Logical Reasoning and Seating Arrangements
Logical reasoning tests your ability to decode patterns, decode relationships, and organize information systematically. Seating arrangement problems are particularly time-intensive — the key is to draw a clear diagram from the start.
🟢 Lite — Quick Review (1h–1d)
Core techniques:
- Blood relations: Use family tree diagrams. Identify genders and generations clearly.
- Directions: North-South-East-West. Right/Left turns change orientation.
- Coding-decoding: Look for letter/shift patterns, positional values, or symbol substitution.
- Syllogisms: Check if the conclusion necessarily follows from the premises.
- Seating arrangements: Draw circular/linear diagrams with given clues, place fixed positions first.
⚡ GATE exam tip: In blood relation problems, if the question says “A is B’s sister” — you know A is female. But “A is B’s sibling” — gender unknown. Watch these qualifiers.
⚡ Quick trick — directions: After facing North and turning right, you face East. After turning left, you face West. A 180° turn reverses direction.
⚡ Common trap — syllogisms: “All A are B” + “All B are C” → “All A are C” (valid). But “Some A are B” + “Some B are C” → “Some A are C” (INVALID — doesn’t follow).
🟡 Standard — Regular Study (2d–2mo)
Blood Relations
Family tree convention: Draw generations vertically (ancestors upward, descendants downward). Use standard symbols: □ = male, ○ = female.
Key relations:
- Sibling: Brother or sister
- Cousin: Child of your uncle/aunt
- Nephew/Niece: Child of sibling
- Grandparent/Grandchild: Two generations up/down
- Maternal/Paternal: Mother side / Father side
GATE Example: Pointing to a photograph, a man says, “The sister of this man’s mother is my aunt.” How is the person in the photograph related to this man?
Solution: Man’s mother has a sister (aunt). That aunt is the grandmother of the person in the photo? Wait: “The sister of this man’s mother is my aunt” → The sister is his maternal aunt → She has a child (the person in the photo) who is the man’s cousin. Answer: Cousin.
Coding relationships: “A + B” means A is B’s brother. “A − B” means A is B’s sister. Combine using the family tree.
Direction and Distance
Cardinal directions: North (N), South (S), East (E), West (W).
Right turn from facing North → East Left turn from facing North → West
Diagonal directions: NE = North-East, SE = South-East, etc.
GATE Example (2020, 1 mark): Ram starts at his house and walks 5 km South, turns right and walks 3 km East, turns right and walks 2 km South. How far is he from his house?
Draw: South 5 → East 3 → South 2. Total South = 7 km, East = 3 km. Distance = √(7² + 3²) = √(49+9) = √58 ≈ 7.62 km.
Clock directions: At 12 o’clock position = directly in front. Clockwise = right turn direction.
Coding-Decoding
Types of coding:
- Letter shift (Caesar cipher): Each letter shifted by a fixed number. A→D means shift +3.
- Positional coding: A=1, B=2… Z=26. Word code is the positional sum/product.
- Mirror coding: Reverse the alphabet (A↔Z, B↔Y).
- Symbol substitution: Each letter replaced by a symbol/number from a given key.
GATE Example: In a certain code, PRODUCT is coded as 16-18-15-3-21-19-20. How is QUANTUM coded?
P=16, R=18, O=15, D=4, U=21, C=3, T=20 (using A=1, Z=26). Wait, 3 is C but D=4 in standard. Let me recheck: P=16, R=18, O=15, U=21, A=1, N=14, T=20, M=13. Pattern: seems like consecutive letters starting from P: P=16, R=18 (skip Q=17), O=15… not linear. Actually: P(16), R(18), O(15), D(4), U(21), C(3), T(20) — these are not simple. Since it’s the first occurrence: In PRODUCT, P=16, R=18, O=15, D=4, U=21, C=3, T=20. These look like 26-n for reverse alphabetical: Z=1, so P(16)=11? No. Let’s use direct: Q=17, U=21, A=1, N=14, T=20, U=21, M=13 → 17-21-1-14-20-21-13.
Syllogisms
Standard premises and conclusions:
| Premise Type | Conclusion Valid? |
|---|---|
| All A are B + All B are C | All A are C ✓ |
| All A are B + Some A are B | Some A are B ✓ (but already known) |
| Some A are B + All B are C | Some A are C ✓ |
| No A is B + All B are C | No A is C ✓ |
| Some A are B + Some B are C | Some A are C ✗ |
GATE Example: Statements: (1) All teachers are educated. (2) Some educated are women. Conclusions: (I) Some teachers are women. (II) Some women are teachers.
Does (1)+(2) support (I)? Not necessarily — teachers and women could be disjoint within the educated group. INVALID. Does (2) say “Some women are teachers”? This is the converse of (2). “Some educated are women” → by conversion: “Some women are educated” — still no info about teachers. INVALID.
Only conclusion that necessarily follows goes in the answer.
Seating Arrangements — Circular
Convention: In circular seating (facing center):
- Left of A = clockwise neighbor
- Right of A = anticlockwise neighbor
Key steps:
- Draw a circle
- Place someone with a fixed position (or start from a reference point)
- Use “to the left of” = clockwise in circular facing center
- Place remaining people
- Answer the questions
GATE Example Pattern: 6 people A, B, C, D, E, F sit around a circular table. A sits between D and E. B is second to the right of E. C and F are not adjacent.
Solution approach: Place A. D and E are neighbors of A. Since B is second to right of E, count 2 positions clockwise from E. Continue placing C, F, and remaining.
Linear Seating Arrangements
Linear arrangements are simpler — just a row:
- Left = position increasing (or decreasing — be consistent)
- Establish leftmost/rightmost first
- Work inward
GATE Example Pattern: 5 people P, Q, R, S, T sit in a row. P sits at one end. Q is next to P. R is in the middle. S is not at any end. Find positions.
🔴 Extended — Deep Study (3mo+)
Complex Blood Relations with Generations
GATE Advanced Example: In a family, A is B’s mother. C is B’s daughter. D is C’s sister. E is D’s grandmother. Who is E’s daughter?
A is B’s mother → B is A’s child. C is B’s daughter → C is A’s granddaughter. D is C’s sister → D is also A’s granddaughter. E is D’s grandmother → E is A’s mother. E’s daughter = A. Answer: A.
Coding-Decoding with Conditional Rules
Some GATE problems give conditional code words: “If in a code, ‘PLAY’ is written as ‘1234’ and ‘WORK’ is written as ‘5678’, how is ‘POOL’ written?” You need to extract letter-to-digit mapping from the two given words.
Logical Sequencing
Arrange events in chronological or logical order based on given clues. Look for “before,” “after,” “immediately preceding,” and “immediately following.”
Key words:
- “Immediately before” = the person/event right before with nothing in between
- “Somewhere between” = there are other people/events between them
Matrix/Puzzle Grid Reasoning
GATE sometimes gives a grid with some cells filled and asks you to deduce the pattern and fill remaining cells. Look for:
- Row patterns (each row has one of each of n symbols)
- Column patterns
- Diagonal patterns
- Symmetry (rotational or mirror)
Statement and Conclusions (More Complex Syllogisms)
Additional valid/invalid patterns:
| Premise | Conclusion | Valid? |
|---|---|---|
| All A are B + Some A are not B | Contradiction (Some A are not B is already known if “All A are B” doesn’t hold) | INVALID unless specific |
| No A is B + Some A are C | Some C are not B | VALID |
| Some A are B + No B is C | Some A are not C | VALID |
⚡ GATE trick: When multiple conclusions are possible but only one necessarily follows, pick the one that is DEFINITELY true, not just possibly true.
Venn Diagram Reasoning
For three-set Venn diagrams (A, B, C), the 7 regions represent:
- Only A, only B, only C
- A∩B only, B∩C only, C∩A only
- A∩B∩C
GATE Example: In a survey, 60 people use product A, 50 use product B, 40 use product C, 20 use A and B, 15 use B and C, 10 use A and C, and 5 use all three. How many use at least one product?
Use inclusion-exclusion: |A∪B∪C| = |A|+|B|+|C| − |A∩B| − |B∩C| − |C∩A| + |A∩B∩C| = 60+50+40 − 20−15−10 + 5 = 110.
Number/Letter Series Completion
GATE DI/logic section sometimes includes series problems:
- Find the next number/letter in the pattern
- Common patterns: arithmetic, geometric, Fibonacci-like, alternating, squared/cubed
- Also: 2, 3, 4-digit series where you need to find the pattern
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