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SSC MTS Counting Figures

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This page covers SSC MTS Counting Figures with complete concept notes, 7 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

7graded MCQs · easy to hard · full solution & trap analysis

All MCQs →
Practice 1medium

How many distinct paths of length 3 (moving only right or down) exist from the top-left corner to the bottom-right corner of a 3×3 grid? A) 6 B) 10 C) 15 D) 20

Practice 2hard

A figure consists of a triangle with all three sides equal (equilateral). From each vertex, a line is drawn to the midpoint of the opposite side. These three lines (medians) intersect at a single point (centroid), dividing the triangle into 6 smaller triangles. Additionally, 2 of these 6 smaller triangles are shaded. How many total triangles of all sizes can be counted in this figure, including both shaded and unshaded triangles?

Practice 3hard

A square is divided into 9 equal smaller squares (3×3 grid). In 4 of these smaller squares, circles are inscribed. In 2 other smaller squares, squares are inscribed. The remaining 3 smaller squares are left empty. How many total distinct shapes (all sizes of squares + circles + inscribed squares + empty regions) are present in the figure?

Practice 4hard

A rectangle is divided by 3 vertical lines and 2 horizontal lines (not including the borders) into smaller rectangular cells. Additionally, 4 of these cells are further subdivided by a diagonal line each. How many total triangles can be counted in this figure?

Practice 5hard

A figure consists of a rectangle with 5 vertical lines and 4 horizontal lines drawn inside it (including the borders). These lines create a grid of smaller rectangles. Additionally, 6 of these smaller rectangles are shaded. How many total rectangles of all possible sizes can be counted in this figure, excluding the shaded ones?

Practice 6hard

In a complex figure, there is a large square divided into 4 equal smaller squares. Each of these 4 squares is further divided into 4 equal smaller squares. Additionally, there are 3 circles inscribed in 3 of the smallest squares, and 2 triangles inscribed in 2 other smallest squares. How many total geometric shapes (squares of all sizes + circles + triangles) are present in the figure?

Practice 7hard

A figure shows a hexagon with all 6 sides equal. Inside, there are 2 diagonals drawn from one vertex to two non-adjacent vertices. Additionally, the hexagon is divided into 3 equal triangular regions by drawing lines from the center to each alternate vertex. How many total triangles of all sizes can be counted in this figure?

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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