The riddle arose when the first showing of Roots - rerun on TV this week - topped even Howard Hughes' will for prompting Americans to trace their genealogies.

A writer named Edward Devol, said in March, 1977, that it's fine to go rooting around for ancestors, say 10 generations ago, but keep in mind that back then, you had 1,024 direct ancestors. (You had two parents, four grandparents. Keep multiplying.)

My mistake was to keep multiplying, one rainy Saturday morning.I discovered that, allowing 25 years per generation, I had 4,096 ancestors when the pilgrims landed; a million before Columbus sailed, and a billion not long ago after the Norman Conquest.

The implications were astonishing, especially since four billion people are alive now. The world, it seemed, once had an infinite population, decreasing by half every 25 years. If true, this meant bad things for the Washington real estate boom.

If false - well, how could it be? Didn't everybody have two parents? And didn't each parent have two parents? The only exception would seem to be virgin births.

I bludgeoned the life out of a couple of 1977 dinner parties with my theory, decided I was ahead of my time and shut up.

Then came the Roots rerun, this week. I decided to get expert advice.

I called Dave Gwatkin, who does international research in population and health, here in Washington.

"Something's got to be wrong," he said, "but I didn't see the program."

So I called Ellen Jamison, an international demographer at the Census Bureau, and explained the problem.

"Somehow that doesn't sound quite right," she said. "Think of it not as going backwards as going forwards. Suppose Adam and Eve have two children, and each of them has two . . ."

"Fine," I said. "But just to make the problem simple I assumed I was the only human alive. Imagine how many people there used to be if you start with four billion."

"This really isn't my field," she said. "Maybe you should talk with Eduardo Arriaga."

"You have to be doing something wrong," said Arriaga, who is chief of the evaluation division at the Census Bureau. "For one thing, you have to remember that several children can have two parents. Say there are four children of two parents. Well, then, each has half a parent. If you have eight, then each has one-quarter of a parent."

"Ahhhh," I said, closing for the pounce. "That would imply, however, that the more children we breed, the smaller the number of people in the past."

"Exactly," Arriaga said, his voice cracking with triumph, like a double-jumping checker. "Compare two populations of 1,000 each. One grows at 3 percent, the other at 1 percent. A year ago, the faster growing one had only 970 people, while the slower one had 990 people."

"Yes," I said. Anything to make him stop it before circuit breakers started slapping shut inside my head.

I called Georgetown University, next, and interrogated Murray Gendell, director of the university's demography program.

"I wish I could give you an answer right off the bat," he said. "But I think the answer lies in mortality rates. Let me check the library."

The next morning his enthusiasm sizzled through the receiver.

"For most of man's history, death rates have nearly balanced birth rates," he said. "The rate of growth in population was exceedingly small until relatively recently, about 15 per million per year."

Inspiration hit again. "The problem," I said, "is that all of my ancestors had to survive long enough to reproduce, or I wouldn't be here.So death rates don't matter."

At this point, I began to see the possibility of a Nobel in science.

Then I went too far.

I called Herman Chernoff, professor of statistics at MIT. He ended it quickly.

"Incest," Chernoff said.

This terrible thought had flickered at the outer edges of my thinking. In fact, I'd even tried to draw what a family tree would look like if everyone was the product of the union of first cousins. What I found was that these people would have six great-grandparents instead of eight. The progression would be slower - I'd have 20 ancestors rather than 1,024, 10 generations back - but it was still increasing.

I tried this gambit on Chernoff.

"Look at it from the reverse point of view, he said. "We start off with two humans, say Adam and Eve. Assume six children per generation. After 10 generations, you'd have a lot of people. If one of the people in the 10th generation looked back, using your system, he'd think he had 1,024 ancestors 10 generations ago, but he'd be wrong."

"Right. But it's entirely possible that I had two completely unrelated parents, and four unrelated grandparents . . ."

"Sure,' Chernoff agreed, with short cheer. "But if you keep multiplying, sooner or later some of those people have to be related. And that means that the increase of ancestors gets smaller, until they begin to decrease all the way back to two."

"Assuming there was not once an infinite number of people in the world?"

"Of course," Chernoff snapped.

My confidence, like that of an over-matched young boxer, had been shattered.

I'd like to think that I've given the world a system for proving you're descended from Aristotle, Attila the Hun, anybody you damn well please. (My personal favorite, among the billions, is Ethelred the Unready, an Anglo-Saxon king.)

But if you use it, you'll have to do the math yourself. And stay away from Chernoff.