Question

A square paper is folded and cut as shown below. The paper is then used as a template to spray-paint a white wall with black paint. Which pattern would be created using this template? 

• A. A
• B. B
• C. C
• D. D

Hint:

The easiest way to solve paper-folding problems is often to use the principle of symmetry. Identify where the cuts are made relative to the final fold lines. A cut on a fold line will be mirrored to create a larger, symmetrical shape. A cut on an open edge will be replicated in multiple symmetrical locations on the unfolded paper.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a paper folding and cutting problem. The best way to solve it is to work in reverse, unfolding the paper step-by-step and observing how the cuts are mirrored across the fold lines.
Step 2: Detailed Explanation (Unfolding Process):
Let's reverse the steps shown in the diagram.
Initial State: A triangular piece of paper with two cuts. `cut 1` is on the longest side (hypotenuse), and `cut 2` is on one of the shorter, vertical sides.
Unfold 3: The third fold was a diagonal one. Unfolding it mirrors the cuts across this diagonal line.

`cut 1`, which is on the fold line itself, will be mirrored to create a diamond or kite shape, with its long axis along the diagonal.
`cut 2` will be mirrored to create a triangular notch on the bottom edge of the resulting square (which is the top-left quadrant of the original paper).
Result after Unfold 3: We have a square (1/4 of the original paper) with a diamond shape pointing towards the bottom-right corner and a triangular notch on its bottom edge.
Unfold 2: The second fold was vertical. Unfolding it to the right mirrors the entire quadrant.

The diamond shape, which is centered on the vertical fold line, will be mirrored to form a larger, symmetrical diamond shape in the center of the resulting rectangle (the top half of the original paper).
The triangular notch on the bottom edge of the left quadrant will be mirrored, creating an identical notch on the bottom edge of the right quadrant. We now have two triangular notches on the bottom edge of the half-sheet.
Result after Unfold 2: A rectangle with a large diamond in the middle of its horizontal centerline, and two triangular notches on its bottom edge.
Unfold 1: The first fold was horizontal. Unfolding it downwards mirrors the entire top half.

The large central diamond, located on the fold line, is mirrored to create an even larger diamond/square shape in the absolute center of the final paper.
The two triangular notches on the top edge (which was the bottom edge of the top half) are mirrored to create two new notches on the bottom edge.
Additionally, the existing pattern of notches needs to be considered. Let's re-examine this step. The two triangular notches from the previous step were on the *bottom edge* of the top-half rectangle. When this is unfolded, this edge becomes the central horizontal fold line. Wait, let's re-trace from the cuts more carefully.
Revised Unfolding Analysis:

Folded state: A triangle. `cut 1` is on the hypotenuse. `cut 2` is on the vertical edge.
Unfold 3 (diagonal unfold): We get the top-left square quadrant. `cut 1` creates a diamond pointing towards the center. `cut 2` creates a triangle cut out from the right edge of this quadrant.
Unfold 2 (vertical unfold): We get the top half of the paper. The diamond on the fold line becomes a larger diamond on the vertical centerline. The triangle cut from the right edge is mirrored, creating a diamond-shaped hole on the right edge of the paper.
Unfold 1 (horizontal unfold): We get the full square. The central diamond on the fold line is mirrored, creating a larger diamond/square in the very center. The diamond-shaped hole on the right edge is mirrored, creating an identical hole on the right edge of the bottom half. Also, the pattern on the left is mirrored downwards.
This creates a pattern with a central shape and symmetrical holes. Let me rethink the location of the cuts.
Symmetry-based analysis (easier method):

The paper is folded into 8 layers (one-eighth of the total area). Any cut will be replicated 8 times through mirroring.
Cut 1: This cut is on an edge that, when unfolded, becomes a line from the center to the midpoint of a side. There are 4 such lines of symmetry. The cut will be mirrored across these lines, creating a star-like shape in the center.
Cut 2: This cut is on an edge that is part of the original paper's edge. This cut will be replicated in the middle of each of the four sides of the square.
Combining these: We get a central star-like/diamond shape, and four triangular patterns pointing inwards from the middle of each side.
This description perfectly matches the pattern in option (B).
Step 3: Final Answer:
By tracing the unfolding process and the symmetry it creates, the resulting pattern from the cuts will be the one shown in option (B).