Paper Folding in the Art Room
In art class, students use an 8-by-8 grid of squares, numbered rows 1–8 from top to bottom and columns 1–8 from left to right. The paper is folded once along the vertical line between columns 4 and 5, with the right half folded onto the left half. It is then folded along the horizontal line between rows 4 and 5, with the bottom half folded up onto the top half. The resulting packet is a 4-by-4 stack. A hole punch is made through the entire packet at row 2, column 3, and a second punch is made at row 4, column 1. The teacher asks students to predict the full pattern when the paper is completely unfolded, without unfolding it first. One student says there will be four holes because there were two punches and two folds; another says eight. The teacher reminds them that a punch goes through every layer and that each fold acts like a mirror.
1. List the coordinates of every hole after unfolding. Explain how each fold transforms the punch location.
Read the worked answer
There are 8 distinct holes.
Use the fold lines as mirrors: the vertical fold is the line between columns 4 and 5, so a column c reflects to 9 − c. The horizontal fold is the line between rows 4 and 5, so a row r reflects to 9 − r.
- Punch at (row 2, column 3):
- original: (2, 3)
- vertical mirror: (2, 6)
- horizontal mirror: (7, 3)
- both mirrors: (7, 6)
- Punch at (row 4, column 1):
- original: (4, 1)
- vertical mirror: (4, 8)
- horizontal mirror: (5, 1)
- both mirrors: (5, 8)
Each punch goes through four layers, so each produces four hole positions. The order of the two folds does not change the final set because the reflections are independent and commute.
2. How many holes would result if the second punch were made exactly at the intersection of the two fold lines? Explain the change.
Read the worked answer
The total would be 5 holes, not 8. The first punch still produces four distinct positions: (2,3), (2,6), (7,3), and (7,6). A punch exactly at the intersection of the two fold lines lies at the center point where all four folded layers meet. When the paper is unfolded, those four layer-holes coincide at the same center point, so they count as one hole. The reason is that a punch on a crease does not create two separated mirror holes; the two half-holes join across the fold. At the intersection of two creases, all four quadrants meet at one point, so the count collapses from four to one. Distinct hole positions, not layers punched, determine the answer.