Counting of Figures: Notes और MCQ Practice

आसान भाषा में:

Counting of Figures में exam में complex diagrams दिए जाते हैं, जिनमें Triangles, Squares या Rectangles की total count बतानी होती है। SSC CGL Tier-1 में 1-2 questions पक्के आते हैं। इन्हें manually एक-एक करके गिनने के बजाय standard patterns और formulas से seconds में solve किया जाता है।

Important Formulas & Rules:

• Triangles (Vertical base division): जब एक main triangle के base को $n$ parts में बांटा जाए:

Total Triangles = 1 + 2 + 3 + ... + n = n × (n + 1) ÷ 2

• Triangles inside Square/Rectangle: जब diagonals cross करें और square $n$ small triangles में बंटे:

Total Triangles = n × 2

• Squares in an n × n Grid (Symmetric):

Total Squares = 1² + 2² + 3² + ... + n²

• Squares in an m × n Grid (Non-symmetric):

Total Squares = (m × n) + (m − 1) × (n − 1) + ... (जब तक कोई एक factor 1 न हो जाए)

• Total Rectangles in a Grid:

Total = (Sum of Row numbers) × (Sum of Column numbers)

2 Quick Tricks:

1. Horizontal Layers Trick: अगर triangle में horizontal partition lines हों, तो base formula से count निकालकर simply Number of Horizontal Lines से multiply कर दें।
2. Junction Points Trick: जब दो cross-diagonal squares आपस में जुड़ते हैं, तो 8 + 8 = 16 के अलावा common base वाले 2 extra hidden triangles हमेशा बनते हैं (Total = 18)। Junction points को directly +2 गिनें।

2 Common Mistakes:

1. Every Square is a Rectangle: भूलें नहीं कि हर Square एक Rectangle होता है। अगर question में "Total Rectangles" पूछा है, तो उसमें Squares included होंगे; "Only Rectangles" पूछे तभी Squares minus करें।
2. Hidden/Combined Triangles छोड़ देना: Base figures को जोड़ते समय जो बड़े triangles boundary lines से बनते हैं, students अक्सर उन्हें count करना भूल जाते हैं।

Counting of Figures

1. Concept

Counting of figures is a non-verbal reasoning topic where you are given a complex geometric figure composed of various shapes (triangles, squares, rectangles, etc.) and asked to count the total number of specific shapes present in it.

The core challenge is that shapes can be simple (made of one part) or complex (made of multiple parts combined). For example, a large square might be divided into four smaller squares. In this case, you must count the four small squares AND the one large square, totaling 5 squares.

Key Principle: Do not just count the smallest individual units. You must identify all possible combinations of units that form the target shape.

2. Important formulas / rules / facts

• Systematic Approach: Always count in a fixed order (e.g., left to right, top to bottom, or by size: smallest to largest) to avoid missing any shape.
• Composite Shapes: A shape formed by combining two or more smaller parts is also counted as one shape of that type.
• Triangles:
• Count single triangles first.
• Count triangles made of 2 parts.
• Count triangles made of 3 parts.
• Continue until the largest triangle is counted.
• Squares/Rectangles:
• In a grid of $m \times n$ squares, the total number of squares is not simply $m \times n$. You must count $1 \times 1$, $2 \times 2$, $3 \times 3$, etc., squares.
• Formula for total squares in an $n \times n$ grid: $1^2 + 2^2 + 3^2 + ... + n^2$.
• Lines: If asked to count lines, distinguish between horizontal, vertical, and slanting lines.
• Intersection Points: Sometimes, counting the number of intersection points helps verify the complexity of the figure.

3. Solved Examples

Example 1: Counting Triangles

Question: Count the number of triangles in the figure below.

(Imagine a large triangle ABC. A line is drawn from vertex A to the midpoint of base BC, dividing the large triangle into two smaller triangles.)

Solution:

1. Identify the smallest triangles: There are 2 small triangles (let's call them ABD and ACD).
2. Identify composite triangles: The two small triangles combine to form the large triangle ABC.
3. Total = 2 (small) + 1 (large) = 3.

Answer: 3

Example 2: Counting Squares in a Grid

Question: How many squares are there in a $2 \times 2$ grid?

(Imagine a large square divided into 4 equal smaller squares by one horizontal and one vertical line in the middle.)

Solution:

1. Count $1 \times 1$ squares (smallest): There are 4.
2. Count $2 \times 2$ squares (largest): There is 1 (the outer boundary).
3. Total = 4 + 1 = 5.

Answer: 5

Example 3: Counting Rectangles

Question: How many rectangles are there in a $2 \times 2$ grid?

(Same figure as Example 2: A large square divided into 4 smaller squares.)

Solution:

1. Note: A square is a special type of rectangle. So, we count all squares as rectangles too.
2. Count $1 \times 1$ rectangles (squares): 4.
3. Count $2 \times 1$ rectangles (horizontal pairs): 2 (top row pair, bottom row pair).
4. Count $1 \times 2$ rectangles (vertical pairs): 2 (left column pair, right column pair).
5. Count $2 \times 2$ rectangle (the whole figure): 1.
6. Total = 4 + 2 + 2 + 1 = 9.

Answer: 9

Example 4: Complex Triangle Counting

Question: Count the triangles in a figure where a triangle is divided by a line from one vertex to the opposite side, and another line connects the midpoints of the other two sides.

(Simpler description: A triangle ABC. Line AD is drawn from A to BC. Line EF is drawn parallel to BC, cutting AB and AC.)

Solution:

1. Small Triangles:

• Top small triangle (AEF): 1
• Bottom left triangle (BDF? No, let's assume standard division). Let's use a clearer standard example: A triangle with a line from vertex to base, and a line parallel to base.
• Let's stick to a very common SSC pattern: A triangle divided into 4 smaller triangles by joining midpoints.
• Small triangles: 4 (Top, Bottom-Left, Bottom-Right, and the central inverted one? No, usually it's 4 upright small triangles if it's a simple subdivision).
• Let's use the Midpoint Triangle rule: Joining midpoints of a triangle creates 4 small congruent triangles.
• Count small triangles: 4.
• Count large triangle: 1.
• Are there any others? No.
• Total = 5.

Answer: 5

Example 5: Counting Lines

Question: How many straight lines are in a simple 'H' shape?

Solution:

1. Vertical lines: 2 (left and right).
2. Horizontal lines: 1 (middle crossbar).
3. Total = 2 + 1 = 3.

Answer: 3

4. Shortcuts and tricks

• Labeling Method: Label all intersection points and vertices with letters (A, B, C, etc.). This makes it easier to refer to specific shapes (e.g., "Triangle ABC") and ensures you don't double-count or miss any.
• Size-Based Counting: Always count by size. First, count all the smallest units. Then, count shapes made of 2 units, then 3 units, etc. This prevents confusion.
• Grid Formula for Squares: For an $n \times n$ grid, use the sum of squares formula: $\sum_{i=1}^{n} i^2$. For $3 \times 3$, it is $1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14$.
• Grid Formula for Rectangles: For an $m \times n$ grid, the total number of rectangles is $\frac{m(m+1)}{2} \times \frac{n(n+1)}{2}$.
• Example: For $2 \times 2$, $\frac{2(3)}{2} \times \frac{2(3)}{2} = 3 \times 3 = 9$.
• Symmetry Check: If the figure is symmetrical, count the shapes on one half and double it, but be careful not to double-count shapes that lie on the axis of symmetry.

5. Common mistakes to avoid

• Missing Composite Shapes: The most common error is counting only the smallest individual shapes and forgetting the larger shapes formed by combining them.
• Double Counting: Counting the same shape twice because you changed your counting order midway. Stick to one systematic order.
• Ignoring Inverted Shapes: In triangle counting, inverted triangles (pointing down) are often missed if the figure is complex.
• Confusing Squares and Rectangles: Remember that every square is a rectangle, but not every rectangle is a square. If the question asks for rectangles, include squares in the count.
• Assuming All Lines are Straight: Some figures may have curved lines. The question will specify if it asks for "straight lines" or "curved lines." Read the question carefully.
• Rushing: These questions require patience. Spending 30-45 seconds to label and systematically count is better than guessing in 5 seconds.