Two Folds, One Punch
A print studio is testing a folding card. A square sheet of paper 8 cm by 8 cm is folded once so that the left edge lands exactly on the right edge, then folded again so that the top edge lands exactly on the bottom edge. The result is a 4 cm by 4 cm packet whose two folded edges meet at one corner. A single hole is punched straight through all layers at a point 1 cm inside the packet from each of the two folded edges. The sheet is then opened flat. The studio needs to know how many holes the customer will see and where they will sit, because the printed design must not be crossed by a hole. A second card is to be punched so that it opens with only two holes, and a third with four holes forming a rectangle rather than a square. The punch position is the only thing that can be varied; the two folds are fixed.
1. How many holes appear on the opened sheet, and where are they? Justify the count and the positions.
Read the worked answer
Four holes.
Unfolding reverses each fold, and each reversal mirrors every existing hole across the corresponding crease. The creases sit 4 cm from the edges, and the punch lies 5 cm from the left edge and 5 cm from the top edge (4 cm to the crease plus 1 cm into the packet).
- Unfold the vertical fold: the hole at 5 cm from the left is mirrored to 3 cm from the left. Two holes.
- Unfold the horizontal fold: both holes are mirrored across the horizontal crease, giving two more at 3 cm from the bottom edge.
The four holes are therefore at the points that are 3 cm from the left or right edge and 3 cm from the top or bottom edge. They form a 2 cm by 2 cm square centred on the sheet. No hole comes closer than 3 cm to an outer edge, because the punch was only 1 cm from two creases, and the mirror images always land at least 3 cm from the outside.
2. Where must the punch go if the opened card is to show exactly two holes? Explain, including the special case of one hole.
Read the worked answer
The punch must lie on exactly one crease, away from the point where the two creases cross.
A hole on a crease is its own mirror image across that crease, so the corresponding unfold produces no new hole.
- Punch anywhere on the vertical crease (4 cm from the left edge) and the vertical unfold adds nothing; only the horizontal unfold doubles it, giving two holes mirrored above and below the horizontal crease.
- Punch anywhere on the horizontal crease and the argument runs the other way, giving two holes mirrored left and right.
In both cases the pair is symmetric about the sheet's centre, and the separation equals twice the punch's distance from the other crease - so a punch 2 cm from the horizontal crease yields two holes 4 cm apart.
Special case: if the punch lands exactly on the corner where the creases meet, both reflections fix the point and only one hole appears, dead centre.
3. The third card needs four holes forming a rectangle that is twice as wide as it is tall, with the closest pair of holes at least 2 cm apart. Is this achievable with the fixed folds? If so, give a punch position.
Read the worked answer
Yes, it is achievable.
The two folds act independently, one on horizontal distance and one on vertical distance, so the four holes always form an axis-aligned rectangle centred on the sheet. Its width is twice the punch's distance from the vertical crease, and its height is twice its distance from the horizontal crease. These two distances are independent, so a square is only one of many possible outcomes.
For a rectangle twice as wide as tall, punch twice as far from the vertical crease as from the horizontal crease. Punching 2 cm from the vertical crease and 1 cm from the horizontal crease gives holes at 2 cm and 6 cm across and at 3 cm and 5 cm down: width 4 cm, height 2 cm, and the closest pair of holes is 2 cm apart, which meets the minimum.
The constraint to watch is the small measurement. Dropping below 1 cm from the horizontal crease would push the closest gap under 2 cm, so the punch must stay at least 1 cm from both creases.