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SSC CGL Counting Figures

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This page covers SSC CGL Counting Figures with complete concept notes, 13 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Counting Figures— Rules & Concept

Core ConceptRead this first — the foundation of the topic
CORE CONCEPT

You are given a complex figure made of triangles, squares, rectangles, lines, or other shapes. Your job is to count ALL shapes of a specific type (or all shapes total) without missing any or double-counting

KEY RULES

Count systematically — start from one corner and move in order (left to right, top to bottom). 2. Count individual shapes first, then combinations of 2, 3, 4 shapes, etc. 3. Overlapping shapes count as separate shapes. 4. Don't recount the same shape twice when moving to combinations. 5.

Use different colors or marks mentally to track what you've counted.

Exam PatternsWhat examiners ask — read before attempting PYQs

— Count total triangles in a figure — Count total squares/rectangles — Count total line segments — Count figures formed by combining smaller shapes — Questions ask: "How many triangles are there?" or "Find the total number of squares." SHORTCUT/TRICK: Use the "Layer Method" — Count shapes by size. First count the smallest individual shapes. Then count shapes made of 2 units, then 3 units, and so on. This prevents overlap and ensures nothing is missed.

Worked ExampleSolve this step-by-step before moving on
1
Step 1

Count 1×1 squares = 4 (the four small squares)

2
Step 2

Count 2×2 squares = 1 (the entire large square)

3
Step 3

Add them up = 4 + 1 = 5 total squares Answer: 5 squares

Exam TrapsCommon mistakes students make — avoid these

Students often count only the smallest individual shapes and forget to count larger shapes formed by combining smaller ones. For example, in a 2×2 grid, many count only the 4 small squares and give 4 as the answer, missing the 1 large square. Always count combinations systematically. FORMULA (For grid-based counting): For an n×n grid of squares: Total = 1² + 2² + 3² + ... + n² Example: For a 3×3 grid = 1 + 4 + 9 = 14 squares

Key Points to Remember

  • Count shapes systematically from one direction to avoid missing or double-counting.
  • Always count individual shapes first, then combinations of 2, 3, 4 units.
  • For an n×n grid of squares, use formula: Total = 1² + 2² + 3² + ... + n²
  • Overlapping shapes and shapes formed by combinations are counted separately.
  • Common error: forgetting to count larger shapes made from combining smaller ones.
  • Mark mentally or use colors to track which shapes you've already counted.

Exam-Specific Tips

  • In a 2×2 square grid, total number of squares = 5 (four 1×1 squares + one 2×2 square).
  • In a 3×3 square grid, total number of squares = 14 (calculated as 1² + 2² + 3² = 1 + 4 + 9).
  • For a figure with straight lines, each intersection point can create multiple segments to count.
  • In triangle counting problems, overlapping triangles of different sizes must be counted as separate entities.
  • The sum formula for square counting in n×n grids is: n(n+1)(2n+1)/6 for advanced calculations.
  • SSC CGL typically presents 2-3 figure counting questions in the Non-Verbal Reasoning section.
  • Time limit per counting figure question is approximately 1-1.5 minutes in actual exam.
  • Grid-based counting (squares and rectangles) appears more frequently than line segment counting.
Practice MCQs

Counting Figures — Practice Questions

13graded MCQs · easy to hard · full solution & trap analysis

All MCQs →
Practice 1medium

A rectangle is divided into 12 equal smaller rectangles arranged in 3 rows and 4 columns. Count the total number of rectangles of all possible sizes that can be formed by combining these smaller rectangles. [A 3×4 grid of rectangles is shown.]

Practice 2medium

In the given figure, a rectangle is divided into smaller sections by horizontal and vertical lines. There are 3 horizontal lines and 4 vertical lines drawn inside the rectangle (including the borders). How many rectangles of all sizes can be formed in this grid?

Practice 3hard

A square grid of 5×5 contains some cells shaded black and others white. A 'region' is defined as a connected group of cells of the same colour (horizontally or vertically adjacent, not diagonally). If the grid has exactly 8 black regions and 6 white regions, what is the minimum number of black cells required?

Practice 4hard

A figure shows a hexagon divided by internal lines into smaller polygons. The hexagon has 2 diagonals drawn from one vertex, creating internal divisions. Additionally, 3 horizontal lines cross the entire hexagon. How many distinct polygons (triangles, quadrilaterals, pentagons, or hexagons) are formed in total?

Practice 5hard

In a complex figure, there is a large triangle subdivided by internal lines into smaller triangles. The internal lines create 3 horizontal divisions and 2 vertical divisions within the triangle. Count the total number of triangles of all sizes visible in the figure.

Practice 6hard

A rectangular grid contains horizontal and vertical lines. There are 5 horizontal lines and 6 vertical lines. How many rectangles of all possible sizes can be formed by these lines?

Practice 7hard

A figure consists of a square subdivided by 2 horizontal lines and 2 vertical lines, creating a 3×3 grid of 9 cells. Additionally, both diagonals of the main square are drawn. How many quadrilaterals (including squares and rectangles) of all sizes are present in the figure?

Practice 8hard

A square is divided into 16 equal smaller squares (4×4 grid). Additionally, there are diagonal lines drawn from two opposite corners of the large square, creating further subdivisions. Count the total number of triangles of all sizes that can be identified in this figure.

Practice 9hard

A hexagon is divided into 6 equilateral triangles by drawing lines from the center to each vertex. Additionally, lines are drawn connecting alternate vertices, creating a smaller hexagon in the center. Count the total number of triangles of all sizes visible in the final figure.

Practice 10hard

A rectangle is divided into a 3×4 grid of smaller rectangles. Additionally, one diagonal is drawn from the top-left corner to the bottom-right corner of the entire large rectangle. Count the total number of triangles formed by this diagonal and the grid lines.

Practice 11hard

A figure shows a cube with all six faces visible. Some faces contain patterns. If the cube is unfolded into a net, and we know that opposite faces never touch in the net, how many distinct nets can be formed from a standard cube?

Practice 12hard

A figure shows a large triangle subdivided by internal lines into smaller triangles. The large triangle has 2 internal parallel lines to its base, creating 3 rows of smaller triangles. The top row has 1 triangle, the middle row has 3 triangles, and the bottom row has 5 triangles. Count the total number of triangles of all sizes (including those formed by combining smaller triangles).

Practice 13hard

A hexagon is divided into smaller regions by drawing 3 lines from the center to alternate vertices, and 2 additional lines parallel to two of its sides. How many triangles of all sizes are formed? (Assume the hexagon is regular and all internal divisions are clearly defined.)

60-Second Revision — Counting Figures

  • Remember: Count systematically by size — small shapes first, then combinations. Never skip larger combined shapes.
  • Formula for n×n square grid: Total = 1² + 2² + 3² + ... + n². For 3×3 grid = 14 squares.
  • Trap: Students miss composite shapes. In a divided square, the outer boundary also counts as one shape.
  • Layer method works best: identify all 1-unit shapes, then 2-unit shapes, then larger combinations.
  • Verify your count: Use a different mental path to recount. If answers match, you're likely correct.
  • Mark shapes mentally as you count to avoid recounting or skipping in complex figures.
  • Practice on grid patterns (2×2, 3×3, 3×4) — these dominate SSC CGL counting questions.
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