Question 1

Step 1 · Exponential Growth in Thickness

Let standard initial paper thickness be $T_0 = 0.1\text{ mm}$. With each fold, thickness doubles:

After 1 fold: $0.1\text{ mm} \times 2^1 = 0.2\text{ mm}$

After 2 folds: $0.1\text{ mm} \times 2^2 = 0.4\text{ mm}$

After 3 folds: $0.1\text{ mm} \times 2^3 = 0.8\text{ mm}$

After 4 folds: $0.1\text{ mm} \times 2^4 = 1.6\text{ mm}$

Step 2 · Calculate Theoretical Thickness After 46 Folds

Given initial thickness $T_0 = 0.1\text{ mm} = 10^{-4}\text{ m}$ and distance to the Moon $\approx 384,400\text{ km} = 3.844 \times 10^8\text{ m}$.

Thickness after $n$ folds is $T_n = T_0 \times 2^n$.

For $n = 46$ folds: $T_{46} = 10^{-4}\text{ m} \times 2^{46}$

Since $2^{10} \approx 10^3$: $2^{46} = 2^6 \times (2^{10})^4 \approx 64 \times (10^3)^4 = 64 \times 10^{12}$

Since $6.4 \times 10^9\text{ m} > 3.844 \times 10^8\text{ m}$, the paper theoretically exceeds the distance to the Moon.

Step 3 · Analyze Practical Physical Limitations

In practice, folding is limited by physical constraints:

• Exponentially increasing thickness: Each fold doubles the thickness and substantially increases the force required to crease.
• Shrinking surface area: Each fold halves the area, leaving insufficient perimeter to wrap around the crease.

For a standard sheet of paper (like A4), the limit is typically $7$ to $8$ folds. Using thinner, larger sheets (like newspaper or tissue paper) allows up to $9$ to $12$ folds.