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Counting Figures

AFCAT · Logical Reasoning

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Core Concept

Read 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.

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Exam Patterns

What 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.

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Worked Example

Solve 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 COMMON MISTAKE: 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

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 Facts

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.

🚀 60-Second Revision

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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Test yourself under real exam conditions

A timed AFCAT mock shows exactly how Counting Figures questions appear in the actual paper — and where you lose marks.

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