Counting Figures Reasoning – Master Reasoning for Competitive Exams
Boost your understanding of counting figures reasoning with proven strategies designed for competitive exams like SSC, UPSC, and Banking.
Counting Figures in Reasoning
Counting Figures is an essential topic in logical reasoning that tests your ability to identify and count specific geometric shapes within complex diagrams. It evaluates your visual perception, pattern recognition, and systematic counting skills - all crucial for solving complex problems in competitive exams.
This topic frequently appears in various Indian competitive examinations with varying difficulty levels. Mastering Counting Figures can give you a significant edge as these questions are typically quick to solve once you understand the patterns and develop systematic counting approaches.
Key exams where Counting Figures is important:
- SSC CGL, CHSL, CPO, MTS, Steno
- UPSC CSAT (Prelims)
- Banking Exams: IBPS PO/Clerk, SBI PO/Clerk, RBI Grade B
- Railway Exams: RRB NTPC, Group D, JE
- State PSCs (UPPSC, MPPSC, BPSC, etc.)
- CAT and other MBA entrance exams
- Defense Exams: CDS, AFCAT, CAPF
Types of Counting Figures Problems
This type involves counting the number of straight lines in a given figure, including both horizontal and vertical lines, as well as diagonal lines in some cases.
Solved Example 1:
How many straight lines are there in the following figure?
——
| |
——
| |
——
Solution:
- 1. Identify all horizontal lines: There are 3 horizontal lines (top, middle, bottom)
- 2. Identify all vertical lines: There are 2 vertical lines on each side (total 4)
- 3. Check for diagonal lines: None in this figure
- 4. Total straight lines = Horizontal + Vertical = 3 + 4 = 7 lines
Solved Example 2:
Count the number of straight lines in this figure from a Delhi Metro map:
/\
/ \
/____\
/ \
/ \
Solution:
- 1. Left diagonal line: 1
- 2. Right diagonal line: 1
- 3. Bottom horizontal line: 1
- 4. Total straight lines = 3
/\
/ \
/____\
| |
| |
\____/
Solution:
- Top left diagonal: 1
- Top right diagonal: 1
- Top horizontal: 1
- Left vertical: 1
- Right vertical: 1
- Bottom left diagonal: 1
- Bottom right diagonal: 1
- Bottom horizontal: 1
- Total straight lines = 8
This type requires counting all possible triangles in a given figure, including smaller triangles that form part of larger ones.
Solved Example 1:
How many triangles are there in the following figure?
/\
/__\
/ \
/______\
Solution:
- 1. Smallest triangles: The upper part has 1 small triangle (top)
- 2. Medium triangle: The whole figure is 1 big triangle
- 3. Combined triangles: The upper triangle with either left or right side forms additional triangles
- 4. Total triangles = 4 (1 small, 2 medium, 1 large)
/\
/__\
/\ /\
/__\/__\
Solution:
- Smallest triangles: 4 (top and bottom pairs)
- Medium triangles: 3 (two at bottom, one at top)
- Large triangle: 1 (the whole figure)
- Total triangles = 8
This involves counting all possible squares in a given figure, including smaller squares within larger ones.
Solved Example 1:
How many squares are there in a standard 2x2 chessboard (like one in Mumbai's local chess club)?
┌───┬───┐
│ │ │
├───┼───┤
│ │ │
└───┴───┘
Solution:
- 1. 1x1 squares: There are 4 small squares
- 2. 2x2 squares: There is 1 large square
- 3. Total squares = 4 (small) + 1 (large) = 5 squares
┌───┬───┬───┐
│ │ │ │
├───┼───┼───┤
│ │ │ │
├───┼───┼───┤
│ │ │ │
└───┴───┴───┘
Solution:
- 1x1 squares: 9
- 2x2 squares: 4
- 3x3 squares: 1
- Total squares = 9 + 4 + 1 = 14 squares
This type involves counting all possible rectangles in a given figure, including squares (which are special rectangles).
Solved Example 1:
How many rectangles are there in this figure from a Delhi school's window design?
┌───────┐
│ ┌───┐ │
│ │ │ │
│ └───┘ │
└───────┘
Solution:
- 1. Inner rectangle: 1
- 2. Outer rectangle: 1
- 3. Combined rectangles (inner + outer boundaries): 2
- 4. Total rectangles = 4
This type requires counting all possible circles in a given figure, including those formed by intersections.
Solved Example 1:
How many circles are there in this Olympic rings-inspired figure used in a Bangalore sports event?
╭───╮ ╭───╮
╭╯ ╰─╮ ╭╯ ╰─╮
╭╯ ╰─╯ ╰╮
╰╮ ╭─╮ ╭╯
╰╮ ╭─╯ ╰╮ ╭─╯
╰───╯ ╰───╯
Solution:
- 1. Complete circles: There are 2 complete circles
- 2. Partial circles: The intersections create additional circular segments
- 3. Total distinct circles = 2 (only complete circles are counted in basic problems)
Step-by-Step Solving Techniques
Systematic Counting Approach
Develop a consistent method to count figures without missing any or double-counting.
- Start from one corner and move systematically
- Count smallest figures first
- Then count combinations of small figures
- Finally count the entire figure
Formula Application
Learn standard formulas for common figures to save time.
- Squares in n×n grid: n(n+1)(2n+1)/6
- Rectangles in m×n grid: m(m+1)n(n+1)/4
- Straight lines formed by n points: n(n-1)/2
Visual Segmentation
Break complex figures into simpler components for easier counting.
- Identify symmetrical parts
- Count identical segments separately
- Multiply by number of identical segments
- Add unique figures
Elimination Method
Eliminate obviously wrong options quickly to save time.
- Estimate reasonable range
- Eliminate too high/low options
- Verify remaining options
Pattern Recognition
Identify repeating patterns in complex figures to simplify counting.
- Look for symmetrical patterns
- Identify smallest repeating unit
- Count figures in one unit
- Multiply by number of units
Progressive Complexity
Start with simpler versions of the figure and build complexity.
- Solve basic version first
- Add one layer of complexity
- Note how count changes
- Extrapolate to given figure
📚 Topic-Wise Practice Worksheets
Master Counting Figures with our structured practice materials
Each worksheet includes detailed solutions and explanations
Triangles Simple Free
10 worksheets available
Simple Triangles Counting problems involve counting triangles in basic triangular figures such as a single triangle, a triangle with one median, a triangle with two medians, or a triangle divided into smaller congruent triangles. These foundational problems build the skills needed for more complex figure counting.
Triangles Grid Free
10 worksheets available
Triangular Grid Counting problems involve counting triangles in a grid of small equilateral triangles arranged in rows. These problems require counting both upward-pointing triangles and downward-pointing triangles of various sizes using combinatorial formulas.
Triangles Star Free
10 worksheets available
Star Figure Triangle Counting problems involve counting triangles in 5-pointed stars (pentagrams) and 6-pointed stars (hexagrams). These figures contain many intersecting lines that create numerous triangles of various sizes and orientations.
Triangles Complex Free
10 worksheets available
Complex Triangle Counting problems involve figures with multiple intersecting lines, cevians, and composite shapes. These problems require systematic counting methods, labeling, and often combinatorial reasoning to count all triangles correctly.
Squares Grid Free
10 worksheets available
Square Grid Counting problems involve counting the total number of squares (of all sizes) in an n×n grid of small squares. The total includes 1×1 squares, 2×2 squares, up to n×n squares. These problems use the formula for sum of squares.
Squares Rectangular Free
10 worksheets available
Rectangular Grid Square Counting problems involve counting squares in an m×n grid (where m and n may be different). The number of squares is limited by the smaller dimension, and the formula sums over square sizes from 1 to min(m,n).
Squares With Diagonals Free
10 worksheets available
Squares with Diagonals problems involve counting squares in figures where one or more squares have diagonals drawn. The diagonals create triangles but do not create new squares. These problems test understanding that diagonals only create triangles, not additional squares.
Rectangles Grid Free
10 worksheets available
Rectangle Grid Counting problems involve counting the total number of rectangles (including squares) in an m×n grid of cells. The formula uses combinations: choose 2 horizontal lines from (m+1) lines and 2 vertical lines from (n+1) lines.
Circles Overlapping Free
10 worksheets available
Overlapping Circles problems involve counting circles or circular regions in figures where circles overlap. These problems often use Venn diagram-style configurations (two or three overlapping circles).
Line Segments Free
10 worksheets available
Line Segment Counting problems involve counting the number of line segments formed by points on a line or within geometric figures. When n points are on a line, the number of segments = C(n,2) = n(n-1)/2.
Parallelograms Free
10 worksheets available
Parallelogram Counting problems involve counting the number of parallelograms formed by a set of intersecting lines (usually horizontal and vertical lines). The number of parallelograms = C(h,2) × C(v,2), where h is the number of horizontal lines and v is the number of vertical lines.
📖 Mixed Practice Worksheets
Comprehensive worksheets combining all problem types for Counting Figures
Perfect for exam simulation and revision
Each worksheet contains 20 mixed questions covering all problem types of Counting Figures, with detailed solutions and answer keys.
Expert Tips & Tricks
💡 Speed & Time Management Hacks:
- Practice with a timer to improve speed - aim for 30-45 seconds per question.
- Learn to quickly identify figure types to apply the right counting method.
- For exams with negative marking, attempt only those questions you're confident about.
- Mark difficult questions and return to them if time permits.
- Develop your own shorthand notations for quick counting.
⚠️ Avoid These Common Traps:
- Double-counting figures - always mark counted figures mentally or with a pencil.
- Missing hidden figures - check behind overlapping lines and in intersections.
- Ignoring the entire figure as one big shape - count the whole figure too.
- Assuming symmetry means equal count - verify each symmetrical part.
- Rushing through easy-looking questions - they might have traps.
✅ Strategies for Success:
- Start with easier counting problems to build confidence.
- Create a mental checklist of things to count for each figure type.
- Practice with real exam papers to understand question patterns.
- Join study groups to learn different counting approaches.
- Review mistakes thoroughly to identify weak areas.
🛑 Crucial Reminders:
- Squares are rectangles, but not all rectangles are squares.
- Intersection points can create additional figures to count.
- Complex figures often contain simple figures within them.
- Counting systematically prevents omission and duplication.
- Regular practice is the only way to develop speed and accuracy.
📚 Frequently Asked Questions About Counting Figures
Counting Figures is a logical reasoning topic that tests your ability to identify and count specific geometric shapes within complex diagrams. It evaluates visual perception, pattern recognition, and systematic counting skills - all crucial for solving complex problems quickly.
In competitive exams, it's important because:
- Tests observation skills under time pressure
- Evaluates systematic thinking ability
- Helps assess spatial intelligence
- Often carries 1-2 marks per question
- Can be solved quickly with practice
Effective preparation strategies include:
- Master the basics: Start with simple figures and gradually increase complexity
- Learn formulas: Memorize counting formulas for common figures
- Develop systematic approaches: Create step-by-step methods for each figure type
- Practice daily: Solve at least 10 different counting problems daily
- Analyze mistakes: Understand why errors occur and how to prevent them
- Time yourself: Gradually reduce time per question to build speed
- Use visual aids: Draw figures to understand patterns better
Counting Figures questions regularly appear in:
- SSC exams: CGL, CHSL, CPO, Steno, MTS
- Banking exams: IBPS PO/Clerk, SBI PO/Clerk, RBI Grade B
- UPSC: CSAT (Prelims)
- Railway exams: RRB NTPC, Group D, JE
- State PSCs: UPPSC, MPPSC, BPSC, WBPSC, etc.
- Defense exams: CDS, AFCAT, CAPF
- CAT and other MBA entrances: Usually in logical reasoning sections
- Teaching exams: CTET, UGC NET
Counting Figures is typically considered:
- Moderate difficulty for most students
- Can become challenging with complex figures
- Considered easier than verbal reasoning by visual learners
- Gets easier with practice as patterns become recognizable
Common pitfalls include:
- Double-counting figures
- Missing hidden or overlapping figures
- Not counting the entire figure as one unit
- Assuming symmetry means equal count without verification
- Rushing through apparently simple questions
The most effective mastery approach combines:
- Conceptual clarity: Understand fundamental principles thoroughly
- Diverse practice: Solve questions of all difficulty levels
- Shortcut methods: Learn and apply time-saving techniques
- Timed tests: Regularly practice under exam conditions
- Mistake analysis: Review errors to identify weak areas
- Pattern recognition: Develop ability to spot common configurations
- Systematic approaches: Create step-by-step methods for each figure type
- Confidence building: Start with easier problems and gradually increase difficulty
Additional tips:
- Practice with previous years' exam papers
- Join online forums to discuss challenging problems
- Teach others to reinforce your own understanding
- Maintain an error log to track progress
Sandeep Nehra
B.Tech (Mech) | MBA (HRM & IB) | Lead Developer & Reasoning Expert (16+ Yrs)
Sandeep is a Mechanical Engineer and dual MBA (HR & International Business) with over 16 years of experience as a Senior Web Architect and Tech Lead. Combining his engineering precision with deep behavioral insights, he founded ReasoningAbility.com to revolutionize competitive exam preparation. His unique methodology — blending logical structuring from engineering with psychological clarity from HRM — helps aspirants crack BITSAT, SSC, and Banking exams faster. His mission remains simple: provide high-quality, free practice resources that turn complex logic into accessible, high-speed solving techniques for students worldwide.