One Pin, Four Layers
A square sheet of paper, 8 cm on each side, is folded twice and then punched once. First the right edge is folded over onto the left edge, so the fold line lies along the sheet's vertical midline and the paper is now 4 cm wide. Then the bottom edge is folded up onto the top edge, so the second fold line lies along the horizontal midline and the packet is 4 cm by 4 cm with four layers.
A single hole is punched straight through all four layers with a pin, at a point 1 cm from the packet's left edge and 1 cm from its top edge. The paper is then opened out flat and the holes are counted.
Note where the fold lines ended up: after both folds, both fold lines form edges of the packet - the vertical fold is the packet's right edge and the horizontal fold is the packet's bottom edge. The packet's left and top edges are the original outer edges of the sheet.
1. How many holes appear when the sheet is opened, and where is each one?
Read the worked answer
Two folds double the layers twice - 1, then 2, then 4 - so a single punch gives four holes, provided the punch is not on a fold line. It is not: the punch sits 1 cm from the packet's outer edges, while both fold lines are 4 cm away on the opposite sides.
Unfold in reverse order:
- Undo the vertical fold. The hole reflects across the vertical midline, giving one hole 1 cm in from the left edge and one 1 cm in from the right edge, both 1 cm below the top edge.
- Undo the horizontal fold. Both holes reflect across the horizontal midline, adding two more 1 cm above the bottom edge.
Result: four holes, one near each corner of the opened sheet, each 1 cm from the two nearest edges. The arrangement is symmetric about both midlines.
2. Suppose the pin had gone through a point exactly on the packet's right edge, 1 cm below the packet's top edge. How many holes would there be, and why is the count different?
Read the worked answer
Two holes, not four.
The packet's right edge is the vertical fold line, so the punch sits on the crease. When the vertical fold is opened, a hole on a fold maps onto itself - it is not duplicated, it simply opens out into one hole straddling the midline. The horizontal fold still duplicates it, because the point is not on that crease.
Positions: both holes on the vertical midline, 4 cm from the left edge and 4 cm from the right edge, one 1 cm below the top edge and one 1 cm above the bottom edge.
The general rule is that each fold the point does not lie on doubles the count. A point on one crease of a two-fold packet gives 2 holes; a point on both creases would give one.
3. If the two folds were made in the opposite order - bottom to top first, then right to left - would the pin mark the same four places? Justify your answer.
Read the worked answer
Yes, unchanged.
The two fold lines are the sheet's vertical and horizontal midlines, and their positions do not depend on which fold is made first. The packet ends up 4 cm by 4 cm either way, with the same outer edges and the same fold edges. The punch is specified relative to the packet's left and top edges, which are the sheet's original outer edges in both orderings, so it marks the same physical point on the sheet. Reflections across two perpendicular midlines commute, so unfolding produces the same four positions.
The only thing the order changes is which layer ends up on top of the packet. That is irrelevant here, because the pin passes through all four layers - but it would matter if the question asked which layer the pin entered first.