Paper Folding & Cutting

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

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Counting layers and holes

⏱ 5 min read🧩 7 question types🎯 14 practice Q

The idea in one minute

Many questions only ask how many holes or layers. Count the layers under each punch. A full fold doubles the layers everywhere. A strip fold doubles them only under the strip.

Count first

Many questions only ask how many holes appear, or how many layers a punch goes through. Even in figure questions, counting first removes wrong options fast.

Rule: Holes after unfolding = the layers under each punch, added up.

Full folds

A full fold lays one half exactly on the other half. The layers double everywhere.

Full folds	Layers	Holes from 1 punch
1	2	2
2	4	4
3	8	8
4	16	16

With $p$ punches and $n$ full folds, holes $= p \times 2^{n}$ .

A diagonal fold of a square, or of a folded square, is also a full fold. The two halves match exactly.

Strip folds

If only a strip is folded over, the layers double only under the strip. Work punch by punch.

Punch under the strip: the old layers doubled.
Punch outside the strip: the old layers.

Example: fold the left quarter over the second quarter. Then punch 2 holes on the doubled part and 1 on the single part. Holes: $2 \times 2 + 1 \times 1 = 5$ .

Three-way fold

Fold a sheet into three equal strips, like a letter. Every part now has 3 layers. Fold that in half and you have 6.

Watch: Not every layer count is a power of 2. A three-way fold gives 3.

Punch on a crease

A punch exactly on a crease joins the two layers that meet there. The copies merge into one hole.

Quarter fold, punch on one crease: 2 holes, not 4.
Quarter fold, punch where both creases meet: 1 hole.

Doubling chain

Write the layers as you read each fold: 1, 2, 4, 8.

For a strip fold, write two chains. One is for the doubled part and one for the single part. Multiply by the punches in each part and add.

Tip: Check every punch for a crease before you multiply.

Then the opened sheet needs a check. The holes must be mirror images round every crease. If a hole has no partner across a crease that passed under it, the count is wrong.

Diagonal folds

Fold a folded square along its diagonal. The two halves match, so the layers double again.

A punch inside the triangle goes through all the layers. Two straight folds and a diagonal give $2 \times 2 \times 2 = 8$ .

If the punch is on the diagonal crease, that fold does not double it.

Worked example

A sheet is folded in half, and in half again. Then 3 holes are punched through all the layers.

Layers: 2 after the first fold, 4 after the second. Holes: $3 \times 4 = 12$ .

If only the left quarter had been folded over first, only punches inside that doubled quarter would double. The count would be smaller.

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

Holes after full folds

How to spot it:

Text only: folded in half n times, p holes punched.

H = p \times 2^{n}

Method

Count the full folds, n.
Layers $= 2^{n}$ .
Multiply by the number of punches.

Why it works:

Every full fold doubles the layers, so each punch becomes $2^{n}$ holes.

Try this

A sheet is folded in half three times and 2 holes are punched. How many holes are there after unfolding?

Show solution

Layers: $2^{3} = 8$ .
Holes: $2 \times 8 = 16$ .

Answer

Type 2common2 practice Q

Holes after a strip fold

How to spot it:

Only a strip or a quarter of the sheet is folded over, and the punches are in different parts.

Method

Mark the part the flap covers.
Layers there: doubled. Layers elsewhere: unchanged.
Add the layers under every punch.

Why it works:

The strip is the only doubled part, so each punch must be counted by where it lies.

Try this

The left quarter of a sheet is folded onto the second quarter. Two punches are made on the doubled part and one on the single part. How many holes are there?

Show solution

Doubled part: $2 \times 2 = 4$ .
Single part: $1 \times 1 = 1$ .
$4 + 1 = 5$ .

Answer

Type 3occasional2 practice Q

Punch on a crease

How to spot it:

The question says the punch is on a fold line or at the folded corner.

Method

Find the creases through the punch.
Do not double for them.
Double for the other folds.

Why it works:

Copies across a crease through the punch merge into one.

Try this

A sheet is folded in half twice to make quarters. A punch is made exactly on the crease that runs across the middle of the folded piece, away from the other crease. How many holes are there?

Show solution

Only one crease passes through the punch. It does not double.
The other fold doubles once.
$1 \times 2 = 2$ .

Answer

Type 4common2 practice Q

Number of layers

How to spot it:

The question asks for layers or thickness at a point, or a three-way fold appears.

Method

List each step: half fold is 2, three-way fold is 3.
Multiply the factors.

Why it works:

Each fold multiplies the layers by its own factor.

Try this

A sheet is folded into three equal strips and then folded in half. How many layers of paper are there?

Show solution

Three-way fold: 3 layers.
Half fold: $3 \times 2 = 6$ .

Answer

Type 5common2 practice Q

Hole count from a folding figure

How to spot it:

The folding steps are drawn and the options are numbers.

Method

Read each step as a full fold, a strip fold or a diagonal fold.
Count the layers under each punch mark.
Add the layers for all the punches.

Why it works:

The count depends only on the layers, so you do not need to draw the opened sheet.

Try this

A sheet is folded into quarters. Two punches are made, neither on a crease. How many holes are there?

Show solution

Layers: $2 \times 2 = 4$ .
Holes: $2 \times 4 = 8$ .

Answer

Type 6occasional

Fold, then a diagonal fold

How to spot it:

A folded square is folded again along its diagonal, giving a triangle.

Method

Count the layers of the folded square.
The diagonal fold doubles them, because the two halves match.
A punch inside the triangle goes through all of them.

Why it works:

A diagonal of a square is a full fold, so it doubles the layers.

Try this

A sheet is folded in half, and in half again. The folded square is folded along its diagonal. One punch is made away from every crease. How many holes are there?

Show solution

Layers: $2 \times 2 = 4$ .
Diagonal fold: $4 \times 2 = 8$ .

Answer

Type 7occasional

Mixed punches on and off creases

How to spot it:

A quarter fold, with one punch near the open corner and one on a crease.

Method

Count the layers for each punch separately.
Punch off the creases: 4 layers.
Punch on one crease: 2 layers.
Add the holes.

Why it works:

Each punch has its own layer count. A crease removes one doubling.

Try this

A sheet is folded into quarters. One punch is made near the open corner. Another is made on one crease, away from the other crease. How many holes are there in all?

Show solution

Open corner punch: $4$ holes.
Crease punch: $2$ holes.
$4 + 2 = 6$ .

Answer

Formula sheet

Holes after full half-folds

H = p \times 2^{n}

p punches through n full half-folds.

Layers after full half-folds

L = 2^{n}

Each full fold doubles the layers.

Shortcuts that save time

⚡ Doubling chain

Write 1, 2, 4, 8 as you read each full fold. Multiply by the number of punches at the end.

Example

A sheet is folded in half three times. Two punches are made. How many holes?

Show solution

Layers: $2^3 = 8$ .
Holes: $2 \times 8 = 16$ .

Answer

⚡ Multiply the steps

Each step multiplies the layers: a half fold by 2, a three-way fold by 3.

Example

A sheet is folded into three equal strips and then in half. How many layers are there?

Show solution

$3 \times 2 = 6$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Adding 2 layers for each fold.

Each full fold doubles the layers. Multiply by 2, do not add 2.

Mistake 02

Doubling a punch that is outside a folded strip.

Only the paper under the strip is doubled. Work punch by punch.

Mistake 03

Counting 4 holes for a punch that sits on a crease.

Do not double for a crease that passes through the punch.

Mistake 04

Thinking the layers are always a power of 2.

A three-way fold gives 3 layers, and folding it in half gives 6.

Mistake 05

Forgetting to multiply by the number of punches.

Holes = punches times layers. Multiply at the end.

Quick revision

Read this the night before the exam.

Holes = the layers under each punch, added up.
$n$ full folds, $p$ punches: $H = p \times 2^{n}$ .
Strip fold: double only under the strip.
Three-way fold: 3 layers. Then in half: 6 layers.
Crease punch: do not double for a crease through the punch.
Diagonal fold of a square: counts as a full fold.

Practice: 14 questions

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

Topic test · 10 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.

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