Missing Numbers in Figures; Data Sufficiency; Input-Output; Decision Making

Missing Numbers in Figures

16.1 Matrix/Grid Problems

Numbers arranged in rows and columns with one missing.

Common Patterns:

• Row sum/product is constant
• Column sum/product is constant

Diagonal sum/product is constant

• Adjacent/opposite number relationship

Approach:

1. Check row sums, then column sums
2. Check diagonal relationships
3. Look for multiplication/division patterns
4. Check if each number is function of adjacent numbers

16.2 Triangle Figure Problems

Numbers at corners and center of a triangle.

Common Patterns:

• Sum of corner values = center value
• Product of corners / some operation = center
• Average of corners = center

16.3 Circle Problems

Numbers around a circle with center value.

Common Patterns:

• Center = Sum of opposite pairs
• Center = Average of all surrounded numbers
• Center = Difference of pairs

16.4 Counting Figures

Counting triangles, squares, rectangles in a given figure.

Method for counting triangles:

1. Count smallest triangles first
2. Then progressively larger combinations
3. Use formula for simple arrangements:
   • n rows of triangle → total triangles = n(n+2)(2n+1)/8

For squares in a grid:

• 1×1 grid has 1 square
• 2×2 grid has 1+4 = 5 squares (1 big + 4 small)
• n×n grid has Σk² for k=1 to n = n(n+1)(2n+1)/6

Data Sufficiency

17.1 Format

A question is given with two statements (I and II). Determine which statement(s) are sufficient to answer the question.

17.2 Answer Choices

17.3 Approach

1. Take Statement I alone. Is date enough to answer?
2. Take Statement II alone. Is it enough?
3. If neither alone → try combining both
4. If still not enough → answer is E

Remember: You don't need to actually solve — just check if ENOUGH information is given.

Input-Output

18.1 Concept

A word/number rearrangement machine. Given the input and a few steps, determine the final output or a specific step.

18.2 Common Arrangements

Ascending/Descending Sort

• Numbers sorted smallest to largest
• Words sorted in alphabetical order
• Alternating: largest number, smallest word, 2nd largest number, 2nd smallest word...

Word-Number Alternating

• Words move to one side, numbers to another
• Words in alphabetical order on left
• Numbers in ascending order on right

18.3 Steps to Solve

1. Look at each step of the example carefully
2. Identify what changes happen in each step
3. Apply the SAME rule to find the required step output

Decision Making

19.1 Types

• Selection criteria: Based on age, qualification, experience → Who qualifies?

Course of action: Given a problem → Which action should be taken?

19.2 Course of Action Rules

A course of action is valid if:

1. It directly addresses the problem
2. It is practically feasible
3. It follows from the problem logically
4. It doesn't introduce new unrelated problems