RRB Group D Reasoning Counting Of Figures By Deepak Sir

For aspirants preparing for the RRB Group D exam, mastering the reasoning section is essential. The Counting of Figures is a crucial topic in the reasoning section of competitive exams, designed to test a candidate's spatial reasoning and analytical abilities.

In this detailed session, Deepak Sir explains the Counting of Figures in a highly structured and exam-oriented manner, focusing on the two most important areas: counting of Lines and Counting of Triangles, as asked in RRB Group D and similar railway exams. 

Instead of relying on guesswork, Deepak Sir emphasizes logical classification, systematic counting methods, and proven shortcuts that help candidates avoid common mistakes. From understanding what truly constitutes a single line to applying smart summation tricks for triangles and decoding frequently asked star-shaped figures, this lesson builds a strong foundation for solving even complex figure-counting questions confidently within limited exam time.

Counting Of Lines

While less common than triangle questions, counting lines remains an important part of the syllabus. Understanding the correct method is crucial to avoid simple errors.

What Defines a Single Line?

A common mistake is counting segments of a straight line as separate entities. A line is defined as a continuous straight path. Adding points or labels along its length does not break it into multiple distinct lines. For example, in a figure with two intersecting straight lines (forming an 'X'), there are only two lines, not four segments. A line is considered a single entity until it changes direction or breaks.

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A Systematic Method for Counting Lines

To count lines accurately and avoid mistakes, you must categorize and count them systematically, never randomly. The process involves counting each type of line separately and then summing the totals:

1. Horizontal Lines: Count all lines that are lying down ("padi rekha").

2. Vertical Lines: Count all lines that are standing ("khadi rekha").

3. Slant Lines: Count all diagonal or slanting lines ("tirchi rekha").

• When counting slant lines, first count all lines slanting in one direction, then count all lines slanting in the other direction to ensure none are missed.

Final Formula: Total Lines = (Number of Horizontal Lines) + (Number of Vertical Lines) + (Number of Slant Lines)

This structured approach helps solve any line-counting problem accurately.

Worked Example 1:

Consider a simple house-like figure:

• Horizontal Lines: 3

• Vertical Lines: 3

• Slant Lines: 5

• Total Lines: 3 + 3 + 5 = 11

Worked Example 2:

For a more complex grid with diagonals:

• Horizontal Lines: 4

• Vertical Lines: 4

• Slant Lines: 8

• Total Lines: 4 + 4 + 8 = 16

Using this H + V + S method drastically reduces errors, even in complex figures.

Counting Of Triangles

Counting triangles is the most important sub-topic in figure counting, historically accounting for approximately 95% of questions in competitive exams.

A Conceptual Approach to Counting Triangles

Before applying shortcuts, it is essential to understand a foundational conceptual method. This approach helps in problems where simple tricks might not apply. It involves categorizing triangles based on their composition.

Applying the Conceptual Method:

• Figure with 2 small triangles: 2 (small) + 1 (middle, combination of 2) = 3 total.

• Figure with 3 small triangles: 3 (small) + 2 (middle, combination of 2) + 1 (large, combination of 3) = 6 total.

• Figure with 4 small triangles: 4 (small) + 3 (middle, combination of 2) + 3 (large, combinations of 3 or 4) = 10 total.

Part 1: Triangles Divided from the Same Point

This is a fundamental and common pattern in triangle counting. While the conceptual method works, a much faster trick is available for this specific case.

The Summation Trick

1. Number the small triangles consecutively from 1, starting from one end of the base.

2. Sum these numbers. The total sum represents the total number of triangles in the figure.

Examples:

• A triangle divided into 2 small parts: 1 + 2 = 3 triangles.

• A triangle divided into 3 small parts: 1 + 2 + 3 = 6 triangles.

• A triangle divided into 4 small parts: 1 + 2 + 3 + 4 = 10 triangles.

• A triangle divided into 8 small parts: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 triangles.
  Practicing this summation technique significantly improves calculation speed in timed exams.

Extension: Adding Horizontal Divisions

When horizontal lines are added to a triangle already divided from a vertex, the counting method is a simple extension.

Method:

1. Calculate the number of triangles formed on the main base using the summation trick.

2. Each additional horizontal line acts as a new "base." Calculate the number of triangles for each of these new bases (by applying the summation trick to the segments on that line).

3. Sum the totals from all the horizontal bases.

Example:

A triangle with its main base divided into 4 parts, and additional horizontal lines:

• Main Base (bottom, spanning 4 parts): 1 + 2 + 3 + 4 = 10 triangles.

• Middle Horizontal Line (spanning all 4 parts): 1 + 2 + 3 + 4 = 10 triangles.

• Top Horizontal Line (spanning only the first 2 parts): 1 + 2 = 3 triangles.

• Another partial line (spanning 3 parts): 1 + 2 + 3 = 6 triangles.

• Total Triangles = 10 + 10 + 3 + 6 = 29.

Part 2: Star-Shaped Figures

Star-shaped figures are a frequently asked question type in competitive exams, presenting two primary configurations.

Classification and Rules for Star Figures: