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.

Formulas / Key Facts

**Rectangle Counting in m×n Grid**: Total rectangles = [m(m+1)/2] × [n(n+1)/2], where m = number of horizontal divisions, n = number of vertical divisions.

**Square Counting in n×n Grid**: For an n×n chessboard-style grid, total squares = 1² + 2² + 3² + ... + n² = n(n+1)(2n+1)/6.

**Triangle in Divided Triangle**: When a large triangle is divided by n parallel lines to one side, total triangles ≈ n(n+2). Exact count depends on configuration.

**Diagonal Rule**: One diagonal in a 4-sided figure → 2 triangles. Two diagonals intersecting inside → 4 small + 4 composite = 8+ triangles depending on shape.

**Counting Order**: Always follow smallest-to-largest, left-to-right, or top-to-bottom to maintain consistency.

**Double-Check Formula**: After manual count, verify if total count follows logical progression. For example, in a figure with clear symmetry, counts should reflect that symmetry.

**Maximum 1 Minute Per Question**: In exam conditions, spend no more than 60–70 seconds on a counting figures question. If stuck, make an educated guess and move on.

Worked Examples

**Example 1: Count Triangles**

*Figure: A large triangle divided by two horizontal lines parallel to the base, creating three rows of smaller triangles.*

**Step 1 — Count smallest triangles (pointing upward)**: Row 1 (top) = 1, Row 2 = 2, Row 3 (bottom) = 3. Total small upward = 6 triangles.

**Step 2 — Count smallest triangles (pointing downward)**: Row 2 has 1, Row 3 has 2. Total small downward = 3 triangles.

**Step 3 — Count medium composite triangles**: Formed by combining 2 small triangles. Count systematically: 3 medium upward triangles.

**Step 4 — Count large composite triangles**: Formed by combining 3+ small triangles. 1 large triangle covering rows 2+3, and 1 covering entire figure = 1 large upward.

**Step 5 — Add all**: 6 + 3 + 3 + 1 = **13 triangles**.

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**Example 2: Count Squares in 3×3 Grid**

*Figure: A 3×3 chessboard-style grid (9 small squares visible).*

**Step 1 — Count 1×1 squares**: Directly visible = 9 squares.

**Step 2 — Count 2×2 squares**: Mentally place a 2×2 overlay. Top-left, top-middle, middle-left, middle-middle = 4 squares (2×2 grid gives 2² = 4).

**Step 3 — Count 3×3 squares**: The entire figure = 1 square.

**Total**: 9 + 4 + 1 = **14 squares**.

**Verification by Formula**: n = 3, so 1² + 2² + 3² = 1 + 4 + 9 = 14. ✓

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**Example 3: Count Rectangles in 2×3 Grid**

*Figure: 2 rows, 3 columns of small rectangles.*

**Using Formula**: m = 2 (horizontal divisions), n = 3 (vertical divisions).

Total rectangles = [2(2+1)/2] × [3(3+1)/2] = [2×3/2] × [3×4/2] = 3 × 6 = **18 rectangles**.

**Manual Verification (optional in exam)**: 1×1 rectangles = 6, 1×2 rectangles = 4, 1×3 rectangles = 2, 2×1 rectangles = 3, 2×2 rectangles = 2, 2×3 rectangles = 1. Total = 6+4+2+3+2+1 = 18. ✓

Common Mistakes

**Mistake 1 — Counting Only Simple Shapes**: Students count only the obvious small shapes and forget composite (larger) shapes. → **Fix**: Always count smallest shapes first, then systematically identify larger combinations. Ask yourself: "Can two or more of these combine to form a bigger shape?"

**Mistake 2 — Double-Counting**: Counting the same composite shape twice from different starting points. → **Fix**: Use a consistent direction (left-to-right, top-to-bottom) and mentally or physically mark sections you've already counted. Never count the same configuration twice.

**Mistake 3 — Confusing Squares and Rectangles**: When asked for squares, students count all rectangles. → **Fix**: Read the question carefully. If it asks for "squares," count only perfect squares (equal sides). If it asks for "rectangles," count all four-sided figures including squares.

**Mistake 4 — Ignoring Orientation**: In triangle problems, forgetting to count inverted (downward-pointing) triangles separately. → **Fix**: Always categorize triangles by orientation first: upward-pointing, downward-pointing, left-facing, right-facing. Count each category separately.

**Mistake 5 — Rushing Without System**: Randomly picking shapes from all over the figure leads to missed or repeated counts. → **Fix**: Adopt a rigid counting sequence. For example, in a grid, always go row 1 left-to-right, then row 2 left-to-right, etc. Never jump around the figure.

Quick Reference

• **Counting Order**: Smallest → Largest. Single shapes → Composite shapes. Left → Right or Top → Bottom.

• **Rectangle Formula (m×n grid)**: [m(m+1)/2] × [n(n+1)/2]. Memorize this — it saves 30+ seconds.

• **Square Formula (n×n grid)**: 1² + 2² + ... + n² = n(n+1)(2n+1)/6. For 4×4 grid = 30 squares.

• **Triangle Tip**: Separate by orientation first (up/down/left/right), then count size-wise within each orientation.

• **Double-Check by Symmetry**: If the figure is symmetric, your count should reflect that symmetry (e.g., equal triangles on both halves).

• **Time Limit**: Never exceed 1 minute. If the figure is too complex and you're stuck at 50 seconds, pick the closest reasonable option and move on — one question doesn't decide your score.