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Backward Unfolding
The Reflection Law
Summary
One crease, one mirror
Unfolding is not a redrawing. Every hole already on the sheet in front of you is in its final position; opening a crease only reveals the holes that were hiding under the flap. What gets revealed is a mirror image.
Here is the smallest case the section can build: one fold, one punch.
The punch sits in row 2, one column to the left of the crease. Unfolded, the square carries two holes — the one you punched, unmoved, and its mirror one column to the right of the crease. Same row, opposite side, equal distance.
Undoing a fold reflects the holes across that fold's crease and keeps both copies. Three consequences follow, and each one guards against a different mistake.
- Holes never move. The half of the sheet that stayed put during the fold never traveled, so its holes are already home. Unfolding never slides a hole across to a tidier position.
- Copies are added. The half that moved swings back out and carries its holes to their mirror positions.
- The set only grows. A crease can add holes; it can never take one away. Four holes in hand means at least four in the answer.
The three creases
Which of a hole's two coordinates changes depends entirely on how the crease runs. The general move is always the same — measure straight out from the crease, and put the copy the same distance on the other side — but it is worth knowing each case cold.
A vertical crease changes the column and keeps the row
That is the figure above: row 2 stays row 2, and column 2 becomes column 3. Across the square's vertical midline the columns pair up 1 with 4 and 2 with 3. Across a crease drawn one column in from the left edge, only columns 1 and 2 pair with each other; the mirror of column 3 is off the square entirely.
A horizontal crease changes the row and keeps the column
The same idea turned through ninety degrees. The punch in row 2, column 3 unfolds to holes in rows 2 and 3, both still in column 3. Reading a horizontal crease as though it moved the columns costs you the item quietly: what you produce is a perfectly ordinary hole pattern, just not this item's.
A 45-degree crease exchanges row and column
This crease runs corner to corner, from the square's top-left corner to its bottom-right. The punch in row 1, column 3 unfolds to a hole in row 3, column 1: the two numbers trade places. The other corner-to-corner diagonal performs the same trade counted from the opposite pair of corners, and a small corner flap performs it shifted by however far its crease lies from the corner. A diagonal is the one crease where you cannot answer "which coordinate changed" — both did.
Where the copy does not appear
"Every crease doubles the holes" is the most attractive false rule in this section. Here are two folds and two holes.
The first fold turned the left-hand column in over its neighbor, nowhere near the punch: reflecting a hole in column 3 across a crease drawn between columns 1 and 2 lands in column 0, which is not on the paper. The second fold laid column 4 directly over column 3, right on top of the punch, so that crease does double. One crease doubled the holes, one added nothing.
A mirrored copy is real only where the sheet actually reached before that fold. The paper always stays inside the edges of the original square, so a mirror position outside the square is dead — and so is one that is inside the square but outside the sheet's own outline at that moment. The same test applies to the holes you already have: if the punch pierced flap that was overhanging bare table, that hole travels back with the flap rather than staying where it is.
This is not a rare footnote. Take every legal fold-and-punch combination the engine can build: with one fold the holes always double, but with two folds three unfolds in four already have at least one reflection landing off the sheet, and with three folds more than nine in ten do. Doubling cleanly at every crease happens in roughly one three-fold unfold in seventy.
The takeaway
Opening a crease reflects the hole set across it and keeps both copies: holes never move, copies are only ever added, and the set never shrinks. A vertical crease changes the column and keeps the row, a horizontal crease does the reverse, and a 45-degree crease trades row for column. A copy exists only where the sheet reached before that fold, which is why expecting the holes to double every time will cost you an answer.
Practice Questions
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