If you fold a piece of paper 50 times, will it reach the Moon?

This post was prompted by @mikemcsharry 's tweet of a similar nature the other day. But is it true? For the pedants out there, I'm assuming that we're folding our imaginary sheet of paper exactly in half each time.

Stuff we need to know:

• How far away is the Moon?

It's about 384,400 kilometres away, and counting.

• How thick is a sheet of paper?

That all depends on the quality of paper and the manufacturer. Typical office paper (80 "gsm"*) is about 0.1mm thick, I think. As it makes sense to have everything in the same units, 0.1 mm is 0.0000001 kilometres.

Answering the question, part 1: How thick is a piece of paper folded 50 times?
I've seen this question tackled before, and the biggest mistake is to rush straight in and say "if we fold the paper 50 times, the paper will be 50 times thicker".

Think about it like this (feel free to grab a sheet of paper and try it):

• Fold a piece of paper in half once. It's twice as thick as it was, right?
• Fold it in half again. It's now twice as thick as it was last time- if you've got the paper in front of you, you can count the layers: it's four times as thick as the original piece of paper.
• Fold it in half again. if you count the layers, you'll see that it's now eight times thicker than the original.

By now, it's fairly easy to see what's happening: each time you fold, you're doubling the thickness of your lump of paper. That means it's fairly easy to see the long, slow way of calculating the thickness of the paper after 50 folds:

Start off with the thickness of one sheet, and then double it. Double it again. Double once more. Double again. And again. And again. Keep going until you've done it 50 times. If you're into button-mashing** on a calculator, you could type this in:

• 0.0000001 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 112,589,991

Because the original number (0.0000001) was in kilometres, we know that the answer must be. So if we fold a piece of paper 50 times, it would end up being 112,589,991 kilometres thick!

Answering the question, part 2: Would this reach the Moon?

This is the easy bit: we just have to compare the two numbers:

• Distance to Moon: 384,400 km
• Height of paper folded 50 times: 112,589,991 km

The second number is bigger (and, importantly, given in the same units), so we have our answer: yes, it would!

But it'd be boring to stop there...

• 112,589,991 / 384,400 = 292.897999...

That means that our folded paper would reach nearly 300 times further than the Moon! So what's that far out?

• Travelling inwards from Earth to the Sun, we'd cross the orbit of Venus just after climbing a bit more than a third of our stack of paper.
• About 80% of the way up our stack, we'd cross Mercury's orbit.
• We'd be about 3/4 of the way to the Sun when we were standing on top of our paper tower.
• Travelling outwards from the Earth away from the Sun, we'd cross Mars's orbit a bit more than a tenth of the way up the stack.
• We wouldn't quite reach the main asteroid belt, and we'd need more than 400 more of these paper towers to reach the next planet, Jupiter.

Why don't we do it, then?!

Give it a try! If you get past 8 folds (remember, you're folding the paper exactly in half each time!), I'll happily send you a Mars bar. Any more than 10 and I could probably be persuaded to give you my car.****

* Why they can't use standard notation, I don't know. "gsm" stands for 'grams per square metre', which would properly be written as "g/m²"
** I'm not into button mashing. There has to be an easier way - we can use powers:

• 2 x 2 means the same as 2².
• 2 x 2 x 2 means that same as 2³.
• 2 x 2 x 2 x 2 means that same as 2⁴.

See the pattern? So,

• 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 means the same as 2⁵⁰.

This means that we can write our calculation much more easily and accurately*** as:

• 0.0000001 x 2⁵⁰ 

*** We're typing less, so there's less chance of' 'user error'.
**** As a disclaimer, you're not getting my car.