Figure series, mirrors, embedded shapes and paper folding.
About Paper Cutting
A paper cutting question folds a sheet, punches a hole through the folded layers, and then asks which pattern of holes the opened sheet shows. Opening the sheet reflects every hole in the fold line, so the answer is found by mirroring the punched hole rather than by guessing at the pattern.
What you need to understand
Folding the sheet in half puts two layers under the punch, so one hole becomes two when the sheet is opened.
Opening the sheet reflects each hole in the fold line. The two holes are the same distance from that line, on opposite sides of it.
The fold line runs down the middle of the sheet when the sheet is folded in half, so the two holes are mirror images of each other in the middle line.
A hole punched exactly on the fold line gives only one hole, because the two halves of it coincide when the sheet is opened.
Folding twice at right angles puts four layers under the punch, so one hole becomes four.
The height of a hole does not change when the sheet is opened, if the fold runs vertically. Only the distance across the fold changes.
A hole has a shape as well as a position, and the shape is mirrored too. A triangular punch leaves a triangle pointing the other way.
How to work through these questions
Find the fold line and decide whether the sheet was folded across it vertically or horizontally.
Count the layers under the punch. One fold means two layers and therefore two holes.
Mark where the punched hole sits and measure how far it is from the fold line.
Mirror it: place a second hole the same distance on the other side of the fold line.
Check the count. Two holes for one fold, four for two folds at right angles, one if the punch sat on the fold line.
Compare with the options, discarding any that shows the wrong number of holes.
Between the options that remain, check the distance from the fold line, which is what separates a correct mirror from a wrong one.
Mistakes that cost marks
Showing only the hole that was punched and forgetting that it goes through every layer.
Mirroring the hole about the edge of the sheet instead of about the fold line.
Mirroring about the other axis, which puts both holes in the wrong places while keeping the count right.
Giving four holes when the sheet was folded only once.
Placing the second hole at the same distance from the edge of the sheet rather than from the fold line.
Forgetting that the shape of the hole is mirrored as well, so a triangle comes out reversed.
Worked example
A square sheet of paper is folded in half with its right half laid exactly on its left half. A hole is punched through both layers 10 cm from the left edge and 5 cm from the top. Which holes are seen when the sheet is opened out flat?
The fold line is the vertical line down the middle of the sheet, since the right half was brought across to the left.
Two layers lie under the punch, so one hole becomes two holes when the sheet is opened.
The punched hole is 10 cm from the left edge, and the sheet is 20 cm wide, so the fold line is 10 cm from the left edge.
The hole is therefore 0 cm from the fold line, which means it sits on the fold, and the two halves of it coincide so only one hole appears. Change the position so it is 6 cm from the left edge instead.
Now the hole is 4 cm from the fold line, so the second hole appears 4 cm on the other side, which is 14 cm from the left edge.
Both holes are 5 cm from the top, because folding vertically does not move anything up or down.
The opened sheet shows two holes, both 5 cm from the top, one 6 cm from the left edge and the other 14 cm from the left edge.
Answer: Two holes, both at the same height, the same distance on either side of the fold line.
Practice questions with answers
A few Paper Cutting questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
A square sheet of paper is folded in half so that its bottom half lies exactly on its top half. A single hole is then punched through both layers at the position marked, and the sheet is opened out flat again.
Which pattern of holes does the opened sheet show?
A
B
C
D
Answer: Option A — with explanation
Opening the sheet reflects the punched hole in the fold, which is the horizontal line across the middle of the sheet. The hole therefore appears twice: where it was punched, and again at the same distance on the other side of that line.
Nothing else on the sheet is disturbed, so the opened sheet shows exactly those holes and no others.
Common mistakes
- opened the sheet about the vertical line down the middle of the sheet instead of about the horizontal line across the middle of the sheet
- opened the sheet along both lines, when it was folded along one
- left the sheet folded, so only the hole that was punched is shown
Question 2
A square sheet of paper is folded in half so that its bottom half lies exactly on its top half. A single hole is then punched through both layers at the position marked, and the sheet is opened out flat again.
Which pattern of holes does the opened sheet show?
A
B
C
D
Answer: Option C — with explanation
Opening the sheet reflects the punched hole in the fold, which is the horizontal line across the middle of the sheet. The hole therefore appears twice: where it was punched, and again at the same distance on the other side of that line.
Nothing else on the sheet is disturbed, so the opened sheet shows exactly those holes and no others.
Common mistakes
- opened the sheet along both lines, when it was folded along one
- opened the sheet about the vertical line down the middle of the sheet instead of about the horizontal line across the middle of the sheet
- left the sheet folded, so only the hole that was punched is shown
Question 3
A square sheet of paper is folded in half so that its bottom half lies exactly on its top half. A single hole is then punched through both layers at the position marked, and the sheet is opened out flat again.
Which pattern of holes does the opened sheet show?
A
B
C
D
Answer: Option C — with explanation
Opening the sheet reflects the punched hole in the fold, which is the horizontal line across the middle of the sheet. The hole therefore appears twice: where it was punched, and again at the same distance on the other side of that line.
Nothing else on the sheet is disturbed, so the opened sheet shows exactly those holes and no others.
Common mistakes
- opened the sheet about the vertical line down the middle of the sheet instead of about the horizontal line across the middle of the sheet
- left the sheet folded, so only the hole that was punched is shown
- opened the sheet along both lines, when it was folded along one
Question 4
A square sheet of paper is folded in half so that its bottom half lies exactly on its top half. A single hole is then punched through both layers at the position marked, and the sheet is opened out flat again.
Which pattern of holes does the opened sheet show?
A
B
C
D
Answer: Option B — with explanation
Opening the sheet reflects the punched hole in the fold, which is the horizontal line across the middle of the sheet. The hole therefore appears twice: where it was punched, and again at the same distance on the other side of that line.
Nothing else on the sheet is disturbed, so the opened sheet shows exactly those holes and no others.
Common mistakes
- opened the sheet along both lines, when it was folded along one
- left the sheet folded, so only the hole that was punched is shown
- opened the sheet about the vertical line down the middle of the sheet instead of about the horizontal line across the middle of the sheet
Question 5
A square sheet of paper is folded in half so that its right half lies exactly on its left half. A single hole is then punched through both layers at the position marked, and the sheet is opened out flat again.
Which pattern of holes does the opened sheet show?
A
B
C
D
Answer: Option B — with explanation
Opening the sheet reflects the punched hole in the fold, which is the vertical line down the middle of the sheet. The hole therefore appears twice: where it was punched, and again at the same distance on the other side of that line.
Nothing else on the sheet is disturbed, so the opened sheet shows exactly those holes and no others.
Common mistakes
- opened the sheet along both lines, when it was folded along one
- opened the sheet about the horizontal line across the middle of the sheet instead of about the vertical line down the middle of the sheet
- left the sheet folded, so only the hole that was punched is shown
Frequently asked questions
Why does one hole become two?
Because the punch goes through every layer that is stacked at that point. Folding the sheet in half puts two layers there, and each layer keeps its own hole when the sheet is opened.
How far apart are the two holes?
Twice the distance from the punched hole to the fold line, measured across the fold. If the hole is 4 cm from the fold line, the two holes are 8 cm apart.
What if the hole is punched on the fold line itself?
Then the two halves of the hole coincide when the sheet is opened and only one hole appears. A punch on the fold line is the one case that gives a single hole.
What happens when the sheet is folded twice?
Two folds at right angles put four layers under the punch, so one hole becomes four when the sheet is opened. Two folds along the same line put three layers there and give three holes.
Does the shape of the hole matter?
Yes. Each hole is a mirror image of the punched hole, so a shape with a direction to it comes out reversed in the copy.
Take the Paper Cutting test
Two timed papers on the same syllabus — sit the foundation paper first,
then the advanced one. Both use the real exam paper format with a full
step-by-step review of every question once you submit.
