Take a regular A4 sheet of paper. How many times do you think you can fold it at most? Just give it a simple try, and you'll find that your estimate is likely too high. In fact, the maximum number of folds you can make is no more than seven. Why does a piece of paper become increasingly difficult to fold after just a few folds? Let's look at the relationship between the thickness of the paper and the number of folds. If the thickness of an A4 sheet is 0.1 mm, then after one fold, its thickness becomes 0.2 mm; after two folds, it becomes 0.4 mm... The thickness of the paper increases proportionally with the number of folds. At the same time, the width of the paper decreases proportionally. The long side of an A4 sheet is about 300 mm. After the first fold, it becomes 150 mm; after the second fold, it becomes 75 mm... Thus, if you fold a 0.1-mm-thick, 300-mm-long piece of paper seven times (if you could actually do it!), you'll end up with a stack that's 12.8 mm thick and 2.34 mm wide. After the eighth fold, the paper would be 25.6 mm thick and 1.17 mm wide. The thickness would be more than 20 times the width. Obviously, no one could achieve such a feat.
If you don't fold the paper in a conventional manner, but constantly change the direction of folding; if you choose a larger or longer piece of paper, could we fold the paper more times? In other words, is the popular saying that "no piece of paper can be folded more than seven times" correct? A foreign "MythBusters" program once challenged this claim. The experimenters used a 0.1-mm-thick piece of paper the size of a football field to start the experiment, and eventually completed 11 folds with mechanical tools.
In 2001, an American high school girl provided a more comprehensive answer to this question. She carefully studied both the method of alternately changing the direction of folding and the method of continuously folding in one direction. She ultimately concluded that if you want to fold a piece of paper many times, it's best to choose a long but narrow piece of paper and keep folding it. She also derived the relationship between the number of folds, the minimum length of the paper, and the thickness of the material in this case. According to her formula, if you want to successfully fold a piece of paper 12 times, you need to choose a suitable toilet paper roll that's at least 1.2 kilometers long.
In January 2002, accompanied by her parents, she entered a large shopping mall to begin her challenge. She spread out the long toilet paper roll and started folding it. After 7 hours, she folded it 11 times, obtaining a 80-cm-long, 40-cm-high thick board. After one more fold, she finally succeeded in folding the long piece of paper 12 times.
In 2011, a math teacher and 15 students used more than 3.2 kilometers of toilet paper to set a new record of 13 folds in 4 hours. Does this mean that as long as you have longer paper and greater patience, the record can keep being broken? A simple calculation shows that the thickness of a 0.1-mm-thick piece of paper after 20 folds would reach 10^5 meters, and after 30 folds, it would reach 10^7 kilometers. And after 51 folds, its thickness would exceed the distance between the Earth and the Sun! Indeed, it's hard to imagine that such an astonishing thickness could be achieved after just folding a thin piece of paper dozens of times. In the example of folding paper, we see a counterintuitive fact: by constantly doubling, you can produce enormous numbers after a limited number of folds.