The image to the left indicates why I could not resist my title. It shows a kink put into a long pipeline to prevent the pipe from fracturing when it expands in hot weather.
Which brings me to the mathematics. Imagine there is a stretch of pipe a mile long, laid in cold weather, firmly anchored at both ends. During hot weather, the pipe expands by a foot over its entire one-mile length, causing it to buckle upwards in an arch. Roughly how high will the arch be off the ground at its center (highest) point?
Before I give the answer, I should say that my point is to illustrate how bad we can be at some quantitative estimation tasks. An example that I think is better known than the pipeline puzzle is the one about wrapping a belt around the Equator. To wrap exactly around, the length of the belt has to be the circumference of the Earth at the Equator. But what if the belt is 10ft too long? How far off the surface of the Earth will the belt be if you string it from equally tall poles all the way around? Since an excess of 10ft seems tiny when spread over the entire circumference of the Earth, our intuitions tell us the height of the belt will be too small to measure. But in fact it is about 1.5ft.
Mathematically, the issue is straightforward. An increase in the circumference of length L corresponds to an increase in diameter of length L/PI, so the 10ft excess in the belt needs a circle of diameter 10/PI more, or about 3.2ft, to accommodate the excess length, putting the belt itself about 1.6ft above the ground.
Now back to the expanding pipeline. Primed with the answer to the circumventing-belt puzzle, what is your first guess about the height of that buckle? Remember, the pipe expands by a mere 1ft over its entire one mile length.
The answer is, I think, even more surprising than in the previous puzzle. The pipe forms an arch over 50ft high. Here's the calculation.
For simplicity, assume the pipe has a clean bend in the center, with each half remaining straight. The diagram below shows one half of the bent pipe, with the units in feet.
so h = 51.4 (approx).
Finally, I'll leave you with this puzzle. To the best of our knowledge, our species Homo sapiens is 200,000 years old. For a celebration of human evolution, you decide to line up a group of people to represent your entire personal lineage in the species, with you at one end holding hands with your father next to you, his father holding hands with your father, etc.
Sometimes, numbers just surprise us.