The Ultimate GRE/GMAT/CAT Cheatsheet

# Step-by-Step SE Method
1. READ the full sentence carefully
2. PREDICT a word for the blank based on context
3. SCAN the 6 choices for synonyms of your predicted word
4. TEST both synonyms in the sentence — must produce SAME meaning
5. ELIMINATE traps: words that fit individually but don't form a synonym pair

# Transition Signals
- SAME DIRECTION:  and, moreover, furthermore, similarly, likewise, additionally
- OPPOSITE DIRECTION: however, but, although, despite, nevertheless, whereas, yet

# Common Traps
- Two words that fit the blank but are NOT synonyms → wrong
- Near-synonyms with different connotations (e.g., pragmatic vs utilitarian)
- Words with secondary definitions being tested

# High-Yield Greek/Latin Roots

BEN/BON = good         MAL = bad           BELL = war
BREV = short           CHRON = time        CIDE = kill
CRED = believe         DICT = say          EQU = equal
FER = carry            FID = faith         GEN = birth/origin
GRAV = heavy           JECT = throw        JUD = judge
LUC/LUM = light        METER = measure     MIT/MISS = send
MORB = death           NOM = name          PATH = feeling
PHIL = love            PLAC = calm         PORT = carry
SCRIB/SCRIPT = write   SPECT = look        TANG = touch
TEMP = time            TRACT = pull/drag   VEN = come
VID/VIS = see          VIV = life          VOC = voice
VOL = wish/will        VERT/VERS = turn

# Common Prefixes
AB/ABS = away from     ANTI = against      AUTO = self
BEN = well             BI = two            CIRCUM = around
CO/COM/CON = together  CONTRA = against    DE = down/away
DIS/DIF = apart/not    EX/E = out of       EXTRA = beyond
IN/IM = not/into       INTER = between     INTRA = within
MIS = wrong            NON = not           OVER = excessive
PER = through          POST = after        PRE = before
PRO = forward          RE = again          SEMI = half
SUB/SUP = under        SUPER = above       TRANS = across
UN = not               WITH = against

# Common Suffixes
-ABLE = capable of     -ATE = to make      -ATION = process
-EN = to make          -FUL = full of      -IBLE = capable
-IC = relating to      -IFY = to make      -ILY = in manner of
-ISM = belief          -IST = one who      -ITY = state of
-IVE = tending to      -IZE = to make      -LESS = without
-LOGUE = speech        -MENT = act of      -NESS = state of
-OR = one who          -OUS = full of      -TION = act of

# Types of Numbers
Integers:    ..., -3, -2, -1, 0, 1, 2, 3, ...
  Positive Integers: 1, 2, 3, ... (counting numbers / natural numbers)
  Non-negative: 0, 1, 2, 3, ...
  Even: ... -4, -2, 0, 2, 4, ... (divisible by 2)
  Odd:  ... -3, -1, 1, 3, 5, ... (not divisible by 2)

Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
  - 2 is the ONLY even prime
  - 1 is NOT prime (or composite)
  - Every integer > 1 is prime or can be factored into primes uniquely

# Divisibility Rules
2: last digit even       3: sum of digits divisible by 3
4: last 2 digits ÷ 4     5: last digit 0 or 5
6: divisible by 2 AND 3  8: last 3 digits ÷ 8
9: sum of digits ÷ 9    10: ends in 0
11: (sum of odd-pos digits) - (sum of even-pos digits) is 0 or ÷ 11

# Remainders & Factorization
If N = qk + r, then r = N mod k (remainder when N divided by k)
N! = 1 × 2 × 3 × ... × N
Trailing zeros in N! = floor(N/5) + floor(N/25) + floor(N/125) + ...
Number of factors of N = (a+1)(b+1)(c+1) if N = p^a × q^b × r^c
Sum of factors of N = [(p^(a+1)-1)/(p-1)] × [(q^(b+1)-1)/(q-1)] × ...

# Fraction Operations
a/b + c/d = (ad + bc) / bd
a/b × c/d = ac / bd
a/b ÷ c/d = a/b × d/c = ad / bc
Mixed number: p q/r = (pr + q) / r

# Percent Conversions
Percent = (Part / Whole) × 100
Percent Change = [(New - Old) / Old] × 100
  - If result is negative → percent decrease
  - If result is positive → percent increase
Successive Percent Changes: NOT additive!
  If ×r1 then ×r2: net factor = r1 × r2
  e.g., +20% then -20%: 1.2 × 0.8 = 0.96 → -4% net (NOT 0%)
Simple Interest: I = P × R × T
Compound Interest: A = P(1 + r/n)^(nt)

# Ratios & Proportions
If a:b:c = 2:3:5, then a = 2k, b = 3k, c = 5k
Part:Part ratio vs Part:Whole ratio
If A:B = 3:5, then A/(A+B) = 3/8, B/(A+B) = 5/8
Direct Proportion: y = kx
Inverse Proportion: xy = k

# Linear Equations
ax + b = cx + d → x = (d - b) / (a - c)
Systems of 2 equations → substitution or elimination
  x + y = S,  x - y = D  →  x = (S+D)/2,  y = (S-D)/2

# Quadratic Equations
ax² + bx + c = 0
Discriminant: Δ = b² - 4ac
  Δ > 0 → 2 real roots:  x = (-b ± √Δ) / 2a
  Δ = 0 → 1 repeated root: x = -b / 2a
  Δ < 0 → no real roots
Sum of roots = -b/a     Product of roots = c/a
Factoring: a² - b² = (a+b)(a-b),  a² + 2ab + b² = (a+b)²

# Inequalities
If a < b and c > 0 → ac < bc  (direction SAME)
If a < b and c < 0 → ac > bc  (direction FLIPS!)
|x| < a  → -a < x < a  (AND condition)
|x| > a  → x < -a OR x > a  (OR condition)
|x - h| = r → circle center (h,0), radius r

# Exponents & Roots
a^m × a^n = a^(m+n)     a^m ÷ a^n = a^(m-n)
(a^m)^n = a^(mn)        (ab)^n = a^n × b^n
a^0 = 1 (a ≠ 0)         a^(-n) = 1/a^n
√a × √b = √(ab)        √a + √b ≠ √(a+b)  ← common trap!
(√a + √b)(√a - √b) = a - b

# Triangles
Sum of angles = 180°
Area = (1/2) × base × height
Pythagorean Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
  (and multiples: 6-8-10, 10-24-26, etc.)
Isosceles: 2 equal sides, 2 equal angles
Equilateral: all sides equal, all angles 60°, area = (s²√3)/4
Triangle Inequality: any side < sum of other two sides
Similar triangles: angles equal, sides proportional
  Area ratio = (side ratio)²

# Circles
Area = πr²         Circumference = 2πr
Arc length = (θ/360) × 2πr
Sector area = (θ/360) × πr²
Tangent is perpendicular to radius at point of tangency
Inscribed angle = (1/2) × central angle (same arc)

# Coordinate Geometry
Distance: d = √[(x₂-x₁)² + (y₂-y₁)²]
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope: m = (y₂-y₁)/(x₂-x₁)
  Parallel: m₁ = m₂        Perpendicular: m₁ × m₂ = -1
Line equation: y = mx + b  (slope-intercept)
  or ax + by + c = 0       (standard form)
Parabola: y = ax² + bx + c → vertex at x = -b/(2a)

# Descriptive Statistics
Mean (Average) = Sum / Count
Median = middle value (odd count) or avg of two middle (even count)
Mode = most frequent value
Range = Max - Min
Standard Deviation = √[Σ(xi - μ)² / n]  (measure of spread)

# Weighted Average
WA = (w₁x₁ + w₂x₂ + ... + wₙxₙ) / (w₁ + w₂ + ... + wₙ)

# Probability
P(A) = favorable outcomes / total outcomes
P(A or B) = P(A) + P(B) - P(A and B)     [General Addition Rule]
P(A and B) = P(A) × P(B)                 [if independent]
P(A or B) = P(A) + P(B)                  [if mutually exclusive]
Complement: P(not A) = 1 - P(A)
At least one: P(at least 1) = 1 - P(none)

# Combinations & Permutations
nCr = n! / [r!(n-r)!]  (order does NOT matter)
nPr = n! / (n-r)!       (order matters)
With repetition: n^r choices for r positions from n items
Circular permutations: (n-1)! arrangements

# Overlapping Sets (2-Set Venn)
|A ∪ B| = |A| + |B| - |A ∩ B|
Only A = |A| - |A ∩ B|
Only B = |B| - |A ∩ B|
Neither = Total - |A ∪ B|

# Overlapping Sets (3-Set Venn)
|A ∪ B ∪ C| = |A| + |B| + |C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|

# GRE Issue Essay Template (30 min)

## Paragraph 1: Introduction (~60-80 words)
- Restate the prompt in your own words
- Clearly state your thesis (agree, disagree, or qualified position)
- Provide a brief roadmap of your reasons

## Paragraph 2: Reason 1 with Example (~120-150 words)
- Topic sentence: State your first reason clearly
- Elaboration: Explain WHY this reason supports your thesis
- Evidence: Provide a specific, real-world example
  - Historical events, scientific discoveries, personal experience, literature

## Paragraph 3: Reason 2 with Example (~120-150 words)
- Topic sentence: State your second reason
- Elaboration: Explain the reasoning
- Evidence: A DIFFERENT type of example (diversify: don't use two similar examples)

## Paragraph 4: Counterargument & Rebuttal (~100-120 words)
- Acknowledge a potential opposing viewpoint
- Explain WHY it is less compelling or incomplete
- This demonstrates critical thinking (scores higher!)

## Paragraph 5: Conclusion (~60-80 words)
- Restate thesis (in different words)
- Summarize key points briefly
- End with a broader implication or final thought

# GRE Argument Essay Template (30 min)
NOTE: Do NOT give your opinion — analyze the logical flaws!

## Paragraph 1: Introduction (~60-80 words)
- Summarize the argument (conclusion + key evidence)
- State that the argument is flawed/unconvincing
- Preview the main flaws you'll discuss

## Paragraph 2: Flaw 1 (~100-120 words)
- Identify the specific logical fallacy (e.g., questionable assumption)
- Explain WHY it weakens the argument
- Suggest what additional evidence would help

## Paragraph 3: Flaw 2 (~100-120 words)
- Identify a second distinct flaw
- Could be: small sample, correlation ≠ causation, vague terms,
  false analogy, unexamined alternatives, survey reliability
- Explain the impact on the argument's validity

## Paragraph 4: Flaw 3 or Strengthening (~100-120 words)
- Either: identify a third flaw
- Or: discuss what additional information is needed to evaluate
  the argument more effectively

## Paragraph 5: Conclusion (~60-80 words)
- Summarize why the argument is unconvincing
- Briefly suggest what would make it stronger
- Do NOT introduce new points

# 1. Plugging In Numbers
When variables appear in answers: substitute simple values (e.g., x=2, y=3)
- Always verify with a second set of numbers
- Choose values that are easy to work with
- Test edge cases: 0, 1, negatives, fractions

# 2. Back-Solving
Start with answer choice C (middle value)
- If too small → eliminate smaller choices
- If too big → eliminate bigger choices
- Usually eliminates 2-3 choices in one step

# 3. Estimation
When answer choices are spread apart, estimate rather than calculate exactly
- Round to nearest friendly number
- Use benchmark fractions: 1/2, 1/3, 1/4, 1/5

# 4. Picking Numbers for Percent Problems
Total = 100 → makes percentages easy
- 30% of 100 = 30, increase by 20% → 120, etc.

# 5. Number Properties Quick Checks
Even × Even = Even      Even × Odd = Even     Odd × Odd = Odd
Even + Even = Even      Odd + Odd = Even       Even + Odd = Odd
Positive × Negative = Negative
Positive + Positive = Positive
Negative + Negative = Negative

# THE GOLDEN RULE OF DATA SUFFICIENCY
"You don't need to SOLVE — you need to determine if it IS solvable"

# AD/BCE Method (Elimination Strategy)

Step 1: Evaluate Statement (1) ALONE
  - If SUFFICIENT → answer is A or D
  - If NOT sufficient → answer is B, C, or E

Step 2: Evaluate Statement (2) ALONE
  - If (1) was sufficient AND (2) is sufficient → D
  - If (1) was sufficient AND (2) is NOT sufficient → A
  - If (1) was NOT sufficient AND (2) IS sufficient → B
  - If (1) was NOT sufficient AND (2) is NOT sufficient → C or E

Step 3: If both alone are NOT sufficient
  - Try TOGETHER: If (1) + (2) sufficient → C
  - If (1) + (2) still NOT sufficient → E

# Common DS Traps
- "Is x positive?" → x=0 is NOT positive (0 is neither positive nor negative)
- Integer trap: x is "a number" doesn't mean integer
- Figure NOT drawn to scale in DS
- Both statements give the same info → answer is D or E, never C
- Overlapping information means the combined case won't help

# DS Answer Choices (memorize!)
A: Statement (1) ALONE is sufficient
B: Statement (2) ALONE is sufficient
C: BOTH TOGETHER are sufficient
D: EACH ALONE is sufficient
E: NOT sufficient, even together

# The Negation Test for Assumptions
If you negate an assumption and the argument FALLS APART,
then that assumption was necessary (correct answer).

Example:
Argument: "Company X's profits increased because they launched a new product."
Assumption candidate: "No other factor caused the profit increase."

Negation: "Some OTHER factor caused the profit increase."
→ If true, the argument falls apart. So this IS the correct assumption.

# Common Logical Flaws on GMAT
1. Correlation ≠ Causation
   "A and B occur together" → doesn't mean A causes B (could be B→A or C→both)

2. Sample Bias
   "Survey of 500 people found..." → was the sample representative?

3. Scope Shift / Equivocation
   Using a term with a different meaning than intended

4. False Analogy
   "Because X worked in situation A, it will work in situation B"

5. Necessary vs Sufficient
   Confusing "necessary condition" with "sufficient condition"
   Rain is necessary for a rainbow, but rain alone isn't sufficient

6. Circular Reasoning
   Conclusion restated as a premise

7. Ad Hominem / Appeal to Authority
   Attacking the person rather than the argument

# Divisibility & Remainders
- Divisibility by 2,3,4,5,6,8,9,10,11 (same rules as GRE)
- Euler's Theorem: a^φ(n) ≡ 1 (mod n), when gcd(a,n)=1
  where φ(n) = Euler's totient function
- Fermat's Little Theorem: a^(p-1) ≡ 1 (mod p), when p is prime
- Wilson's Theorem: (p-1)! ≡ -1 (mod p), when p is prime

# Chinese Remainder Theorem (CRT)
If x ≡ a₁ (mod m₁) and x ≡ a₂ (mod m₂), and gcd(m₁,m₂)=1,
then there exists a unique solution mod (m₁×m₂).

# HCF and LCM
HCF × LCM = a × b  (for two numbers)
LCM(a,b,c) = abc × HCF(a,b,c) / [HCF(a,b) × HCF(b,c) × HCF(c,a)]
For HCF of fractions: HCF of numerators / LCM of denominators
For LCM of fractions: LCM of numerators / HCF of denominators

# Base System
To convert N from base b₁ to base b₂:
  N (base b₁) → Decimal → N (base b₂)
In base b: digits range from 0 to b-1
Last digit of a^b in base 10 = last digit of a^(b mod 4)

# Digital Root (Sum of Digits until single digit)
Digital Root of N = N mod 9 (if N mod 9 ≠ 0), else 9
Digital Root of product = Digital Root of (digital roots)

# Percentages
Successive % changes: A × (1 + r₁/100)(1 + r₂/100) = B
Percentage point difference = actual % difference
x% of y = y% of x (commutative)
If price increased by x% to maintain same expenditure:
  consumption must decrease by [100x/(100+x)]%
If price decreased by x%:
  consumption increases by [100x/(100-x)]%

# Profit & Loss
SP = CP × (1 + P/100)     where P = profit%
SP = CP × (1 - L/100)     where L = loss%
Marked Price: MP × Discount% = SP
  If CP = a, MP = b: markup = (b-a)/a × 100

# Time, Speed, Distance
Speed = Distance / Time
Average Speed = Total Distance / Total Time
  (NOT average of speeds!)
If distances are equal: Avg Speed = 2ab/(a+b)  (harmonic mean)
If times are equal:    Avg Speed = (a+b)/2     (arithmetic mean)
Relative Speed:
  Same direction: |a - b|
  Opposite directions: a + b
Circular track: meetings at LCM of times
Trains: Time to cross = (L₁ + L₂) / Relative Speed

# Time & Work
Work = Rate × Time
If A takes m days, B takes n days:
  Together: 1/m + 1/n = 1/t → t = mn/(m+n)
Man-days concept: If 10 workers do a job in 6 days,
  then Work = 60 man-days. 15 workers take 4 days.
Pipes & Cisterns:
  Inlet: positive rate; Outlet: negative rate
  Net rate = sum of inlets - sum of outlets

# Progressions
Arithmetic Progression (AP):
  aₙ = a + (n-1)d
  Sₙ = n/2 × [2a + (n-1)d] = n/2 × (a + aₙ)
  Arithmetic Mean of n numbers = (first + last) / 2

Geometric Progression (GP):
  aₙ = a × r^(n-1)
  Sₙ = a(rⁿ - 1)/(r - 1)   for r ≠ 1
  S∞ = a/(1 - r)             for |r| < 1
  Geometric Mean = √(ab)

Harmonic Progression (HP):
  If a, b, c are in AP → 1/a, 1/b, 1/c are in HP
  Harmonic Mean = 2ab/(a+b)

AM ≥ GM ≥ HM (for positive numbers, equality when a=b)

# Logarithms
log(ab) = log a + log b
log(a/b) = log a - log b
log(aⁿ) = n log a
logₐb = log b / log a (change of base)
a^(logₐb) = b

# Permutations & Combinations
nCr = n! / [r!(n-r)!]
nPr = n! / (n-r)!
nCr + nCr-1 = (n+1)Cr
nC0 + nC1 + ... + nCn = 2ⁿ
nC1 + nC2 + ... + nCn = 2ⁿ - 1
Circular permutations: (n-1)!
  If clockwise ≠ anticlockwise: (n-1)!/2

# Probability
P(E) = n(E) / n(S)
P(E') = 1 - P(E)
P(A∪B) = P(A) + P(B) - P(A∩B)
P(A∩B) = P(A) × P(B|A) = P(B) × P(A|B)
Independent: P(A∩B) = P(A) × P(B)
Odds in favor = P(E) : P(E') = a : (1-a)

# Triangles
Area = (1/2) × b × h = √[s(s-a)(s-b)(s-c)]  (Heron's)
  where s = (a+b+c)/2 (semi-perimeter)
Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41
Median divides into two equal areas
Angle bisector: divides opposite side in ratio of adjacent sides
  AD is angle bisector → BD/DC = AB/AC
Inradius: r = Area / s
Circumradius: R = abc / (4 × Area)

# Quadrilaterals
Parallelogram: opp sides equal, opp angles equal, diagonals bisect
Rhombus: all sides equal, diagonals perpendicular, Area = (d₁×d₂)/2
Rectangle: Area = l×b, Diagonal = √(l²+b²)
Square: Area = a², Diagonal = a√2
Trapezium: Area = (1/2)(a+b)h

# Circles
Area = πr², Circumference = 2πr
Length of tangent from external point P = √(d² - r²)
  where d = distance from P to center
Inscribed angle = (1/2) × central angle
  Angle in semicircle = 90°

# Coordinate Geometry
Area of triangle from coordinates:
  |(x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂))| / 2
Distance from point (x₀,y₀) to line ax+by+c=0:
  |ax₀ + by₀ + c| / √(a²+b²)
Section formula: (mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)

# Mensuration
Cube: SA=6a², V=a³, Diagonal=a√3
Cuboid: SA=2(lb+bh+hl), V=lbh, Diag=√(l²+b²+h²)
Cylinder: CSA=2πrh, TSA=2πr(r+h), V=πr²h
Cone: CSA=πrl, TSA=πr(r+l), V=(1/3)πr²h
Sphere: SA=4πr², V=(4/3)πr³
Hemisphere: CSA=2πr², TSA=3πr², V=(2/3)πr³

# CAT RC Approach — "First-Read Method"

## First Read (2-3 minutes per passage):
1. Read the FIRST paragraph thoroughly → contains author's thesis
2. Read the FIRST and LAST sentence of each body paragraph
3. Read the LAST paragraph thoroughly → contains conclusion
4. Build a mental map: Main Idea → Support → Counter → Conclusion

## During Questions:
1. Read the question stem FIRST
2. Return to the relevant part of the passage
3. Find the answer in the TEXT, not from outside knowledge
4. Eliminate options using process of elimination

## Key Principles:
- CAT RC does NOT require outside knowledge
- The CORRECT answer is always SUPPORTED by the passage
- Be careful of: extreme language (all, never, always), scope creep,
  half-right answers (one part correct, one part wrong)
- The "most appropriate" answer may not be perfect, just the best option

## Common RC Question Types on CAT:
1. Main Idea / Central Theme
2. Author's Purpose / Tone
3. Specific Detail / Factual
4. Inference (what must be true based on passage)
5. Application (if the author were to agree with X, which would it be?)
6. "According to the passage" / "The author mentions X to..."
7. Exception questions ("All of the following EXCEPT...")

# Para Jumble Approach (4-5 sentences)

## Step 1: Find Mandatory Pairs
Look for:
- Pronoun reference: "He/This/These" → must follow the sentence with the noun
- Chronological markers: "First," "Then," "Finally," "Later"
- Cause-Effect: "Because of this," "Consequently," "As a result"
- Definition → Example pairs: General statement followed by specific instance
- Contrast connectors: "However," "But," "On the other hand"

## Step 2: Find Opening Sentence
Good opening sentences:
- Introduce the main topic/subject
- Are general statements (not starting with "However" or "Also")
- Don't contain pronouns (he, it, this, those) without antecedents
- Often provide definition or broad context

## Step 3: Find Closing Sentence
Good closing sentences:
- Contain concluding words: "Therefore," "Thus," "Hence"
- Summarize or provide a final thought
- May link back to the opening (creating a loop)

## Step 4: Arrange Middle Sentences
- Use logical flow: general → specific, cause → effect
- Check for transitional consistency

## TITA Para Jumbles (type the order)
- No options to test against — must be confident in your arrangement
- Often easier than MCQ PJs because options don't create confusion
- Strategy same, but double-check your arrangement

# Summary Question Strategy
Choose the option that BEST captures the essence of the passage.

## What a GOOD summary does:
- Captures the MAIN IDEA (not details or examples)
- Is CONCISE but complete
- Maintains the author's TONE and perspective
- Does NOT introduce new information
- Does NOT contradict the passage

## What a BAD summary looks like:
- Too specific (focuses on a detail/example, not the main idea)
- Too broad (so vague it could apply to any passage)
- Introduces outside information or opinions
- Contradicts the passage
- Misses the author's main point entirely

## Elimination Process:
1. Eliminate options with details/examples (too specific)
2. Eliminate options that contradict the passage
3. Eliminate options with wrong tone (critical passage → positive summary = wrong)
4. Between remaining options, pick the most complete one
5. If two seem equal, pick the one that better captures the "why" (not just "what")

# DI Problem-Solving Framework

## Step 1: SCAN the data (30 seconds)
- What TYPE of data? (table, chart, caselet)
- What are the VARIABLES? (rows, columns, categories)
- What are the UNITS? (lakhs, crores, %, thousands)
- Are there any NOTES or FOOTNOTES?

## Step 2: READ the question (30 seconds)
- What SPECIFICALLY is being asked?
- What CALCULATION is needed?
- Can you ESTIMATE instead of calculate exactly?

## Step 3: SOLVE efficiently
- Use approximation for close-enough answers
- Write down intermediate values
- For ratio problems: simplify before calculating
- For percentage problems: use fraction equivalents

## Key Techniques:
1. Percentage to Fraction Conversions:
   50% = 1/2,  33.33% = 1/3,  25% = 1/4,  20% = 1/5
   16.67% = 1/6,  12.5% = 1/8,  10% = 1/10
   66.67% = 2/3,  75% = 3/4,  60% = 3/5

2. Quick Multiplication Tricks:
   ×2: double;  ×5: ×10/2;  ×25: ×100/4;  ×125: ×1000/8

3. Approximation:
   498 ≈ 500,  999 ≈ 1000,  3.01 ≈ 3
   Only approximate if answer choices are far apart

4. Averages from Table:
   No need to calculate every value — use deviation method
   If mean ≈ 50, and values are 48, 53, 51, 47, 51:
   Deviations: -2, +3, +1, -3, +1 → net deviation = 0 → mean = 50

# Arrangement Puzzle Framework

## Step 1: Create a representation
Linear: draw positions 1-N in a row
Circular: draw N positions in a circle
Matrix: create a grid (rows = people, columns = attributes)

## Step 2: Extract ALL constraints
- Direct: "A sits next to B"
- Negative: "C does NOT sit at the end"
- Conditional: "If D is in position 3, then E is in position 5"
- Relative: "F sits 2 places to the left of G"
- Reference point: "H is in the middle"

## Step 3: Fill definite information first
- Place any person with a fixed position
- Then place those with only one possible position
- Re-evaluate constraints after each placement

## Step 4: Handle uncertainty
- Create branches for ambiguous cases
- Look for constraints that eliminate branches quickly
- If too many branches, you may have missed a constraint

## Common Puzzle Types:
1. Linear Arrangement (N people in a row, facing N/S)
2. Circular Arrangement (N people in a circle)
3. Floor/Puzzle (N people on N floors with attributes)
4. Blood Relations (family tree)
5. Grouping & Distribution (teams, subjects, cities)
6. Scheduling (days/time slots with constraints)
7. Matrix Matching (people × attributes, one-to-one)
8. Sequential Ordering (ranking, ordering events)

# Round Robin Tournament
N teams, each plays every other team once
Total matches = N(N-1)/2
Points: Win = W, Draw = D, Loss = L
Maximum points per team = (N-1) × W
Minimum points = 0

# Key Deductions:
- If total wins = total losses (in pure win/loss, no draws)
- Maximum possible points for a team = matches × win points
- If a team has lost all matches: all opponents got at least 1 win against them
- If two teams are tied: use tiebreaker (head-to-head, NRR, etc.)

# Knockout Tournament
N teams → N-1 matches total
Each match eliminates exactly 1 team
Quarter-finals: 8 teams → 4 matches
Semi-finals: 4 teams → 2 matches
Finals: 2 teams → 1 match

# Points Table Analysis
- Champion team: has maximum possible OR can be deduced from wins
- Eliminated team: has minimum possible OR cannot catch up
- To find if qualification is possible: calculate maximum achievable
  points and compare with current standings
- "Minimum wins to guarantee qualification" = check worst-case tie scenarios

# Knights and Knaves (Binary Logic)
Knights ALWAYS tell truth, Knaves ALWAYS lie

## Approach:
- Assume one person is a Knight (truth-teller)
- Follow the chain of implications
- Check for consistency (no contradictions)
- If contradiction → assumption was wrong → person is Knave

## Matrix Method:
Create a table: Person × (Statement 1, Statement 2, ...)
Mark each as True/False for Knight case and Knave case
Find the assignment where all statements are consistent

## Multiple Statement Puzzles:
"At least one of us is a Knave" → If Knight says this, must be true
  → At least one Knave exists (could be the speaker if Knave)
"If I am a Knight, then X is a Knave" → Knight: X is Knave; Knave: can't evaluate (self-reference)

# Einstein Puzzles (Zebra Puzzles)
N categories × M attributes, each attribute used exactly once
Approach: Create a grid/matrix, fill definites, use constraints to narrow down
Key: Process constraints from most specific to most general

# How to Get the Most from Practice Tests

## Before Each Mock:
- Take at the SAME TIME as your real test (morning for GRE/CAT)
- Recreate test conditions: quiet room, no breaks (except scheduled ones)
- Use official scratch paper / whiteboard
- Do NOT pause the test

## During Each Mock:
- Time yourself strictly per section
- If stuck > 2-3 min → guess and move on
- Mark uncertain questions for review
- Do NOT change answers unless very confident

## After Each Mock (MOST IMPORTANT):
1. Score the test immediately
2. Review EVERY question (not just wrong ones)
3. Categorize mistakes:
   - Concept gap → revisit concept
   - Calculation error → practice speed
   - Trap identified too late → note the trap type
   - Time pressure → need more timed practice
4. Maintain an ERROR LOG:
   - Question | Topic | Mistake Type | Key Learning
5. Track progress over time

## Mock Test Schedule:
- Start mocks 3-4 weeks before exam
- 2-3 mocks per week initially, tapering to 1/week before exam
- Last mock 3-4 days before exam (NOT the day before)
- Review time should be 2× the test time