Study guide
8.1 Number Series, Letter Series, Coding-Decoding & Relationships
Chhattisgarh State Eligibility Test CG SET
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1 Chapter 1: Exam Structure, Blueprint & Preparation Strategy
1.1 Exam Blueprint, Marking Scheme & Qualifying Cutoffs 1.2 Eligibility Requirements, Master's Degree Criteria & Domicile Policy 1.3 Paper I Strategic Study Plan, Time Allocation & OMR Tactics
2 Chapter 2: Teaching Aptitude — Concepts, Objectives & Learner Characteristics
2.1 Concepts, Objectives & Levels of Teaching 2.2 Learner Characteristics & Individual Differences 2.3 Factors Influencing Teaching Effectiveness & Instructional Facilities
3 Chapter 3: Teaching Aptitude — Methods, Support Systems & Evaluation
3.1 Teacher-Centered vs. Learner-Centered Methods & Modern Pedagogy 3.2 Teaching Support Systems & Online Digital Platforms (SWAYAM, MOOCs) 3.3 Evaluation Systems, CBCS & Assessment Innovations
4 Chapter 4: Research Aptitude — Paradigms, Types & Methodologies
4.1 Meaning, Objectives, Characteristics & Types of Research 4.2 Epistemological Approaches: Positivism vs. Post-Positivist Philosophies 4.3 Core Research Methods: Descriptive, Experimental & Historical
5 Chapter 5: Research Aptitude — Process, Ethics & Scholarly Writing
5.1 Systematic Steps of Research & Hypothesis Testing 5.2 Thesis, Dissertation & Scholarly Article Writing 5.3 Application of ICT in Research & Research Ethics
6 Chapter 6: Reading Comprehension — Critical Analysis & Interpretation
6.1 Critical Reading Strategies: Structure, Main Ideas & Skimming vs. Scanning 6.2 Inferential Analysis, Contextual Vocabulary & Author's Tone 6.3 Question-Solving Tactics & Eliminating Distractors in Academic Passages
7 Chapter 7: Communication in Education & Mass Media
7.1 Meaning, Types & Characteristics of Communication 7.2 Effective Classroom Communication: Verbal/Non-Verbal Dynamics & Immediacy 7.3 Communication Barriers & The Role of Mass Media in Society
8 Chapter 8: Mathematical Reasoning & Quantitative Aptitude
8.1 Number Series, Letter Series, Coding-Decoding & Relationships 8.2 Fractions, Ratios, Proportions & Percentage Mastery 8.3 Profit & Loss, Interest, Discounting, Speed-Distance & Averages
9 Chapter 9: Logical Reasoning — Western Deductive & Inductive Logic
9.1 Structure of Arguments & Categorical Propositions 9.2 Formal & Informal Fallacies & Connotation vs. Denotation 9.3 Classical Square of Opposition & Venn Diagrams for Validity
10 Chapter 10: Logical Reasoning — Indian Logic & Nyaya Epistemology
10.1 Indian Logic: Six Pramanas (Means of Valid Knowledge) 10.2 Structure and Types of Anumana (Inference), Vyapti & Hetu 10.3 Hetvabhasa: Fallacies of Inference in Nyaya Logic
11 Chapter 11: Data Interpretation & Quantitative Representation
11.1 Sources, Acquisition & Classification of Quantitative and Qualitative Data 11.2 Graphical Representation: Bar Charts, Histograms, Pie Charts & Line Graphs 11.3 Table Charts, Cross-Tabulation & Data Interpretation for Governance 11.4 Mapping of Data & Data in Governance
12 Chapter 12: Information & Communication Technology (ICT)
12.1 Foundational ICT Terminology, Acronyms & Computer Architecture 12.2 Internet Architecture, Intranets, Email & Video Conferencing Mechanics 12.3 National Digital Initiatives in Higher Education & ICT Governance
13 Chapter 13: People, Development & Environment — Goals & Pollution
13.1 Development and Environment: Millennium & Sustainable Development Goals 13.2 Anthropogenic Impacts & Environmental Pollution Types 13.3 Health Impacts of Pollutants, Heavy Metals & Climate Dynamics
14 Chapter 14: People, Development & Environment — Energy, Hazards & Treaties
14.1 Renewable & Non-Renewable Energy Resources and India's Energy Transition 14.2 Natural Hazards, Disasters & Disaster Management Architecture 14.3 Environmental Legislation, NAPCC & International Conventions
15 Chapter 15: Higher Education System — Historical Evolution & Modalities
15.1 Higher Learning & Education in Ancient India 15.2 Evolution of Higher Learning in Post-Independence India & NEP 2020 15.3 Oriental, Conventional, Non-Conventional & Vocational Education
16 Chapter 16: Higher Education System — Governance, Ethics & Policy
16.1 Value Education & Environmental Education in Academic Curricula 16.2 Regulatory Landscape: UGC, Professional Councils, NAAC & NIRF 16.3 Higher Education Governance & Administration in Chhattisgarh State
8.1 Number Series, Letter Series, Coding-Decoding & Relationships
Key Takeaways Explain This Topic AI
Unit 5 of CG SET Paper I systematically evaluates inductive reasoning, algebraic progressions, and symbolic pattern recognition across number series, letter codes, direction tests, and family relationships.
Number series follow identifiable mathematical laws: arithmetic differences (including second differences), geometric ratios, exponential squares and cubes ( n 2 ± k n^2 \pm k n 2 ± k , n 3 ± k n^3 \pm k n 3 ± k ), prime sequences, and interleaved alternating series.
Positional letter coding relies on forward numbering ( A = 1 A=1 A = 1 to Z = 26 Z=26 Z = 26 ), reverse numbering ( A = 26 A=26 A = 26 to Z = 1 Z=1 Z = 1 ), and the complementary Rule of 27 ( Forward + Reverse = 27 \text{Forward} + \text{Reverse} = 27 Forward + Reverse = 27 ) to identify opposite letter pairs.
Direction sense problems require constructing a two-dimensional Cartesian plane with cardinal and intercardinal coordinates, applying clockwise/counter-clockwise angular turns, and utilizing the Pythagorean theorem ( Displacement = Δ x 2 + Δ y 2 \text{Displacement} = \sqrt{\Delta x^2 + \Delta y^2} Displacement = Δ x 2 + Δ y 2 ).
Blood relationship analysis is solved deterministically by building generational family tree diagrams with standard gender ( + + + for male, − - − for female) and marital ( = = = ) conventions, neutralizing deceptive linguistic distractors.
Last updated: October 2026
8.1 Number Series, Letter Series, Coding-Decoding & Relationships
Core Principle: Mathematical reasoning in competitive examinations is not a test of arbitrary calculation tricks or intuitive guesswork. It is a systematic diagnostic of pattern recognition, algebraic symmetry, spatial coordinate tracking, and relational network deduction. By mastering standardized diagnostic protocols, candidates transform seemingly opaque sequences and coded puzzles into deterministic mathematical operations.
1. Mathematical and Logical Reasoning in CG SET Paper I: Blueprint and Epistemological Scope
In the Chhattisgarh State Eligibility Test (CG SET) Paper I, Unit 5: Mathematical Reasoning and Aptitude occupies a pivotal deterministic role. CG SET follows the UGC Paper I syllabus, and this unit evaluates an aspirant's foundational computational fluency, structural logic, and capacity to abstract relationships from symbolic data. Under the UGC note of five 2-mark questions per unit, Unit 5 usually accounts for 5 questions (10 marks) , representing 10% of the total Paper I score.
While general awareness units require extensive empirical recall of environmental pacts or educational commissions, mathematical reasoning is completely algorithmic and self-verifiable . A candidate who methodically deconstructs the underlying rule can achieve an absolute 100% accuracy rate (10/10 marks), providing a crucial competitive buffer for Paper I qualification.
Core Cognitive Modalities Tested in Unit 5
Questions in Unit 5 test four primary cognitive modalities:
Inductive Numerical Pattern Recognition: Identifying the mathematical progression governing an ordered sequence of numbers to extrapolate a missing term or identify an erroneous outlier.
Symbolic and Positional Decoupling: Translating alphabetic characters into numerical coordinates, manipulating modular cyclic shifts, and deciphering linguistic cipher algorithms.
Spatial Vector Analysis: Mapping spatial trajectories across two-dimensional cardinal coordinates, tracking angular rotations, and computing net Euclidean displacement.
Relational Network Deduction: Deconstructing multi-generational genealogical statements into formal family-tree networks and decoding symbolic kinship operators.
Types of Reasoning
The UGC syllabus opens Unit 5 with types of reasoning . The main types are:
Type Direction of thought Example
Deductive From general rules to a certain conclusion about a case All primes above 2 are odd; 17 is a prime above 2; so 17 is odd
Inductive From observed cases to a probable general rule In 2, 6, 18, 54 each term triples the last, so the next is probably 162
Abductive From an observation to the most likely explanation Attendance fell on Monday; a festival holiday best explains it
- Analogical From similarity in known respects to similarity in another Doctor : Hospital :: Teacher : School
- Mathematical (quantitative) Applying number rules and operations Percentages, ratios, interest
- Verbal and non-verbal Reasoning with words, or with figures and patterns Coding-decoding; figure series
Number and letter series questions are mainly inductive , because you infer the rule from the terms given. Blood-relation and direction questions are deductive , because the answer follows necessarily from the statements. Section 9.1 covers deductive, inductive, abductive and analogical reasoning in more depth.
2. Number Series Mastery & Diagnostic Workflow
A number series is an ordered sequence of real numbers generated according to a specific, invariant mathematical rule. In CG SET Paper I, questions typically present a sequence with one omitted value indicated by a question mark ( ? ? ? ) or ask candidates to detect the single "wrong term" that violates the governing rule.
Primary Architectural Patterns of Number Series
Candidates must be thoroughly conversant with the eight primary mathematical archetypes encountered in examination series:
1. Arithmetic Progressions (AP) & Linear Difference Series
The difference between successive terms is a constant scalar d d d :
T n = T n − 1 + d ⟹ T n = a + ( n − 1 ) d T_n = T_{n-1} + d \implies T_n = a + (n - 1)d T n = T n − 1 + d ⟹ T n = a + ( n − 1 ) d
Example: 7 , 13 , 19 , 25 , 31 , … 7, 13, 19, 25, 31, \dots 7 , 13 , 19 , 25 , 31 , … (Common difference d = + 6 d = +6 d = + 6 ).
2. Arithmetic of Second Differences (Polynomial / Triangular Series)
The primary differences between consecutive terms ( Δ 1 \Delta_1 Δ 1 ) are not constant, but the differences of those differences ( Δ 2 \Delta_2 Δ 2 ) form a constant arithmetic progression. These series reflect underlying quadratic polynomials of the form T n = a n 2 + b n + c T_n = an^2 + bn + c T n = a n 2 + bn + c .
- Example: 3 , 8 , 17 , 30 , 47 , … 3, 8, 17, 30, 47, \dots 3 , 8 , 17 , 30 , 47 , …
- First differences ( Δ 1 \Delta_1 Δ 1 ): 5 , 9 , 13 , 17 5, 9, 13, 17 5 , 9 , 13 , 17
- Second differences ( Δ 2 \Delta_2 Δ 2 ): + 4 , + 4 , + 4 +4, +4, +4 + 4 , + 4 , + 4 (Constant)
Next Δ 1 = 17 + 4 = 21 ⟹ Next term = 47 + 21 = 68 \Delta_1 = 17 + 4 = 21 \implies \text{Next term} = 47 + 21 = 68 Δ 1 = 17 + 4 = 21 ⟹ Next term = 47 + 21 = 68 .
3. Geometric Progressions (GP) & Multiplicative Series
Each successive term is produced by multiplying the preceding term by a constant non-zero common ratio r r r :
T n = T n − 1 × r ⟹ T n = a ⋅ r n − 1 T_n = T_{n-1} \times r \implies T_n = a \cdot r^{n-1} T n = T n − 1 × r ⟹ T n = a ⋅ r n − 1
Example: 4 , 12 , 36 , 108 , 324 , … 4, 12, 36, 108, 324, \dots 4 , 12 , 36 , 108 , 324 , … ( r = 3 r = 3 r = 3 ).
Fractional / Decimal Multipliers: Series frequently involve multipliers such as × 0.5 , × 1.5 , × 2.5 , … \times 0.5, \times 1.5, \times 2.5, \dots × 0.5 , × 1.5 , × 2.5 , … (e.g., 8 , 4 , 6 , 15 , 52.5 , … 8, 4, 6, 15, 52.5, \dots 8 , 4 , 6 , 15 , 52.5 , … ).
4. Hybrid Arithmetic-Geometric Series ( a ⋅ T n − 1 ± k a \cdot T_{n-1} \pm k a ⋅ T n − 1 ± k )
Each term is generated by multiplying the prior term by a factor and then adding or subtracting a constant or progressive integer:
- Example: 2 , 5 , 12 , 27 , 58 , … 2, 5, 12, 27, 58, \dots 2 , 5 , 12 , 27 , 58 , …
- 2 × 2 + 1 = 5 2 \times 2 + 1 = 5 2 × 2 + 1 = 5
- 5 × 2 + 2 = 12 5 \times 2 + 2 = 12 5 × 2 + 2 = 12
- 12 × 2 + 3 = 27 12 \times 2 + 3 = 27 12 × 2 + 3 = 27
- 27 × 2 + 4 = 58 27 \times 2 + 4 = 58 27 × 2 + 4 = 58
Next term: 58 × 2 + 5 = 116 + 5 = 121 58 \times 2 + 5 = 116 + 5 = 121 58 × 2 + 5 = 116 + 5 = 121 .
5. Exponential Square and Cube Series ( n 2 ± k , n 3 ± k n^2 \pm k, n^3 \pm k n 2 ± k , n 3 ± k )
Terms are anchored to integer powers. Mastery requires instant mental recognition of the first 25 squares and the first 12 cubes:
Value ( n n n ) n 2 n^2 n 2 n 3 n^3 n 3 n 2 − 1 n^2 - 1 n 2 − 1 n 2 + 1 n^2 + 1 n 2 + 1 n 3 − 1 n^3 - 1 n 3 − 1 n 3 + 1 n^3 + 1 n 3 + 1 n 3 − n n^3 - n n 3 − n
- 1 1 1 0 2 0 2 0
- 2 4 8 3 5 7 9 6
- 3 9 27 8 10 26 28 24
- 4 16 64 15 17 63 65 60
- 5 25 125 24 26 124 126 120
- 6 36 216 35 37 215 217 210
- 7 49 343 48 50 342 344 336
- 8 64 512 63 65 511 513 504
Classic Exam Sequence: 0 , 6 , 24 , 60 , 120 , 210 , ? 0, 6, 24, 60, 120, 210, ? 0 , 6 , 24 , 60 , 120 , 210 , ? . Recognizing the n 3 − n n^3 - n n 3 − n benchmark immediately reveals the next term for n = 7 n=7 n = 7 as 7 3 − 7 = 343 − 7 = 336 7^3 - 7 = 343 - 7 = 336 7 3 − 7 = 343 − 7 = 336 .
6. Alternating / Interleaved Sequences
Two distinct mathematical series are interleaved into a single visible sequence, occupying alternate positions. Odd indices follow one rule while even indices follow an entirely independent rule.
Example: 4 , 20 , 7 , 17 , 10 , 14 , 13 , 11 , ? 4, 20, 7, 17, 10, 14, 13, 11, ? 4 , 20 , 7 , 17 , 10 , 14 , 13 , 11 , ?
Sub-series 1 (Odd positions: 1st, 3rd, 5th, 7th, 9th): 4 , 7 , 10 , 13 , ? 4, 7, 10, 13, ? 4 , 7 , 10 , 13 , ? ⟹ \implies ⟹ Common difference + 3 +3 + 3 . Next term = 16 = 16 = 16 .
Sub-series 2 (Even positions: 2nd, 4th, 6th, 8th): 20 , 17 , 14 , 11 20, 17, 14, 11 20 , 17 , 14 , 11 ⟹ \implies ⟹ Common difference − 3 -3 − 3 .
7. Prime Number Progressions
Sequences governed by consecutive prime numbers: 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , … 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, \dots 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , …
Critical Caution: 1 1 1 is neither prime nor composite. 2 2 2 is the only even prime number. 9 , 15 , 21 , 25 , 27 , 33 , 35 9, 15, 21, 25, 27, 33, 35 9 , 15 , 21 , 25 , 27 , 33 , 35 are composite and must never be mistaken for primes.
8. Fibonacci & Recursive Additive Series
Each term is the sum of the preceding two (or three) terms: T n = T n − 1 + T n − 2 T_n = T_{n-1} + T_{n-2} T n = T n − 1 + T n − 2 .
Standard: 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , … 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , …
Modified: 2 , 3 , 5 , 8 , 13 , 21 , 34 , … 2, 3, 5, 8, 13, 21, 34, \dots 2 , 3 , 5 , 8 , 13 , 21 , 34 , …
Tribonacci: T n = T n − 1 + T n − 2 + T n − 3 T_n = T_{n-1} + T_{n-2} + T_{n-3} T n = T n − 1 + T n − 2 + T n − 3 (e.g., 1 , 2 , 3 , 6 , 11 , 20 , 37 , … 1, 2, 3, 6, 11, 20, 37, \dots 1 , 2 , 3 , 6 , 11 , 20 , 37 , … ).
Systematic 5-Step Diagnostic Protocol for Unknown Number Series
When confronted with an unfamiliar series under exam pressure, candidates should avoid random arithmetic attempts and execute this structured 5-step diagnostic workflow: Step 1: Inspect Growth Trajectory
- ├── Steep / Exponential Growth ──> Test Geometric (× r) or Powers (n², n³)
- └── Moderate / Linear Growth ──> Test Arithmetic Differences
- Step 2: Calculate Level 1 Differences (Δ₁)
- ├── Constant Difference? ──> Pure Arithmetic Progression identified
- └── Non-constant? ──> Proceed to Step 3
- Step 3: Calculate Level 2 Differences (Δ₂)
- ├── Constant Second Difference? ──> Quadratic Polynomial series (an² + bn + c)
- └── Patterned (e.g., +2, +4, +6)?──> Triangular progression solved
- Step 4: Benchmark Power Test
- └── Check distance from nearest perfect squares (n² ± k) or cubes (n³ ± k)
- Step 5: Structural & Positional Tests
- ├── Values oscillating up & down? ──> Interleaved / Alternating series
- ├── Digits summing / reversing? ──> Internal digit manipulation rule
- └── Unbroken prime sequence? ──> Prime number progression
- 3. Letter Series and Alpha-Numeric Progressions
Letter series test the candidate's understanding of the English alphabet as a closed cyclic mathematical group of order 26 (modulo 26 arithmetic).
3.1 Forward and Reverse Positional Values
Every letter holds two intrinsic numerical coordinates:
Forward Positional Value: A = 1 , B = 2 , … , Z = 26 A = 1, B = 2, \dots, Z = 26 A = 1 , B = 2 , … , Z = 26 .
Reverse Positional Value (Backward Rank): A = 26 , B = 25 , … , Z = 1 A = 26, B = 25, \dots, Z = 1 A = 26 , B = 25 , … , Z = 1 . Forward: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
- Rank: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
- Reverse: 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
- The Rule of 27 (Opposite Letter Invariant)
A fundamental mathematical property links the forward and reverse coordinates of any letter:
Forward Position + Reverse Position = 27 \text{Forward Position} + \text{Reverse Position} = 27 Forward Position + Reverse Position = 27
Consequently, two letters are diametrically opposite in the 26-letter alphabet if their forward positional values sum to exactly 27:
Pos ( L 1 ) + Pos ( L 2 ) = 27 \text{Pos}(L_1) + \text{Pos}(L_2) = 27 Pos ( L 1 ) + Pos ( L 2 ) = 27
- Pair Forward Values Sum Mnemonic Anchor
- A – Z 1 + 26 1 + 26 1 + 26 27 A to Z (Alphabet extremes)
- B – Y 2 + 25 2 + 25 2 + 25 27 B o Y
- C – X 3 + 24 3 + 24 3 + 24 27 C ru X / C o X
- D – W 4 + 23 4 + 23 4 + 23 27 D e W / D own- W ard
- E – V 5 + 22 5 + 22 5 + 22 27 EV ening / EV ery
- F – U 6 + 21 6 + 21 6 + 21 27 FU ll / F or U
- G – T 7 + 20 7 + 20 7 + 20 27 G . T . Road (Grand Trunk)
- H – S 8 + 19 8 + 19 8 + 19 27 H igh S chool
- I – R 9 + 18 9 + 18 9 + 18 27 I ndian R ailways
- J – Q 10 + 17 10 + 17 10 + 17 27 J ungle Q ueen / J ack- Q ueen
- K – P 11 + 16 11 + 16 11 + 16 27 K urukshetra P ost / K an P ur
- L – O 12 + 15 12 + 15 12 + 15 27 LO ve
- M – N 13 + 14 13 + 14 13 + 14 27 M a N
- High-Speed Positional Anchors: The EJOTY and CFILORUX Systems
Rather than counting fingers from A during the exam, memorize the five-interval EJOTY benchmark:
- E = 5 E = 5 E = 5
- J = 10 J = 10 J = 10
- O = 15 O = 15 O = 15
- T = 20 T = 20 T = 20
- Y = 25 Y = 25 Y = 25
To identify any letter, anchor to the nearest EJOTY landmark (e.g., S S S is one before T = 20 ⟹ S = 19 T = 20 \implies S = 19 T = 20 ⟹ S = 19 ; M M M is two before O = 15 ⟹ M = 13 O = 15 \implies M = 13 O = 15 ⟹ M = 13 ).
Alternatively, utilize the three-interval CFILORUX system: C ( 3 ) , F ( 6 ) , I ( 9 ) , L ( 12 ) , O ( 15 ) , R ( 18 ) , U ( 21 ) , X ( 24 ) C(3), F(6), I(9), L(12), O(15), R(18), U(21), X(24) C ( 3 ) , F ( 6 ) , I ( 9 ) , L ( 12 ) , O ( 15 ) , R ( 18 ) , U ( 21 ) , X ( 24 ) .
3.2 Modulo 26 Cyclic Shifts
Letter series frequently loop around the end of the alphabet. Under modulo 26:
Moving forward past Z Z Z : 25 ( Y ) → 26 ( Z ) → 1 ( A ) → 2 ( B ) 25(Y) \to 26(Z) \to 1(A) \to 2(B) 25 ( Y ) → 26 ( Z ) → 1 ( A ) → 2 ( B ) .
Moving backward past A A A : 3 ( C ) → 2 ( B ) → 1 ( A ) → 26 ( Z ) → 25 ( Y ) 3(C) \to 2(B) \to 1(A) \to 26(Z) \to 25(Y) 3 ( C ) → 2 ( B ) → 1 ( A ) → 26 ( Z ) → 25 ( Y ) .
4. Coding-Decoding Paradigms
Coding-decoding evaluates the ability to infer an encryption cipher applied to a known string and apply that exact cipher to encrypt or decrypt a target string.
4.1 Letter-to-Letter Coding
Constant Scalar Shift: Every character is shifted by + k +k + k or − k -k − k positions (e.g., each letter + 3 +3 + 3 ).
Progressive Positional Shift: Successive letters shift by increasing intervals: 1st letter + 1 +1 + 1 , 2nd letter + 2 +2 + 2 , 3rd letter + 3 +3 + 3 , etc.
Reverse Order Inversion: The word is reversed end-to-end, or divided into two halves and each half reversed internally.
Opposite Letter Coding: Every letter is replaced by its Rule of 27 counterpart ( A → Z , B → Y , C → X A \to Z, B \to Y, C \to X A → Z , B → Y , C → X ).
Cross-Pair Shifts: In a 6-letter word, letters 1 and 2 swap with shifts, letters 3 and 4 swap, and letters 5 and 6 swap.
4.2 Number and Symbol Coding
Direct Positional Summation: The code equals the arithmetic sum of forward positional values.
Example: CAT = 3 + 1 + 20 = 24 \text{CAT} = 3 + 1 + 20 = 24 CAT = 3 + 1 + 20 = 24 .
Reverse Positional Summation: The code equals the sum of reverse values.
Example: CAT = ( 27 − 3 ) + ( 27 − 1 ) + ( 27 − 20 ) = 24 + 26 + 7 = 57 \text{CAT} = (27 - 3) + (27 - 1) + (27 - 20) = 24 + 26 + 7 = 57 CAT = ( 27 − 3 ) + ( 27 − 1 ) + ( 27 − 20 ) = 24 + 26 + 7 = 57 .
Positional Product or Digit Sum: Multiplying coordinates or summing individual digits of coordinates.
Direct Symbol Assignment: Characters are mapped one-to-one to specific punctuation marks or digits without mathematical operations (e.g., R = 5, E = #, A = 9, D = $ ).
4.3 Substitution Coding (Fictional Nomenclature)
In substitution coding, names of everyday objects are arbitrarily mapped to other words (e.g., "If blue is called red, red is called green, green is called white, and white is called yellow, what is the color of clear sky?" ).
The Direct Resolution Rule: Determine the true real-world factual answer first ( Clear sky is blue ). Then identify the single code word that blue is called ( Blue is called red ). The correct answer is red . Never chain substitutions through multiple levels.
4.4 Deciphering Message / Sentence Codes (Common Word Elimination)
In sentence coding, multi-word statements are encrypted into nonsense syllable codes. The assignment of code syllables to words is scrambled.
Given:
- "pic vic nic" means "winter is cold"
- "to nic re" means "summer is hot"
- "re pic maj" means "winter was hot"
Elimination Protocol:
Compare (1) and (2): Common word is "is" ; common code is "nic" ⟹ is = nic \implies \text{is} = \text{nic} ⟹ is = nic .
Compare (1) and (3): Common word is "winter" ; common code is "pic" ⟹ winter = pic \implies \text{winter} = \text{pic} ⟹ winter = pic .
From (1), remaining word is "cold" and remaining code is "vic" ⟹ cold = vic \implies \text{cold} = \text{vic} ⟹ cold = vic .
Compare (2) and (3): Common word is "hot" ; common code is "re" ⟹ hot = re \implies \text{hot} = \text{re} ⟹ hot = re .
From (2), remaining word is "summer" and code is "to" ⟹ summer = to \implies \text{summer} = \text{to} ⟹ summer = to .
From (3), remaining word is "was" and code is "maj" ⟹ was = maj \implies \text{was} = \text{maj} ⟹ was = maj .
5. Direction and Distance Sense Tests
Direction sense questions test orientation in two-dimensional Euclidean space. All movements must be modeled upon a standardized Cartesian coordinate plane. North (+y)
- │
- North-West │ North-East
- (NW) │ (NE)
- │
- West (-x) ───────────────┼─────────────── East (+x)
- │
- South-West │ South-East
- (SW) │ (SE)
- │
- South (-y)
- 5.1 Cardinal Directions and Angular Vectors
Four Cardinal Directions: North ( 0 ∘ / 360 ∘ 0^\circ / 360^\circ 0 ∘ /36 0 ∘ ), East ( 90 ∘ 90^\circ 9 0 ∘ ), South ( 180 ∘ 180^\circ 18 0 ∘ ), West ( 270 ∘ 270^\circ 27 0 ∘ ).
Four Intercardinal (Ordinal) Directions: North-East ( 45 ∘ 45^\circ 4 5 ∘ ), South-East ( 135 ∘ 135^\circ 13 5 ∘ ), South-West ( 225 ∘ 225^\circ 22 5 ∘ ), North-West ( 315 ∘ 315^\circ 31 5 ∘ ).
Turning Rules:
Right Turn: Always corresponds to a 90 ∘ 90^\circ 9 0 ∘ clockwise rotation.
Left Turn: Always corresponds to a 90 ∘ 90^\circ 9 0 ∘ counter-clockwise (anti-clockwise) rotation.
Facing North: Right = East ( + x +x + x ), Left = West ( − x -x − x ).
Facing South: Right = West ( − x -x − x ), Left = East ( + x +x + x ) (Direction orientation reverses!) .
Facing East: Right = South ( − y -y − y ), Left = North ( + y +y + y ).
Facing West: Right = North ( + y +y + y ), Left = South ( − y -y − y ).
5.2 Shortest Displacement via the Pythagorean Theorem
To compute the net shortest distance between a traveler's terminal position and their origin, sum all horizontal displacements ( Δ x \Delta x Δ x ) and vertical displacements ( Δ y \Delta y Δ y ):
Net Horizontal Displacement ( Δ x ) = ∑ x East − ∑ x West \text{Net Horizontal Displacement } (\Delta x) = \sum x_{\text{East}} - \sum x_{\text{West}} Net Horizontal Displacement ( Δ x ) = ∑ x East − ∑ x West Net Vertical Displacement ( Δ y ) = ∑ y North − ∑ y South \text{Net Vertical Displacement } (\Delta y) = \sum y_{\text{North}} - \sum y_{\text{South}} Net Vertical Displacement ( Δ y ) = ∑ y North − ∑ y South
The straight-line Euclidean distance D D D is given by the Pythagorean theorem:
- D = ( Δ x ) 2 + ( Δ y ) 2 D = \sqrt{(\Delta x)^2 + (\Delta y)^2} D = ( Δ x ) 2 + ( Δ y ) 2
- Common Pythagorean Triples in Examination Problems
Recognizing integer triples eliminates long square-root calculations:
( 3 , 4 , 5 ) (3, 4, 5) ( 3 , 4 , 5 ) and its multiples: ( 6 , 8 , 10 ) (6, 8, 10) ( 6 , 8 , 10 ) , ( 9 , 12 , 15 ) (9, 12, 15) ( 9 , 12 , 15 ) , ( 12 , 16 , 20 ) (12, 16, 20) ( 12 , 16 , 20 )
- ( 5 , 12 , 13 ) (5, 12, 13) ( 5 , 12 , 13 ) and its multiples: ( 10 , 24 , 26 ) (10, 24, 26) ( 10 , 24 , 26 )
- ( 8 , 15 , 17 ) (8, 15, 17) ( 8 , 15 , 17 )
- ( 7 , 24 , 25 ) (7, 24, 25) ( 7 , 24 , 25 )
- ( 9 , 40 , 41 ) (9, 40, 41) ( 9 , 40 , 41 )
- 6. Blood Relationships & Family Tree Modeling
Blood relationship problems evaluate deductive reasoning over kinship structures. To avoid cognitive overload from complex narrative descriptions, candidates should map relationships into a standardized visual Family Tree Diagram .
6.1 Standardized Family Tree Notational Conventions
- Kinship Relationship Visual Representation Graphical Syntax
- Male Person [ + ] [+] [ + ] Enclose in brackets or annotate with + + +
- Female Person [ − ] [-] [ − ] Enclose in brackets or annotate with − - −
- Marital Bond (Husband = Wife) A = B A = B A = B Double horizontal line connecting spouses
- Sibling Bond (Brother / Sister) A − B A - B A − B Single horizontal line connecting siblings
Generational Transition Vertical line Upper tier = Parent; Lower tier = Child Grandfather [+] ════ Grandmother [-]
- │
- ┌──────────────────────┴──────────────────────┐
- │ │
- Father [+] ════ Mother [-] Uncle [+]
- │
- ┌──────────────┴──────────────┐
- │ │
- Self [+] Sister [-] ════ Brother-in-law [+]
- 6.2 Key Relational Nomenclature and Distinctions
Paternal vs. Maternal Lines:
Paternal grandfather/grandmother: Father's parents.
Maternal grandfather/grandmother: Mother's parents.
Paternal uncle: Father's brother ( Chacha ).
Maternal uncle: Mother's brother ( Mama ).
Nephew vs. Niece:
Nephew: Brother's or sister's son.
Niece: Brother's or sister's daughter.
In-Law Relationships:
Father-in-law / Mother-in-law: Spouse's father / mother.
Brother-in-law: Spouse's brother OR sister's husband.
Sister-in-law: Spouse's sister OR brother's wife.
Cousin: Child of an aunt or uncle (independent of paternal/maternal distinction).
6.3 Coded Blood Relations
In advanced SET problems, kinship is expressed via symbolic algebraic operators (e.g., P + Q P + Q P + Q means P P P is father of Q Q Q ; P − Q P - Q P − Q means P P P is sister of Q Q Q ; P × Q P \times Q P × Q means P P P is brother of Q Q Q ; P ÷ Q P \div Q P ÷ Q means P P P is mother of Q Q Q ).
The Two-Step Elimination Protocol for Coded Relations
The Gender Filter: If the question asks for a male relationship (e.g., "A is the uncle of B" ), immediately inspect the symbol following A A A . If the operator defines A A A as female, eliminate that option instantly without drawing the tree.
The Generation Gap Calculator: Assign numerical generation values:
- Parents, Uncles, Aunts: + 1 +1 + 1
- Grandparents: + 2 +2 + 2
- Siblings, Spouses, Self, Cousins: 0 0 0
- Children, Nephews, Nieces: − 1 -1 − 1
Grandchildren: − 2 -2 − 2 Sum the generation values across the expression. If the target relationship requires a net generation gap of + 1 +1 + 1 , any expression yielding 0 , − 1 0, -1 0 , − 1 , or + 2 +2 + 2 is mathematically impossible.
7. Exam Traps & Common Errors in Unit 5
Trap 1: The First Difference Fallacy in Number Series: Stopping analysis after calculating first differences ( Δ 1 \Delta_1 Δ 1 ) and assuming no pattern exists. Always compute second differences ( Δ 2 \Delta_2 Δ 2 ) or test distances from adjacent squares ( n 2 ± k n^2 \pm k n 2 ± k ) and cubes ( n 3 ± k n^3 \pm k n 3 ± k ).
Trap 2: The Left/Right Inversion Trap When Facing South or West: When a problem states "traveling South, they turned right" , candidates instinctively map right as East. In reality, facing South ( − y -y − y ) inverts orientation: a right turn points West ( − x -x − x ).
Trap 3: Presuming Gender from Names or Indirect Association: In blood relationships, never assume a person is male or female based on cultural names (e.g., "Kiran" , "Suman" ). Gender is determined strictly if explicitly designated by a kinship title (e.g., "father" , "sister" , "wife" ) or coded gender operator.
Trap 4: Forgetting the Rule of 27 in Letter Coding: Manually counting letters backward from Z Z Z wastes critical seconds. Utilize Reverse Rank = 27 − Forward Rank \text{Reverse Rank} = 27 - \text{Forward Rank} Reverse Rank = 27 − Forward Rank for instant, error-free reverse mapping.