Question

A piece of paper is folded according to the sequence of steps given below. After folding, square cuts are made as shown in the last step. Which of the options will result when the paper is unfolded? 

• A. A
• B. B
• C. C
• D. D

Hint:

In paper cutting problems, first determine the number of layers in the final folded piece. Multiply the number of cuts by the number of layers to find the total number of holes. This can quickly eliminate most incorrect options. Then, consider the symmetry created by the folds to choose the correct pattern.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a paper folding and cutting problem. The key principles are: 1. A cut made through the folded paper will appear on every layer it passes through. 2. Unfolding the paper mirrors the existing pattern of cuts across the fold line.
Step 2: Key Formula or Approach:
A systematic way to solve this is to count the number of layers of paper in the final folded state and analyze the symmetry. This allows us to predict the total number of holes and their arrangement.
Step 3: Detailed Explanation:
1. Count the Layers: - Step 1: The paper is folded in half diagonally. This creates 2 layers. - Step 2: The top corner is folded down. The area of the folded section now has 4 layers. - Step 3 & 4: The left and right corners are folded in. The central area where the cuts are made now consists of 8 layers of paper. 2. Calculate the Total Number of Holes: - Three square holes are cut in the final step. - Since these cuts go through all 8 layers of paper, the total number of holes on the unfolded paper will be \(3 \times 8 = 24\).
3. Analyze Hole Orientation and Pattern Symmetry: - The folds create symmetry. The first diagonal fold creates symmetry along that diagonal. Subsequent folds create more complex symmetries. The final pattern will be symmetrical about both diagonals and also the horizontal and vertical midlines.
- The final folded piece is oriented at a 45-degree angle to the original edges of the square paper. Therefore, the square cuts will appear as diamonds (squares rotated by 45 degrees) when the paper is unfolded. This eliminates options A and C, which show axis-aligned squares.
4. Evaluate the Remaining Options:
- We are left with options B and D, both of which show diamond-shaped holes and have the required symmetry. - We must now use our hole count. Let's count the holes in B and D. - Option B: Has an inner ring of 4 holes and an outer ring of 8 holes, for a total of \(4 + 8 = 12\) holes. - Option D: Has an inner ring of 8 holes and an outer ring of 16 holes, for a total of \(8 + 16 = 24\) holes.
Step 4: Final Answer:
Our calculation predicted 24 holes. Option D is the only one with 24 holes. Therefore, D is the correct unfolded pattern.