Counting Figures — Study Notes
Overview
Counting Figures is a pure visual reasoning topic that tests your ability to systematically identify and count geometric shapes embedded within complex diagrams. In SSC MTS Paper 1, you'll encounter questions asking "How many triangles/squares/rectangles are there in the given figure?" The catch: shapes overlap, share sides, and form larger composite shapes that also count.
This topic appears in 1–2 questions in the Non-Verbal Reasoning section. While it seems straightforward, most students lose marks due to double-counting, missed shapes, or lack of a systematic approach. The key to mastering this is developing a foolproof counting method that eliminates guesswork. Unlike other reasoning topics that require pattern recognition or logical deduction, this is purely about visual analysis and methodical enumeration. Master the technique once, and you can handle any figure the exam throws at you.
Questions range from simple 4–5 triangles to complex diagrams with 20+ shapes. Time management is crucial — spending over 90 seconds on a single counting question is inefficient. Your goal: develop a system that's both accurate and quick.
Key Concepts
• **Simple vs. Composite Shapes**: A simple shape has no internal divisions (just the boundary lines). A composite shape is formed by combining two or more simple shapes. Both count separately.
• **Shared Sides Principle**: When two shapes share a common side or vertex, they can form a larger shape. For example, two adjacent small triangles can form one large triangle — all three count individually.
• **Systematic Counting Method**: Never count randomly. Use a fixed order: start with the smallest shapes, then progressively larger ones. For triangles, count by pointing direction (upward, downward, left, right). For rectangles/squares, count row-wise or column-wise.
• **Labeling Technique**: In practice (not in exam unless time permits), mentally or physically label vertices or sections (A, B, C, etc.) to track which combinations you've already counted. This prevents double-counting.
• **The Rectangle-Square Relationship**: Every square is a rectangle, but not every rectangle is a square. When the question asks for "squares," count only equilateral rectangles. When it asks for "rectangles," count all rectangles including squares.
• **Diagonal Lines Create Triangles**: A single diagonal in a rectangle creates 2 triangles. Two diagonals in a rectangle create 4 triangles plus additional composite triangles from overlapping.
• **Grid Formula Shortcut**: For an m×n grid of squares/rectangles without diagonals, the number of rectangles = [m(m+1)/2] × [n(n+1)/2]. For a 2×2 grid: [2×3/2] × [2×3/2] = 3 × 3 = 9 rectangles.
• **Verification by Formula (Triangles)**: For standard patterns like a triangle divided into smaller triangles by parallel lines, use formula: If n lines divide one side, total triangles ≈ n(n+1)(n+2)/6 for simple configurations. However, complex figures need manual counting.