Unfolding in Reverse Order

One item, three folds, one punch — taken apart backward, a single crease at a time. The reflection you learned in the last topic does the work at every step; the only new question is which crease to open first.

The last fold is the first one undone

The punch is made after the final fold, through whatever stack of paper happens to be sitting under that spot. Before the final fold, that stack does not exist. The item engine picks the punch position from the finished sheet and from nowhere else, which means there is no forward route through the problem: you cannot mark a punch on the flat square, because you do not yet know which parts of the square will end up beneath it.

Backward, the difficulty evaporates. You start from the one thing the item hands you — the punch on the finished sheet — and give the folds back in the reverse of the order they were made. Last fold first, first fold last. Every step is one reflection, and you never guess.

Start

→

Fold 1

→

Fold 2

→

Fold 3

→

Punch

The item: three folds, then one punch through five layers. Two teaching additions the test does not print — the flat square shown first, and the extra dashed outline on each fold panel, which is the sheet's shape before that fold.

Reading the creases off the panels

No crease is ever drawn for you. Each panel gives a solid outline — the paper's actual edge at that moment — sitting inside a dashed square that never changes from panel to panel. Because that dashed square is a fixed reference and not a record of the last fold, the crease has to come from somewhere else: compare one panel's solid outline against the previous panel's. The new straight edge that was not there before is the fold line, and the paper on the far side of it is what moved.

Three creases in this item:

Fold 1 — a vertical line one column in from the left edge. The left-hand column turned in over its neighbor.
Fold 2 — a horizontal line one row up from the bottom edge. The bottom row turned up over the row above it.
Fold 3 — a 45-degree line cutting off the sheet's bottom-left corner, lying along the square's own top-left to bottom-right diagonal.

The punch is on the last panel, in row 2, column 3 of the original grid, driven through five layers of paper.

Backward, one crease at a time

Step through it. Each grid holds every hole known at that point in the unfold, and the creases come off in the order 3, 2, 1.

Unfolding the item, one crease per step Step 1 of 4

1. The punch, on the folded sheet

One position: row 2, column 3. That is where the punch sits after all three folds, and it is the first hole you know about on the original square — every other hole in the answer is a reflection of it.

Row 2, column 3: where the punch sits, and the first known hole on the original square.

2. Undo fold 3 — the 45-degree crease

The last fold is the first one back. Its crease lies on the square's main diagonal, so the reflection trades the two coordinates: row 2, column 3 gains a partner at row 3, column 2. Two holes.

After undoing the 45-degree crease: row and column trade places.

3. Undo fold 2 — the horizontal crease

This crease sits between rows 3 and 4, so rows reflect and columns hold still. Row 3, column 2 gains row 4, column 2. The hole in row 2 would reflect to row 5, which is off the square, so it gains nothing. Three holes.

After undoing the horizontal crease: one hole reflects, one has no paper to reflect onto.

4. Undo fold 1 — the vertical crease

This crease sits between columns 1 and 2, so columns reflect and rows hold still. Both holes in column 2 gain partners in column 1. The hole in column 3 would reflect to column 0, off the square again, so it gains nothing. Five holes, and the unfold is done.

After undoing the vertical crease: the finished answer, five holes.

All four steps are printed below one another if this stepper does not respond.

Two habits are worth taking from those four steps. Reflect the whole set every time. At step 4 both holes in column 2 mirror across, not only the one added most recently; dropping the older holes is the commonest way to hand in an answer one or two holes short. And keep the sheet's earlier outline in view. At steps 3 and 4 a reflection landed off the paper and contributed nothing, which is why the count ran 1, 2, 3, 5 instead of 1, 2, 4, 8.

Checking the answer

Start

1 layer

→

Fold 1

2 layers

→

Fold 2

4 layers

→

Fold 3

5 layers

→

Punch

5 layers

→

Unfolded

5 holes

The same sequence computed end to end by the item engine — five layers under the punch, five holes in the answer.

That is the same fold sequence walked end to end by the item engine, and its last panel is the grid the stepper finished on — the same five positions, arrived at without any reflecting by hand. Use the layer count as your independent check: the punch went through five layers, so the answer had to carry five holes, and it does.

When the two disagree, trust the count and re-run the unfold. A total that comes in low means a crease that should have doubled its holes did not; a total that comes in high means you mirrored a hole into a position the sheet had already folded away from. Either way the fault is in one identifiable step, not in the whole solve.

The takeaway

Give the folds back in reverse: the last fold made is the first one undone. There is no forward route, because a punch position only means anything on the finished stack. Recover each crease by comparing one panel's outline with the one before it, reflect the entire hole set across it, and stop when the first fold has been returned. Then check the total against the layers you counted under the punch. Then run the whole loop, item after item, in the hole punching generator.

Unfolding in Reverse Order

Summary

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