Three Folds, Two Punches
A square sheet of paper, 16 cm on a side, is folded three times: the left half onto the right half, then the bottom half up onto the top half, then the resulting small square along its diagonal. The three creases therefore all pass through the centre of the original sheet.
A hole is punched through all layers at a point 2 cm from the vertical crease and 3 cm from the horizontal crease. Nobody checks whether the punch crosses a fold.
Later, on an identical fresh sheet, the same three folds are made and a single punch is made exactly on the diagonal crease, 1 cm from the corner where all three creases meet.
1. How many separate holes does the first punch produce when the sheet is unfolded? Justify the number rather than asserting it.
Read the worked answer
Eight holes.
Each fold doubles the number of layers that the punch passes through: 1 fold gives 2 layers, 2 folds give 4, 3 folds give 8. Because the punch is placed away from every crease, no two of those eight layer-positions coincide when the paper is opened out, so eight layers means eight distinct holes.
The eight are not scattered at random. Unfolding applies one reflection per crease, and each of the three creases passes through the centre of the sheet, so the holes form four mirror-image pairs about the vertical crease, the horizontal crease and the diagonal. That symmetry is the check on the count: if a proposed arrangement is not symmetric about all three creases, the count or the positions are wrong.
2. How many holes does the second punch produce, and what shape is each? Explain why the naive answer of eight half-holes is wrong.
Read the worked answer
Four holes, each a full circle the same size as the punch.
A punch on a crease sits astride the fold, so the two layers joined at that crease receive what looks like two half-punches but is really one punch across continuous paper. When the sheet is opened, the two halves join back up and the hole closes into a single full circle. Eight layer-positions collapse into four distinct locations.
The 'eight half-holes' answer treats the crease as a place where material is missing. It is not: after unfolding, the crease is ordinary paper again, and a punch centred on it removes a shape symmetric about that line. A circular punch centred on a line produces a full circle, not a crescent.
This is the general rule worth carrying: a punch that lies on a fold reduces the hole count and leaves intact shapes; a punch on the outer edge leaves notches on the boundary.
3. A student says: 'The punch's own image must be the hole closest to the centre of the sheet, since it was made nearest the corner.' Evaluate the claim using the 2 cm and 3 cm figures.
Read the worked answer
The claim is false: all eight holes are the same distance from the centre.
Take the sheet's centre as the origin. The punch is 2 cm from one crease and 3 cm from the other, so its image sits at an offset of (2, 3), at distance sqrt(4 + 9) = sqrt(13), about 3.6 cm, from the centre.
Every other image is produced by reflecting across a crease that passes through the centre. A reflection about a line through the centre preserves distance to the centre, so no image can be nearer or farther. The reflections send (2, 3) to (-2, 3), (2, -3), (-2, -3) and the diagonal swaps the coordinates, giving (3, 2), (-3, 2), (3, -2), (-3, -2). All eight lie on a circle of radius sqrt(13) about the centre.
The student has confused two different distances - distance to the creases (2 cm and 3 cm, which the reflections do swap and permute) with distance to the centre (which none of them changes).