Paper Folding & Cutting
🔒 Log in to trackFolding and punching: unfolding the pattern
🔒 Log in to trackWhen a folded sheet is punched, the punch goes through every layer under that spot. Unfolding is done in reverse order: undo the last fold first, and each time every existing hole gets a mirror copy across that fold line (on the part that was folded over).
Checklist:
- Note the order of folds and the final punch position.
- Undo the last fold: reflect each hole across its line.
- Repeat for each earlier fold.
- Compare positions (not just the count) with the options.
A hole lands only where the folded-over flap actually covered the punch: if a narrow strip is folded over, holes punched outside the strip stay single.
Detailed notes
What happens when you fold and punch
A punch goes through every layer of paper under it. When the sheet is opened, each layer shows its own hole. So the job is to find where each layer came from.
The fold line acts like a mirror. A point on the part that was folded over lands at the same distance on the other side of the fold line. Opening the fold sends it back. So every unfold gives each hole a mirror copy across that fold line.
The method: unfold in reverse order
- Read the folds in order and mark the final punch position(s).
- Undo the last fold first. For every hole that lies on the part that was folded over, draw its mirror copy across that fold line.
- Undo the fold before it in the same way, and so on, until the sheet is fully open.
- Now compare positions, not just the number of holes.
Each full fold doubles the holes: 1 fold gives 2 holes per punch, 2 folds give 4, 3 folds give 8.
Where the copies go
- Vertical fold line (left-right fold): the copy keeps its height and moves to the other side. A hole near the top-right gets a partner near the top-left.
- Horizontal fold line (top-bottom fold): the copy keeps its left-right position and moves up or down.
- Diagonal fold along the line from top-left to bottom-right: a point at (x, y) goes to (y, x). A hole near the middle of the top edge gets a partner near the middle of the left edge.
- Other diagonal (top-right to bottom-left) on a square of side s: (x, y) goes to (s − y, s − x).
Quarter fold
Folding in half and then in half again makes 4 layers. One punch unfolds into 4 holes, one in each quarter, placed like mirror images around the two centre lines. If the punch is near the folded corner (the centre of the sheet), the 4 holes cluster near the centre. If it is near the open corners, they sit near the 4 corners of the sheet.
Partial (strip) folds
If only a strip is folded over, only the part of the sheet under that strip has two layers. A punch inside the doubled strip makes 2 holes; a punch outside it makes only 1 hole. This is the most common trap in option questions.
Punch on a fold line
A punch placed exactly on a crease does not make two separate holes across that crease; the two copies meet at the same spot and open as one hole on the crease. A punch at the centre of a sheet folded into quarters (where both creases meet) opens as just 1 hole. A punch on only one crease of a quarter fold opens as 2 holes.
Cutting instead of punching
Cutting works the same way. A notch cut from a folded edge opens into a shape that is symmetric about that fold line. A cut on a corner where two folds meet repeats in all four quarters. Draw the cut piece, then mirror it across each fold line in reverse order.
Common traps
- Rotating instead of reflecting: copies must be mirror images across the fold line, not a half turn of the flap.
- Undoing folds in the original order instead of the reverse order.
- Forgetting the last unfold, which halves the true count.
- Doubling a punch that lies outside a folded strip.
Worked example
A sheet is folded left over right, then top down onto the bottom, and punched once near the bottom-right corner of the folded piece. Undo the top-down fold: a copy appears near the top-right. Undo the left-right fold: both holes get copies on the left. Result: 4 holes, one near each corner of the sheet.
Quick revision
- A punch goes through all layers; each layer gives one hole.
- Unfold in reverse order; each unfold mirrors the holes across that fold line.
- Vertical fold keeps height, horizontal fold keeps left-right position, main diagonal swaps x and y.
- Full folds: 1 → 2 → 4 → 8 holes per punch.
- Strip fold: double only the punches inside the strip.
- Punch on a crease: the copies merge into one hole on that crease.
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: Single half fold, then punchvery common2 practice Q
One dashed line through the middle of the sheet and one arrow. One or two punches on the folded half.
Each punch makes 2 holes: the one you see and its mirror copy across the fold line, at the same distance on the other side.
Example: Q. A square sheet is folded left over right and punched once near the top-right corner. Where are the holes after unfolding?
One hole near the top-right and one near the top-left, both at the same height.
Type 2: Quarter fold, then punchvery common2 practice Q
Two folds: first one half onto the other, then the result halved again at right angles.
4 layers under the punch, so 4 holes per punch. Undo the second fold first, then the first. The 4 holes are mirror images around both centre lines.
Example: Q. A sheet folded into quarters is punched once near the open corner of the folded square. How many holes, and where?
4 holes, one near each corner of the sheet.
Type 3: Diagonal foldcommon2 practice Q
A dashed line from corner to corner. The folded sheet is a triangle.
Reflect across the diagonal. For the diagonal from top-left to bottom-right, (x, y) goes to (y, x): a hole near the top edge gets a partner near the left or right edge.
Example: Q. A square is folded along the diagonal from top-left to bottom-right, the top-right half moving over. A punch is made near the middle of the left edge of the folded triangle. Where is the second hole?
Near the middle of the bottom edge. (x, y) becomes (y, x): (1, 4) goes to (4, 1).
Type 4: Strip (partial) foldcommon2 practice Q
The dashed line is not through the middle. Only a narrow strip is folded over.
Only the part covered by the strip has 2 layers. Punches in the covered part make 2 holes; punches elsewhere make 1 hole.
Example: Q. The left quarter of a sheet is folded over to the right. One punch is made on the doubled part and one on the single part. How many holes?
2 + 1 = 3 holes.
Type 5: Three foldsoccasional2 practice Q
Three folding steps: usually a quarter fold followed by a diagonal fold, or three half folds.
8 layers under a punch away from the creases. Undo the diagonal (last) fold first, then the two straight folds.
Example: Q. A sheet is folded in half, then in half again, then along a diagonal. One punch is made. How many holes?
2 × 2 × 2 = 8 holes.
Type 6: Punch on a fold lineoccasional2 practice Q
The punch mark sits exactly on an edge of the folded piece where the paper bends.
Copies across that crease land on the same spot, so they merge into one hole on the crease. Count only the folds whose crease does not pass through the punch.
Example: Q. A sheet folded into quarters is punched exactly at the folded corner (the centre of the sheet). How many holes?
Only 1 hole, at the centre, because both creases pass through the punch.
Shortcut tricks
⚡ Reflect, do not rotate
Each unfold is a mirror reflection across the fold line — a hole near the top-right, unfolded across a vertical midline, reappears near the top-left at the same height.
Example: Q. A sheet is folded left-over-right and punched near the top-right corner. Where are the holes after unfolding?
Sol. Reflect across the vertical midline: one hole top-right, one top-left, both at the same height.
Same distance from the fold line on the other side.
⚡ Layer count shortcut
Holes in the final sheet = punches × layers at the punched spot. Each full half-fold doubles the layers: 1 fold → 2, 2 folds → 4, 3 folds → 8.
Example: Q. A sheet folded in half twice is punched once. How many holes?
Sol. Layers = 2 × 2 = 4 → 4 holes.
Count first to eliminate options, then check positions.
Where students lose marks
Unfolding in the original order instead of the reverse order.
Rotating hole positions instead of reflecting them.
Assuming every punch doubles, even where the flap does not reach.
Practice sets — 22 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 9 min · wrong answers go to your mistake notebook automatically.