Paper Folding & Cutting

medium importance~0-1 Q in Tier 12 formulas⚡ 4 shortcuts2 subtopics

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Folding and punching: unfolding the pattern

⏱ 6 min read🧩 8 question types🎯 22 practice Q

The idea in one minute

When a folded sheet is punched, the punch goes through every layer under it. To open the sheet, undo the folds in reverse order. Each unfold gives every hole a mirror copy across that fold line. Compare the positions of the holes with the options, not just their number.

What a punch does

A punch goes through every layer of paper under it. When you open the sheet, each layer shows its own hole.

So the job is to find where each layer came from.

The fold line works like a mirror. A point on the folded-over part sits at the same distance from the line, on the other side.

Rule: Open the folds in reverse order. Each unfold gives every hole a mirror copy across that fold line.

The method

Read the folds in order. Mark the punch on the last drawing.
Undo the last fold first. Draw a mirror copy of every hole across that fold line.
Undo the fold before it in the same way.
Stop when the sheet is flat. Compare the positions of the holes with the options.

Each full fold doubles the holes. One fold gives 2 holes per punch, two folds give 4, three folds give 8.

Where the copies go

Vertical fold line: the copy keeps its height and moves to the other side.
Horizontal fold line: the copy keeps its left-right place and moves up or down.
Diagonal fold line: the copy swaps sides. A hole near the middle of the left edge gets a copy near the middle of the top edge.

Example: a sheet is folded left over right. A hole near the top-right corner gets a copy near the top-left corner, at the same height.

Quarter fold

Fold in half, then in half again. Now there are 4 layers. One punch opens into 4 holes, one in each quarter.

Punch near the open corner: the 4 holes sit near the 4 corners of the sheet.
Punch near the folded corner: the 4 holes sit close together round the centre.

Tip: The four holes are mirror images round both centre lines. If your figure is not symmetric, it is wrong.

Strip folds

Sometimes only a strip is folded over. Only the paper under the strip has two layers.

Punch inside the doubled strip: 2 holes.
Punch outside the strip: 1 hole.

This is the most common trap in option questions. Check where the flap lies before you double.

Punch on a fold line

A punch exactly on a crease does not make two holes across that crease. The two copies meet at the same place, so one hole opens on the crease.

Quarter fold, punch at the centre where both creases meet: 1 hole.
Quarter fold, punch on one crease only: 2 holes.

Watch: Count only the folds whose crease does not pass through the punch.

Cutting instead of punching

Cutting works the same way. A cut piece opens into a shape that is symmetric about the fold line.

A notch cut with one side on the fold opens into one block, twice as wide. A notch cut away from the fold opens into two separate pieces.

Worked example

A square sheet is folded left over right, then top over bottom. It is punched once near the open corner of the folded square.

Undo the top-over-bottom fold: a copy appears above the first hole. Undo the left-over-right fold: both holes get copies on the left. The result is 4 holes, one near each corner of the sheet.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Single half fold, then punch

How to spot it:

One dashed line through the middle of the sheet and one arrow. One or two punches on the folded half.

Method

Note the fold line and the punch.
Reflect the hole across the fold line.
Keep the copy at the same distance from the line.

Why it works:

Each punch goes through 2 layers, so it makes 2 holes: the one you see and its mirror copy.

Try this

A square sheet is folded left over right and punched once near the top-right corner. Where are the holes after unfolding?

Show solution

The fold line is vertical, so height stays the same.
The mirror copy is on the left side, same distance from the line.

Answer

One hole near the top-right corner and one near the top-left corner

Type 2very common2 practice Q

Quarter fold, then punch

How to spot it:

Two folds: one half onto the other, then the result halved again at right angles.

H = p \times 2^{n}

Method

Undo the second fold first: copy the hole across the second crease.
Undo the first fold: copy both holes across the first crease.
You now have 4 holes, mirror images round both centre lines.

Why it works:

There are 4 layers under the punch, so one punch makes 4 holes.

Try this

A square sheet is folded in half left over right, then in half top over bottom. It is punched once near the open corner of the folded square. How many holes are there, and where?

Show solution

Layers: $2 \times 2 = 4$ .
The open corner of the folded square is a corner of the sheet.
The mirror copies land near the other three corners.

Answer

4 holes, one near each corner of the sheet

Type 3common2 practice Q

Diagonal fold

How to spot it:

A dashed line from corner to corner. The folded sheet is a triangle.

Method

Draw the diagonal line.
Reflect the hole across it. Distance to the line stays the same.
A hole near the middle of one edge moves to the middle of the neighbouring edge.

Why it works:

The diagonal is the mirror. Edges that meet at the corner on the line swap places.

Try this

A square sheet is folded along the diagonal from top-left to bottom-right. The top-right half is folded over. A punch is made near the middle of the left edge of the folded triangle. Where is the second hole?

Show solution

The mirror line runs from top-left to bottom-right.
The left edge and the top edge meet at the top-left corner and swap places.
The mirror copy is at the same distance from the corner, along the top edge.

Answer

Near the middle of the top edge

Type 4common2 practice Q

Strip (partial) fold

How to spot it:

The dashed line is not through the middle. Only a narrow strip is folded over.

Method

Mark the part of the sheet the flap covers.
Punches inside the covered part make 2 holes.
Punches outside it make 1 hole.

Why it works:

Only the covered part has two layers. The rest is still one layer.

Try this

The left quarter of a sheet is folded over to the right, on top of the second quarter. One punch is made on the doubled part and one on the single part. How many holes are there after unfolding?

Show solution

Doubled part: 2 layers, so 2 holes.
Single part: 1 layer, so 1 hole.

Answer

3 holes

Type 5occasional2 practice Q

Three folds

How to spot it:

Three folding steps: often a quarter fold followed by a diagonal fold, or three half folds.

H = p \times 2^{3}

Method

Write the layers after each fold: 2, 4, 8.
Undo the last fold first, then the second, then the first.
Check that the 8 holes are mirror images round every crease.

Why it works:

Each full fold doubles the layers. Three folds make 8 layers.

Try this

A sheet is folded in half, then in half again, then along a diagonal of the folded square. One punch is made away from every crease. How many holes are there?

Show solution

Layers: $2 \times 2 \times 2 = 8$ .

Answer

8 holes

Type 6occasional2 practice Q

Punch on a fold line

How to spot it:

The punch sits exactly on an edge of the folded piece where the paper bends.

Method

Find which creases pass through the punch.
Do not double for those creases.
Double for every other fold.

Why it works:

Copies across a crease through the punch land on the same spot, so they merge.

Try this

A sheet folded into quarters is punched at the folded corner, which is the centre of the sheet. How many holes are there?

Show solution

Both creases pass through the punch.
No fold doubles it.

Answer

1 hole, at the centre

Type 7common

Cut a shape from the fold

How to spot it:

A small piece is cut out of the folded paper, often a notch or a corner.

Method

Draw the cut shape on the folded paper.
Undo the last fold: mirror the shape across that crease.
A shape touching the crease joins its copy into one piece.

Why it works:

Cutting goes through every layer, just as a punch does. The opened shape is symmetric about each crease.

Try this

A square sheet is folded in half, left over right. A small square notch is cut with one side along the fold line. After unfolding, how many separate cut-outs are there, and how wide is the cut-out compared with the notch?

Show solution

The notch touches the crease, so it joins its mirror copy.
The copy is on the other side of the crease and is the same size.

Answer

One cut-out, twice as wide as the notch

Type 8very common

Pick the correct opened figure

How to spot it:

The folded sheet and punch are drawn. Four opened sheets are given as options.

Method

Count the layers. Remove options with the wrong number of holes.
Check the symmetry: the holes must mirror across every crease.
Check the positions, near a corner or near the centre.

Why it works:

A correct opened figure is symmetric about every fold line.

Try this

A square sheet is folded in half left over right, then in half top over bottom. One punch is made near the open corner. Which opened figure is correct? (1) four holes, one near each corner of the sheet (2) two holes near the top corners (3) four holes close together near the centre (4) one hole near the bottom-right corner

Show solution

Two folds give 4 layers, so 4 holes. Options 2 and 4 are wrong.
The punch is near the open corner, a corner of the sheet, not the centre. Option 3 is wrong.

Answer

Option 1

Shortcuts that save time

⚡ Reflect, do not rotate

Each unfold is a mirror. The copy is at the same distance from the fold line, on the other side. A half turn of the flap is wrong.

Example

A sheet is folded left over right and punched near the top-right corner. Where are the holes after unfolding?

Show solution

The fold line is vertical, so height stays the same.
The copy is on the left, at the same distance from the fold line.

Answer

One hole near the top-right and one near the top-left, at the same height

⚡ Count first, then check positions

Holes = punches times layers under each punch. Use the count to remove wrong options, then check positions.

Example

A sheet is folded in half twice and punched once away from the creases. How many holes?

Show solution

Layers: $2 \times 2 = 4$ .
One punch goes through 4 layers.

Answer

4 holes

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Undoing the folds in the original order.

Undo the last fold first. Work backwards.

Mistake 02

Turning the flap half a turn instead of reflecting it.

Copy each hole as a mirror image across the fold line.

Mistake 03

Doubling a punch that lies outside a folded strip.

Only the paper under the strip has two layers. Check where the flap lies.

Mistake 04

Making 2 or 4 holes for a punch that sits on a crease.

Copies across a crease through the punch land on the same spot. They merge into one hole.

Mistake 05

Checking only the number of holes.

Two options can have the same number of holes. Check the positions too.

Quick revision

Read this the night before the exam.

A punch goes through all layers. Each layer makes one hole.
Unfold in reverse order. Each unfold mirrors the holes across that fold line.
Vertical fold keeps height. Horizontal fold keeps left-right place.
Full folds: 1, 2, 4, 8 holes per punch.
Strip fold: double only the punches under the strip.
Punch on a crease: the copies merge into one hole.

Practice: 22 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

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