TACHS: Paper folding: one horizontal fold
Learn to unfold a single side-to-side crease by flipping each hole straight up or down across it, keeping its column and its distance from the crease.
Where do the holes end up when a sheet folded along a horizontal crease is opened?
Short answer
Each punch makes two holes. Flip each hole straight across the crease: same column, 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 horizontal crease flips holes up and down
A horizontal fold runs side to side across the middle of the sheet. The top half folds down onto the bottom half, or the bottom half folds up onto the top. Either way the folded paper is two layers thick, so each punch makes two holes.
When you unfold, every hole gets a partner straight above or below it, on the other side of the crease, at the same distance from the crease.
| Hole in | Distance from crease | Partner in |
|---|---|---|
| Row 1 | 2 rows | Row 4 |
| Row 2 | 1 row | Row 3 |
| Row 3 | 1 row | Row 2 |
| Row 4 | 2 rows | Row 1 |
Picture the sheet as a 4-by-4 grid and name squares as (row, column).
Step 2 of 5
Flip each hole, keep its column
Worked example: Two punches, one horizontal fold
A square sheet is a 4-by-4 grid. The bottom half folds up onto the top half, so the crease runs between rows 2 and 3 and the folded paper covers rows 1 and 2. Holes are punched at (2, 2) and (1, 4). Which picture shows the opened sheet?
- A(1, 4), (2, 2), (3, 4), (4, 2)
- B(1, 1), (1, 4), (2, 2), (2, 3)
- C(1, 4), (2, 2), (3, 2)
- D(1, 4), (2, 2), (3, 2), (4, 4)
- E(1, 4), (2, 2), (3, 3), (4, 4)
- 1
Find the crease and count
The crease is between rows 2 and 3. Two layers everywhere, two punches: 4 holes in all.
- 2
Flip the first hole
(2, 2) is 1 row above the crease. Its partner is 1 row below it, in row 3, same column: (3, 2). The two holes sit right next to each other, one on each side of the middle line.
- 3
Flip the second hole
(1, 4) is 2 rows above the crease, at the top edge. Its partner is 2 rows below it, at the bottom edge: (4, 4).
- 4
See why the other choices are there
(1, 4), (2, 2), (3, 4), (4, 2) slides each hole down 2 rows without flipping it. (1, 1), (1, 4), (2, 2), (2, 3) reflects across a vertical line. (1, 4), (2, 2), (3, 2) forgets one partner. (1, 4), (2, 2), (3, 3), (4, 4) moves a partner sideways.
Answer
Holes at (1, 4), (2, 2), (3, 2) and (4, 4).
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | · | · | · | hole |
| Row 2 | · | hole | · | · |
| Row 3 | · | hole | · | · |
| Row 4 | · | · | · | hole |
Each column with holes is balanced across the middle line.
Find the wrong step
The top half of a 4-by-4 sheet folds down onto the bottom half. A student unfolds a hole punched at (3, 1), just below the crease.
- 1
Finds the crease between rows 2 and 3.
- 2
Moves the hole up by the height of the flap, 2 rows, and puts the partner at (1, 1).
What went wrong
That slides the hole. (3, 1) is only 1 row from the crease, so its partner is 1 row above the crease.
- 3
Keeps the partner in column 1.
The fix
The partner is at (2, 1), right beside the crease, because the hole was right beside the crease.
Step 3 of 5
Unfold on your own
Check yourself · Question 1
The top half of a 4-by-4 sheet folds down onto the bottom half. One hole is punched at (4, 2), along the bottom edge. Which holes appear when the sheet is opened?
Answer: C
(4, 2) is 2 rows below the crease. Its partner is 2 rows above it, in row 1, same column.
Check yourself · Question 2
The bottom half of a 4-by-4 sheet folds up onto the top half. Holes are punched at (2, 4) and (1, 3). Which holes appear when the sheet is opened?
Answer: D
(2, 4) is 1 row above the crease, so its partner is (3, 4). (1, 3) is 2 rows above, so its partner is (4, 3). Columns never change.
Trap answer: (1, 3), (2, 4), (3, 3), (4, 4)
Slide instead of flip
- Why it looks right
- It has four holes in the right columns, and shifting each hole by the flap's height feels like undoing the fold.
- Why it's wrong
- The flap turned over as it moved. A hole at the top edge pairs with the bottom edge, and a hole beside the crease pairs beside it.
- The right answer
- (1, 3), (2, 4), (3, 4), (4, 3).
Common mistake
I put the crease at the edge of the folded paper I could see, not at the middle of the full sheet.
- Why it's tempting
- The folded paper looks like a whole sheet, so its edge feels like the place to measure from.
- Do this instead
- Find the middle of the full sheet first. For one fold in half, the crease is always the middle line.
Step 4 of 5
What to remember
Remember
- Each punch through a single fold makes two holes.
- Flip each hole straight across the crease: same column, same distance from the crease, other side.
- A horizontal crease never moves a hole sideways, so rule out any choice that does.
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, which row was the punch in, and how many rows from the crease did that put its partner?