Paper Folding & Cutting
🔒 Log in to trackCounting layers and holes
🔒 Log in to trackMany questions only need a count. Track the number of layers under each punch:
- a full half-fold (vertical, horizontal or diagonal through the centre) doubles the layers everywhere on the folded result;
- a partial fold (a strip folded over) doubles layers only inside the covered strip.
Total holes = Σ (layers under each punch). With n full half-folds and p punches, holes = p × 2ⁿ.
Detailed notes
Count first, then check positions
Many questions only ask how many holes appear, or how many layers of paper a punch goes through. Even in figure questions, counting first removes most wrong options quickly.
Rule: holes after unfolding = the sum, over all punches, of the number of layers under that punch.
Full folds
A full fold lays one half exactly on the other half, so the number of layers doubles everywhere:
| Full folds | Layers | Holes from 1 punch |
|---|---|---|
| 1 | 2 | 2 |
| 2 | 4 | 4 |
| 3 | 8 | 8 |
| 4 | 16 | 16 |
So with p punches and n full folds: . A diagonal fold of a square or of a folded square also counts as a full fold, because the two halves match exactly.
Partial folds
If only a strip is folded over, layers double only inside the strip. Work punch by punch:
- punch inside the doubled strip: 2 layers (or double what was there before);
- punch outside it: the old number of layers.
Example: the top quarter is folded down, then the whole sheet is folded in half left over right. Where the flap lies the paper has 2 × 2 = 4 layers; below the flap it has 2. One punch in each part gives 4 + 2 = 6 holes.
Three-way (letter) fold
A sheet folded into three equal strips, like a letter going into an envelope, has 3 layers everywhere. Folding that in half again gives 6 layers. Not every count is a power of 2.
Punches on a crease
A punch exactly on a crease joins the two layers that meet there, so the copies merge into one hole. Count the folds whose crease does not pass through the punch, and double only for those.
- Quarter fold, punch on one crease: 2 holes (not 4).
- Quarter fold, punch at the point where both creases meet: 1 hole.
Doubling chain
Write the count as you read each step: 1 → 2 → 4 → 8. For a strip fold, write two chains, one for the doubled part and one for the single part. Multiply by the punches in each part and add.
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 × 4 = 12. But if only the left quarter had been folded over first, only punches made inside that doubled quarter would double, and the count would be smaller.
Common mistakes
- Adding 2 for each fold instead of doubling.
- Doubling a punch that is outside a folded strip.
- Counting 4 holes for a punch that sits on a crease.
- Thinking layers must be a power of 2 after a three-way fold.
Quick revision
- Holes = sum of the layers under each punch.
- n full folds, p punches: H = p × 2ⁿ.
- Strip fold: double only inside the strip.
- Three-way fold: 3 layers; then halved: 6 layers.
- Crease punch: do not double for a crease through the punch.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Holes after full foldsvery common2 practice Q
Text only: 'folded in half n times', 'p holes punched'.
Every full fold doubles the layers, so each punch becomes 2ⁿ holes.
Example: Q. A sheet is folded in half three times and 2 holes are punched. How many holes after unfolding?
2 × 2³ = 16.
Type 2: Holes after a strip foldcommon2 practice Q
Only a strip or a quarter of the sheet is folded over, and punches are in different parts.
Find the layers in each part separately (doubled only where the flap lies). Add the layers under each punch.
Example: Q. The left quarter of a sheet is folded onto the next quarter. 2 punches are made on the doubled part and 1 on the single part. Holes?
2 × 2 + 1 × 1 = 5.
Type 3: Punch on a creaseoccasional2 practice Q
The question says the punch is made exactly on a fold line or at the folded corner.
Copies across a crease through the punch merge. Double only for creases that do not pass through the punch.
Example: Q. A sheet folded into quarters is punched exactly at the folded corner. Holes?
1 hole: both creases pass through the punch.
Type 4: Number of layerscommon2 practice Q
Asked for layers or thickness at a point, or a three-way fold appears.
Doubling for full folds; a three-way fold gives 3 layers; multiply the factors of each step.
Example: Q. A sheet is folded into three equal strips and then folded in half. How many layers?
3 × 2 = 6.
Type 5: Hole count from a figurecommon2 practice Q
Folding steps are drawn; the options are numbers.
Read each drawn step as full, strip or diagonal; count layers under each punch mark; add.
Example: Q. The figure shows a quarter fold and 2 punches. How many holes?
2 × 4 = 8, unless a punch sits on a crease.
Formulas
p punches through n full half-folds.
Shortcut tricks
⚡ Doubling chain
Write 1 → 2 → 4 → 8 as you read each full fold; multiply by the number of punches at the end.
Example: Q. Folded in half three times, 2 punches. Holes?
Sol. 2 × 2³ = 16.
p × 2ⁿ for full folds.
Where students lose marks
Adding 2 per fold instead of doubling.
Doubling a punch that sits outside a partially folded strip.
Practice sets — 14 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 10 questions
Suggested time 6 min · wrong answers go to your mistake notebook automatically.