posted much too late at: http://ronin-kakuhito.livejournal.com/257809.html
So, instead of sleeping...
It turns out that absent bastardy and immigration, our pateralinnear system of naming would eventually winnow down our family names until there was only one.
Assumption (false): Every child is either legitimate or takes his or her father's last name.
Assumption (false): in the case of marriage, every woman takes her husband's last name.
Assumption (False): Across a population, you will get the same number of girls as boys.
Assumption (False): In the case of marriage, before getting married the subjects will have different last names.
Based on these four assumptions, you can show that eventually any given family name will die out with a single exception. That exception is whichever family name ends up being the universal name.
Let's take a look:
For Abe Smith, half of his descendants will be boys and half will be girls. With the four assumptions above, the girls don't matter for carrying on Mr. Smith's last name. None of their children will be named Smith.
For the boys, all of their children will be named Smith. The sticking point is that not every boy will end up having children, and some of the boys will have only female children.
Here, let's look at an example. I'll be using coins and dice. Mr Smith and his descendents will each have 1d6-1 children and each of those children will have a i in 2 chance of being female. Abe, being odd, started a tradition within his family. Each child carries his or her father's first name as his middle name. I'll bold each child who ends a line of the Smiths.
Abe Smith 2 children
Barry Abe Smith(2), Bianica Abe Smith
Carl Barry Smith(1), Christine Barry Smith
Darla Carl Smith
Hum.. that was less than satisfying... lets try again.
Abe Smith (4)
Barry Abe Smith (2), Bianica Abe Smith, Bethany Abe Smith, Betty Abe Smith
Carl Barry Smith (4), Christine Barry Smith(4)
Dennis Carl Smith(5), David Carl Smith(3), Douglas Carl Smith(2), Dominique Carl Smith(0)
Epheram Dennis Smith (0) , Ezekial Dennis Smith (2), Ernest Dennis Smith (3), Eustice Dennis Smith(1), Erika David Smith (2), Elsie David Smith(2), Ephemeria David Smith (1), Evan Douglas Smith (3), Erin Douglas Smith (3)
Francine Ezekial Smith(0), Fatima Ezekial Smith (4), Frank Ernest Smith (0), Fred Ernest Smith (4), Felicity Ernest Smith (0) , Frank Evan Smith (2), Falstaff Evan Smith(1), Francis Evan Smith(2)
Gregory Fred Smith(0), Galahad Fred Smith(1), George Fred Smith(2), Giselle Fred Smith(4), Giddion Frank Smith(5), Gabrielle Frank Smith (4), Gabe Falstaff Smith(0), Gia Francis Smith(5), Gloria Francis Smith 1
Isabella Galahad Smith(3), Isaac George Smith(0), Irma George Smith(4), Irving Giddion Smith(5), Ira Giddion Smith(2), Ima Giddion Smith(5), Ionia Giddion Smith(0), Irene Giddion Smith(3),
John Irving Smith(3), James Irving Smith(0), Janice Irving Smith(5), Jessica Irving Smith(4), Jem Irving Smith(5), Jurt Ira Smith(4), Joan Ira Smith(1)
Ken John Smith(0), Key John Smith(1), Kelly John Smith(3), Kyle Jurt Smith(0), Kelsie Jurt Smith(3), Kyra Jurt Smith(5), Konstance Jurt Smith<(5)
Lisa Key Smith(4)
In this case, Abe isn't going to be alive to meet the last member of his family to carry his name, since she is his Great Great Great Great Great Great Great Great Granddaughter, but she is still there, the final bearer of the noble Smith name. (She also has an incredible passel of assorted cousins, on the order of 9500 of them (the average of a d6-1 is 2.5, and this is 10 generations later, so 9527 people in the last generation, and 15900 descendants total.)
