Paper Folding and Cutting

Pattern Name

Type

Non-verbal Reasoning

Explanation

Paper Folding & Cutting

In these problems, a sheet of paper (typically square or circular) is folded along dotted lines, and then cuts or punches are made in its folded state. The objective is to determine the shape and pattern of the paper when it is completely unfolded.

Key Principles

Mirror Symmetry: Each fold crease acts as a line of symmetry. When the paper is unfolded, the cuts made on one side of a crease are reflected as mirror images on the other side.
Layer Multiplication: If a paper is folded $n$ times, it creates $2^n$ layers. A single punch made through all layers will result in up to $2^n$ holes in the unfolded sheet, provided the punch is not directly on a fold line.
Crease Cuts: If a cut or hole is punched directly along a fold line, the resulting shape will merge symmetrically across the fold, creating a single, combined shape when the paper is unfolded.

Solving Strategy

Solving Strategy

Reverse Unfolding: Mentally unfold the paper in the exact reverse sequence of the folding steps. Always begin by mirroring the cuts across the last fold line applied.
Hole Count Elimination: Estimate the total number of holes. For $n$ complete folds, a single punch through all layers will typically create $2^n$ holes. Use this to immediately eliminate options that do not match the expected hole count.
Quadrant Analysis: If the paper was folded multiple times (e.g., twice into a quadrant), divide the unfolded options into imaginary quadrants. The final pattern in the folded quadrant must precisely match the original cuts in both shape and orientation.
Mirror vs. Translation: Carefully ensure that the shapes in adjacent sections, created by unfolding, are true reflections (mirror images) of each other, not merely identical shapes positioned side-by-side (translations).

Difficulty

Hard