The Puzzle
You may have heard of the claim "I bet you can't fold a piece of paper more than eight times ... and you can use whatever size paper you want". This is the underlying issue that is 'The Paper Fold Puzzle".
For a long time the claim that this couldn't be done stood along w/ the ugly mathematical proofs that showed a piece of paper folded onto itself fifty times (if you could actually do that) would have a thickness greater than half the distance between the earth and the sun (paper will be 100 Million km thick ... the distance between Earth and Sun is approximately 150 Million km). The simple reason (and I'm pretty simple) is that you have a exponential progression in which your doubling the folded thickness each time you make a fold. If you're skeptical you are welcome to crank thru the many formulas below and in the link section.
Some Strategies
Many different stategies have been forwarded as the best approach to this problem. The classic format for the puzzle is to fold the paper in one direction ... turn it 90 degrees ... and fold it again ... then another 90 degrees ... and another fold ... etc ... etc ... etc.
It turns out that if you stick w/ that strategy you're not likely to succeed w/ any normal piece of paper. It might be possible to use some other material whose characteristics make it super thin (much thinner than normal paper). In fact, the woman (Britney Gallivan) who ultimately solved the puzzle and currenlty holds the record for number of folds initially took on this challenge using gold foil which has a nominal thickness of 0.05 mm.
After much analysis of the issue Britney came up w/ a mathematical model that relates the number of folds, width of the starting sheet, and the folding material's thickness (see the section titled 'The Math' or the first link in the links section to Britney's site)
The Solution
As was stated in the opening section above ... the one who solved the puzzle and currently holds the record for number of folds is Britney Gallivan. Britney was first presented with this problem as a high school student in 2001 and was offered extra credit in one of her math classes if you could solve the problem of folding a piece of paper 12 times (the classic form of the problem is 8 times).
Britney originally used gold foil w/ an unbelievably thin guage (thickness) in a 10 cm square and was able to successfully fold the piece of foil twelve times. Seems like a solution (emperical) but it was not accepted by her teacher since it wasn't technically done with paper.
So Britney sits down and after a careful analysis determines that the classic method of 'fold ... turn 90 degrees ... fold ... turn 90 degrees ... wasn't likely to work. Based on what she had determined from her close study she noticed that there was a key relationship between the number of folds, the width of the starting sheet, and the thickness of the material your using (thickness of the sheet of paper unfolded). You can see the details of the math below in addition to many references in the links section which have mathematical discussions for all types of paper folding (including origami). The image shown is her solution at the 11th fold.
The Math
Be careful of what you ask for. This is not for the faint of heart and for some just the sight of the formula could be enough to trigger seizures. I can't explan it so I'll leave it to the reader to persue on their own ... there are lots of links below that cover the math exhaustively for those of you who like this type of thing.
[Here's the first of two formulas that describe the most efficient way to fold the paper]
[Here's the second of the two formulas. I have no doubt that if there were six forumulas ... they'd all have the factor pi in there somewhere.]
