6.3 Matrix & Category Puzzles (Department/Profession Allocation & Constraint Elimination)

6.3 Matrix & Category Puzzles (Department/Profession Allocation & Constraint Elimination)

Matrix and category allocation puzzles represent a cornerstone of the Reasoning Ability section in SBI PO examinations. These puzzles require mapping a set of 7 to 9 individuals into a smaller number of predefined categories (such as departments, cities, banks, or streams) alongside multi-variable attributes like laptop brands, vehicle models, or specializations.

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1. Group Capacity Bounds & Mathematical Constraints

Unlike floor or linear puzzles where each position has a unique capacity of one person, category puzzles distribute multiple individuals into shared groups subject to capacity constraints.

Standard SBI PO Capacity Rules

• Bounded Group Size: Prompts typically state: "Not less than two and not more than four persons work in any department."
• Sum of Group Sizes: If $N$ individuals are divided into $k$ departments ($D_1, D_2, \dots, D_k$), then: $$\sum_{i=1}^{k} |D_i| = N$$ Example: Distributing 7 persons across 3 departments with a minimum of 2 per department forces the distribution of group sizes to be either (3, 2, 2) or (2, 3, 2) or (2, 2, 3).

Key Phrase Deductions

• "Person $A$ works with only Person $B$": This statement contains two critical pieces of information:
  1. $A$ and $B$ are in the same department.
  2. The capacity of that department is exactly 2 persons ($A$ and $B$). No other person can be assigned to that department.
• "Person $X$ works in the same department as the one who likes Red": $X$ does not like Red, but shares a department with the Red enthusiast.

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2. Grid Setup & Constraint Elimination Methodology

Constructing the Master Table

When solving category puzzles, fix the Categories (Departments/Cities) as the primary table rows, or list the Persons as fixed rows and create columns for Category and Secondary Attributes.

+---------+------------+---------------+--------------------+
| Person  | Department | Laptop Brand  | Negative Exclusions|
+---------+------------+---------------+--------------------+
| A       | Credit     | Dell          | Not HR, Not Apple  |
| B       | Risk       | HP            | Not Wealth         |
+---------+------------+---------------+--------------------+

The Elimination Matrix Workflow

1. Fixed Group Anchor Assignment: Fill in explicit assignments (e.g., "$A$ works in Credit and uses Dell").
2. Exclusive Group Closure: Lock down closed departments immediately when their capacity limit is reached.
3. Negative Constraint Purge: Cross out disallowed combinations in the candidate options grid.
4. Dual Option Pairing: When an attribute has only two candidate slots left, branch into micro-cases.

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3. Fully Worked Step-by-Step Example: SBI PO Category Allocation

Problem Statement

Seven officers—A, B, C, D, E, F, and G—work in a bank across three departments: Credit, Risk, and Wealth Management. At least two officers work in each department. Each officer uses a different brand of laptop: Dell, HP, Lenovo, Apple, Asus, Acer, and MSI.

Clues:

1. $A$ works in Credit with the officer who uses Dell.
2. The officer who uses Apple works in Wealth Management.
3. $E$ works with only $F$. $E$ does not use Apple.
4. $B$ uses HP and does not work in Credit.
5. $C$ uses Lenovo and works in the same department as $G$.
6. $D$ does not work in Wealth Management.
7. The officer who uses Asus works in Risk.
8. $G$ does not use MSI.

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Step-by-Step Deduction & Case Resolution

Step 1: Department Capacity Analysis

• Total officers = 7. Departments = 3. Minimum per department = 2.
• Clue 3 states: "$E$ works with only $F$." This means $E$ and $F$ share a department whose total capacity is exactly 2 persons.
• Therefore, the 7 officers must be distributed as (3, 2, 2) across the three departments.

Step 2: Placing E and F

• $E$ and $F$ work together in a department of capacity 2.
• $E$ does not use Apple. Apple is in Wealth Management $\implies E$ and $F$ do not work in Wealth Management.
• Can $E$ and $F$ be in Credit? Clue 1 says $A$ works in Credit with the officer who uses Dell. If $E$ and $F$ are in Credit, Credit has size 2 ($E$ and $F$), leaving no room for $A$. So $E$ and $F$ cannot be in Credit.
• Thus, $E$ and $F$ work in Risk Management (Size of Risk = 2, completely full!).

Step 3: Determining Department Capacities & Member Allocations

• Since Risk has size 2 ($E, F$), the remaining sizes for Credit and Wealth Management must be 3 and 2 (or 2 and 3).
• Clue 5: $C$ (Lenovo) works in the same department as $G$.
• Clue 6: $D$ does not work in Wealth Management. Risk is full ($E, F$). So $D$ must work in Credit.
• Clue 1: $A$ works in Credit. So Credit already has $A$ and $D$.
• If $C$ and $G$ worked in Credit, Credit would have $A, D, C, G$ (4 persons), exceeding the max department size. Thus, $C$ and $G$ must work in Wealth Management.
• Wealth Management has $C$ and $G$. But Clue 2 says Apple is in Wealth Management. Clue 4 says $B$ does not work in Credit. Risk is full, so $B$ (HP) must work in Wealth Management.
• Thus, Wealth Management has $B, C, G$ (Size 3).
• Credit has $A, D,$ and one more person $\implies$ Wait, officers assigned so far: Risk ($E, F$), Wealth ($B, C, G$), Credit ($A, D$). Total = $2 + 3 + 2 = 7$ officers!

Department Membership Summary

• Credit (Size 2): $A, D$
• Risk (Size 2): $E, F$
• Wealth Management (Size 3): $B, C, G$

Step 4: Laptop Brand Matching

• Wealth Management ($B, C, G$):
  • $B$ uses HP (Clue 4).
  • $C$ uses Lenovo (Clue 5).
  • Clue 2: Apple is in Wealth Management $\implies$ $G$ uses Apple (since $B$=HP, $C$=Lenovo).
• Risk ($E, F$):
  • Clue 7: Asus is in Risk. Clue 3: $E$ does not use Apple. $E$ vs $F$ for Asus:
  • If $F$ uses Asus, $E$ uses MSI/Acer.
• Credit ($A, D$):
  • Clue 1: $A$ works in Credit with the officer who uses Dell $\implies$ $D$ uses Dell (since $A$ works with the Dell user).
  • $G$ does not use MSI (already Apple).
  • Remaining brands: Asus, Acer, MSI.
  • $E$ uses Asus, $F$ uses MSI, $A$ uses Acer.

Final Allocation Table

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4. Exam Strategies & High-Yield Avoidance Traps

1. The 'Works With Only' Trap: Always instantly lock a department's capacity to 2 when you see "$X$ works with only $Y$". No further officers can enter that department.
2. Confusing 'Shares Department' with 'Identical Attribute': "$A$ works with the one who uses Dell" means $A \neq \text{Dell}$. $A$ and Dell are two distinct people in the same department.
3. Neglecting Department Min/Max Boundaries: Frequently track the running count of assigned members per department to spot forced allocations early.