TACHS: Paper folding: two folds, four holes
Learn to undo two perpendicular folds one at a time, last fold first, so a single punch becomes four holes placed by two mirror lines.
How do you unfold a sheet that was folded in half one way and then in half the other way?
Short answer
Undo the last fold first and flip every hole across that crease, then undo the first fold and flip all the holes again. A punch away from both creases makes four holes, one in each quarter.
Know first: Unfolding a single vertical or horizontal fold, Rows and columns in a grid
Step 1 of 5
Two folds make four layers
When a square sheet is folded in half one way and then in half the other way, it becomes a small square a quarter of the full size and four layers thick. One punch goes through all four layers, so it makes four holes.
The safe method is to read the fold arrows in order, then undo them in reverse. Each time you undo a fold, flip every hole you have so far across that crease.
| Stage | Layers under the punch | Holes |
|---|---|---|
| Folded twice | 4 | 1 punch |
| After undoing the last fold | 2 | 2 |
| After undoing the first fold | 1 | 4 |
A punch exactly on a crease is its own mirror image, so it makes fewer separate holes.
Step 2 of 5
Undo one fold at a time
Worked example: A punch at the edge of a folded quarter
A 4-by-4 sheet is folded twice. First the right half folds over onto the left half (crease between columns 2 and 3). Then the top half folds down onto the bottom half (crease between rows 2 and 3). The folded paper is the bottom-left quarter: rows 3 and 4, columns 1 and 2. A hole is punched at (4, 2). Which picture shows the opened sheet?
- A(1, 1), (1, 4), (4, 1), (4, 4)
- B(1, 2), (1, 3), (4, 2), (4, 3)
- C(1, 2), (4, 2)
- D(4, 2), (4, 3)
- E(1, 3), (2, 1), (3, 4), (4, 2)
- 1
Undo the last fold
The last crease is between rows 2 and 3. (4, 2) is 2 rows below it, so its partner is 2 rows above it: (1, 2). Holes so far: (1, 2) and (4, 2).
- 2
Undo the first fold
The first crease is between columns 2 and 3. Column 2 is 1 column left of it, so each hole gets a partner in column 3. Holes: (1, 2), (1, 3), (4, 2), (4, 3).
- 3
Check the pattern
Two holes side by side in the middle of the top edge, two in the middle of the bottom edge. It mirrors across both middle lines.
- 4
See why the other choices are there
(1, 1), (1, 4), (4, 1), (4, 4) spreads the holes to the corners instead of measuring from the creases. (1, 2), (4, 2) undoes only the last fold. (4, 2), (4, 3) undoes only the first fold. (1, 3), (2, 1), (3, 4), (4, 2) turns the hole around the center like a pinwheel, but unfolding flips holes.
Answer
Holes at (1, 2), (1, 3), (4, 2) and (4, 3).
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | · | hole | hole | · |
| Row 2 | · | · | · | · |
| Row 3 | · | · | · | · |
| Row 4 | · | hole | hole | · |
Wrong: Four holes always go in the four corners.
Right: Four holes go wherever two mirror flips put them.
Each partner sits the same distance from its crease as the hole it came from. A punch near the middle of the folded quarter makes four holes near the middle of the sheet.
Step 3 of 5
Unfold two folds on your own
Check yourself · Question 1
A 4-by-4 sheet is folded twice. First the left half folds over onto the right half. Then the bottom half folds up onto the top half, leaving the top-right quarter (rows 1 and 2, columns 3 and 4). A hole is punched at (2, 4). Which holes appear when the sheet is opened?
Answer: C
Undo the last fold: (2, 4) is 1 row above the horizontal crease, so add (3, 4). Undo the first fold: column 4 is 2 columns right of the vertical crease, so add (2, 1) and (3, 1).
Check yourself · Question 2
A 4-by-4 sheet is folded twice. First the bottom half folds up onto the top half. Then the right half folds over onto the left half, leaving the top-left quarter. Holes are punched at (1, 1) and (2, 2). Which holes appear when the sheet is opened?
Answer: E
Each punch makes four holes. (1, 1) is at the outer corner of the folded quarter, so its holes are the four corners of the sheet. (2, 2) is beside both creases, so its holes are the four center squares. That is 8 holes in all.
Trap answer: (1, 1), (1, 3), (2, 2), (2, 4), (3, 1), (3, 3), (4, 2), (4, 4)
Slide instead of flip
- Why it looks right
- It has the right number of holes, 8, and looks evenly spread.
- Why it's wrong
- Sliding moves a hole by the width of the flap. Flipping keeps its distance from the crease, so (2, 2), beside both creases, keeps all its partners beside the creases.
- The right answer
- The four corners plus the four center squares.
Common mistake
I unfolded the first fold first because it was the first one I read.
- Why it's tempting
- Reading order feels like the natural order to work in.
- Do this instead
- Undo the last fold first. With these two folds the result happens to match, but with flaps that cover only part of the paper, the wrong order gives the wrong holes. Build the habit now.
Step 4 of 5
What to remember
Remember
- Two perpendicular folds make four layers, so one punch away from the creases makes four holes.
- Undo the last fold first, flip every hole across that crease, then do the same for the first fold.
- Measure each partner from its crease. Corner patterns and checkerboards are the usual wrong pictures.
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 fold did you undo first, and how many holes did you have after each undo?