Paper Folding & Cutting

Subject: Mental Ability | Frequency: 1 question per APPSC paper | Time: 45-60 sec/question

Introduction

Paper folding questions show a piece of paper being folded and then cut or punched. You must determine what the paper looks like when unfolded. The key principle: each fold creates a line of symmetry, and the 2^n rule (n folds = up to 2^n copies of each cut) solves the counting aspect of most problems.

Core Method

For Paper Cutting:

Track each fold — note direction and number of folds
Identify the cut position — where on the folded paper is the cut made?
Unfold in REVERSE order — unfold last fold first
Mirror the cut — each unfold mirrors the cut across the fold line
Count final holes — use the 2^n rule

For Transparent Sheet Folding:

Observe the paper shape and content
Identify fold direction — horizontal, vertical, or diagonal
Visualize overlap — transparent paper shows both layers
Note reversed positions — elements appear mirrored after fold

Critical Rule: Number of Symmetric Copies

Folds	Maximum Copies
1 fold	2 (1 original + 1 mirror)
2 folds	4
3 folds	8
n folds	2^n

Key: Cut on the fold line doubles the shape (semicircle on fold = full circle when unfolded).

Worked Examples — Easy

Q1: Square paper folded in half vertically (left to right). Small circle cut from top-right corner. What appears when unfolded?

Cut at top-right of folded paper
Unfold: mirrors across vertical fold → top-right AND top-left
Answer: Two circles at the top, one on each side

Q2: Square paper folded in half horizontally (top to bottom). Triangle cut from bottom-left corner. What appears when unfolded?

Cut at bottom-left of folded paper
Unfold: mirrors across horizontal fold → bottom-left AND top-left
Answer: Two triangles on the left side, one top, one bottom

Q3: Square paper folded in half vertically. Semicircle cut from the fold line at center. What appears when unfolded?

Cut ON the fold line → shape doubles into full shape
Answer: A full circle at the center of the paper

Worked Examples — Medium

Q4: Square paper folded vertically, then horizontally. Small square cut from bottom-right corner. What appears when unfolded?

2 folds = 4 copies
Unfold horizontal: bottom-right AND top-right
Unfold vertical: each mirrors to the left side
Answer: 4 small squares, one at each corner

Q5: Square paper folded diagonally. Notch cut from hypotenuse center. What appears when unfolded?

Diagonal fold: cut mirrors across the diagonal
Notch on hypotenuse center → diamond-shaped hole at center
Answer: Diamond-shaped hole at the center

Q6: Circular paper folded in half. Triangle cut at fold + triangle at curved edge. What appears when unfolded?

Triangle at fold → full diamond/rhombus at center
Triangle at curved edge → two symmetric triangles at the circular edge
Answer: One diamond at center + two triangles at the edge

Worked Examples — Hard

Q7: Paper folded three times: vertical, horizontal, then diagonal. A corner is cut off. How many holes when fully unfolded?

3 folds = 2^3 = 8 potential copies
One corner cut → up to 8 holes when fully unfolded
Positions: distributed symmetrically across all three fold axes

Q8: Square paper folded vertically. Semicircle cut from top edge, triangle cut from bottom-right corner. What appears when unfolded?

Semicircle at top: mirrors → two semicircles at top (or full circle if cut at fold)
Triangle at bottom-right: mirrors to bottom-left
Answer: Two semicircles at top + two triangles at bottom corners

Q9: Paper folded into quarters (vertical then horizontal). V-shape cut from center point where all folds meet. What appears when unfolded?

V-shape at center, 2 folds = 4 copies
4 V-shapes meeting at center form a star/X pattern
Answer: X-shaped or star-shaped hole at the center

Shortcuts & Tricks

Shortcut	When to Use
2^n rule	n folds = max 2^n symmetric copies of each cut
Unfold last fold first	Always reverse the folding order
Fold line = symmetry axis	Cut on fold line → shape doubles (semicircle → circle)
Corner cuts multiply	Corner cut on doubly-folded paper → 4 corner holes
Eliminate by count	Count expected holes first, eliminate options with wrong count
Center cuts	Cuts at fold intersection → radially symmetric patterns

Common Mistakes

Wrong unfolding order — must unfold in REVERSE order of folding
Forgetting mirror symmetry — each unfold creates a mirror, not a copy
Miscounting holes — not applying 2^n rule correctly
Confusing fold direction — left-to-right vs top-to-bottom produce different results
Transparent vs opaque confusion — transparent sheets show both layers; opaque only top

Exam Strategy

Practice the 2^n counting rule and reverse unfolding order — these two rules solve 80%+ of questions
APPSC typically asks single or double fold problems (not triple)
Time: 45-60 seconds per question
Start by counting: how many folds? → expected number of holes = 2^n per cut
Then check positions by mentally unfolding in reverse

Practice Questions

Paper folded once vertically, circle cut at center of fold. Unfolded? → Full circle at center
Paper folded once horizontally, corner cut at bottom-right. Unfolded? → Two cuts: bottom-right and top-right
Paper folded twice (V then H), one hole punched in center. How many holes when unfolded? → 4
Paper folded once diagonally, triangle cut from the folded corner. Unfolded? → Two symmetric triangles
Paper folded 3 times, one punch made. Maximum holes when unfolded? → 8

Key Terms / Formulas

Term	Meaning
Fold line	The line along which paper is folded; acts as symmetry axis
2^n rule	n folds produce up to 2^n copies of each cut
Reverse unfold	Unfold last fold first, then second-to-last, etc.
Mirror symmetry	Each unfold creates a mirror image across the fold line
Transparent sheet	Both layers visible when folded