Paper Cutting
Paper Cutting questions involve situations where a sheet of paper is folded first (once or multiple times), and then cuts are made along the folds. The challenge is to determine what the paper will look like when it is completely unfolded.
These problems test your ability to mentally visualize symmetry, repetition, and the effect of cuts across folds.
Types of Paper Cutting Problems
| Type | Description | Example |
|---|---|---|
| Single Fold Cutting | Paper folded once and cut → the cut mirrors across the fold line. | Fold vertically → a triangle cut on the left edge appears on both left and right when unfolded. |
| Multiple Fold Cutting | Paper folded more than once → cuts replicate multiple times symmetrically. | Fold vertically then horizontally → one cut replicates into four cuts after unfolding. |
| Diagonal Fold Cutting | Paper folded diagonally, then cut → cuts mirror along the diagonal axis. | Fold along diagonal, cut a small square → two mirrored squares after unfolding. |
| Complex Multi-Fold Cutting | Several folds + multiple cuts → intricate patterns appear after unfolding. | Folded into quarters, cut in one area → when unfolded, the cut appears in all quadrants. |
How to Solve Paper Cutting Questions
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Identify fold directions: Vertical, horizontal, diagonal, or a combination.
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Understand symmetry: Every cut mirrors across fold lines.
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Count replications: Each fold doubles the number of appearances of a cut.
- 1 fold = 2 replications
- 2 folds = 4 replications
- 3 folds = 8 replications
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Visualize unfolding step by step: Imagine reflecting the cut across each fold.
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Eliminate impossible options: Discard figures where cuts are missing, asymmetric, or outside the paper boundary.
Conceptual Tips and Common Mistakes
- Folds create symmetry, not rotation: Cuts reflect across folds, they don’t rotate.
- Number of cuts multiplies: More folds mean exponential replication of cuts.
- Diagonal folds are tricky: Remember cuts will reflect diagonally, not horizontally or vertically.
- Placement matters: A cut near an edge in folded paper will appear on both edges after unfolding.
- Don’t confuse with Paper Folding: Folding = predicts hole positions; Cutting = predicts shapes/edges after unfolding.
Examples
Example 1 — Single Vertical Fold
Paper folded vertically. A small triangle is cut at the left edge.
Answer: When unfolded, two triangles appear, one on the left and one mirrored on the right.
Example 2 — Single Horizontal Fold
Paper folded horizontally. A semicircle cut is made on the bottom edge.
Answer: On unfolding, two semicircles appear, one at the bottom and one mirrored at the top.
Example 3 — Double Fold (Vertical + Horizontal)
Paper folded vertically and then horizontally. A diamond shape is cut in one quadrant.
Answer: On unfolding, four diamonds appear, one in each quadrant.
Example 4 — Diagonal Fold
Paper folded along a diagonal. A star shape is cut near the folded edge.
Answer: On unfolding, two star shapes appear, mirrored along the diagonal axis.