Two Folds, One Punch: Predicting the Unfolded Holes
During a small-group spatial reasoning session, a teacher gives each student a 16 cm by 16 cm paper square and reads three steps aloud. Step 1: fold the left edge over to meet the right edge, crease, and leave the paper folded, so the crease is now the left edge of the packet. Step 2: fold the top edge down to meet the bottom edge, crease, and leave it folded, so the crease is now the top edge of the packet. Step 3: punch one hole through all layers, 4 cm from the creased left edge and 4 cm from the creased top edge. Students predict the unfolded pattern before checking. Priya predicts two holes. Marcus predicts four holes forming a square 8 cm on a side, each hole 4 cm from the two nearest outer edges. Aiden predicts four holes bunched within 4 cm of the punched corner. The teacher wants a sixty-second check that settles the disagreement without unfolding a second sheet.
1. How many holes are there, and where? Trace the unfolding and say whose prediction is right.
Read the worked answer
Two folds produce four layers, and one punch through four layers makes four holes.
Set the origin at the top-left corner of the flat square. The vertical fold puts a crease at 8 cm, and the horizontal fold puts a crease at 8 cm. In the folded packet the punch sits 4 cm from each crease, so in paper coordinates it is at 12 cm from the top and 12 cm from the left.
Unfold the horizontal fold first: the hole at (12, 12) copies to (12, 4). Unfold the vertical fold: both holes copy across the crease at 8 cm, giving (4, 12) and (4, 4).
The result is four holes at (4, 4), (4, 12), (12, 4) and (12, 12): a square 8 cm on a side, each hole 4 cm from the two outer edges it is nearest. Marcus's prediction is correct. Priya undercounted the layers, and Aiden placed the copies inside the original quadrant instead of across the creases.
2. Explain Priya's error precisely and give a check that catches it in ten seconds.
Read the worked answer
Priya counted holes produced by folds rather than layers crossed by the punch. One punch through n layers always makes n holes, and each fold doubles the number of layers at the punch point, so two folds give four layers and four holes.
The ten-second check is to unfold in imagination one crease at a time and double as you go: one hole in the packet, unfold the second fold and it becomes two, unfold the first fold and it becomes four. This ties the count to a physical action rather than to a remembered formula, so it also catches the case of three folds, where the count becomes eight.
The check has one important exception, which Priya should hear at the same time: a punch that lands exactly on a crease does not double, because the hole and its mirror image coincide.
3. What if the punch landed exactly on the creased left edge? Predict the pattern and explain the general rule.
Read the worked answer
A punch on a crease is its own mirror image, so the two copies merge into a single hole straddling the crease rather than two separate holes.
If the punch sits on the vertical crease but still 4 cm from the horizontal crease, the horizontal fold still doubles it. The unfolded sheet shows two distinct holes, both centered on the vertical midline, 4 cm apart from each other and mirrored across the horizontal midline, each drawn as an elongated slot rather than a circular hole.
The general rule: each fold that the punch does not touch doubles the hole count, and each fold the punch lies on contributes no doubling. With two folds the counts are four when the punch misses both creases, two when it lies on one crease, and one when it sits on the intersection of both. That progression is a compact way to check any paper-folding answer.