While procrastinating around the web I stumbled on a page that contained the image below, from cracked.com.
I can't help but believe that this is false… Even though the article header says:
22 Statistics That Will Change The Way You See the World
My question: is what the image below implies a mathematical impossibility? (…Just for procrastination's sake…)
If you could fold a piece of A4 paper just 42 times it would be thick enough to reach the moon
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The statement is true in two different senses. As Sabyasachi shows, the intended sense that $2^{42}$ times the thickness of a sheet of paper is greater than the distance to the moon is correct. In the spirit of achille hui's comment, the sentence is an implication with a false antecendent, so it is true in that sense as well. It is also true to say "If you could fold a piece of A4 paper 42 times then the moon is made of green cheese."
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Even if the sheet of paper were infinitely foldable, the answer is that no, you can't reach the moon by folding a sheet of A4 paper any number of times, for a reason that bears calling out (and in fact explains why a sheet of paper that size can only be folded a certain number of times — that is, why it's impossible to fold it 42 times in the first place): consider the last fold and imagine looking at the sheet in a cross-section perpendicular to this fold. The 'faces' of the folded paper that are at the top and at the bottom after the last fold must be connected along the fold edge, since they were part of a single 'face' before the fold — but this means that the distance along the paper between the top and bottom must be at least as long as the distance 'through' the paper on a straight line between them. In other words, you need to start with a sheet of paper that's at least 385,000km along at least one direction (using Sabyasachi's numbers) to be able to reach that far, regardless of what sequence of folds you use.
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Mathematically, it is possible. however realistically you will loss paper every time you fold, because you have to count the width of the paper being folded.
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I have been reading all these theoretical answers, but no one made a comment about taking a piece of paper and actually doing the folding.
I bet that you can not fold a standard sheet of paper (75 g/cm2) with your bare hands more than 6-7 times. And you will end up with a total height of about 1cm.
If you ask a group, most of them will think we can fold the paper 20, 30, 40 or even more times.
This exercise is a good one to show the disparity between the physical world and the abstraction of it inside our mind.
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Just for the sake of discussion, lets consider how skinny the paper would get after folding it 42 times.
A sheet of A4 paper is 30 cm long. If you fold it in half 42 times and alternate directions, you'll get down to a length of 30 cm / 2^21 = 1430 angstroms. ("cut in half" might be more accurate.) Your paper would mathematically reach the moon, but since paper is made of long cellulose fibers (thousands of units), it wouldn't really be paper any more. The dimensions of the paper would be under the length of a single cellulose fiber.
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I think the question has been misunderstood by those who offer a straight "no".
As already pointed out, it has been proven that you can only fold paper 7 times, if you always fold it in half every other direction.
You can fold it 11 times if you always fold it in the same direction, see here:
http://en.wikipedia.org/wiki/Britney_Gallivan
But if you fold it like an accordion, as demonstrated in the sketch, well, who says you cannot reach the moon? :-)
I don't think we have the machinery to perform such a fine folding task--see also @blah's comment. Anyone interested?
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Unless you tear the paper while you fold it, no two points of the paper can become farther from each other (in three dimensions) after folding than when the paper was flat.
Okay, perhaps there is some give in the paper, so let's generously say the folded paper forms a Lipschitz continuous embedding of the original flat paper into physical space, with Lipschitz constant $2$.
This still means that no two points on the folded (or scrunched or whatever) A4 paper can be farther apart than twice the diagonal of the flat paper, or about 72 centimeters. That's a far way from the distance to the moon.
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It depends on whether the sheet can be compressed.
The other answers all assume that the sheet has a constant thickness. If this is the case, then consider "folding" the paper in such a way that we don't care if the edge of the crease rips; i.e., we ignore the paradox demonstrated in Steven Stadnicki's answer. So our "folded" stack is really equivalent to cutting the paper into tiny rectangles and stacking them, as long as with each cut we separate every rectangle into two new rectangles (i.e. we double the number of rectangles each time). (This is a pretty loose definition of "folding," of course, but we're trying to reach the moon with a piece of paper, so that's hardly surprising.) If we use this definition of "folding", and we're able to perform the cuts at the atomic level and ensure that all the rectangles are perfectly stacked on top of one another, and the rectangles still have the same width as the original piece of paper (which is, at this point, a ridiculous assumption; see blah's answer), then yes, we'll reach the moon (as per Sabyasachi's answer).
If, however, the pressure created by making the folds (and cutting the paper into tiny rectangles and whatnot) compresses the paper so that it becomes less than ~0.1mm thick, then our exponent will no longer be valid. Say that during the cutting process, the cellulose fibers unravel somewhat, leaving only two layers of fiber. Since the fibers are 2-20 nm in diameter, let's say that the two-fiber-layer sheets are about 10nm thick. $2^{42} \times 10nm = 43,980 m$, which, according to Wolfram Alpha, is about five times the height of Mount Everest. Impressive, but only about 1.14% of the distance to the moon.
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There are $2$ more problems most people did not adress here.
If you folded a paper $42$ times, its area would decrease accordingly, and after $42$ folds, the area in contact with earth would be $1.4\cdot 10^{-14}$ metres. This means you would have a column of paper roughly $0.1$ of a micrometer in width (if it was square). If you always folded paper over the same axis (keeping the width constant in one direction), this would mean the paper is still less than $10^{-13}m$ thick in the other dimension. This is smaller than one atom.
A column of paper reaching to the moon would have to spin along with the earth. Its centre of gravity would be $219,902,325$ kilometres above the surface of the earth, way above the geostacionary satelite height. This means the paper would actually be pulled upwards by the centrifugal force of earth's spinning.
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Look at the pattern-
If you fold the paper $1$ time,you get $2$ folds ($2$ papers one below the other).
If you fold $2$ times you get $4$ folds.
If you fold it $3$ times,you get $8$ folds.
Now,surprisingly it is in the form of a G.P. with common ratio $2$.
We also know,nth term of a GP=$a_n=ar^{n-1}$(a=first term,r=common ratio,n=nth term)
Now,here $n=42$, we have $a_n=2\times2^{42-1}=2\times2^{41}=2^{42}$.
So,if we fold a paper $42$ times we will have a total of $2^{42}$ folds.
Assuming one fold has $0.1mm$ (nearly),you can get the thickness of our resulting paper as $2^{42}\times0.1=439804Km$(approx.) which is more than enough to reach the moon.
In my experience a standard sheet of paper, has thickness around $0.1$ mm.
Folding $42$ times, the thickness is,
$$2^{42}\times0.1\approx 439804 \,km$$
Wolfram Alpha tells us that the average distance is, $385000$ kilometers which makes the claim most certainly valid.