Puzzles

1. Foundation & Types of Puzzles

1.1 What is a Puzzle?

A logical puzzle is a narrative-based problem where participants are given a set of fragmented conditions about persons, objects, or events. The objective is to reconstruct the complete scenario—such as who likes what, who lives where, or the order of a sequence—using deductive reasoning.

Key Tools for Solving:

• Systematic Tables: For attribute matching (e.g., Person vs. Hobby).
• Linear Diagrams: For ranking, scheduling, or floor arrangements.
• Symbolic Logic: Using >, <, and = to chain comparative data.

1.2 Common Puzzle Categories

1. Order & Ranking: Arranging entities based on height, weight, or marks.
2. Scheduling/Day-Based: Mapping events to specific days, months, or time slots.
3. Floor Arrangement: Placing persons into a multi-story building template.
4. Tabular / Matrix: Interlinking multiple independent attributes (e.g., "A likes Apple and wears Blue").
5. Selection / Grouping: Forming teams based on conditional inclusions (e.g., "If A is chosen, B cannot be").

1.3 General Methodology

To solve any puzzle efficiently, follow the D.E.R. approach:

1. Deduce the Type: Is it a row? A table? A building? Choose your diagram.
2. Extract Clues: Convert sentences into shorthand (e.g., $A > B$, $C \neq \text{Monday}$).
3. Reconstruct:
   • Start with Definite Clues (e.g., "A lives on Floor 4").
   • Use Negative Clues to narrow down possibilities.
   • Link clues using common denominators.
   • Verify the final arrangement against all initial statements.

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2. Order & Ranking Puzzles

2.1 Core Concepts

Order & Ranking puzzles focus on linear comparisons. You must determine the relative or absolute position of an entity based on comparative descriptors like taller, heavier, faster, or more expensive.

Logic Essentials:

• Comparative Chains: Chaining A > B and B > C to conclude A > C.
• Branching: When two chains exist (e.g., A > B and C > B) but A and C are not compared.
• Relative Ranks: "A is 3rd from the top."
• Adjacency: "A is immediately above B."

2.2 The Fundamental Ranking Equations

When working with single rows or lists, use these proven formulas:

$$ \text{Total Persons} = (\text{Rank from Top/Left}) + (\text{Rank from Bottom/Right}) - 1 $$

$$ \text{Persons Between X and Y} = |\text{Position X} - \text{Position Y}| - 1 $$

2.3 Worked Examples: Order & Ranking

Example 1: Multiple Comparative Branches
Question: 6 persons (P-U) have different heights. U > P. P > Q. Q > R. S > T. U > S. R > S. Arrange them in descending order.

1. Step 1: Chain 1 (U to R): $U > P > Q > R$.
2. Step 2: Chain 2 (R to T): $R > S > T$.
3. Step 3: Combined Chain: Since $R > S$, we can merge the chains: U > P > Q > R > S > T.
4. Answer: U is the tallest.

Example 2: Rank Calculation
Question: In a class of 40, Rajan is 15th from the top. Pradhan is 10th from the bottom. How many students are between them?

1. Step 1: Convert Pradhan to "Rank from Top".
   $$ \text{Top Rank} = \text{Total} - \text{Bottom Rank} + 1 = 40 - 10 + 1 = 31\text{st} $$
2. Step 2: Calculate students between.
   $$ \text{Between} = |31 - 15| - 1 = 16 - 1 = 15 $$
3. Answer: 15 students are between Rajan and Pradhan.

2.4 Tabular Puzzles (Attribute Matching)

For puzzles with multiple variables (Person, Item, Feature), use a grid to mark positive ($\checkmark$) and negative ($\times$) relations.

2.5 Practice Set: Order & Ranking

1. Q: A is older than B. B is older than C. D is older than E. C is older than D. Who is the youngest?
   Solution: A > B > C > D > E. Youngest = E.
2. Q: In a row of 30, X is 12th from left. Y is 8th from right. How many between them?
   Solution: Y(Left) = 30-8+1 = 23. Between = (23-12)-1 = 10.
3. Q: 5 friends lift different weights. A < B. C < B. E < C. B < D. Who is the heaviest?
   Solution: D > B > C > E. Since B > A as well, D is greater than all. Answer: D.

Summary: Puzzle Foundations

• Tables are your best friend for complex matching.
• Ladders work perfectly for height/weight ranking.
• Formulas prevent "off-by-one" errors in ranking.
• Negative Clues are the key to breaking stalemates.

3: Scheduling / Day-Based Puzzles

3.1 Core Concepts

Scheduling puzzles involve assigning items (persons, events, tasks) to a fixed set of time slots (e.g., days of the week, dates of a month, or specific time periods). The slots are ordered, making the puzzle a linear arrangement in time.

Example of a linear weekly timeline mapping.

Key terminology:

• Direct assignment: "A is on Wednesday."
• Relative position: "B is two days after A."
• Gap constraints: "Exactly one day between G and H."
• Immediate neighbors: "E is immediately after F" (consecutive).

3.2 Methodology

1. Index the Slots: Represent days as numbers (Mon=1, Tue=2, etc.).
2. Symbolic translation:
   • "B is 2 days after A" $\rightarrow$ $B = A + 2$.
   • "E is before F" $\rightarrow$ $E < F$.
3. Place fixed clues: Fill in any "definitely on X day" clues first.
4. Case elimination: If a "gap" clue (e.g., $|G - H| = 2$) has multiple positions, list the cases and filter them using other constraints.

3.3 Worked Example: Weekday Scheduling

Problem: Six persons (A, B, C, D, E, F) have meetings from Monday to Saturday.

• A is on Wednesday.
• B is two days after C.
• D is on Friday.
• E is immediately before F.
• C is after A.

Step-by-Step Solution:

1. Fixed: A = 3 (Wed), D = 5 (Fri).
2. Relative: C > A $\rightarrow$ C > 3. Since D (5) is taken, C must be 4 or 6.
3. Relative: B = C + 2. If C=4, B=6. If C=6, B=8 (out of range).
4. Placement: C = 4 (Thu), B = 6 (Sat).
5. Remaining: E and F need days 1 and 2. Since E = F - 1, E = 1 (Mon) and F = 2 (Tue).
6. Final Schedule: Mon: E, Tue: F, Wed: A, Thu: C, Fri: D, Sat: B.

Answer: B's meeting is on Saturday.

3.4 Practice Set: Scheduling

1. Q: P, Q, R, S are on Mon-Thu. P is before R. Q is after S. S is on Tuesday. Who is on Monday?
   Solution: S=2. Since Q>S, Q is 3 or 4. P<R. Since S=2, P and R must be 1 and 3/4. Since P<R, P must be 1. Answer: P.
2. Q: 3 friends meet in Jan, Mar, May. A is before B. C is after B. Who is in May?
   Solution: A < B < C. Answer: C.

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4: Selection / Grouping Puzzles

4.1 Core Concepts

Selection puzzles involve choosing a subset of individuals from a pool subject to logical constraints (Inclusions, Exclusions, and Implications).

Visualizing the transformation from pool to committee via implications (A ➔ B).

Conditional Logic:

• Inclusion: "If A is selected, B must be selected" ($A \rightarrow B$).
• Exclusion: "A and B cannot be together" ($A \rightarrow \neg B$).
• Contrapositive: If $A \rightarrow B$ is true, then $\neg B \rightarrow \neg A$ (If B isn't picked, A cannot be picked).

4.2 Methodology

1. Track In/Out: Use a simple T-table or list.
2. Fixed Selections: Mark "definite" members first.
3. Chain Reactions: Use implication arrows ($P \rightarrow Q \rightarrow R$) to see who is pulled in or pushed out.
4. Total Count: Ensure the final group size matches the requirement (e.g., "Select exactly 3").

4.3 Worked Example: Committee Selection

Problem: Select a committee of 3 from P, Q, R, S, T.

• P must be selected.
• Q cannot be selected.
• If R is selected, then S is selected.
• T and S cannot be together.

Solution:

1. Fixed: P = IN, Q = OUT.
2. Remaining slots: Need 2 more from {R, S, T}.
3. Hypothesis 1: Pick T. Since T and S cannot be together, S = OUT. If S = OUT, then R must be OUT (because R $\rightarrow$ S). This leaves no one to fill the 3rd slot. (Invalid)
4. Hypothesis 2: Pick R. Since R $\rightarrow$ S, S = IN. This fills all 3 slots {P, R, S}. Check "T and S" rule: T is not in, so it passes.
5. Final Group: {P, R, S}.

4.4 Practice Set: Selection

1. Q: Choose 2 from A, B, C. If A is picked, B is picked. A and C cannot be together. B must be picked. Can A be picked?
   Solution: If A is in, B is in. That meets the "size 2" requirement. A and C rule is satisfied (C is out). Answer: Yes, {A, B} is valid.
2. Q: Divide A, B, C, D into 2 groups of 2. A and B must be together. Can C be with A?
   Solution: A and B are Group 1. C and D must be Group 2. Answer: No.

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5: Arrangement Puzzles (Tabular / Grid)

5.1 Core Concepts

Tabular or grid puzzles involve a set of entities (persons, items) each having several attributes (e.g., city, profession, hobby). You use a matrix to track relationships and systematically eliminate impossible matches.

Deduction rule: A single checkmark (✓) eliminates all other options in its row and column.

Key Rules:

• One Match per Category: Usually, each person has exactly one unique city, one unique profession, etc.
• Negative Clues: "Bob is not the Engineer" is just as valuable as a positive match.
• Propagated Logic: If the Doctor lives in Delhi, and A is the Doctor, then A lives in Delhi.

5.2 Methodology

1. Construct the Grid: List persons as rows and attributes as columns.
2. Mark Definite Clues: Use $\checkmark$ for "Is" and $\times$ for "Is Not".
3. Cross-Eliminate: Once you place a $\checkmark$, mark $\times$ in the rest of that row and column for that attribute set.
4. Deduce Hidden Links: If a row has only one empty cell left, it must be the $\checkmark$.

5.3 Worked Example: Three-Set Matching

Problem: A, B, C, D are from Mumbai, Delhi, Chennai, Kolkata and are Doctor, Engineer, Lawyer, Teacher.

• A is from Mumbai.
• The Engineer is from Delhi.
• C is a Lawyer.
• B is not a Doctor.
• D is from Kolkata.
• The Teacher is not from Chennai.

Solution:

1. Cities: A=Mumbai, D=Kolkata. Remaining Delhi/Chennai for B/C.
2. Linking: The Engineer is from Delhi. Since C is a Lawyer, C cannot be the Engineer.
3. Placement: Therefore, B must be from Delhi and is the Engineer.
4. Remaining City: C is from Chennai.
5. Professions: B=Engineer, C=Lawyer. Remaining Doctor/Teacher for A/D.
6. Refining: Since the Teacher is NOT from Chennai (C), this is consistent. If we add "A is not a Teacher", then A=Doctor and D=Teacher.

5.4 Practice Set: Grid Puzzles

1. Q: P, Q, R are logic, math, art. P is not math. Q is art. Who is math?
   Solution: Q=Art. P=Logic (not math). R=Math.
2. Q: 3 men in Red, Blue, Green shirts. P doesn't wear Red. The one in Blue is Q. What color is R?
   Solution: Q=Blue. P must be Green (not Red). R must be Red.

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6: Mixed / Interlinked Puzzles

6.1 Core Concepts

Mixed puzzles integrate multiple reasoning domains: Spatial (Arrangement), Relational (Family), and Logical (Attributes). Success depends on building a "Master Diagram".

The intersection of different reasoning types requires a multi-layered approach.

6.2 Methodology

1. Categorize Clues: Separate "Family" clues from "Seating" or "Ranking" clues.
2. Solve the Sub-Problem: Usually, solving the Family Tree first simplifies the names for the Seating arrangement.
3. Map Common Entities: Identify the person who links two sets (e.g., "The Father (Family) is at the Left End (Spatial)").
4. Check Consistency: Re-read all specific constraints once the map is complete.

6.3 Worked Example: Family + Seating

Problem: A, B, C sit in a row. A is the father of B. The father is at the left end. B is to the right of C.
Solution:

1. Family: A is the Father.
2. Spatial: Father (A) is at Pos 1.
3. Layout: Remaining Pos 2 and 3 for B and C. Since B is to the right of C, C is at 2 and B is at 3.
4. Order: A - C - B.

6.4 Practice Set: Mixed Puzzles

1. Q: A, B, C are 3 generations (Grandpa, Dad, Son). The oldest is at the back. The youngest is in front. B is the Dad. A is the Son. Who is in the middle?
   Solution: Grandpa > Dad (B) > Son (A). Back to Front: C - B - A. Middle = B.

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Summary: Mastering Puzzles

• Visualization is half the battle: Never solve "in your head".
• Eliminate until only truth remains: Logic is the art of excluding the impossible.
• Interlink your maps: A name on a family tree is the same name on a seating chart.

Summary: Complex Puzzles

• Logic first, diagram second: Map your conditions before sketching.
• Scheduling is just math on a timeline (Day + 2, Month - 1).
• Selection relies heavily on the Contrapositive (If B is out, A must be out).

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