Figure series, mirrors, embedded shapes and paper folding.
About Paper Folding
A paper folding question puts a mark on a sheet of paper, folds the sheet once, and asks where the mark ends up. The half that is carried over takes its marks with it and reflects them in the fold line, while the half that stays put keeps its marks exactly where they were.
What you need to understand
Folding a sheet in half puts the fold line down the middle, so the two halves of the sheet are mirror images in it.
A mark on the half that moves is carried across the fold line and comes to rest the same distance on the other side of it.
The distance from the fold line does not change, only the side the mark is on.
A mark keeps its place along the fold line. Folding a sheet across a vertical line moves a mark sideways and does not move it up or down at all.
A mark on the half that stays put does not move, even though the other half of the sheet has come to rest on top of it.
A mark exactly on the fold line does not move either, because a point on the line is already on both halves.
The mark always finishes inside the folded packet, since that is where the paper now lies.
How to work through these questions
Find the fold line. Folding in half puts it through the middle of the sheet.
Decide which half moved, from the shaded area that shows where the folded half came to rest.
Say which side of the fold line the mark starts on, and whether that side moved.
If the mark is on the half that stays put, it does not move at all and the answer is where it started.
If the mark is on the half that moves, measure its distance from the fold line and place it that same distance on the other side.
Keep its position along the fold line unchanged, so a sideways fold moves the mark only sideways.
Check the answer lies inside the shaded area, which is where the paper now sits.
Mistakes that cost marks
Moving the mark when it is on the half that does not move.
Reflecting the mark about the other axis, which keeps the distance right but puts it in the wrong direction.
Moving the mark by its distance from the fold line rather than to the mirrored position, so it lands on the fold line instead of across it.
Changing the mark position along the fold line as well as across it.
Assuming the mark always moves, when a mark on the stationary half does not.
Putting the mark outside the shaded area, where there is no paper after folding.
Worked example
A square sheet 20 cm wide carries a mark 15 cm from its left edge and 4 cm from its top edge. Its right half is folded over onto its left half. Where does the mark end up?
The sheet is folded in half, so the fold line is the vertical line 10 cm from the left edge.
The right half was folded onto the left half, so the right half is the part that moves.
The mark is 15 cm from the left edge, which puts it on the right half, so it is on the part that moves.
It is 5 cm to the right of the fold line, because 15 - 10 = 5.
After folding it lies 5 cm to the left of the fold line, which is 10 - 5 = 5 cm from the left edge.
Folding across a vertical line does not move the mark up or down, so it stays 4 cm from the top edge.
The mark ends up 5 cm from the left edge and 4 cm from the top edge.
Answer: 5 cm from the left edge and 4 cm from the top edge.
Practice questions with answers
A few Paper Folding 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 carries a mark, shown by the dot. The sheet is then folded once, and the shaded area shows where the half that moved came to rest.
Where does the mark end up?
A
B
C
D
Answer: Option A — with explanation
The shaded area shows that its right half was folded over onto its left half, so the fold line is the vertical line through the middle of the sheet.
The mark begins on the half that moves, so it is carried across that line and comes to rest the same distance on the other side of it, at the same height.
Common mistakes
- folded the mark about the other axis of the sheet
- carried the mark to the opposite corner instead of across the fold
- left the mark where it was, as though the sheet had not been folded
Question 2
A square sheet of paper carries a mark, shown by the dot. The sheet is then folded once, and the shaded area shows where the half that moved came to rest.
Where does the mark end up?
A
B
C
D
Answer: Option B — with explanation
The shaded area shows that its bottom half was folded up onto its top half, so the fold line is the horizontal line through the middle of the sheet.
The mark begins on the half that moves, so it is carried across that line and comes to rest the same distance on the other side of it, at the same position across.
Common mistakes
- folded the mark about the other axis of the sheet
- left the mark where it was, as though the sheet had not been folded
- carried the mark to the opposite corner instead of across the fold
Question 3
A square sheet of paper carries a mark, shown by the dot. The sheet is then folded once, and the shaded area shows where the half that moved came to rest.
Where does the mark end up?
A
B
C
D
Answer: Option B — with explanation
The shaded area shows that its bottom half was folded up onto its top half, so the fold line is the horizontal line through the middle of the sheet.
The mark begins on the half that moves, so it is carried across that line and comes to rest the same distance on the other side of it, at the same position across.
Common mistakes
- folded the mark about the other axis of the sheet
- left the mark where it was, as though the sheet had not been folded
- carried the mark to the opposite corner instead of across the fold
Question 4
A square sheet of paper carries a mark, shown by the dot. The sheet is then folded once, and the shaded area shows where the half that moved came to rest.
Where does the mark end up?
A
B
C
D
Answer: Option C — with explanation
The shaded area shows that its top half was folded down onto its bottom half, so the fold line is the horizontal line through the middle of the sheet.
The mark begins on the half that moves, so it is carried across that line and comes to rest the same distance on the other side of it, at the same position across.
Common mistakes
- folded the mark about the other axis of the sheet
- carried the mark to the opposite corner instead of across the fold
- left the mark where it was, as though the sheet had not been folded
Question 5
A square sheet of paper carries a mark, shown by the dot. The sheet is then folded once, and the shaded area shows where the half that moved came to rest.
Where does the mark end up?
A
B
C
D
Answer: Option B — with explanation
The shaded area shows that its right half was folded over onto its left half, so the fold line is the vertical line through the middle of the sheet.
The mark begins on the half that moves, so it is carried across that line and comes to rest the same distance on the other side of it, at the same height.
Common mistakes
- carried the mark to the opposite corner instead of across the fold
- folded the mark about the other axis of the sheet
- left the mark where it was, as though the sheet had not been folded
Frequently asked questions
How do I know which half moved?
The shaded area in the diagram shows where the folded half came to rest. Whichever half of the sheet is left uncovered is the half that stayed put.
Does the mark always move when the sheet is folded?
No. A mark on the half that stays put does not move, and a mark on the fold line does not move either. Only marks on the moving half are carried across.
Does the mark move up or down when the sheet is folded sideways?
No. Folding across a vertical line only changes the distance from the left edge. The distance from the top edge is unchanged.
How far is the mark from the fold line after folding?
The same distance as before, but on the other side. Folding is a reflection, and a reflection keeps distances the same.
Can the mark finish outside the folded packet?
No. After folding, the whole sheet lies within the packet, so every mark on it lies there too. An answer outside the shaded area can be discarded at once.
Take the Paper Folding 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.
