June 14, 1996

Suppose that a man has two children, say in the year 2000. And say that each of the children has two children 25 years later, and that each of those children has two children 25 years later, and so on. In the year 3000, how many new descendants are born?

Answer. The father has 2 children (the first generation), then 2x2 = 2^{2} = 4 grandchildren (second generation), and 2x2x2 = 2^{3} = 8 great grandchildren (third generation). In the year 3000 (at the 41st generation), he will have 2^{41}, or 2,199,023,255,552, ever-so-great grandchildren. That's about 2.2 trillion or 2.2 thousand billion new descendants.

But that answer seems too big. There would hardly be room on the earth for them all. The whole population didn't grow nearly that fast from the year 1000 to the year 2000. Today's global population is only about 5.7 billion. If every couple had two children, just to replace itself, the world population should not grow. So what's wrong?

The mistake comes from double counting, since after many centuries the descendants will run out of enough other people to marry and will have to marry each other. It's not fair to let them marry each other and have four children.

If everyone had two children, however, population would stay relatively constant, say for simplicity at 3 billion per generation. If descendants of our original father married other descendants or nondescendants at random (admittedly an oversimplification), after 31 generations or 775 years, more than half of all newborns would be descendents of his. From the 29th century our original father will hear a universal chorus from his three billion grandchildren:

"Happy Father's Day!"

Challenge question. Suppose every husband and wife keep having children until they have a girl and then stop. Assuming boys and girls are equally likely, will this produce more baby boys or more baby girls in the whole population?

Send answers, comments and new questions to: Math Chat, Bronfman Science Center, Williams College, Williamstown, MA 01267 or by email: Frank.Morgan@williams.edu. The best submissions will receive a copy of the classic *Flatland*, which explains higher dimensional spaces.

--1996 Frank Morgan