Paper Folding and Cutting — Study Notes

Overview

Paper folding and cutting is a spatial reasoning topic that tests your ability to visualize how a flat piece of paper transforms when folded, cut, and then unfolded. This is a staple in the Logical Reasoning section of NSO, usually appearing in 1–2 questions per paper. The concept is simple: you're shown a sequence of folds, a cut pattern, and must predict what the unfolded paper looks like.

Mastering this topic requires mental visualization rather than memorization. You must track how many layers exist at each fold, where the cut penetrates those layers, and how symmetry creates the final pattern. Students who practice methodically — using real paper initially, then moving to mental visualization — typically score full marks here. The key challenge is avoiding mirror-image confusion and accurately counting the number of holes or shapes that appear after unfolding.

This topic directly connects to non-verbal reasoning skills tested across many competitive exams. It sharpens your ability to manipulate 3D objects mentally, a skill useful in geometry, Figure Matrix problems, and even real-world engineering thinking.

Key Concepts

• **Folding creates layers**: Each fold doubles the number of paper layers at that location. One fold = 2 layers, two folds = 4 layers (if folded in the same region), and so on.

• **Cuts replicate across layers**: A single cut through folded paper creates identical holes in every layer beneath. If you cut through 4 layers, you create 4 identical holes in corresponding positions.

• **Symmetry is your compass**: Folds create axes of symmetry. A vertical fold produces left-right symmetry; a horizontal fold produces top-bottom symmetry. Diagonal folds create diagonal symmetry axes.

• **Unfold in reverse order**: To predict the final pattern, mentally unfold the paper in the exact reverse sequence of how it was folded. Each unfold step mirrors the cut pattern across the fold line.

• **Corner folds are special**: Folding corners (triangular folds) creates radial or rotational patterns rather than simple mirror patterns. The cut position relative to the corner determines the final design.

• **Number of holes = number of layers**: If you make one cut through N layers, you'll see N holes when fully unfolded. The spatial arrangement follows the symmetry of the folds.

Formulas / Key Facts

1. **Total holes after unfolding** = Number of layers at cut location. For example, paper folded twice (4 layers) and cut once → 4 holes.

2. **Symmetry axes from folds**: 1 fold = 1 symmetry line; 2 perpendicular folds = 2 symmetry lines (quadrant symmetry); 3 folds = up to 3 symmetry lines.

3. **Mirror reflection rule**: Each unfold step creates a mirror image of existing cuts across the fold line.

4. **Position preservation**: The relative position of a cut from the fold line determines where mirrored copies appear when unfolded.

5. **Punch/cut shapes replicate**: If you cut a triangle through 4 layers, you get 4 triangles (in symmetrical positions) when unfolded.

6. **Edge cuts vs center cuts**: Cuts near folded edges produce clustered patterns; cuts near the center (far from folds) produce widely spaced patterns.

7. **Diagonal fold formula**: Paper folded along diagonal creates 45° rotational patterns; two diagonal folds can create 8-way symmetry.

Worked Examples

**Example 1: Single Vertical Fold with Circular Cut**

Problem: A square paper is folded vertically in half (left onto right), then a circular hole is punched near the folded edge. What does the unfolded paper show?

Solution:

• Folding vertically creates 2 layers.
• The circular cut goes through both layers.
• When unfolded, the cut near the fold edge creates 2 circles symmetrically placed about the vertical center line.
• Since the cut was near the fold, the circles are close to each other, straddling the middle.

Answer: Two circles symmetrically positioned around the vertical center of the paper.

**Example 2: Two Perpendicular Folds with Triangle Cut**

Problem: A paper is folded horizontally (bottom to top), then folded vertically (left to right). A small triangle is cut from the corner where all folds meet. Predict the unfolded pattern.

Solution:

• First fold (horizontal) creates 2 layers.
• Second fold (vertical) on already-folded paper creates 4 layers total.
• Cutting the corner where folds meet means cutting through all 4 layers at the center region of the original paper.
• First unfold (vertical): Creates 2 triangles side-by-side (left-right symmetry).
• Second unfold (horizontal): Each of those 2 triangles now mirrors top-bottom, giving 4 triangles total.
• The triangles form a cross or plus pattern around the center point.

Answer: Four triangles arranged symmetrically around the center, forming a cross pattern.

**Example 3: Diagonal Fold with Rectangular Cut**

Problem: A square paper is folded diagonally (corner to opposite corner), then a small rectangle is cut from the middle of the folded edge. What appears when unfolded?

Solution:

• Diagonal fold creates 2 triangular layers.
• The fold line is a diagonal of the square.
• A cut on the folded edge (the diagonal) means the cut is along the paper's diagonal axis.
• When unfolded, the diagonal fold creates mirror symmetry across that diagonal.
• One rectangular cut through 2 layers produces 2 rectangles mirrored across the diagonal.
• The rectangles touch or overlap along the diagonal line.

Answer: Two rectangles symmetrically placed along the diagonal of the square.

Common Mistakes

**Mistake 1: Forgetting to count all layers** Wrong thinking: "One fold means one extra hole." Correct fix: Each fold doubles layers in that region. Two folds usually mean 4 holes, not 2. Always track layer count at the exact cut location.

**Mistake 2: Unfolding in the wrong order** Wrong thinking: "I can unfold in any order and get the same result." Correct fix: You must unfold in reverse order of folding. If you folded vertically then horizontally, unfold horizontally first, then vertically. Order determines which symmetry applies first.

**Mistake 3: Confusing mirror axis direction** Wrong thinking: "A horizontal fold creates left-right symmetry." Correct fix: A horizontal fold (top-bottom motion) creates top-bottom symmetry when unfolded. Vertical fold creates left-right symmetry. The fold line is the symmetry axis.

**Mistake 4: Ignoring cut position relative to folds** Wrong thinking: "All cuts produce centered patterns." Correct fix: A cut near a fold edge creates closely spaced holes. A cut far from folds creates widely spaced holes. Distance from fold line determines spacing in the final pattern.

**Mistake 5: Assuming single cut = single hole after unfolding** Wrong thinking: "One cut with scissors means one hole in the unfolded paper." Correct fix: One cut through N layers creates N holes. If paper is folded twice (4 layers) and you make one cut, the unfolded paper has 4 holes arranged symmetrically.

Quick Reference

• **Layer count = Hole count**: One cut through 4 layers → 4 holes when unfolded.
• **Symmetry axis = Fold line**: Vertical fold → left-right mirror; Horizontal fold → top-bottom mirror.
• **Unfold in reverse order**: Last fold made is first fold undone.
• **Corner cuts → Centered patterns**: Cutting folded corners produces designs near the paper's center.
• **Edge cuts → Repeated boundary patterns**: Cuts at folded edges create patterns near original edges.
• **Practice with real paper first**: Physically fold and cut scrap paper to internalize the logic before attempting mental visualization.