Module 2 · SSC CGL Reasoning
Non-Verbal Reasoning
Mirror Images, Water Images, Embedded Figures, Paper Folding, and Figure Counting.
Mirror Images · Water Images · Embedded Figures · Paper Folding · Figure Counting
Learning Objectives
- Understand fundamental concepts of non-verbal reasoning
- Apply shortcuts to solve mirror/water image, embedded figures, and counting problems
- Practice with exam-level questions to build speed and accuracy
Key Concepts
1. Mirror Images — Lateral Inversion
A mirror image is the reflection of an object in a mirror. In SSC CGL, the mirror is usually placed on the right or left side. The image is laterally inverted: left ↔ right swap. The top and bottom remain unchanged.
Left Mirror (Mirror on Right Side): The left side of the object appears on the right side of the image, and vice versa.
Right Mirror (Mirror on Left Side): The right side appears on the left in the image.
Fig 1: Lateral inversion in a mirror — PQR becomes RQP (right mirror)
Tip 1: Mirror Image = Lateral Inversion
Mirror image swaps left and right. The top and bottom stay the same. For letters, check if the letter is symmetric (A, H, I, M, O, T, U, V, W, X, Y have vertical symmetry — their mirror image is identical). Asymmetric letters like B, D, E, F, G change direction.
Tip 2: Mirror on Right vs Left
If mirror is on the RIGHT side → image is formed by swapping left-right of the object. If mirror is on the LEFT side → image is formed by swapping right-left. SSC CGL usually places the mirror on the right side (the image is to the left of the object).
Example: Letter Mirror Image
If the mirror is placed on the right, what is the mirror image of "SSC CGL"?
a) SSC CGL b) LGC CSS c) LGC 2S2 d) LGƆ SSƆ
a) SSC CGL b) LGC CSS c) LGC 2S2 d) LGƆ SSƆ
Solution: d) LGƆ SSƆ. Each letter is laterally inverted. S mirrors to Ɔ (reversed S), C mirrors to Ɔ (reversed C), G mirrors to O... actually S→Ɔ, C→Ɔ, G→O reversed. The sequence of letters is reversed and each letter is mirrored.
2. Water Images — Vertical Inversion
A water image is the reflection of an object in water. Unlike a mirror image (lateral), a water image is vertically inverted: top ↔ bottom swap. The left and right remain unchanged.
Tip 3: Water Image = Vertical Inversion
Water image swaps top and bottom. Left and right stay the same. Letters with horizontal symmetry (B, C, D, E, H, I, K, O, X) look the same in water. Asymmetric letters like P, R, F, G, L, J appear upside down.
Fig 2: Water image — vertically symmetric letters remain unchanged
Example: Water Image
What is the water image of "PAGE"?
a) PAGE b) ƎGAP c) ƎGⱯᖷ d) ƎƸⱯᖷ
a) PAGE b) ƎGAP c) ƎGⱯᖷ d) ƎƸⱯᖷ
Solution: c) ƎGⱯᖷ. P → ᖷ, A → Ɐ, G → G (symmetric), E → Ǝ. The sequence is reversed vertically, and letters are inverted top-to-bottom.
3. Embedded Figures — Find the Hidden Shape
In embedded figure questions, you are given a simple figure (the 'problem figure') and a set of complex figures. You must find which complex figure contains the simple figure as a hidden part (not necessarily in the same orientation).
Tip 4: Embedded Figure Tracing Technique
Trace the outline of the problem figure mentally over each option. Look for: (a) same number of sides/vertices, (b) same internal lines or curves, (c) same relative proportions. If the figure is rotated, mentally rotate the problem figure to match.
Tip 5: Look for Distinctive Features
Check for unique features of the problem figure: a specific angle, a curve, a particular intersection point. Scan each option for that feature first. If it's absent, eliminate that option immediately.
Example: Embedded Figure
Which of the following figures contains the shape shown? (Problem: A triangle with a small circle inside)
a) A hexagon with internal lines b) A square with a triangle and circle c) A complex polygon with a triangular section containing a circle d) A rectangle with diagonal lines
a) A hexagon with internal lines b) A square with a triangle and circle c) A complex polygon with a triangular section containing a circle d) A rectangle with diagonal lines
Solution: c) The figure contains a triangle with a circle inside it, matching the problem figure. Option b has a triangle and circle but they may not be arranged the same way.
Tip 6: SSC CGL Embedded Figure Shortcut
In SSC CGL, the problem figure is usually embedded in only ONE of the options. If you find the figure in more than one option, check orientation — most SSC CGL questions preserve relative orientation (figure is not rotated).
4. Paper Folding and Unfolding
In paper folding questions, a paper is folded 1-3 times, and holes are punched. You must identify the pattern of holes when the paper is unfolded. Alternatively, you may be shown a folded paper with punched holes and asked to match it to the unfolded version.
Tip 7: Paper Folding — Unfold from Last Fold First
Reverse the folding process mentally. Start from the final folded state, note where holes are punched, then unfold in reverse order. Each fold line creates a mirror image of the hole pattern on the other side. One fold = 2x the holes, two folds = 4x, three folds = 8x.
Fig 3: One fold creates a mirror hole across the fold line
Tip 8: Mark Fold Lines
Draw imaginary fold lines. Each fold creates symmetry. If the paper is folded along a vertical line, the holes will be mirrored horizontally. If folded along a diagonal, the holes mirror diagonally. Count holes and check symmetry.
Example: Paper Folding
A square paper is folded in half vertically and then horizontally. A hole is punched at the top-right corner of the folded paper. How many holes appear when unfolded?
a) 2 b) 4 c) 6 d) 8
a) 2 b) 4 c) 6 d) 8
Solution: b) 4. Two folds = 2² = 4 holes. The hole is mirrored across the vertical fold (2 holes), and then both are mirrored across the horizontal fold (4 holes total).
5. Figure Counting — Triangles, Squares, Rectangles
Figure counting questions ask you to count the total number of triangles, squares, or rectangles in a given figure. These require systematic counting using formulas for complex overlapping figures.
Triangle Counting Formulas:
- Triangle divided by n lines from vertex: n(n+1)/2 triangles
- Triangle with intersecting lines inside: count systematically by size (1-unit, 2-unit, etc.)
- Star/pentagram patterns: each intersection point creates multiple triangles
Square Counting:
- n × m grid: total squares = Σ(i=1 to n) (n-i+1)(m-i+1) for i ≤ min(n,m)
- For n × n grid: n(n+1)(2n+1)/6 squares
Rectangle Counting:
- n × m grid: total rectangles = n(n+1)/2 × m(m+1)/2
- This counts all rectangles including squares
- To get non-square rectangles: subtract square count from rectangle count
Fig 4: Counting squares and rectangles in a 3x3 grid
Tip 9: Systematic Triangle Counting
Label each small triangle area. Count triangles by size: first count all 1-unit triangles, then 2-unit (made of 2 combined), then 3-unit, etc. Sum all sizes. For complex figures, use the node method: count lines from each vertex.
Tip 10: Rectangle Counting Formula
For an n × m grid of squares, the number of rectangles = n(n+1)/2 × m(m+1)/2. For 3×3: 3×4/2 × 3×4/2 = 6×6 = 36 rectangles. This includes squares. For non-square rectangles: 36 - 14 = 22.
Tip 11: Square Counting Formula
For an n × n grid, squares = n(n+1)(2n+1)/6. For 3×3: 3×4×7/6 = 84/6 = 14 squares. For an n × m grid (n≤m): squares = Σ(i=1 to n) (n-i+1)(m-i+1).
Example: Counting Triangles
How many triangles are there in a triangle divided by 3 lines from each vertex?
a) 10 b) 13 c) 15 d) 18
a) 10 b) 13 c) 15 d) 18
Solution: b) 13. With 3 lines from a single vertex to the opposite side: n(n+1)/2 = 3×4/2 = 6. But with lines from multiple vertices intersecting, the total is 13 (counted systematically).
Example: Counting Squares
How many squares are in a 4×4 grid?
a) 16 b) 25 c) 30 d) 36
a) 16 b) 25 c) 30 d) 36
Solution: c) 30. Formula: n(n+1)(2n+1)/6 = 4×5×9/6 = 180/6 = 30. Breakdown: 1×1 squares = 16, 2×2 = 9, 3×3 = 4, 4×4 = 1. Total = 30.
Tip 12: SSC CGL Common Counting Patterns
Pattern 1: Triangle with internal lines from one vertex → n(n+1)/2. Pattern 2: Triangle divided by parallel lines to base → n². Pattern 3: Grid of squares → use formula. Pattern 4: Overlapping rectangles → count by individual rectangles first.
Tip 13: Overlapping Figures — Mark and Count
For complex overlapping figures, number each region. Then count combinations: single regions, double regions (two adjacent regions combined), triple, etc. This systematic approach prevents missing or double-counting.
SSC CGL Shortcut Techniques
Tip 14: Mirror Image in 5 Seconds
Check only the first and last characters of the word. If the mirror is on the right, the image is reverse order of characters with each character laterally inverted. For letter-based questions, focus on the first letter (becomes last in image) — if it doesn't match, eliminate.
Tip 15: Water Image in 3 Seconds
Check top and bottom of the word/figure. In water images, top becomes bottom. For words, look for letters that change when inverted vertically: p→b, b→p, d→q, q→d, 6→9, 9→6. If these letters are present, check carefully.
Tip 16: Embedded Figure — Edge Matching
Compare the OUTER boundary of the problem figure with each option. The embedded figure's outer boundary must be a subset of the option's lines. If a line in the problem figure is not present in the option, eliminate it.
Tip 17: Paper Punch — Hole Count Check
Remember: n folds = 2^n holes per punch. If paper is folded 3 times and 2 holes are punched, total holes = 2 × 2³ = 16 holes. This quick check eliminates most wrong options immediately.
Tip 18: Figure Counting — The Label Method
Label every junction point with a letter. Then list all possible shapes by their vertices. For triangles: (A,B,C), (C,D,E), etc. This takes longer but is 100% accurate for complex figures. Use it as a verification method.
Tip 19: SSC CGL Non-Verbal Time Management
Mirror/water images: 15-20 seconds per question. Embedded figures: 30-40 seconds. Paper folding: 30-45 seconds. Figure counting: 45-60 seconds. Don't spend more than 1 minute on any single non-verbal question.
Tip 20: Practice with Alphabet Chart
Create a chart of letters with their mirror and water images. A→A (mirror=water=same), B→B (water same, mirror differs). Memorize tricky pairs: p↔q, b↔d in water. For mirror: R→Ⅾ (reversed), S→Ƨ (reversed). Quick reference saves precious seconds.
Solved Examples
Solved Example 1 (Mirror Image)
If a mirror is placed on the right side, find the mirror image of "CGL2025".
a) 5202LGƆ b) 5202GLC c) CGL2025 d) 5202LCƆ
a) 5202LGƆ b) 5202GLC c) CGL2025 d) 5202LCƆ
Solution: a) 5202LGƆ. The sequence is reversed (5202LGC) and each character is laterally inverted: C→Ɔ, G→G(vertical symmetry), L→⅂ reversed, 2→2(symmetric), 0→0, 2→2, 5→5(symmetric). Note: 2 and 5 are mirror-symmetric vertically.
Solved Example 2 (Water Image)
Find the water image of "B6P".
a) B6P b) B9q c) q9B d) P9B
a) B6P b) B9q c) q9B d) P9B
Solution: c) q9B. Water image inverts top-bottom. B→B (horizontal symmetry), 6→9 (inverted), P→q (inverted). In water image, the sequence order stays the same but letters invert vertically: B→B, 6→9, P→q. So B6P → B9q when inverted vertically... wait, P becomes q. But in water image, the left-right order stays the same, just each letter is flipped vertically. So B6P remains B→B, 6→9, P→q = B9q. Actually wait - the order should be reversed? No, water image preserves left-right order. The answer should be B9q. But option c says q9B which is reversed order. Let me reconsider — actually in water image, the word is reflected as a whole. The leftmost stays leftmost, rightmost stays rightmost. B→B (horizontal symmetry), 6→9, P→q. So B6P → B9q. Option a is B6P (original), b is B9q (this matches), c is q9B (reversed), d is P9B. So the answer should be b) B9q.
Solved Example 3 (Embedded Figure)
A problem figure shows a right-angled triangle. Which of the following complex figures contains this triangle?
a) A hexagon with a diagonal line forming two triangles b) A square with both diagonals forming 4 triangles c) A rectangle with one diagonal d) A circle with inscribed triangle
a) A hexagon with a diagonal line forming two triangles b) A square with both diagonals forming 4 triangles c) A rectangle with one diagonal d) A circle with inscribed triangle
Solution: b) A square with both diagonals forms 4 right-angled triangles at the centre. Each is a right-angled triangle matching the problem figure.
Solved Example 4 (Paper Folding)
A paper is folded in half along a vertical line. A hole is punched at the top-right corner. After unfolding, where are the holes?
a) Top-right only b) Top-right and top-left c) Top-right and bottom-right d) All four corners
a) Top-right only b) Top-right and top-left c) Top-right and bottom-right d) All four corners
Solution: b) Top-right and top-left. The vertical fold creates a mirror image of the hole across the fold line. The hole at top-right mirrors to top-left. Only 2 holes (one fold = 2x holes).
Solved Example 5 (Figure Counting — Triangles)
How many triangles are in a square divided by both diagonals?
a) 4 b) 6 c) 8 d) 10
a) 4 b) 6 c) 8 d) 10
Solution: c) 8. The diagonals divide the square into 4 small triangles. Additionally, pairs of adjacent small triangles form 4 larger triangles (each half of the square). Total = 4 + 4 = 8.
Solved Example 6 (Mirror Image — Clock)
A clock shows 3:15. If a mirror is placed on the right, what time does the mirror image show?
a) 8:45 b) 9:45 c) 8:15 d) 9:15
a) 8:45 b) 9:45 c) 8:15 d) 9:15
Solution: a) 8:45. Mirror time formula: Mirror Time = 12:00 - Given Time (for analog clocks). 11:60 - 3:15 = 8:45. Shortcut: Subtract from 12:00 (borrow 1 hour as 60 min).
Solved Example 7 (Water Image — Clock)
A clock shows 6:30. What is its water image?
a) 6:30 b) 5:30 c) 12:30 d) 7:30
a) 6:30 b) 5:30 c) 12:30 d) 7:30
Solution: a) 6:30. At 6:30, the hour hand is between 6 and 7, minute hand at 6. The vertical axis passes through 12-6. Since both hands lie on or near the vertical axis, the water image appears nearly the same. For exact: Water Image Time = 12:00 - Given Time, but for vertical axis symmetry it can be the same.
Solved Example 8 (Embedded Figure)
Find the figure that contains a small circle located inside a triangle that is inside a square.
a) Square containing a triangle with a circle b) Circle inside a square with no triangle c) Triangle inside a circle inside a square d) Square with circle and separate triangle
a) Square containing a triangle with a circle b) Circle inside a square with no triangle c) Triangle inside a circle inside a square d) Square with circle and separate triangle
Solution: a) The outer shape is a square, inside it is a triangle, and inside the triangle is a small circle — exactly matching the problem description.
Solved Example 9 (Paper Folding)
A paper is folded in half diagonally. A hole is punched near the folded corner. After unfolding, how many holes appear?
a) 1 b) 2 c) 3 d) 4
a) 1 b) 2 c) 3 d) 4
Solution: b) 2. One diagonal fold creates a mirror image of the hole across the diagonal line. Two holes appear — one on each side of the diagonal fold line.
Solved Example 10 (Figure Counting — Rectangles)
How many rectangles are in a 2×3 grid?
a) 12 b) 15 c) 18 d) 20
a) 12 b) 15 c) 18 d) 20
Solution: c) 18. Formula: n(n+1)/2 × m(m+1)/2 = 2×3/2 × 3×4/2 = 3 × 6 = 18 rectangles.
Solved Example 11 (Mirror Image — Figure)
A figure shows an arrow pointing to the top-left. What is its mirror image with mirror on the right?
a) Arrow pointing top-left b) Arrow pointing top-right c) Arrow pointing bottom-left d) Arrow pointing bottom-right
a) Arrow pointing top-left b) Arrow pointing top-right c) Arrow pointing bottom-left d) Arrow pointing bottom-right
Solution: b) Arrow pointing top-right. Mirror (lateral) inversion swaps left and right. Top-left becomes top-right. The upward direction remains unchanged.
Solved Example 12 (Water Image — Figure)
A figure shows an arrow pointing top-left. What is its water image?
a) Arrow pointing top-left b) Arrow pointing top-right c) Arrow pointing bottom-left d) Arrow pointing bottom-right
a) Arrow pointing top-left b) Arrow pointing top-right c) Arrow pointing bottom-left d) Arrow pointing bottom-right
Solution: c) Arrow pointing bottom-left. Water (vertical) inversion swaps top and bottom. The left direction remains unchanged, but top becomes bottom.
Solved Example 13 (Figure Counting — Triangles)
A triangle has 4 horizontal lines parallel to its base. How many triangles can be counted?
a) 10 b) 12 c) 15 d) 16
a) 10 b) 12 c) 15 d) 16
Solution: d) 16. Formula: (n+1)² where n = number of parallel lines to base. 4 lines → (4+1)² = 25. Alternatively, if the lines are from vertex to opposite side: n(n+1)/2 = 4×5/2 = 10. The correct answer depends on the arrangement. For horizontal parallel lines dividing the triangle into smaller triangles: 4 lines create 5 rows of small triangles. Total = 1+3+5+7 = 16.
Solved Example 14 (Paper Folding — Complex)
A paper is folded in half vertically, then horizontally. A hole is punched at the folded corner (top-right of folded paper). How many holes appear when fully unfolded?
a) 2 b) 4 c) 6 d) 8
a) 2 b) 4 c) 6 d) 8
Solution: b) 4. Two folds = 2² = 4 holes. The corner of the folded paper corresponds to the original centre. After unfolding vertically: 2 holes (mirrored). After unfolding horizontally: 4 holes (both mirrored).
Solved Example 15 (Figure Counting — Squares)
How many squares are in a 5×5 grid?
a) 25 b) 45 c) 55 d) 65
a) 25 b) 45 c) 55 d) 65
Solution: c) 55. Formula: n(n+1)(2n+1)/6 = 5×6×11/6 = 330/6 = 55. Breakdown: 25 + 16 + 9 + 4 + 1 = 55.
Summary of SSC CGL Non-Verbal Shortcuts
- Mirror Image: Lateral inversion (left↔right). Mirror on right → reverse sequence + invert each character. Symmetric letters: A, H, I, M, O, T, U, V, W, X, Y
- Water Image: Vertical inversion (top↔bottom). Sequence stays same, each character flips vertically. Many symmetric letters: B, C, D, E, H, I, K, O, X
- Embedded Figure: Trace boundary first, then internal lines. Look for distinctive features (angles, curves, intersections)
- Paper Folding: n folds = 2ⁿ holes per punch. Reverse unfold step by step, mirroring holes across fold lines
- Figure Counting: Use formulas for grids. For triangles, count by size systematically. Label regions for complex figures