Paper Cut/Fold

 Paper Folding and Cutting problems are based on the visual imagination of folding a paper in a particular way (half, quarter, etc.), performing one or more cuts, and then unfolding it to predict the resulting pattern.

📚 Types of Questions

Type	Description
📄 Paper Folding	Predict the shape after unfolding a folded paper
✂️ Paper Cutting	Predict cut patterns after unfolding
🔳 Hole Punching	Predict positions of punched holes after unfolding
🌀 Crease Pattern	Identify folds (not cuts) and their effects

🎯 Key Concepts to Remember

🔹 Symmetry is the Key

• Folding creates mirror images

• A fold along a line creates a symmetrical reflection

🔹 Types of Fold Lines

Fold Type	Result on Unfolding
Vertical Fold (↕️)	Horizontal symmetry
Horizontal Fold (↔️)	Vertical symmetry
Diagonal Fold (⤡)	Diagonal mirror image
Multiple Folds	Combined mirror effects

🧠 Important Methods to Solve

✅ Step-by-Step Strategy

1. Identify Fold Direction
   Top, bottom, left, right, or diagonal

2. Count Number of Folds
   One fold = 2 parts
   Two folds = 4 parts (like quarter-fold)

3. Mark the Cut Position
   Understand where cut/punch is made after folding

4. Apply Mirror Logic
   Replicate cut symmetrically for each fold

5. Unfold Step by Step
   Visualize each unfolding and mirror duplication

🧩 Solved Examples with Diagram Descriptions

🔷 Example 1: Vertical Fold with One Cut

Instruction:
A square paper is folded vertically (left to right). A triangle is cut on the right edge.

Fold → ▲ cut → Unfold

✅ Solution:
When unfolded, the cut triangle will mirror on the left side.

✅ Answer: Two symmetric triangles on left and right edges.

🔷 Example 2: Horizontal Fold with One Hole Punch

Instruction:
A square is folded horizontally top to bottom. A circular hole is punched near the bottom edge.

✅ Solution:
When unfolded, a symmetric circle appears at the top edge too.

✅ Answer: Two circles — one on top, one on bottom

🔷 Example 3: Two Folds and One Cut

Instruction:

• First folded vertically, then horizontally (quarter fold)

• One small square is cut at bottom-right corner

• Unfold completely

✅ Solution:
Each fold mirrors the cut, so 1 cut becomes 4 identical cuts in all corners

✅ Answer: 4 squares in 4 corners

🔷 Example 4: Diagonal Fold with Cut

Instruction:

• Folded diagonally from top-left to bottom-right

• One triangle cut along the open edge

✅ Solution:
When unfolded, the cut triangle mirrors along the diagonal

✅ Answer: Two triangles in a symmetric diagonal pattern

📊 Diagrams (Described Visually)

Imagine a square like this:

+---------+
|         |
|         |
|         |
+---------+

• Vertical fold: left meets right → mirror cuts left-right

• Horizontal fold: top meets bottom → mirror cuts up-down

• Diagonal fold: corner meets opposite corner → diagonal mirror

💡Shortcut Tips

Trick	Use When
Mirror logic	Any single fold
4× replication	Double fold (quarter)
Trace cut position mentally	For exact placement after unfolding
Eliminate asymmetry	Wrong options often have asymmetry

📘 Practice Pattern Examples

Fold Type	Cut Shape	Unfold Result
Vertical Fold	▲	Two triangles at left and right
Horizontal Fold	○	Two circles at top and bottom
Vertical + Horz	⬛	Four small squares in all corners
Diagonal Fold	♦	Two diamonds mirrored diagonally

📝 Strategy Table

Step	What to Check
1	Number & direction of folds
2	Where the cut/hole is placed
3	Apply symmetry rules
4	Unfold one fold at a time mentally

📘 Practice Questions – Paper Cut/Fold

Q1: Single Horizontal Fold + Punch

Problem:
A square paper is folded horizontally (bottom edge up to top edge), then a hole is punched as shown. What’s the unfolded pattern?

Diagram:

Folded:    Punched:  
┌───┐      ┌───┐  
│   │      │ ● │  
└───┘      └───┘  

Answer:
Two holes, symmetric across the horizontal midline.

Unfolded:  
┌───┐  
│ ● │  
│   │  
│ ● │  
└───┘  

Q2: Diagonal Fold + Corner Cut

Problem:
A square is folded diagonally (top-right to bottom-left), then the top corner is cut off. Unfolded shape?

Diagram:

Folded:    Cut:  
  ◇          ◇  
 / \        /   
/   \      /    

Answer:
Two cut corners symmetric across the diagonal.

Unfolded:  
□◆  
◆□  

Q3: Double Fold (Vertical + Horizontal)

Problem:
Paper is folded vertically, then horizontally, and a hole is punched at the centre. Unfolded result?

Diagram:

Folded:    Punched:  
┌ ┬ ┐      ┌ ┬ ┐  
│ │ │      │ ● │  
└ ┴ ┘      └ ┴ ┘  

Answer:
Four holes at the centre of each quadrant.

Unfolded:  
● │ ●  
──┼──  
● │ ●  

Q4: Triangle Fold + Edge Punch

Problem:
A square is folded into a right triangle (bottom-left to centre), then punched at the folded edge. Unfolded pattern?

Diagram:

Folded:    Punched:  
┌────┐     ┌────┐  
│ /  │     │ /● │  
└────┘     └────┘  

Answer:
Two holes along the diagonal and its mirror position.

Unfolded:  
□●  
●□  

Q5: Hole Punch After Z-Fold

Problem:
Paper is folded in a Z-shape (top 1/3 down, bottom 1/3 up), then punched through all layers at the centre. Unfolded result?

Diagram:

Folded:    Punched:  
┌─┐        ┌─┐  
│ │        │●│  
├─┤        ├─┤  
│ │        │●│  
└─┘        └─┘  

Answer:
Three holes in a vertical line.

Unfolded:  
●  
●  
●  

Q6: Corner Fold + Semi-Circle Cut

Problem:
The top-right corner is folded to the centre, then a semi-circle is cut from the folded edge. Unfolded shape?

Diagram:

Folded:    Cut:  
□          □  
└─┐        ╲  
  │         ╲  

Answer:
Two quarter-circles forming a semi-circle.

Unfolded:  
□○  

Q7: Double Diagonal Fold

Problem:
Paper is folded along both diagonals, then a hole is punched where they intersect. Unfolded pattern?

Diagram:

Folded:    Punched:  
  ✧          ✧  
 / \        / \  
●   ●      ●   ●  
 \ /        \ /  

Answer:
Four holes at the midpoints of each edge.

Unfolded:  
● □ ●  
□   □  
● □ ●  

Q8: Rectangular Fold + Corner Punch

Problem:
A rectangle is folded vertically (left to right), then the top-right corner is punched. Unfolded result?

Diagram:

Folded:    Punched:  
┌──┐       ┌──┐  
│  │       │  ■  
└──┘       └──┘  

Answer:
Two holes at both top corners.

Unfolded:  
■  ■  

Q9: Hexagon Fold + Centre Punch

Problem:
A hexagon is folded into a 60° sector, then punched at the centre. Unfolded pattern?

Diagram:

Folded:    Punched:  
  ◢          ◢  
 /          /●  
└────┘     └────┘  

Answer:
Six holes at the vertices.

Unfolded:  
 ●  
●  ●  
 ●  

Q10: Accordion Fold + Edge Cut

Problem:
Paper is folded like an accordion (4 layers), then a triangle is cut from the open edge. Unfolded shape?

Diagram:

Folded:    Cut:  
┌─┐        ┌─┐  
│ │        │◿  
└─┘        └─┘  

Answer:
Four triangles along the edge.

Unfolded:  
◿◿◿◿