Fold, Punch, Unfold: Counting Layers, Not Folds
A student, Ana, is working through a paper-folding item with a square sheet 16 cm on a side. The instructions are: fold the left half onto the right half, so the crease falls on the vertical midline; then fold the top half down onto the bottom half, so a second crease falls on the horizontal midline; then punch one hole straight through the folded packet at a point 3 cm to the right of the vertical crease and 2 cm below the horizontal crease; then unfold completely.
Ana answers "four holes, one near each corner of the sheet." She is confident about the count but sketches the positions at the corners. Her notes say: "two folds, so the sheet is doubled twice - four thicknesses, four holes. The holes will be symmetric, so they go at the corners." She has previously scored full marks on rotation items and loses points only on folding items involving more than one crease.
1. State the number of holes and their positions, with the reasoning that produces both.
Read the worked answer
Four holes, at the four points 3 cm horizontally and 2 cm vertically from the centre of the sheet.
Set the origin at the top-left corner, with x running right and y running down. The folds create creases at x = 8 and y = 8, and the packet occupies the 8 cm by 8 cm square from (8,8) to (16,16). The punch is at (11,10).
- Layer count: each fold doubles the layers only where the punch lies inside the folded flap. Both folds double at (11,10) - it is 3 cm from one crease and 2 cm from the other, comfortably inside the packet - so 2 × 2 = 4 layers and 4 holes.
- Positions: unfolding reflects the punch across each crease in turn. Across x = 8: (11,10) and (5,10). Across y = 8: (11,6) and (5,6).
So the holes sit at (5,6), (11,6), (5,10) and (11,10) - a rectangle 6 cm wide and 4 cm tall, centred on the middle of the sheet, nowhere near the corners.
2. Ana's count is right and her layout is wrong. Diagnose the layout error and explain why the count reasoning happens to work.
Read the worked answer
Her layout applies the wrong symmetry. "Symmetric, so the corners" is point symmetry about the centre - a 180-degree rotation - or an assumption that the folds ran corner to corner. Neither is what happened. The folds are straight creases through the midlines, and each one produces a reflection across its own line, not a rotation about the centre.
Reflections across perpendicular midlines place the images at the mirror distances: 3 cm to either side of the vertical midline and 2 cm above and below the horizontal midline. The two distances never travel to the edge, because 3 cm and 2 cm are far smaller than the 8 cm half-width. Corners would require the punch to be near the crease intersection's opposite extreme.
The count survives because it depends on layer number, not on symmetry type: two folds that both cover the punch point give 2 × 2 = 4 layers regardless of where the reflections land. Ana has learned the doubling rule as a slogan and never tied it to the geometry, which is why her counting items pass and her placement items fail.
3. Suppose the punch had been made exactly on the vertical crease, 2 cm below the horizontal crease. How many separate holes appear after unfolding, and what shape are they?
Read the worked answer
Two holes, both circular, of the same radius as the punch, centred at (8,10) and (8,6).
The layer count does not change - the punch still passes through four thicknesses - but the vertical reflection now maps the punch point onto itself, because a point on a crease is its own mirror image. Reflection across x = 8 contributes nothing new; only the horizontal crease produces a second position, at (8,6).
The shape question is the part most solvers miss. A punch centred exactly on the fold removes a half-disc from the layer above the crease and a half-disc from the layer below. Those two halves are neighbouring regions of the same sheet, so when the sheet is unfolded they join along the crease into one complete disc. No scissor-cut semicircle appears, and no double-width hole appears either - the unfolded hole is a circle of the punch's radius, sitting astride the midline.
So the general principle: count positions by distinct images, then check whether any images coincide and merge.