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Ancient Mathematical Mystery Highly Cherished

One may find it easy to provide more than one solutions to the one-degree equation "x+y+z=15," with "x," "y" and "z" representing unknown numbers.

But how about building a 3x3 array of the numbers one through to nine, in which the numbers in each row, each column and each of the two diagonals have the same sum of 15?

This is Luoshu (Book of Luohe River), the unique perfect magic square of order three, which appeared in China about 4,000 years ago according to legends and has since been studied and revered for its "magical" properties.

Interpreted as a supernatural sign of the power of the universe, it dominated Chinese cosmological thinking by the 19th century when it was reduced in status to a mathematical curiosity, said Frank Swetz, professor of mathematics with the Pennsylvania State University in his book Legacy of the Luoshu.

As mathematicians and historians today go on with studies of the magic square, grassroots intellectuals at its place of origin also carry out their "research" on Luoshu, in leisure time when they are not farming, in an effort to preserve the cultural heritage.

Fan Shitou, 71, whose last name means literally "stone," is one of the farmers fascinated by the 3x3 display at Xichangshui Village, Changshui Town of Luoning County.

With a determination as firm as his name, Fan has searched for the meaning of the array for half a century.

He has filled the small brick house of two rooms, where he and his family live, with related books ranging from those of ancient Chinese history, to historical documents of the county and the province.

According to Chinese legends, a tortoise emerged from the water bearing the pattern of Luoshu with the numbers encoded in dots on its shell as Yu the Great, an early ancestor of the Chinese, stood on the banks of the Luohe River in central China about four millennia ago.

Inspired by Luoshu, Yu drew up a constitution named Hong Fan Jiu Chou, allegedly the first of its kind in China, to rule the country, according to early Confucianism literature, the Collection of Ancient Texts (Shang Shu).

It was also recorded in Book of Changes (Yijing) that the 3,000-year-old Chinese literature of philosophy was inspired by the magic square.

And the part of the Luohe River running by the Xichangshui Village was arguably where the tortoise and Luoshu made their appearances, Fan said.

"There's been a tradition at the village to peep into the mystery of Luoshu. I inherited the historical documents from my father and he from my grandfather," said the 71-year-old. "I go often to the county's library and I have copied with a pen more than 200 pages of documents related to Luoshu," he said.

Fan has been a member of the Luoshu Research Society of the county since it was established in 1994. He climbs over a hill to the county seat to attend the regular meeting of the society every month.

"Four of my fellow villagers have become members of the society at my recommendation, and we often discuss Luoshu with each other," he said.

Fan said he is most interested in how Luoshu influenced the planning of ancient Chinese capitals.

"The Luoshu pattern relates to the ancient Chinese ideal of a perfect world made up of nine divisions, with eight cardinal directions and the Son of Heaven at the centre. Qi (energy of life) is believed to flow smoothly in such an arrangement of perfect balance," he said.

"The city of Chang'an, capital of the Tang Dynasty (AD 618-907), was designed in the shape of a near-perfect square comprised of nine smaller squares within it -- a configuration of Luoshu. The Luoshu pattern has also impacted on the designs of Luoyang, when it was built as capital of the Eastern Zhou Dynasty (770-256 BC) and re-built in the Han (206 BC-AD 220) and Wei (AD 220-265) dynasties," he added.

When Fan dug into historical files, and his fellow villagers Qu Maiwang and Qu Shaobo travelled around the area to find relics related to the magic square.

They sailed upstream along the Luohe River, berthed their small wooden boat at the foot of a hill about 50 meters tall, and climbed along its stiff slope as they snapped vines grown along the way.

Over the hill there is a small pond, and on a giant stone beside the pond is a carved 28-character poem titled Written on a Visit to the Den of the Divine Tortoise.

At the end of the poem the writer claimed to be Liu Wucheng who was born in western Sichuan, but became "a governmental official in Guangdong."

"With the help of scholars of the county, we find that Liu was a government official in the Hongzhi Administration (1488-1505) of Ming Dynasty (1368-1644). He came a long way from Guangdong in the south of the country to our village to visit the site," Qu Maiwang said proudly.

Relevant cultural relics at the village also include an ancient Temple of Yu the Great, and two stone steles at the bank of the Luohe River.

One of the two, erected in 1724, bear four large characters written by Zhang Han, then governor of Henan, reading Luo Shu Chu  Chu (Where Luoshu Emerged Out of the River).

The other, with only a large character Luo on it, was allegedly erected more than 1,700 years ago, said Fan.

In the past five decades it has been under the protection of Fu Jianlin, 73, in front of whose house the stele stands.

When villagers were to build a bridge over a stream in 1953, some suggested that the millennium-old stele would make an excellent bridge seat, but Fu persuaded them to change that idea.

In 1954, the bridge was completed and some villagers wanted to carve on the ancient stele names of those who contributed to the construction of the bridge, and Fu again succeeded in his persuasions not to do so.

In 1959, when the country was having a "Great Leap" and steel making furnaces were being built everywhere, Fu protected the stele another time from being broken into pieces. It would have otherwise contributed to the construction of a furnace.

For 10 years from 1965, Fu and his wife made a large pile of straw and tree branches above the stele to hide it in the national movement to "wipe out the old."

In 2001, he built a wooden hut to protect the stele from erosion by wind and rain.

Today the county government is having a glass house built for the stele.

"I heard the old talk about the mystery of Luoshu in my childhood, so I think I have to keep it intact for people to wonder about in future years," he said.

(China Daily January 18, 2005)

'Magic Squares' Long Revered as Order in the Universe
Numerical Arrays Hold Clues to Cultural Origin
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