If you fold a paper 42 times, will it reach the moon?
The thing about doubling
Fold paper 42 times – The art of doubling.
We have a limited brain. At least most of us have. We are often more prone to use common sense than to operate mathematical theorems. Even if maths sometimes is as obvious as everyday street smartness.
The old folded paper problem isn’t all that strange if you look at it. It’s all about doubling, and doubling builds up fast. In the beginning, it’s slow but once you reach important figures, the doubling becomes impressive. Double 2 and you get 4, but double 100.000.000… Yes, you see the trick here.
The mathematical explanation
The mathematical explanation looks like this
h=o x 2^(n) where h is the height of the folded paper, o is the thickness, and n is the total number of folds.
If the first fold gives you 0,2 mm, the second will give you 0,4, and the third 0,8. Already at 8 folds, you’re at an inch, 25,60 mm. And 8 is about the maximum folds you can accomplish. Disclaimer: No, you can’t fold a paper 42 times. This is all a theoretical experiment. Mythbusters folded a paper 11 times with the help of a forklift, but that’s an exceptional deed.
At 11 folds you have 204.80 mm, almost a foot. At 15 folds, 3.276.80mm. At 20, 104.857.60 and now we have to start counting in kilometers. At 25 we reach 3,35 kilometers, more than 2 miles and we’re heading out of the atmosphere. At 30 folds, we’re 107 km (67 miles), At 35 folds, almost 3500 km ( more than 2000 miles). Closing in on the moon, at 40 folds we are about 110.000 km (68.000 miles) out in space, and two more folds we arrive at the final number… The total of 439.804.651.110 mm 439.804.65 km (273.281 miles and 653,39 yards). The average distance to the moon is 384 400 km
Another famous story
The magic of doubling has been used many times. By mathematicians, but also as entertaining stories.
The board game, Chess, was invented in northwest India sometime around the 3rd or 4th century. From there it spread to China and when it was introduced to the Emperor, he wanted to honor the merchant who brought this wonderful game to the knowledge of the son of Heaven. And he wanted to reward him with a gift.
– What do you want, he asked.
– Son of Heaven, said the merchant lying on his belly with his face to the ground. I am a poor man, and all I want is some rice.
– So be it. But how much do you want? asked the Emperor.
– I want only one single grain on the first square of the chess board, and two on the second, four on the third, and so on until the last square.
– Your wish shall come true, said the Emperor, who wasn’t very knowledgeable in science and mathematics.
The fact is that what he had agreed to give to the merchant was 18.446.744.073.709.551.615 grains of rice, with a weight of 1199.000.000.000 metric tons. That’s 1700 times the world production of rice today and probably more than the total of all rice that has ever been produced in all the world since the beginning of time. The Emperor wasn’t happy.
The second half of the chessboard.
“The second half of the chessboard” is a phrase and concept, introduced by Ray Kurzweil. The idea is to look ahead and understand the power of exponentiality. If the first half of the chessboard is very big, then the second half is so much bigger, that it can be difficult to even grasp. The first square of the second half is already bigger than the total of the first 32 squares. And the total of the second half is 4.000.000.000 times bigger than the total of the first half.
This idea has a significant impact on the business strategy of a company or an organization.
In the end, it’s also important to remember that, just as it would be impossible to fold a paper 42 times, the exponential growth only works within a certain interval. The merchant in China didn’t get all he was owed. But instead, he got his head chopped off.
Yes, if you could fold a paper that is 0,1 mm thick 42 times, it would reach the moon and more.
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