5$)

Quick Decision Matrix

• No eliminations: Skip (EV too low)
• 1 elimination: Only if very confident
• 2+ eliminations: Always guess
• 3+ eliminations: Strong guess

🔍 Advanced Elimination Strategies

Algebraic Elimination

• Substitute values: Test specific numbers
• Check special cases: $x = 0, 1, -1$
• Boundary conditions: Test limits and extremes
• Symmetry: Look for symmetric properties

Geometric Elimination

• Visual inspection: Does answer look right?
• Scale: Is the answer the right size?
• Angles: Are angles reasonable?
• Coordinates: Do coordinates make sense?

Number Theory Elimination

• Divisibility: Check divisibility rules
• Prime factors: Look for prime patterns
• Modular arithmetic: Use small moduli
• Digit sums: Check digit sum properties

⚡ Quick Elimination Checklist

Before Guessing, Ask:

• Can I eliminate any obviously wrong answers?
• Do the units/dimensions make sense?
• Is the order of magnitude reasonable?
• Does the answer satisfy basic constraints?
• Are there any symmetry or pattern clues?

Elimination Priority:

1. Units/dimensions (highest priority)
2. Order of magnitude
3. Parity/odd-even
4. Boundary conditions
5. Special cases

🎯 Problem-Specific Strategies

Algebra Problems

• Substitute simple values: $x = 0, 1, -1$
• Check degree: Polynomial degree should match
• Test symmetry: Even/odd functions
• Boundary cases: Test limits

Geometry Problems

• Visual inspection: Does it look right?
• Scale: Is it the right size?
• Angles: Are angles reasonable?
• Coordinates: Do they make sense?

Number Theory Problems

• Divisibility: Check divisibility rules
• Last digits: Look for patterns
• Modular arithmetic: Use small moduli
• Prime factors: Check prime patterns

Counting/Probability Problems

• Range check: Probability between 0 and 1
• Symmetry: Look for symmetric cases
• Boundary cases: Test extreme values
• Combinatorial sense: Does count make sense?

🚨 Common Elimination Mistakes

Avoid These Traps:

• Over-elimination: Don’t eliminate too many choices
• Under-elimination: Don’t miss obvious eliminations
• Confirmation bias: Don’t eliminate answers you don’t like
• Rushing: Take time to check eliminations

Red Flags:

• Eliminating all choices: You made an error
• Only one choice left: Double-check your work
• Eliminating based on complexity: Don’t eliminate “too simple” answers
• Ignoring units: Always check units first

📊 Expected Value Calculator

Quick Mental Math:

• 2 eliminations: EV = 2.0 (guess)
• 3 eliminations: EV = 3.0 (guess)
• 4 eliminations: EV = 6.0 (strong guess)
• No eliminations: EV = 1.2 (skip)

Decision Tree:

1. Can I eliminate 2+ choices? → Guess
2. Can I eliminate 1 choice? → Only if confident
3. Can’t eliminate any? → Skip

🎯 Final Tips

Strategic Principles:

• Elimination first: Always try to eliminate before guessing
• Expected value: Use math to make decisions
• Confidence matters: Factor in your confidence level
• Time management: Don’t spend too long on eliminations

Mental Approach:

• Stay calm: Don’t panic when stuck
• Be systematic: Follow elimination checklist
• Trust the math: Expected value is your friend
• Move on: Don’t get trapped on one problem

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