TACHS: Paper folding: diagonal folds and layers
Learn to unfold a corner-to-corner crease by trading a hole's row and column, and to check how many layers a punch really goes through when a second fold covers only part of the paper.
How do you unfold a diagonal fold followed by a second fold that covers only part of the paper?
Short answer
Check whether the second flap lies under the punch, then undo the second fold first. Undo the diagonal fold by trading each hole's row and column.
Know first: Unfolding a single vertical fold, Rows and columns in a grid
Step 1 of 5
A diagonal crease trades rows and columns
In these questions the first fold runs corner to corner. On a 4-by-4 grid with the crease from the top-left corner to the bottom-right corner, unfolding sends the square (row, column) to (column, row). A hole at (4, 1), in the bottom-left corner, gets a partner at (1, 4), in the top-right corner. A hole near the left edge always reflects to a spot near the top edge.
A diagonal fold is a flip, like any other fold. The trading rule is just what that flip does to a grid.
| Fold | What moves | What you have after |
|---|---|---|
| 1. Diagonal, top-left to bottom-right | The upper-right triangle folds down onto the lower-left triangle | A triangle, two layers thick |
| 2. Right to left on the center line | The part of the triangle right of center folds onto the left half | The left half below the diagonal, with a four-layer flap over its lower-right part |
On a 4-by-4 grid, (4, 2) sits under the flap. (2, 1) and (3, 1) do not.
Step 2 of 5
Check the layers, then unfold
Worked example: A punch under the flap
A 4-by-4 sheet is folded along the diagonal from the top-left corner to the bottom-right corner, with the upper-right triangle folding onto the lower-left. Then the part right of the center line folds over to the left. A hole is punched at (4, 2). Which picture shows the opened sheet?
- A(2, 4), (4, 2)
- B(4, 2), (4, 3)
- C(2, 1), (2, 4), (4, 2), (4, 3)
- D(2, 4), (3, 4), (4, 2), (4, 3)
- E(2, 1), (3, 1), (4, 2), (4, 3)
- 1
Count the layers
(4, 2) is in the lower-right part of the folded shape, under the flap from fold 2. The punch goes through four layers, so expect 4 holes.
- 2
Undo fold 2
The crease is between columns 2 and 3. Column 2 is 1 column left of it, so the partner is in column 3: (4, 2) and (4, 3).
- 3
Undo fold 1
Trade row and column for each hole. (4, 2) pairs with (2, 4), and (4, 3) pairs with (3, 4).
- 4
See why the other choices are there
(2, 4), (4, 2) treats the punch as two layers and skips fold 2. (4, 2), (4, 3) stops after fold 2. (2, 1), (2, 4), (4, 2), (4, 3) undoes the diagonal first. (2, 1), (3, 1), (4, 2), (4, 3) reflects across the other diagonal.
Answer
Holes at (2, 4), (3, 4), (4, 2) and (4, 3): two along the bottom edge and two along the right edge.
| Column 1 | Column 2 | Column 3 | Column 4 | |
|---|---|---|---|---|
| Row 1 | · | · | · | · |
| Row 2 | · | · | · | hole |
| Row 3 | · | · | · | hole |
| Row 4 | · | hole | hole | · |
The holes mirror across the diagonal from the top-left corner to the bottom-right corner.
Wrong: Across the top-left to bottom-right diagonal, (4, 3) pairs with (4, 2).
Right: Across the top-left to bottom-right diagonal, (4, 3) pairs with (3, 4).
A diagonal crease trades the row and the column. Keeping the row is what a vertical crease does.
Step 3 of 5
Unfold diagonal folds on your own
Check yourself · Question 1
A 4-by-4 sheet is folded once, along the diagonal from the top-left corner to the bottom-right corner. A hole is punched at (3, 1). Where is its partner when the sheet is opened?
Answer: C
Unfolding across this diagonal trades row and column, so (3, 1) pairs with (1, 3).
Check yourself · Question 2
Same two folds as the worked example: the diagonal fold, then the right part folds over to the left. A hole is punched at (2, 1), near the left edge. Which holes appear when the sheet is opened?
Answer: C
The flap from fold 2 does not reach (2, 1), so the punch goes through only two layers and undoing fold 2 adds nothing. Undoing the diagonal trades row and column: (2, 1) pairs with (1, 2).
Trap answer: (1, 2), (2, 1), (2, 4), (4, 2)
Four layers everywhere
- Why it looks right
- The worked example made four holes, and unfolding every fold every time feels thorough.
- Why it's wrong
- Fold 2 only covers the lower-right part of the folded shape. A punch at (2, 1) never went through that flap.
- The right answer
- (1, 2), (2, 1): two holes.
Common mistake
I unfolded the diagonal first because it was the first fold.
- Why it's tempting
- The diagonal is the big, dramatic fold, so it feels like the one to handle first.
- Do this instead
- Undo the last fold first. Undoing the diagonal first sends holes to squares the punch never touched.
Step 4 of 5
What to remember
Remember
- Across the top-left to bottom-right diagonal, (row, column) unfolds to (column, row).
- Before unfolding, check whether the second flap lies under the punch: four layers under it, two elsewhere.
- Undo the last fold first, then the diagonal.
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 many layers did the punch go through, and how did you decide?