Three Napkins, Three Punches: Counting Holes After Folding
In art, Mr. Osei gives every child a square paper napkin that measures 12 cm by 12 cm. The folding rule is the same for everyone: fold the napkin in half, then fold it in half again, so it becomes a small square with four layers. The folded packet has two folded edges and two open edges. Ana has three spare napkins and a hole punch. On napkin A she punches one hole in the middle of the small square, far from every edge. On napkin B she punches one hole exactly on the left folded edge, halfway down. On napkin C she punches one hole exactly at the bottom-left corner, where the two folded edges meet. Then she opens all three napkins and counts the holes. Before she opens them, Ana says, 'Each one should have 4 holes, because four layers means four holes.' Mr. Osei asks her to think about what happens to a hole that lands on a fold, and to make a prediction for each napkin before she unfolds anything.
1. Predict the number of holes for napkin A and explain the reasoning.
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Napkin A gives 4 holes.
The paper is four layers thick, and the punch passes through all four layers at once, so it cuts one hole in each layer. The punch is in the middle of the small square, away from every edge, so no two layers are joined together at that point.
When the napkin is unfolded, the four holes sit apart from one another, one in each quarter of the napkin.
A quick way to picture it: each fold doubled the number of layers, 1 then 2 then 4, and each doubling also doubled the number of copies of the hole. Ana's rule is right for this napkin: 4 layers, 4 holes.
2. Predict the number of holes for napkins B and C. Explain what happens to a hole that lands on a fold.
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Napkin B gives 2 holes. Napkin C gives 1 hole.
The rule is that a hole lying on a fold is shared. When the paper is unfolded along that fold, the two half-holes join into one hole, so a fold makes the count go down.
For B, the punch sits on the left folded edge. It passes through 4 layers, but the layers are joined in pairs at that fold, so the half-holes on each side of the crease join together. That gives two holes, one above the other, each straddling the vertical centre line of the opened napkin.
For C, the punch is at the corner where both folds meet, which is the centre of the original napkin. All four layers meet at that single point, so the four quarter-holes join into one round hole in the centre of the napkin.
3. Ana's rule 'four layers means four holes' is sometimes right and sometimes wrong. State the general rule that covers all three napkins. Then answer: would a bigger hole punch change any of the counts?
Read the worked answer
The general rule is: count the layers, then subtract the joins.
- A punch that misses every fold makes one hole per layer: 4 layers, 4 holes.
- A punch that sits on one fold makes half as many, because the holes pair up across the crease: 4 layers, 2 holes.
- A punch that sits where two folds cross makes a quarter as many, because all four layers meet at one point: 4 layers, 1 hole.
A bigger punch does not change the counts. A wider punch still makes exactly one hole per layer, or per joined group of layers, as long as it stays on the paper and does not run off an open edge. Size changes how big each hole is, not how many separate holes there are.
The number of holes is decided by the folds, not by the punch.