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OK, so I made an origami balloon using 4 sheets of peper (it is quite large). But then I thought, how many sheets of A4 paper are needed, uncut, so that Y sheets longways = x sheets shortways. This would create a square of paper, allowing me to create a huge origami crane.

AX (where X is a number) is an ISO standard, where the ratio of the height (h) of the paper to the width is sqrt(2) : 1

So, assuming that the sheet of paper is 1cm wide (which it isnt, but it is irrelevant), then the height is sqrt(2)cm.

Therefore, the height of the square = y(sqrt(2))

and the width = x

(Where x is the number of sheets put side to side, and y is the number of sheets placed end-to-end)

eg (an ASCII art diagram)
|_| |_| |_| |_|
|_| |_| |_| |_|
|_| |_| |_| |_|

y = 3, x = 4

So, when the paper is arranged in a perfect square, sqrt(2)y = x

I dont however, know how to go on from there. IF I could come up with two eqations in x, then maybe I could solve them simultaneously. Any ideas?

I am going to try and write a PHP program to brute-force the answer for me, but it's not a very elgeant solution.

If sqrt(2) is an irrational number, I dont think it will work (?) so I suppose I should check that out first...

Feel free to suggest I get out more... (But a huge origami crane would be kinda cool [creane in japanese is "tsuru"])

DS2K3 (BIG file)

methinks that sqrt(2) IS irrational, (2 is a prime number after all), so I will just have to take a best guess as to a] how accurate they were when they cut the paper, and b] how accurate I can be when folding, and therefore how accurate the sqaure needs to be.

I think round(sqrt(2), 20) should be more than accurate enough.

Answer: 1.4142135623 7309504880

So, I need to find when X = 1.4142135623 7309504880 * Y
Have you also taken into consideration the possibility of having to place sheets after rotating them 90. That is, to place the long side of the sheet against the short side of the sheet above or below it.
Wait a minute, I'm probably completely wrong here, but...

If (210 x 297 = 62,370) and vise versa basically, using standard legal letter orientation, the pattern would be 297 sheets wide and 210 sheets high, for a total of 62,370 sheets. Using LCM we reduce this to 20,790 sheets. As for the pattern, number of sheets high and wide, that's beyond me right now.

Edit, to use20,790 sheets would mean to have a pattern of 165x126 which doesn't work. However the first pattern does work by having a number of sheets high equal to the width of the paper in mm and vise versa. This works because we can use whole numbers for height and width of A4 paper thanks to millimeters.
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