Introduction
Punched-hole and pattern folding problems test spatial reasoning. A sheet is folded, holes are punched, and you must identify the unfolded pattern. These are common in SSC CGL Tier-I and require careful step-by-step reversal.
Basics of Folding and Punching
When a paper is folded, each fold creates symmetry. A punch through all layers creates multiple holes when unfolded. The number of holes equals the number of layers at the punch point, unless holes overlap. To solve, track the fold lines and reflect the punched hole across each fold line in reverse order.
Key Points
- Each fold doubles the layers in the folded region.
- Unfold by reflecting the hole across the most recent fold line first.
- Holes on fold lines may merge, reducing the total count.
Solved Example 1
Question: A square paper is folded in half vertically, then in half horizontally. A hole is punched through all layers at the center of the folded paper. How many holes appear when unfolded?
Solution: 4 holes
Explanation: Two folds create 4 layers. Punching through all layers gives 4 holes. Since the punch is at the center, the holes are at the four quadrants' centers, not overlapping.
Symmetry and Reflection Method
Unfolding is the reverse of folding. If the last fold was along a vertical line, reflect the hole horizontally across that line. If the last fold was horizontal, reflect vertically. Continue until all folds are undone. This method works for any number of folds and punch positions.
Key Points
- Always reverse the folding order: last fold first.
- Reflection across a vertical fold line changes the x-coordinate; across a horizontal fold line changes the y-coordinate.
- Use a grid or coordinates to avoid confusion.
Solved Example 1
Question: A rectangular paper is folded along a vertical line 2 cm from the left edge. A hole is punched 1 cm to the right of the fold line. Where does the hole appear when unfolded?
Solution: Two holes: one at 1 cm right of the fold, and one at 1 cm left of the fold (i.e., 1 cm from the left edge).
Explanation: The fold line is at x=2. The punch is at x=3. Reflecting across x=2 gives x=1. So holes at x=1 and x=3.
Practice Questions
Q1. A square paper is folded diagonally from top-left to bottom-right. A hole is punched at the center of the folded triangle. How many holes appear when unfolded?
Answer: 2 holes
Explanation: One diagonal fold creates 2 layers. A punch through both layers gives 2 holes, symmetric about the diagonal. The center of the folded triangle corresponds to two points on the original square, symmetric about the diagonal.
Q2. A paper is folded in half horizontally, then in half vertically. A hole is punched at the top-right corner of the folded paper. How many holes appear when unfolded?
Answer: 4 holes
Explanation: Two folds create 4 layers. The punch at the corner of the folded paper corresponds to the four corners of the original paper. So 4 holes at the four corners.
Q3. A circular paper is folded in half, then in half again (quarter). A hole is punched near the center of the folded quarter. How many holes appear when unfolded?
Answer: 4 holes
Explanation: Two folds create 4 layers. The punch is not on any fold line, so it creates 4 distinct holes, one in each quadrant, symmetric about both fold lines.
Common Mistakes
- Forgetting to reverse the order of folds when unfolding. Always undo the last fold first.
- Assuming holes always double with each fold. If a hole lies on a fold line, it may merge with its reflection, reducing the count.
- Misidentifying the fold line location. Draw a diagram and label coordinates to avoid errors.
Quick Revision
- Number of layers = 2^(number of folds) if folds are independent and full.
- Unfold by reflecting the hole across the most recent fold line, then repeat for earlier folds.
- Holes on fold lines may overlap; check for symmetry and merging.