Puzzle stumped 96% of America's top maths students, twenty years ago. World-wide only 10% of students got it right and only 4% in the U.S. So says that article but I doubt it.
They tell you the string wraps around the cylinder 4 times, the cylinder is 4cm circumference and 12cm length. How long is the string?
This is so easy it burns because they tell you the numbers. You don't even have to solve the circumference from the ratio from the diameter, no pi, no volume of a cylinder to solve, no nothing. They veritably tell you how long the line. Then ask you how long the string. It's insane.
If you had to solve for the circumference of the cylinder, then that would be a different story, but they tell you so that part is done for you. 4. And exactly 4 of those loops around so 16.
Forget cm. It can be anything, inches, feet, yards, meters, snakes, dung beetles, anything. Anything at all. Forget cylinder, think: toilet paper tube or paper towel tube, 4 circumference times 4 loops and 12 length.
Presently the loops are a coil 16 somethings in length how much must the coil stretch to go one end of the tube to the other? 12 somethings. How much will be added to the coil? 12 somethings? No, because it's like a spring, and not a straight line plus the loops. The loops themselves form part of the length because they are not proper loops but stretch to the next stretched loop, but how much? How much to add? This is the core of the problem. This is where intuition kicks in. Let your mind go fuzzy and the answer pops out. You have 16 somethings X 12 somethings with a string that goes from one corner to the other. How long is the string? 20.
Why? Because, not as mathematician but rather as artist you know that a 16*12 square has a 20-something cross inside it. Any such square with sides so lovely as 4:3-somethings will have an X shape of two lines of 5-something inside it. Boom. You can see it in your mind.
From doing pop-up cards and such things a origami and certainly framing. By folding squares in half and measuring the resulting diagonal, by wrapping handles with rope, or sticks with string, or pulling a spring around a post, by wrapping string for a kite. You just know to add a pinch for each incomplete loop and that pinch is 1-something for each loop to stretch to the next stretched loop and that total to add is 4-somethings more. Double intuition check, 4-somethings loop with 1-something extra times 4 loops equals 20. The string is 20.
The mathematician divides it down to 4 Pythagorean triangles added back up, but it's really one big triangle. That's how to see it.