Paper Folding

🟢 Lite — Quick Review (1h–1d)

Paper Folding questions test your ability to visualise the outcome when a sheet of paper is folded, cut, or punched, and then unfolded. This topic appears in SSC CGL Tier-I Non-Verbal Reasoning. There are three main types: (1) determining the unfolded pattern after cuts are made in folded paper, (2) identifying how a punched hole pattern appears when the paper is unfolded, and (3) selecting the correct unfolded shape from given folded-and-cut figures.

Core Principles:

Principle 1 — Symmetry of Folds: When a paper is folded exactly in half, the two halves are mirror images of each other. A cut on one half creates a symmetric cut on the other half when unfolded.

Principle 2 — Punch/Hole Through All Layers: If a hole is punched through all layers of a folded paper, the unfolded paper shows as many holes as there are layers penetrated.

Principle 3 — Rotation Doesn’t Change Position: If the paper is folded and then rotated before cutting, the pattern in the unfolded paper accounts for the rotation.

Principle 4 — Adjacent vs Opposite Folds:

• Folding into 2 layers: each cut makes 2 identical marks (one on each layer).
• Folding into 4 layers: each cut makes 4 identical marks.
• Folding into 8 layers: each cut makes 8 identical marks.

⚡ Exam Tip: The most common trick in SSC CGL paper folding questions: a cut made near the folded edge (crease line) of the paper results in a symmetric pattern centred on the crease line in the unfolded sheet. Always visualise the crease as a line of symmetry. When in doubt, count the number of layers and multiply the visible cut by the number of layers.

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🟡 Standard — Regular Study (2d–2mo)

Paper Folding — SSC CGL Study Guide

Core Concept: Paper folding problems in SSC CGL require spatial reasoning about layers. When paper is folded, each layer is a mirror image of the adjacent layer across the fold line. When the paper is cut in the folded state, the cut goes through all layers simultaneously. When unfolded, each layer’s cut becomes a mark on the unfolded paper. The key is tracking which layers are cut and where they appear in the final unfolded sheet.

Worked Examples:

Example 1 — Single Fold, Single Cut: A square sheet of paper is folded in half diagonally. A hole is punched at the centre of the folded triangle. How many holes appear when the paper is unfolded?

• Folded into 2 layers (triangle from diagonal fold)
• One hole punched through the centre of the folded triangle
• Since there are 2 layers, the hole appears in 2 places when unfolded Answer: 2 holes, symmetrically placed on opposite sides of the diagonal crease line.

Example 2 — Fold Twice, Cut Once: A rectangular paper is folded first horizontally (to make 2 layers) and then vertically (to make 4 layers). A hole is punched at the centre of the folded rectangle. How many holes in the unfolded paper?

• 1 horizontal fold → 2 layers
• 1 vertical fold → 2 × 2 = 4 layers
• 1 hole punched → 4 holes in unfolded paper
• Arrangement: 4 holes at the corners of the original rectangle’s centre region, symmetrically placed Answer: 4 holes, arranged in a 2×2 grid at the centre.

Example 3 — Unfolded Pattern to Folded: Which of the following shows the correct folded-and-cut figure? Given an unfolded pattern with 4 symmetric holes arranged in a square at the centre, which folded state produces this?

• 4 symmetric holes at centre → 2 folds (horizontal and vertical) with 1 hole at the centre of the folded square
• If folded in half twice (making 4 layers) and a hole punched at the centre of the folded piece, the 4 holes in the unfolded state would be at the 4 corners of the central square. Answer: The folded state is a small square with a hole at its centre (from the perspective of one layer, at the corner where all layers meet).

Example 4 — Pattern Recognition: A paper is folded and a shape is cut. The unfolded paper shows a pattern. Which cut was made?

• If the unfolded pattern shows two identical shapes on opposite sides of a line, the fold was along that line.
• If the unfolded pattern shows four identical shapes in four quadrants, the paper was folded twice (horizontal and vertical).
• If the pattern shows shapes along a diagonal, the fold was diagonal.

Example 5 — Cut at Edge: A paper folded in half has its open edge on the left. A cut is made parallel to the fold, 1 cm away from the fold line. How does this appear when unfolded?

• The cut on one layer, when unfolded, creates two cuts — one on each side of the fold line, symmetrically placed.
• Both cuts are 1 cm away from the fold line, on opposite sides. Answer: Two parallel cuts, each 1 cm from the centre fold line.

Common Student Mistakes:

• Forgetting that a cut near the fold affects both halves symmetrically.
• Counting the wrong number of layers after multiple folds.
• Assuming that a diagonal fold creates the same symmetry as a horizontal/vertical fold (it doesn’t — diagonal fold lines are diagonal symmetry axes).
• Confusing the direction of the fold (which side is folded over which).

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🔴 Extended — Deep Study (3mo+)

Paper Folding — Comprehensive SSC CGL Notes

Theoretical Foundation: Paper folding questions test deductive reasoning about spatial transformations. When a paper is folded, the layers are stacked in a specific order. A cut through the folded paper penetrates every layer at that point. When unfolded, the cuts on each layer create marks that are mirror images of each other across each fold line. The sequence of folds determines the final arrangement of marks.

Types of Paper Folding Questions in SSC CGL:

Type 1 — Unfolding to Determine Cut: A paper is folded in a described manner and cuts are made. You must identify the correct unfolded pattern from four options.

Type 2 — Folding to Match Pattern: Given an unfolded pattern, determine the sequence of folds and cuts required to produce it.

Type 3 — Counting Holes/Figures: Given the number of folds and cuts, calculate the number of marks in the unfolded paper.

Type 4 — Hole Position Identification: Where does a particular hole appear in the unfolded paper based on where it was punched in the folded state?

Folding Sequence Analysis:

Case 1 — Single Fold (Folded in Half):

• Paper folded once → 2 layers
• Cut or hole at position (x, y) from the fold line on the visible side
• When unfolded: marks appear at (x, y) and at (x, −y) (mirror position across the fold)
• If cut is on the fold line itself → mark is a line along the fold

Case 2 — Two Folds (Horizontal + Vertical):

• Paper folded twice → 4 layers (forming a quadrant)
• Fold line 1: horizontal centre line
• Fold line 2: vertical centre line
• Cut at centre of folded quadrant (point where all 4 layers meet)
• When unfolded: 4 marks appear — one in each quadrant, arranged symmetrically
• If cut is at (a, b) from centre of folded piece (where a, b are small distances from the inner corner), the unfolded paper shows marks at (±a, ±b) in all four quadrants.

Case 3 — Three Folds (Multiple Folds):

• Paper folded 3 times → 8 layers possible
• Each additional fold doubles the number of layers
• Total marks = 2^(number of folds) for cuts at the convergence point of all layers
• Cuts away from the convergence point affect fewer layers

Number of Layers Formula: If a paper is folded n times:

• Maximum layers = 2^n (when folded so that all layers are aligned and stacked)
• Each fold can potentially double the layers along the fold line

Special Fold Types:

Diagonal Fold:

• Paper folded along diagonal → 2 triangular layers
• A hole at the centre of the folded triangle: when unfolded → 2 holes on opposite sides of the diagonal line
• A hole near one corner: when unfolded → 2 holes near opposite corners on either side of the diagonal

Accordion Fold (Multiple Parallel Folds):

• Paper folded like a fan (multiple parallel folds in alternating directions)
• Each fold direction alternates
• Cuts visible in alternating layers affect different sets of final positions

Reverse Fold:

• When a fold is reversed (folding back over itself), some layers are inverted
• The spatial orientation of cuts must account for which side of each layer was facing up when cut

Worked Examples — Complex Cases:

Example 1: A paper of size 10 cm × 10 cm is folded in half (horizontally), then folded in half again (vertically), so that a quarter-sized folded piece is formed. A hole is punched 1 cm from the left edge and 1 cm from the bottom edge of this quarter piece. How many holes appear when unfolded, and at what positions?

• 2 folds → 4 layers converging at the corner of the quarter piece
• Hole punched at (1, 1) cm from the inner corner
• When unfolded, the 4 layers correspond to 4 quadrants
• Hole appears at: (1, 1), (−1, 1), (1, −1), (−1, −1) relative to the paper centre
• On a 10×10 paper with centre at (5, 5): holes at approximately (4, 4), (6, 4), (4, 6), (6, 6) cm Answer: 4 holes in a 2×2 grid at the centre of the paper.

Example 2: A paper is folded twice (horizontal then vertical), then a cut is made along a line parallel to the horizontal fold, 2 cm above it, cutting through all 4 layers on the right half. What does the unfolded paper look like?

• The cut goes through 4 layers (all layers are stacked on the right side of the vertical fold)
• When unfolded: 4 parallel cuts appear — 2 on the left half and 2 on the right half
• The cuts are symmetric about the vertical fold line
• There are 2 cuts on each side of the vertical fold (since each half has 2 layers)

Example 3: A paper is folded once (horizontally). A cut is made at the centre of the bottom layer (the layer farthest from you when folded). A hole is punched at a point 2 cm from the left edge of the folded piece. How does this appear when unfolded?

• 2 layers
• Bottom layer is the one folded under (not visible)
• Cut at centre of bottom layer → when unfolded, this cut is in the bottom half of the paper
• Hole at 2 cm from left edge, on the visible top layer → when unfolded, 2 holes appear at 2 cm from the left edge, one in top half and one in bottom half
• The hole on the top layer maps to the top half; the hole on the bottom layer (through the fold) maps to the bottom half Answer: 2 holes, each 2 cm from the left edge, one in the top half and one in the bottom half.

SSC CGL PYQ Pattern (2019-2023):

• 2023 Tier-I: 1 paper folding question — Type 1 (unfolding to determine cut)
• 2022 Tier-I: 1 paper folding question — Type 3 (counting holes after multiple folds)
• 2021 Tier-I: 1 paper folding question — Type 4 (hole position identification)
• Most common: Two-fold problems (4 layers) with hole at centre
• Less common: Diagonal fold problems
• Difficulty: Moderate; requires careful mental visualisation

Visualisation Strategy:

1. Draw the folded shape (usually a rectangle or triangle)
2. Mark the layers (label which parts of the unfolded paper correspond to which layers)
3. Mark the cut/hole on the folded shape
4. Unfold step by step, tracking each layer’s cut position
5. Verify the symmetry of the final pattern

⚡ Advanced Exam Tip: In questions where you must choose the correct unfolded pattern from four options, first identify the axis/axes of symmetry in the pattern. Each fold creates one axis of symmetry in the unfolded paper. A pattern with one line of symmetry came from one fold; with two perpendicular lines of symmetry came from two folds (horizontal + vertical); with diagonal symmetry came from a diagonal fold.

⚡ Second Advanced Tip: When cuts are made away from the convergence point of all layers (i.e., not at the centre of the folded piece), only some layers are cut. Draw the folded piece as a smaller rectangle. Identify which of the n layers the cut actually penetrates. For a 4-layer fold (2 folds), cuts near the edges affect only 2 layers (the ones that extend to that edge), while cuts near the centre of the folded piece affect all 4 layers.

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