- A square piece of paper is folded in half diagonally, then folded in half diagonally again. Two small triangular holes are cut from the folded paper: one near the center and one near an edge. When the paper is completely unfolded, how many holes will appear?
Answer: 8 holes total
When a square is folded diagonally twice, it creates 4 layers of paper. Each cut made through all layers will create 4 holes when unfolded - one for each layer. Since 2 cuts were made, this results in 2 × 4 = 8 holes total. Choice A incorrectly assumes only one diagonal fold. Choice B miscounts the layering effect. Choice D incorrectly multiplies by 6 instead of 4.
- A circular piece of paper is folded in half three times along different diameters. A small square hole is cut from the folded paper. After unfolding, the holes form a specific pattern. If the same circular paper were folded just twice along different diameters and the same cut were made, how would the resulting pattern differ?
Answer: The pattern would have half as many holes in the same relative positions
Three folds along diameters create 8 layers, so one cut produces 8 holes. Two folds along diameters create 4 layers, so the same cut would produce 4 holes - exactly half as many. The relative positions would be the same because both folding patterns maintain the radial symmetry of the circle. Choice A is wrong about positions. Choice C is wrong about the number. Choice D incorrectly suggests more holes with fewer folds.
- A triangular piece of paper is folded along its altitude, then folded again along a line parallel to the base. Two cuts are made: a small circle through all layers and a small triangle through only the top layer. When unfolded, what will be visible?
Answer: 4 circles and 1 triangle in predictable symmetric positions
The two folds create 4 layers. The circle cut through all layers will create 4 circles in symmetric positions when unfolded. The triangle cut through only the top layer will appear as just 1 triangle. The positions will be predictable based on the folding pattern. Choice B undercounts the circles. Choice C incorrectly assumes the triangle was cut through all layers. Choice D undercounts both shapes.
- A rectangular piece of paper is folded in half lengthwise, then folded in half widthwise. A circular hole is punched through the center of the folded paper, and a star-shaped hole is cut near one corner. What pattern will appear when the paper is unfolded?
Answer: 4 circles in symmetric positions and 4 stars in symmetric positions
Two perpendicular folds create 4 layers of paper. Any cut through all layers will create 4 identical shapes in symmetric positions when unfolded. Both the circle and star were cut through all 4 layers, so both will appear 4 times in symmetric positions. Choices A and C incorrectly assume some cuts didn't go through all layers. Choice B incorrectly assumes only the star cut went through all layers.
- A hexagonal piece of paper is folded in half along a line connecting two opposite vertices, then folded in half again along a line connecting the midpoints of two opposite sides. A diamond-shaped hole is cut near the center of the folded paper. Which statement best describes the unfolded pattern?
Answer: 4 diamonds arranged in a rectangular pattern reflecting both fold lines
Two perpendicular folds create 4 layers, resulting in 4 diamonds when unfolded. The pattern will reflect both fold lines, creating a rectangular arrangement rather than following the hexagon's natural 6-fold symmetry. Choice A incorrectly suggests a linear arrangement. Choice C confuses the number of folds with the hexagon's natural symmetry. Choice D undercounts the resulting shapes.
- A rectangular piece of paper is accordion-folded into 4 equal sections, then folded in half perpendicular to the accordion folds. A heart-shaped cut is made through all layers at the intersection of the folds. When unfolded, what pattern emerges?
Answer: 8 hearts arranged in two rows of 4 hearts each
When you encounter folding and cutting problems, think systematically about how each fold multiplies the number of cuts that appear when unfolded.
Let's trace through this step-by-step. First, the paper is accordion-folded into 4 equal sections. This creates 3 fold lines that divide the paper into 4 parts. Next, it's folded in half perpendicular to these accordion folds, creating one additional fold line that runs across the width. Now you have a small folded packet.
When you cut a heart at the intersection of all folds, you're cutting through multiple layers. The accordion folds mean your single cut will create 4 hearts in a row when those folds are opened. The perpendicular fold means each of those 4 hearts will be duplicated, creating a second row. This gives you 8 hearts total, arranged in two rows of 4 hearts each.
Choice A misses the effect of the perpendicular fold—it only accounts for the accordion folding. Choice B incorrectly assumes each fold quadruples the pattern rather than doubles it, leading to an impossible 16 hearts. Choice C dramatically undercounts by ignoring most of the folding effects entirely.
The key insight is that each fold doubles the number of cuts that appear when unfolded. With accordion folds creating 4 sections and one perpendicular fold, you get 4 × 2 = 8 hearts arranged in the grid pattern that matches the fold structure.
Remember: count the sections created by folds, then multiply by how the perpendicular fold affects the pattern. Each fold doubles your cuts.
- A square piece of paper is folded along both diagonals to form a smaller triangle, then folded in half along the altitude of that triangle. Two different cuts are made: a small square cut through all layers and a small circle cut through only the outermost 2 layers. What will appear when the paper is unfolded?
Answer: 8 squares and 4 circles arranged in symmetric patterns
Folding along both diagonals and then along the altitude creates 8 layers total. The square cut through all layers produces 8 squares. The circle cut through only the outermost 2 layers produces 4 circles when unfolded. Both will be in symmetric patterns but different ones due to the different cutting depths. Choice B undercounts both shapes. Choice C incorrectly assumes both cuts went through all layers. Choice D has the counts reversed.
- A square piece of paper undergoes the following sequence: folded diagonally, then folded along the perpendicular diagonal, then folded in half along a line parallel to one edge. Three small holes are punched through all layers. How many holes will appear when the paper is completely unfolded?
Answer: 24 holes arranged symmetrically across the entire square
Three folds create 8 layers of paper (2×2×2). Each of the 3 punched holes will appear 8 times when unfolded, giving 3×8 = 24 holes total. The holes will be arranged symmetrically due to the systematic folding pattern. Choice A undercounts by using only the number of folds. Choice C incorrectly suggests overlapping. Choice D undercounts and incorrectly limits the pattern to diagonal lines.
- A paper is folded into a complex pattern: first folded in half, then each half is folded in half again in the opposite direction (creating a Z-fold), then the entire Z-fold is folded in half perpendicular to the previous folds. A single star-shaped hole is cut through the center. How many stars will appear when unfolded?
Answer: 8 stars arranged in two parallel rows of 4 stars each
The Z-fold creates 4 layers (folded in half, then each half folded again). The final perpendicular fold doubles this to 8 layers. One cut creates 8 stars arranged in two rows of 4, reflecting both the Z-fold structure and the final perpendicular fold. Choice A undercounts the layers. Choice B ignores the Z-fold complexity. Choice D overcounts and incorrectly suggests overlapping.
