To the Moon!

If you have a piece of paper that is 0.1 mm thick, then how many times will you have to fold it in half in order for it to become tall enough to reach the moon?

Note: The distance from the Earth to the Moon is 384,400 km.

Round your answer to its ceiling.

(Source: Brilliant (problem page))

Imagine being able to put a perfect crease in that paper, which is assumed by this problem, although not feasible in practice. Once it is folded in half, it will be 0.2 mm thick. Then, when that piece of paper is folded in half, it will be 0.4 mm thick, and so it goes. Converting the distance in kilometers to millimeters, the question comes down to finding \(n\) in:

\[ 0.1 \times 2^n \ge 3.844 \times 10^{11} \]

In other words, to what power does two have to be raised in order to reach (at least) the Moon from Earth when multiplied by 0.1 mm? Solving:

\begin{align*}
2^n &\ge 3.844 \times 10^{12} \\
\log\left(2^n\right) &\ge \log\left(3.844 \times 10^{12}\right) \\
n \times \log\left(2\right) &\ge \log\left(3.844\right) + 12 \times \log\left(10\right) \\
n &\ge \frac{\log\left(3.844\right) + 12 \times \log\left(10\right)}{\log\left(2\right)}
\end{align*}

The right-hand result is approximately 41.81. First integer solution above this is 42. Nice reference (or at least nice coincidence), but don't panic about forgetting your towel just yet, because, as said, real paper can't be used to make a space elevator like this. (Here's why.)