# Math Chat: World Population and Four Dice

Old challenge

Estimate what fraction of the people who have ever lived on Earth are alive on Earth today.

Aubrey Dunne assumes that each generation is twice as large as the one before. So the previous generation was just 1/2 as large as the current generation, the one before that just 1/4 as large as the current generation, the one before that just 1/8 as large as the current generation, and so on.

Since 1/2 + 1/4 + 1/8 + 1/16 + = 1, all of the previous generations add up to just one current generation, and half of all humanity is alive today. Actually humanity has not always been growing that fast.

Jan Smit and Chuck Gahr assume slower growth, so that as you move backward in time the population decreases more slowly, and they conclude that just about 10 percent of humanity is alive today.

John Goekler cites a speculative scholarly article on "How many people have lived on Earth" by Carl Haub in the February 1995 issue of Population Today, which finds much slower growth in earlier periods and conjectures only about 5 percent or so of humanity is alive today, out of a grand total of more than 100 billion people.

Roughly a quarter have lived since 1400, roughly half since AD 1, and only about 1 billion before 8000 BC. Modern Homo sapiens may have appeared more than 50,000 years ago. Edward Deevey in "The Human Population" (Scientific American, September, 1960), estimated 36 billion primitive Paleolithic hunters and gatherers going back a million years. Of course the evidence is sparse and estimates vary widely.

Question on graphics characters (S. H. Logue)

Ann learned that her old dot-matrix printer can create many graphics characters, each composed of any combination of 96 dots in a small rectangle 8 dots high by 12 dots wide. Ann decides to print out one of every possible character pattern, ranging from all-dots-white to all-dots-black, on a long tape, 10 patterns per inch. How long is the finished tape?

There are two possibilities (white or black) for the first dot, times two possibilities for the second dot, times two possibilities for the third dot, and so on, for a total of 296 possible patterns. The length of the tape is 296 times 1/10 inch, which works out to be 21 billion light years (where a light year is the huge distance that light travels in a year, about 6 trillion miles), or about the diameter of the entire universe.

New dice challenge (S. H. Logue)

Here are four magic dice, named A,B,C,D. The six faces of each cube have these numbers on them:

A 1 1 5 5 5 5

B 4 4 4 4 4 4

C 3 3 3 3 7 7

D 2 2 2 6 6 6

You and your opponent each choose one die to use throughout the game. Each round you both roll, and the high number wins. Which is the best die to have?

* Send answers and new questions to:

Math Chat

Bronfman Science Center Williams College Williamstown, MA 01267

or by e-mail to:

Frank.Morgan@williams.edu

The best submissions will receive the classic book "Flatland," which explains higher dimensional spaces. Math Chat is based on the live call-in TV show in Williamstown, Mass.

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