20  Number series, Letter series, Codes and Relationships

20.1 What the Syllabus Covers

This unit bundles four high-frequency NTA Paper-I question types:

  1. Number series — find the missing or next term in a numerical sequence.
  2. Letter series — find the missing or next letter or pair.
  3. Codes — substitution / coding–decoding puzzles.
  4. Relationships — blood relations and direction sense.

Each problem rewards pattern recognition over computation. The candidate trains a small vocabulary of standard patterns and a systematic checking procedure.

20.2 Number Series

A number series is a sequence following a rule. The task is to identify the rule and predict the missing or next term.

20.2.1 The Eight Standard Patterns

TipEight Standard Number-Series Patterns
Pattern Example Rule
Arithmetic progression (AP) 2, 5, 8, 11, ? Add constant difference (+3)
Geometric progression (GP) 3, 6, 12, 24, ? Multiply by constant ratio (×2)
Squares 1, 4, 9, 16, ? n²
Cubes 1, 8, 27, 64, ? n³
Fibonacci 1, 1, 2, 3, 5, 8, ? aₙ = aₙ₋₁ + aₙ₋₂
Prime numbers 2, 3, 5, 7, 11, ? Only divisible by 1 and itself
Mixed operation (alternating) 5, 8, 6, 9, 7, 10, ? +3, −2, +3, −2 …
Increasing difference 2, 3, 5, 8, 12, ? +1, +2, +3, +4 …

20.2.2 Worked Examples

TipWorked Numbers — Step by Step

Q. Next term: 4, 9, 16, 25, ? Differences: 5, 7, 9 → next difference = 11 → next term = 25 + 11 = 36 ✓. Alternative read: These are 2², 3², 4², 5² → next = 6² = 36.

Q. Missing: 6, 11, 21, 36, ? Differences: 5, 10, 15 → next = 20 → 36 + 20 = 56.

Q. Next: 1, 4, 13, 40, ? Rule: Each term × 3 + 1 → 40 × 3 + 1 = 121.

20.2.3 A 4-Step Recipe

TipFour-Step Recipe for Number Series
  1. Find the first-level differences between consecutive terms.
  2. If constant → AP. If ratio constant → GP.
  3. If neither, take second-level differences or look at operations alternating (+, ×, +, ×).
  4. Check special families — squares, cubes, primes, Fibonacci, factorial.

20.2.4 Special Series to Memorise

TipSpecial Series Worth Memorising
  • First 10 squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
  • First 10 cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
  • First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
  • Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.
  • Factorials: 1, 2, 6, 24, 120, 720, 5040.
  • Triangular numbers: 1, 3, 6, 10, 15, 21 — n(n+1)/2.

20.2.5 “Wrong / Odd Term” Variant

NTA often asks which term is wrong in a series. Method: verify pattern at first few terms, then test the suspect.

20.3 Letter Series

A letter series is a sequence of letters that follows a rule. Letter-series problems reduce to position-in-alphabet arithmetic.

20.3.1 The Alphabet Lookup Table

TipAlphabet Positions (Forward) — Memorise

A=1 · B=2 · C=3 · D=4 · E=5 · F=6 · G=7 · H=8 · I=9 · J=10 · K=11 · L=12 · M=13 · N=14 · O=15 · P=16 · Q=17 · R=18 · S=19 · T=20 · U=21 · V=22 · W=23 · X=24 · Y=25 · Z=26.

TipEJOTY — A Lifesaver Mnemonic

E = 5, J = 10, O = 15, T = 20, Y = 25. Five anchor letters, five-apart. To find any letter’s position, find the nearest anchor and add/subtract.

20.3.2 Reverse-Alphabet Positions

Often asked: “From the right end, what position is N?” Formula: Reverse position = 27 − forward position. So N (= 14) is 27 − 14 = 13 from the right.

20.3.3 Standard Letter-Series Patterns

TipStandard Letter-Series Patterns
Pattern Example Rule
Skip-by-constant A, C, E, G, ? +2 each
Skip-by-growing A, B, D, G, K, ? +1, +2, +3, +4
Pair series AB, CD, EF, ? Pairs at +2
Alternating A, Z, B, Y, C, X, ? Forward and backward together
Vowel/consonant A, E, I, O, ? Vowels only
Group-of-three ABC, EFG, IJK, ? Skip one then take three

20.3.4 Worked Examples

TipLetter-Series — Worked

Q. Next: A, D, G, J, ? Gaps: +3 → next = J + 3 = M.

Q. Missing: AC, EG, IK, MO, ? Each pair starts at a +4 letter, +2 within: Next pair starts at Q → QS.

Q. Next: AZ, BY, CX, DW, ? Forward pair 1, 2, 3, 4 … with backward pair Z, Y, X, W … → next = EV.

20.4 Coding-Decoding

A code is a rule that maps a known word/letter/number to another. Decoding reverses the rule.

20.4.1 Six Standard Code Types

TipSix Standard Code Types
  1. Letter shift / Caesar code — each letter shifted by a constant. DOG → EPH (+1).
  2. Reverse alphabet — each letter mapped to its mirror. A↔︎Z, B↔︎Y, C↔︎X …. CAT → XZG.
  3. Number substitution — each letter → its position (or reverse position). BAD → 2-1-4.
  4. Pair / chunk swap — letters swapped in pairs. CAT → ACT (1↔︎2).
  5. Mathematical operation code — each letter shifted by varying amounts. CAT → DCW (+1, +2, +3).
  6. Word-relationship code — entire word remapped per rule (e.g., reverse the word).

20.4.2 Worked Examples

TipCoding — Worked

Q. In a code, MUMBAI is written as NVNCBJ. How is DELHI coded? Rule: Each letter +1. D→E, E→F, L→M, H→I, I→J = EFMIJ.

Q. If PEN = 35, how much is INK? P = 16, E = 5, N = 14 → sum = 35. Apply same: I = 9, N = 14, K = 11 → 34.

Q. FACE → IDF H: each letter +3. So MIND → M+3=P, I+3=L, N+3=Q, D+3=G → PLQG.

20.4.3 Tips and Traps

TipTips and Traps
  • Many codes use ±1, ±2, ±3 shifts.
  • Reverse-alphabet codes use the identity position + reverse-position = 27.
  • Check case patterns — sometimes vowels are coded differently from consonants.
  • Number codes may add the digits, multiply them, or square them.

20.5 Blood Relations

A blood relation problem describes a family tree from limited clues. Always draw the diagram.

20.5.1 Standard Symbols

TipStandard Blood-Relation Symbols
  • + Male / − Female (some textbooks use ☐ / ◯).
  • Horizontal line = spouse.
  • Vertical line down = parent-to-child.
  • Generations stack downward.

20.5.2 Common Relationship Vocabulary

TipVocabulary to Memorise
Relation Means
Father’s father Paternal grandfather
Mother’s mother Maternal grandmother
Brother’s son Nephew
Sister’s son Nephew
Brother’s daughter Niece
Mother’s brother Maternal uncle
Father’s brother Paternal uncle
Father’s sister Paternal aunt
Mother’s sister Maternal aunt
Wife’s father Father-in-law
Sister’s husband Brother-in-law
Brother’s wife Sister-in-law
Uncle’s son Cousin
Grandfather’s son Father or uncle
Daughter’s son Grandson

20.5.3 Worked Examples

TipBlood-Relation Worked Example

Q. Pointing to a man, Sita says, “He is the son of my mother’s only daughter.” How is the man related to Sita?

Method: Sita’s mother’s only daughter = Sita herself. So the man is Sita’s son.

Q. Pointing to a woman, Ravi says, “She is the mother of my father’s only son.” How is the woman related to Ravi?

Method: Ravi’s father’s only son = Ravi himself. So the woman is Ravi’s mother.

Q. A is B’s brother. C is B’s mother. D is C’s father. How is D related to A?

Method: D is C’s father → D is B’s grandfather → since A and B are siblings, D is also A’s grandfather.

20.5.4 Quick Tricks

TipQuick Blood-Relation Tricks
  • “Only son / only daughter” → that person is themselves the speaker’s child or sibling.
  • Always list out the family tree on paper.
  • Pointing-to-a-photograph puzzles — convert “the photo” to “this person”.
  • When the speaker says “my mother’s son” and they have no brothers, it means themselves (if speaker is male).

20.6 Direction Sense

A direction-sense puzzle gives a sequence of movements and asks for the final position, distance, or direction. Always draw the path.

20.6.1 Cardinal Directions

flowchart TB
  N["N (North)"]
  S["S (South)"]
  E["E (East)"]
  W["W (West)"]
  N -.- E
  E -.- S
  S -.- W
  W -.- N
    classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

TipDirection Facts
  • Opposite directions: N↔︎S, E↔︎W.
  • Right-hand rotation from N: N → E → S → W → N.
  • Left-hand rotation from N: N → W → S → E → N.
  • Sunrise side: East. Sunset side: West. Shadow at sunrise falls to the west.

20.6.2 Standard Patterns

TipStandard Direction-Sense Patterns
  • Net displacement — opposing distances subtract; perpendicular distances combined via Pythagoras.
  • Final direction relative to start — track each turn.
  • Shortest distance — use Pythagoras for L-shaped or staircase paths.

20.6.3 Worked Example

TipDirection Worked Example

Q. A man walks 5 km north, then 4 km east, then 3 km south. How far is he from his starting point?

Net: 5 km N − 3 km S = 2 km north. Plus 4 km east. Distance = √(2² + 4²) = √20 = ≈ 4.47 km. Direction = north-east (slightly east of due north-east).

20.6.4 Clock / Angle Rotation Variant

TipClock-Angle Direction Variant
  • A person facing N, turns 90° clockwise → faces E.
  • A person facing S, turns 45° anti-clockwise → faces SE.
  • 180° turn = facing opposite direction.

20.7 Common-PYQ-Variant Puzzles

TipOther PYQ Variants
  • Seating arrangement — circular, linear, double-row.
  • Ranking / order — “A is taller than B but shorter than C”.
  • Calendar / day of week — what day will it be 100 days from today?
  • Clock — angle between hour and minute hand.
  • Cube / dice — opposite faces and adjacent faces.
  • Statement-Conclusion / Statement-Assumption.
  • Course of action / Cause-and-effect.

20.8 The Mental-Habits Behind Pattern Recognition

TipSix Mental Habits for Pattern Problems
  1. Slow down on the first 2-3 terms — most patterns reveal themselves there.
  2. Compute differences and ratios before guessing.
  3. Eliminate options instead of generating the answer from scratch.
  4. Test the rule on all given terms before locking in.
  5. Draw for blood relation and direction problems — always.
  6. Time-box — if a problem takes > 90 seconds, skip and come back.

20.9 Theory Anchors

TipConcepts Worth Naming
Concept Origin Use
Arithmetic progression (AP) Pythagoreans / classical Constant difference
Geometric progression (GP) Classical Constant ratio
Fibonacci sequence Leonardo of Pisa (Fibonacci), 1202 aₙ = aₙ₋₁ + aₙ₋₂
Pythagorean theorem Pythagoras / Baudhāyana a² + b² = c² (direction puzzles)
Caesar cipher Julius Caesar Constant letter shift
EJOTY mnemonic Indian competitive-exam tradition Five-apart anchors in alphabet
Triangular numbers Pythagoreans n(n+1)/2
Prime sieve (Eratosthenes) 3rd c. BCE Generating primes

20.10 Practice Questions

Q 01 AP Easy

Next term: 3, 7, 11, 15, ?

  • A17
  • B18
  • C19
  • D21
View solution
Correct Option: C
AP with d = 4. 15 + 4 = 19.
Q 02 GP Easy

Next term: 2, 6, 18, 54, ?

  • A108
  • B162
  • C216
  • D324
View solution
Correct Option: B
GP with r = 3. 54 × 3 = 162.
Q 03 Squares Medium

Next term: 1, 4, 9, 16, 25, ?

  • A30
  • B34
  • C36
  • D42
View solution
Correct Option: C
Squares: 1², 2², 3², 4², 5², 6² = 36.
Q 04 Fibonacci Medium

Next term: 1, 1, 2, 3, 5, 8, ?

  • A11
  • B12
  • C13
  • D15
View solution
Correct Option: C
Fibonacci: 5 + 8 = 13.
Q 05 Mixed Op Hard

Next term: 5, 11, 23, 47, ?

  • A71
  • B82
  • C95
  • D100
View solution
Correct Option: C
Rule: each term × 2 + 1. 47 × 2 + 1 = 95.
Q 06 Odd Term Hard

Find the WRONG term: 3, 7, 15, 30, 63, 127

  • A7
  • B15
  • C30
  • D63
View solution
Correct Option: C
Rule: × 2 + 1. 3→7, 7→15, 15→31 (not 30), 31→63, 63→127. 30 should be 31.
Q 07 Letter Skip Easy

Next letter: A, D, G, J, ?

  • AK
  • BL
  • CM
  • DN
View solution
Correct Option: C
+3 each. J + 3 = M.
Q 08 Letter Pair Medium

Next pair: AZ, BY, CX, DW, ?

  • AEV
  • BFV
  • CEW
  • DFU
View solution
Correct Option: A
Forward: A, B, C, D, E. Backward: Z, Y, X, W, V. Pair = EV.
Q 09 Alphabet Position Medium

From the right end of the alphabet, which letter is in the 13th position?

  • AM
  • BN
  • CL
  • DO
View solution
Correct Option: B
Forward position = 27 − 13 = 14 = N.
Q 10 Caesar Medium

If MUMBAI is coded as NVNCBJ, how is DELHI coded?

  • AEFMIJ
  • BFEMIJ
  • CEFMHJ
  • DDEMIJ
View solution
Correct Option: A
Each letter +1. D→E, E→F, L→M, H→I, I→J = EFMIJ.
Q 11 Reverse Code Hard

If CAT is coded as XZG (each letter replaced by its mirror in alphabet), how is DOG coded?

  • AWLT
  • BWML
  • CWLU
  • DWLR
View solution
Correct Option: A
Mirror: D(4)↔W(23), O(15)↔L(12), G(7)↔T(20). DOG → WLT.
Q 12 Number Code Medium

If PEN = 35 (sum of positions), then INK = ?

  • A32
  • B33
  • C34
  • D36
View solution
Correct Option: C
I (9) + N (14) + K (11) = 34.
Q 13 Blood Relation Medium

Pointing to a photograph, Ravi says, "He is the son of the only daughter of my grandfather." How is the man related to Ravi?

  • ABrother
  • BCousin
  • CUncle
  • DFather
View solution
Correct Option: A
Grandfather's only daughter = Ravi's mother (if mother is daughter of grandfather). Her son = Ravi himself OR Ravi's brother. Since the photo is of "the man", it is the brother.
Q 14 Blood Relation Hard

A is B's wife. C is B's brother. D is C's father. E is D's mother. How is E related to A?

  • AGrandmother-in-law
  • BMother-in-law
  • CAunt
  • DCousin
View solution
Correct Option: A
D is C's father → D is B's father → E (D's mother) is B's grandmother. A is B's wife → E is A's grandmother-in-law.
Q 15 Direction Medium

A boy walks 4 km north, then 3 km east. How far is he from his starting point?

  • A5 km
  • B6 km
  • C7 km
  • D8 km
View solution
Correct Option: A
Pythagoras: √(4² + 3²) = √25 = 5 km.
Q 16 Direction Net Hard

A man walks 7 km east, then 3 km north, then 4 km west, then 1 km south. How far is he from his start?

  • A√13 km
  • B5 km
  • C√17 km
  • D√29 km
View solution
Correct Option: A
Net east: 7 − 4 = 3. Net north: 3 − 1 = 2. Distance = √(3² + 2²) = √13 km.
Q 17 Rotation Easy

A man facing west turns 90° clockwise, then 180°, then 45° anti-clockwise. He is now facing:

  • ASouth-east
  • BSouth-west
  • CNorth-east
  • DNorth-west
View solution
Correct Option: A
W → +90° CW → N. N → 180° → S. S → 45° ACW → SE.
Q 18 EJOTY Medium

Using EJOTY, what is the position of the letter Q in the alphabet?

  • A16
  • B17
  • C18
  • D19
View solution
Correct Option: B
Nearest anchor: O = 15. Q = O + 2 = 17.
Q 19 Coding Hard

In a code, FACE = IDFH (each letter +3). Encode MIND.

  • APLQG
  • BPLPG
  • CPMQG
  • DQLQF
View solution
Correct Option: A
M(13)+3=P, I(9)+3=L, N(14)+3=Q, D(4)+3=G → PLQG.
Q 20 Trick Hard

Pointing to a woman, a man says, "She is the daughter of the wife of my father's only son." How is the woman related to the man?

  • ASister
  • BDaughter
  • CNiece
  • DCousin
View solution
Correct Option: B
Father's only son = the man himself. Wife of the man = his wife. Their daughter = the man's daughter.

20.11 Quick Recall

ImportantQuick recall
  • Number series patterns: AP · GP · Squares · Cubes · Fibonacci · Primes · Mixed-operation · Increasing-difference.
  • Memorise: First 10 squares (1-100), cubes (1-1000), primes (2-29), Fibonacci (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89), triangular numbers (1, 3, 6, 10, 15, 21).
  • 4-step recipe: differences → AP/GP check → second differences / alternating → special families.
  • Alphabet positions: A=1 … Z=26. EJOTY mnemonic: E5, J10, O15, T20, Y25.
  • Reverse position = 27 − forward position.
  • Letter-series patterns: Skip-constant · Skip-growing · Pair · Alternating · Vowel/consonant · Group-of-three.
  • 6 code types: Caesar shift · Reverse alphabet · Number substitution · Pair swap · Mathematical operation · Word-relationship.
  • Code identity: position + reverse-position = 27.
  • Blood relations: Draw the tree; use “+/−” or “☐/◯”; horizontal = spouse, vertical = parent-child; “only son/daughter” trick = the speaker/their child themselves.
  • Direction sense: N↔︎S · E↔︎W; right-rotation from N: N→E→S→W; sunrise=East; Pythagoras for L-shaped distance.
  • Common combo: net displacement (opposite cancels, perpendicular Pythagoras).
  • Mental habits: examine first 2-3 terms slowly · compute differences and ratios · eliminate · test on full series · draw for relations & directions · time-box 90 sec.