SSC CGL Reasoning Counting of Figures by Jitin Sir

Counting of figures is an important topic in non-verbal reasoning. It is asked in exams like SSC CGL, CHSL, MTS, CPO, and Delhi Police. Many students find this topic confusing at first. This happens because figures look complex and contain many overlapping shapes. However, with the right approach, this topic becomes easy and scoring.

Introduction to Counting of Figures

Counting of figures means finding the total number of shapes present in a given diagram. These shapes can be triangles, squares, rectangles, or other polygons. The difficulty level increases when shapes overlap or are hidden inside bigger figures.

Jitin Sir explains that students should not try to count randomly. A structured approach is always helpful. Breaking the figure into smaller parts is the first step. This makes the counting process simple and manageable.

Counting Triangles: Basic Applications

The concept starts with simple figures. Then the level increases gradually. This helps students build a strong foundation.

Students should always begin with basic patterns. Once the basics are clear, complex figures become easier to solve.

Type 1: Simple Figures

This is the most basic level of triangle counting.

If there is a single triangle, the answer is:

Total triangles = 1

Now consider a triangle where the base is divided into two parts. A point is placed on the base.

In this case:

• Two small triangles are formed

• One larger triangle is formed

So, total triangles = 1 + 2 = 3

This is a standard concept. It is often asked in exams.

Type 2: Figures with Multiple Divided Bases

In this type, more divisions are made on the base. All triangles share a common vertex.

For example, if a figure has 3 base divisions:

• Each division creates 3 triangles

So, total triangles = 3 × (1 + 2) = 9

This method is simple and effective.

In some cases, the figure may look slightly complex. In such cases, the answer may be 11 triangles. The idea is to carefully count each section.

Type 3: Complex Figures with Overlapping Triangles

This type is important for exams. These figures contain overlapping triangles.

Students often make mistakes here. The main issue is double-counting.

Practice Understanding

In a complex figure:

• First, divide the figure into smaller parts

• Count triangles in each part

• Add all counts

However, some triangles may be counted more than once. These are common triangles.

After adjusting for overlaps:
Total triangles = 34

Another Example

In another figure:

• Small and large triangles are counted separately

• All values are added

Final answer = 18 triangles

This shows the importance of a step-by-step approach.

Advanced Triangle Counting Techniques

After understanding basic types, students should move to advanced patterns. These patterns follow certain rules.

Type 4: Symmetrical Divisions from Two Vertices

In this type, lines are drawn from two opposite vertices. The divisions are equal on both sides.

Rule:

If the number of divisions is n, then:

Total Triangles=n3\text{Total Triangles} = n^3Total Triangles=n3

This formula helps in quick calculation.

Example:

If n = 2
Total triangles = 2³ = 8

Concept Understanding

Sometimes manual counting gives a higher number.

For example:

• Each side gives 3 triangles

• Total becomes 12

But 4 triangles are common. These are counted twice.

So, correct answer = 12 − 4 = 8

This shows why adjustment is necessary.

Type 5: Asymmetrical Divisions from Two Vertices

In this type, divisions are not equal. The figure looks irregular.

Method:

• Count triangles from each vertex

• Add all values

• Identify common regions

• Subtract repeated triangles

Example:

Initial count = 21
Common triangles = 6

Final answer = 21 − 6 = 15 triangles

This method is useful for irregular figures.

Type 6: Practice for Advanced Figures

This type includes mixed questions. These questions test full understanding.

Figure 1: Symmetrical Pattern

Number of divisions = 3

Using the formula:
Total triangles = 3³ = 27

Figure 2: Asymmetrical Pattern

Initial count:

• 10 + 10 + 6 + 6 = 32

Common triangles = 8

Final answer = 32 − 8 = 24 triangles

Figure 3: Complex Figure

This figure has multiple overlapping sections.

After careful counting:
Total triangles = 42

Final Total

If all three figures are combined:

• 27 + 24 + 42 = 93 triangles

This type of calculation checks overall understanding.

Key Points to Remember

Students should keep the following points in mind:

• Start counting from smallest triangles

• Move towards larger triangles

• Divide the figure into sections

• Identify overlapping areas

• Subtract common triangles

• Use formulas where possible

These steps make the process clear and systematic.

Core Concepts of Counting Triangles

The most common questions involve counting triangles in complex shapes. Jitin Sir emphasizes breaking large figures into smaller, manageable parts.

• Basic Counting: Start by counting the small triangles in each section.

• Multiplication Trick: For symmetrical figures, you can often multiply the number of sections by the base count.

• Avoiding Errors: Students often count the same triangle twice. You must identify overlapping areas and subtract the duplicates.

Important Tricks and Shortcuts

To score well, you should memorize certain fixed patterns. These shortcuts allow you to answer questions in seconds.

• The Diamond Shape: A standard diamond figure always contains exactly 16 triangles.

• Adding Lines: If you add one extra line to a 16-triangle figure, the count usually jumps to 24.

• The Cube Formula: When lines intersect and create equal divisions, use the power formula. For example, if there are 3 divisions, the total is often 3^3 = 27.

• Complex Patterns: For very large figures, sum the patterns. Jitin Sir showed an example where three patterns combined to create 93 triangles.

Steps for Accurate Counting

Below, we’ve mentioned the steps for accurate counting of the figures:

1. Observe the Figure: Look for basic shapes like squares or triangles hidden inside.

2. Apply Formulas: Use the cube or multiplication tricks for standard sections.

3. Check for Overlaps: Carefully look for "hidden" triangles formed where two shapes meet.

4. Final Verification: Subtract any shapes you may have double-counted.

Practical Tips for Aspirants

Regular practice is the only way to master non-verbal reasoning. Jitin Sir provided 101 homework questions to help students refine their skills.

• Be Regular: Attend both morning and evening classes to cover all topics.

• Use PDFs: Follow the provided notes to revise the formulas and fixed counts.

• Interactive Learning: Participate in polls and quizzes to test your speed against other students.

• Focus on Detail: Pay close attention to small lines, as one extra line can change the total count significantly.