Science Puzzle
How Many Ancestors Do You Have?
You have 2 parents, 4 grandparents and 8 great-grandparents. Every step back, each person in your family tree has two parents of their own, so the number of places in the tree doubles.
A generation is roughly 25 years, so 30 generations takes you back about 750 years, to the late 1200s. How many ancestors does the doubling say you had back then, and why can't that number be right?
The Answer
The doubling says 1,073,741,824: just over a billion ancestors. But in the late 1200s the whole world held only around 400 million people.
Doubling 30 times is 2 multiplied by itself 30 times. Ten doublings give 1,024, twenty give over a million and thirty give over a billion. That's the trouble with doubling: it starts slowly and then runs away. You can't have more ancestors than there were people alive, let alone more than the handful who lived in the right part of the world, so something in the simple tree has to give.
What gives is the idea that every place in the tree holds a different person. Go back far enough and the same people turn up in your family tree again and again, because distant cousins married each other. If a married pair of your great-grandparents were second cousins, the two of them had a set of great-grandparents in common, and those people fill two places in your tree at once. Over 30 generations this happens constantly. Genealogists call it pedigree collapse.
So the billion places in your tree from 750 years ago are filled by a far smaller number of real people, many of whom appear hundreds or even thousands of times. Go back that far and you share ancestors with almost everyone whose family comes from the same region as yours.
The principle: Exponential growth and pedigree collapse. Doubling outruns any real population within a few dozen generations, so family trees must fold back on themselves, with the same ancestors filling many places.