So I was at a friend's house Sunday night playing a game when this odd fact came up in conversation:
If you were to fold a piece of paper in half 42 times, it would reach the moon.
Several of those around the table scoffed at this, exclaiming that a single sheet of paper was simply too thin to have its thickness reach any substantial amount after only a few dozen folds. I pointed out it was entirely possible seeing as how doubling the thickness with each fold would lead to an exponential increase in thickness that would increase slowly at first before quickly getting larger. My friends were clearly imagining a linear increase in thickness.
I also knew that it is pretty much impossible to fold a single sheet of paper more than about 8 times -- though Myth Busters once folded a giant sheet the size of a football field 10 times. The resulting thickness (after hitting it with a bulldozer) was almost a foot tall, though there was quite a bit of air mixed in with the 1,024 sheets. The formula for finding out how many of something you'll have after doubling it N number of times is as follows where O is the original number (or size in our case).
o * 2^(n)
A standard sheet of paper is about 0.1 mm so 42 folds would give us this:
0.1 * 2^(42) = 439,804,651,110 mm
That's 440 billion millimeters, or 439,804 kilometers. The moon on average is 384,400 kilometers from Earth according to Google. I'd say this checks out.
To help visualize the data, I created a quick spreadsheet and graph that tracks the thickness of the paper for each fold.
|# Folds||Thickness (mm)|
And to graph that out in kilometers looks like this: