Four Layers, but Not Always Four Holes
Nia is working through a paper-folding card set. She takes a square sheet, folds it exactly in half so the top edge meets the bottom edge, and creases it. Without unfolding, she folds it in half again so the left edge meets the right edge, and creases. The paper is now a small rectangle with four layers. She punches one hole straight through all four layers, then opens the paper flat.
On Card 1 she punched near the middle of the folded packet, away from every edge. On Card 2 she punched exactly on the crease made by the first fold, away from the second crease. On Card 3 she punched exactly where the two creases cross.
The card set asks for the number of holes on the opened sheet for each card. Nia expects 4 holes every time, because there are 4 layers. Her partner says the layer count is a maximum, not a promise.
1. Card 1: how many holes appear when Nia opens the paper, and where are they? Explain using the creases.
Read the worked answer
4 holes, one in each quarter of the sheet.
A crease works like a mirror. Unfolding the second fold reflects the punched hole across the left-right crease line, so one hole becomes two. Unfolding the first fold reflects both holes across the top-bottom crease line, so two become four.
1 → 2 → 4, doubling at each unfold, which is exactly why the count matches the number of layers.
The four holes sit at mirrored positions: same distance from the left-right crease, same distance from the top-bottom crease. They form a rectangle whose sides are parallel to the creases. If you folded the paper up again, all four holes would stack into one.
2. Cards 2 and 3 give different counts. What are they, and why does punching on a crease change the answer?
Read the worked answer
Card 2: 2 holes. Card 3: 1 hole.
Punching on a crease means the punch removes a half-hole on each side of that crease. When the paper opens, the two halves come together into one whole hole instead of two separate ones.
- Card 2 - the punch sits on the first crease, so the two halves join there and that fold no longer doubles anything. The second crease still mirrors, so the count goes 1 × 2 = 2 holes, placed on opposite sides of the second crease.
- Card 3 - the punch sits on both creases, so every layer's half-hole meets every other one at the centre of the sheet. That is 1 hole.
The layer count says how many pieces of paper the punch passes through. It does not say how many holes you see, because a crease can glue pieces back together.
3. Can Nia end up with exactly 3 holes? Answer for one punch first, then describe a way to get 3 using two punches.
Read the worked answer
One punch: no. With this folding, one punch can only give 4 (off every crease), 2 (on one crease) or 1 (on both creases). Every result is 4 divided by a power of 2, so 3 is impossible from a single punch.
Two punches: yes. Choose:
- one punch exactly on a crease, away from the other crease → 2 holes
- one punch exactly at the crossing of the two creases → 1 hole
2 + 1 = 3 holes.
The trap is to pick a punch that gives 4 and try to "take one away." Punches add holes; they never remove them. To land on an odd number you must use at least one punch that sits on a crease, because that is the only way the count stops coming out even.