A circular paper filter disc is folded in half along its diameter, then folded in half again along its radius into a quarter-circle sector (having $4$ overlapping paper layers). Two distinct triangular notches are punched completely through the sector away from the curved perimeter. When the filter paper is completely unfolded back into the full circle, what is the total number of triangular holes visible in the paper? ________ (answer in integer).
Answer Key:
8
Step-by-Step Solution & Explanation
**Method 1: Fold Multiplicity Principle**
1. Initial disc has 1 layer.
2. First diametral fold produces 2 layers.
3. Second radial fold produces $2 \times 2 = 4$ layers.
4. Each punch penetrates through all 4 layers simultaneously.
$\text{Total Holes} = 2\text{ punches} \times 4\text{ layers} = 8\text{ holes}$
Integer answer: 8.