Two Folds and a Corner Cut
Take a square sheet of paper, 8 inches on a side. Fold the left edge over to meet the right edge, crease, and leave it folded. Then fold the top edge down to meet the bottom edge, crease, and leave it folded. You now hold a 4-inch square packet whose left and top edges are folded (closed) and whose right and bottom edges are open (four free sheet edges). Holding the packet in that orientation, remove a 1-inch by 1-inch square from the bottom-right corner, then unfold the sheet completely and lay it flat. No other cuts are made, and the folds are sharp.
1. Describe the unfolded sheet: how many separate pieces of paper were removed, and where?
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Four 1-inch squares are removed - one from each corner of the original sheet. The sheet stays in one piece with its corners clipped; there are no interior holes.
Why: each fold halves the sheet, so the packet's four corners each stack four layers. Trace the original corners:
- Top-left → after fold 1 lands on top-right → after fold 2 lands on bottom-right.
- Bottom-left → bottom-right.
- Top-right and bottom-right → bottom-right.
All four original corners sit at the packet's bottom-right corner, which is open on both sides (two free edges meeting). A cut there is therefore a corner cut on all four layers at once.
2. Suppose the 1-inch square had been cut from the packet's top-left corner instead. What would the unfolded sheet show, and how large would the feature be?
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One square hole, 2 inches on a side, centred in the sheet.
The packet's top-left corner is the folded-folded corner - the intersection of the two creases. The creases are the sheet's vertical and horizontal centrelines, so this point is the exact centre of the 8-inch sheet, and it is interior to the paper on every layer. A cut there removes material from the middle, not the edge, giving a hole.
Size: the packet is half the sheet in each direction, so unfolding doubles each dimension. A 1-inch cut in the packet becomes a 2-inch × 2-inch hole. Check: the four 1-inch packet squares, mirrored left-right and top-bottom, tile a 2×2 region around the centre.
3. Which packet corner would produce exactly two cuts in the unfolded sheet, and what would they look like? Explain your reasoning.
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Either the bottom-left or the top-right corner - the corners where one folded edge meets one open edge.
Reasoning for bottom-left: its left side is the vertical crease (centre of the sheet) and its bottom side is an open edge (top and bottom edges of the sheet).
- Unfolding left-right mirrors the cut across the vertical centreline, joining the two halves into one notch twice as wide: 2 inches wide.
- Unfolding top-bottom copies it to the opposite edge: same notch, mirrored.
Result: two notches, each 2 inches wide and 1 inch deep, centred on the top edge and on the bottom edge - fewer, larger features than the corner cut, because only one fold doubled the count.