Paper folding with two punches
A square sheet of paper measures 8.0 cm on each side. The sheet is folded once by bringing the top edge down onto the bottom edge, then folded again by bringing the left edge right onto the right edge, producing a 4.0 cm × 4.0 cm packet. The folded packet is held so that its top-left corner is the original center of the sheet. A 1.0 cm diameter circular punch is made through the packet, centered 1.0 cm from the top edge and 1.0 cm from the left edge of the packet. A second punch of the same diameter is centered exactly on the packet's top-left corner. The student then unfolds the sheet and must determine the number, positions, and shapes of the holes. The student also wonders what would happen if the first punch were moved to be centered 0.5 cm from the left edge instead, keeping the same distance from the top edge.
1. How many holes result from the two punches, and where are they located on the unfolded 8 cm × 8 cm sheet? Use coordinates from (0,0) at bottom-left to (8,8) at top-right.
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After the two folds, the packet covers x = 4 to 8 and y = 0 to 4, and its top-left corner is the original center (4,4). A punch 1 cm from the top and 1 cm from the left is at (5,3). Unfolding reflects across the vertical crease x = 4 and the horizontal crease y = 4: (5,3) → (3,3) → (5,5) → (3,5). That is 4 holes.
The second punch is at (4,4), exactly on both creases. Its half-circle mirrors onto itself, producing 1 circular hole at the sheet center. Total = 5 holes: four at (3,3), (5,3), (3,5), (5,5), plus one at (4,4).
2. If the first punch is moved to be centered 0.5 cm from the left edge (same top distance), how do the number and shapes of holes change? Explain with the fold geometry.
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The new center is (4.5,3). Radius is 0.5 cm, so the left edge of the punch is tangent to the vertical crease x = 4; the circle does not cross the crease. Unfolding still gives four holes from this punch, at (4.5,3), (3.5,3), (4.5,5), and (3.5,5). The pair across the vertical crease are tangent at x = 4, and the pair across the horizontal crease are also tangent at y = 4. The total count with the center punch is still 5, but the four outer holes are now tangent pairs rather than separated by a 0.5 cm gap. If the punch were moved closer than 0.5 cm so that it crosses the crease, each mirrored pair would merge into one elongated hole.
3. Explain how you would decide whether a punch that appears to lie on a fold line produces one hole or two after unfolding.
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Compare the punch center with the crease. If the center lies exactly on the crease, the punch is bisected by the fold; unfolding mirrors the half-disc onto the other side of the same crease, giving one circular hole centered on the crease. If the center is offset but the circle crosses the crease, the two half-discs join into one elongated hole, not two. If the circle is tangent to or entirely on one side of the crease, unfolding gives two distinct holes that may touch at a point. The decision rule is therefore based on whether the punch area crosses the crease, not merely on whether it looks close to the fold.