# Folding a piece of paper in half [closed]

How tall would a piece of paper be if you folded it 100 times?

(Assume a normal piece of paper, and that you could theoretically fold the paper perfectly in half.)

• This looks like a homework. Also lateral-thinking in hypothetical situation? – Zikato Mar 23 '15 at 6:43
• Not homework. Just looking for interesting ways people approach this. – TwoShorts Mar 23 '15 at 6:44
• this is not a puzzle and it has been discussed here:iflscience.com/space/…, scienceblogs.com/startswithabang/2009/08/31/…, math.stackexchange.com/questions/729033/… – user9174 Mar 23 '15 at 7:44
• Depends on what you mean by folding 100 times. Do you mean 101 layers of paper or $2^{100}$ layers of paper? – kasperd Mar 23 '15 at 13:25
• Your title states "in half" but in the question you only say "if you folded it 100 times"? Must the paper be folded in half... or just folded? – Digital Chris Mar 23 '15 at 14:51

If you had some sort of super paper that defied all laws of physics, and was $0.05 \text{mm}$ thick, then folding it a hundred times would give a thickness of:
$0.05 \times 2^{100} \text{mm}$
$\approx 6.34\times10^{28} \text{mm}$
Note that the observable universe has a width of around: $8.8\times10^{29} \text{mm}$

Using the paper folding formula $W = \pi t \times 2^{3(n-1)/2}$ where $W$ is the width (of a square piece of paper), $n$ is the number of folds and $t$ is the thickness, we find that the width of the paper would have to be $0.05\pi \times 2^{3(100-1)/2}\text{mm}$ wide, to be foldable 100 times. This is approximately $7.9\times10^{43} \text{mm}$, which is larger than the universe, and hence we can deduce such paper does not exist.

A 'normal piece of paper' cannot be folded more that 7 times.

If any paper can be used, then it can be folded upto 12 times maximum ( Gallivan demonstrated that a single piece of toilet paper 4000 ft (1200 m) in length can be folded in half twelve times.)

Let's consider an ideal case that the paper can be folded 100 times,

Let the length and width of the paper be x and y.

Then after folding 100 times, the length and height will be x/(2^50) and y/(2^50)

Lateral Thinking: The length and width of the paper will remain the same after any number of folds. Only the height and width of the new shape will get reduced.

As we can see

1 fold = 2 layer of paper
2 fold = 4 layer
3 fold = 8 layer
4 fold = 16 layer
5 fold = 32 layer


we can get formula
Number of paper layers = (2)n where n is number of folds.
for 100 folds it will be (2)100
Height will be (2)100 X (Thickness of paper)