TACHS: Paper folding: one vertical fold
Learn to unfold a single up-and-down crease by flipping each hole across it, keeping its height and its distance from the crease.
Where do the holes end up when a sheet folded along a vertical crease is opened?
Short answer
Each punch makes a hole in both layers. Flip each hole across the crease: same row, same distance from the crease on the other side.
Know first: Rows and columns in a grid, Counting squares from a line
Step 1 of 5
A vertical crease is a mirror
In a paper-folding question, a square sheet is folded, holes are punched through every layer, and you choose the picture of the sheet opened up. Official TACHS examples have five answer choices.
With one vertical fold, the crease runs straight down the middle. One half folds over onto the other, so the folded paper is two layers thick. Each punch makes two holes, one in each layer.
| Hole in | Distance from crease | Partner in |
|---|---|---|
| Column 1 | 2 columns | Column 4 |
| Column 2 | 1 column | Column 3 |
| Column 3 | 1 column | Column 2 |
| Column 4 | 2 columns | Column 1 |
Picture the sheet as a 4-by-4 grid. We write a square as (row, column), so (1, 3) is row 1, column 3.
Step 2 of 5
Flip each hole, keep its row
Worked example: Two punches, one fold
A square sheet is a 4-by-4 grid. The left half folds over onto the right half, so the crease runs between columns 2 and 3 and the folded paper covers columns 3 and 4. Two holes are punched: (1, 3) and (3, 4). Which picture shows the opened sheet?
- A(1, 1), (1, 3), (3, 2), (3, 4)
- B(1, 3), (2, 4), (3, 4), (4, 3)
- C(1, 2), (1, 3), (3, 1), (3, 4)
- D(1, 2), (1, 3), (3, 4)
- E(1, 3), (3, 4)
- 1
Find the crease and the layers
The crease is between columns 2 and 3. The folded paper is two layers thick everywhere, so each punch makes 2 holes: 4 in all.
- 2
Flip the first hole
(1, 3) is 1 column right of the crease. Its partner is 1 column left of it, in column 2, still in row 1: (1, 2).
- 3
Flip the second hole
(3, 4) is 2 columns right of the crease, at the right edge. Its partner is 2 columns left, at the left edge, still in row 3: (3, 1).
- 4
See why the other choices are there
(1, 1), (1, 3), (3, 2), (3, 4) slides each hole 2 columns left without flipping it. (1, 3), (2, 4), (3, 4), (4, 3) reflects across a horizontal line, as if the crease ran side to side. (1, 2), (1, 3), (3, 4) forgets one partner. (1, 3), (3, 4) forgets that the punch went through two layers.
Answer
Holes at (1, 2), (1, 3), (3, 1) and (3, 4).
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | · | hole | hole | · |
| Row 2 | · | · | · | · |
| Row 3 | hole | · | · | hole |
| Row 4 | · | · | · | · |
The pattern is balanced across the middle line, which is a quick check.
Wrong: A hole at the right edge unfolds to a partner just left of the crease.
Right: A hole at the right edge unfolds to a partner at the left edge.
Unfolding flips the paper, it does not slide it. Distance from the crease is kept, so an edge hole pairs with an edge hole.
Step 3 of 5
Unfold on your own
Check yourself · Question 1
The right half of a 4-by-4 sheet folds over onto the left half. One hole is punched at (2, 1), along the left edge. Which holes appear when the sheet is opened?
Answer: B
(2, 1) is 2 columns left of the crease. Its partner is 2 columns right of the crease, in column 4, still in row 2.
Check yourself · Question 2
The right half of a 4-by-4 sheet folds over onto the left half. Holes are punched at (1, 2) and (4, 1). Which holes appear when the sheet is opened?
Answer: B
(1, 2) is 1 column left of the crease, so its partner is (1, 3). (4, 1) is 2 columns left, so its partner is (4, 4). Two punches through two layers make four holes.
Trap answer: (1, 2), (1, 4), (4, 1), (4, 3)
Slide instead of flip
- Why it looks right
- The flap really did move 2 columns, so shifting each hole by 2 feels natural.
- Why it's wrong
- The flap turned over as it moved. A hole near the crease stays near it, and a hole at the edge pairs with the far edge.
- The right answer
- (1, 2), (1, 3), (4, 1), (4, 4), each partner the same distance from the crease.
Common mistake
I put a partner in a different row from its hole.
- Why it's tempting
- When there are two punches, the holes get mixed up and the rows blur together.
- Do this instead
- A vertical crease only changes left and right. Unfold one punch at a time, and keep each partner in its own row.
Step 4 of 5
What to remember
Remember
- Each punch through a single fold makes two holes.
- Flip each hole across the crease: same row, same distance from the crease, other side.
- Edge holes pair with edge holes, and holes beside the crease pair beside the crease.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, how far was the punch from the crease, and where did that put its partner?