Science and Health

Ancients, puzzling in math

By Pam Belluck (China Daily)
Updated: 2010-12-19 08:48

Puzzle making is an ancient pastime. Several precocious papyri and other artifacts exist that display Egyptian mathematical ingenuity.

The Rhind Mathematical Papyrus, which dates to 1650 B.C., is one of the oldest. There are also the Moscow Mathematical Papyrus (at the Pushkin State Museum of Fine Arts in Moscow), the Egyptian Mathematical Leather Roll (which along with the Rhind is at the British Museum) and the Akhmim Wooden Tablets (at the Museum of Egyptian Antiquities in Cairo).

They include methods of measuring a ship's mast and rudder, calculating the volume of cylinders and truncated pyramids, dividing grain quantities into fractions and verifying how much bread to exchange for beer.

They even compute a circle's area using an early approximation of pi. (They use 256/81, about 3.16, instead of pi's value of 3.14159....)

It all goes to show that making puzzles is "the most ancient of all instincts," said Marcel Danesi, a puzzle expert and anthropology professor at the University of Toronto, who calls such documents "the first puzzle books in history."

Dr. Danesi says people of all eras and cultures gravitate toward puzzles because puzzles have solutions.

"Other philosophical puzzles of life do not," he continued. "When you do get it you go, Aha, there it is, damn it,' and it gives you some relief."

But the Egyptian puzzles were not just recreational diversions. In the Rhind papyrus, its scribe introduces its roughly 85 problems by saying that he is presenting the "correct method of reckoning, for grasping the meaning of things and knowing everything that is, obscurities and all secrets."

And the documents were practical guides to navigating a maturing civilization and an expanding economy.

"Egypt was going from a centralized, structured world to partially being decentralized," said Milo Gardner, an amateur decoder of Egyptian mathematical texts. "They had an economic system that was run by absentee landowners and paid people in units of grain, and in order to make it fair had to have exact weights and measures."

The Egyptian Mathematical Leather Roll, also from about 1650 B.C., is generally considered a kind of test to learn how to convert fractions into sums of other fractions.

The Rhind papyrus contains geometry problems that compute the slopes of pyramids and the volume of various-shaped granaries.

And the Moscow papyrus, from about 1850 B.C., has about 25 problems, including ways to measure ships' parts and find the surface area of a hemisphere and the area of triangles.

Especially interesting are problems that calculate how efficient a laborer was by how many logs he carried or how many sandals he could make. Or the problems that involve a pefsu, a unit measuring the strength or weakness of beer or bread based on how much grain is used to make it.

One problem calculates whether it's right to exchange 100 loaves of 20-pefsu bread for 10 jugs of 4-pefsu malt-date beer. After a series of steps, the papyrus proclaims: "Behold! The beer quantity is found to be correct."

The problems in these texts are not difficult by modern mathematical standards. The challenge for scholars is in deciphering them and checking their accuracy. Some of the numerical equivalents are written in a symbolic system called the Eye of Horus, based on a drawing representing the eye of the sky god Horus, depicted as a falcon. Sections of the falcon's eye represent fractions: one-half, one-quarter and so on, up to one sixty-fourth.

The equations are considered remarkably accurate.

"What is unsolved about them is the actual thinking in the scribe's head," Mr. Gardner said.

The New York Times