I've got a puzzle for the geeky genealogists.
The number of direct ancestors doubles every generation -- two parents, four grandparents, eight great-grands, etc. I have the names of all sixteen of my great-great-grands. Beyond them there are a few holes in the tree.
So with a doubling every generation in eleven generations we pass a thousand direct ancestors, in 21 generations we pass a million, and in 31 generations we pass a billion.
31 generations will take us back to about 1100 AD (to use round numbers). So in that year I have about a billion ancestors. Except the world population was only about 350 million. Hmm.
In my genealogy databaase there is one case where second cousins married. I can subtract two ancestors out of the great-grandparent generation. There is also the case where an ancestor had two wives. The great-grandson of the first wife married the granddaughter of the second wife. I can subtract one ancestor out of another generation. Those subtractions propagate back through the generations.
In those 31 generations that makes a difference of about 12 million ancestors, but the number of ancestors still tops a billion. (Aren't spreadsheets fun!)
I tried a mathematical experiment. What if a certain number of each generation were duplicates -- their descendants (along my direct line) intermarried and they didn't need to be counted twice in the number of ancestors. Such duplication is plausible in small communities with a lot of intermarrying. I started this in generation 11 and propagated it through generation 31. Yes, it helps -- the number of ancestors is no longer above a billion, but it is still at the level of world population.
I changed the number of duplicates to 4, then to 2. When half the ancestors in each generation were duplicates the number of ancestors in generation 31 were down to 2.5 million. Compared to the population of Europe (71 million in 1350 and where my ancestors are from) this begins to sound plausible in terms of population. Does it make sense in terms of a rate of intermarrying? Alas, my data of direct ancestors ends at about 1600.
The actual formulas:
cell A3 = A2 * 2
Accounting for duplicate ancestors
cell B12 = (B11 * 2) - (B11 / 2)
If I'm generation 1 then I determined generation 12 was born in 1600. For every generation further back I subtracted 25 years, so generation 28 was born in 1200.
Now on to the question that prompted all that musing. If my direct ancestors in 1100 numbered 2.5 million, what are the chances one of them might be a king?