Mythology — A Story & Hour Cultural Series, No. 27
Room 4-2, Yunxi Primary School, Luoyang. A wet Friday in October, radiator ticking, windows fogged at the corners. Ms. Lin has chalk on her sleeve and fifteen minutes before the bell. On the board: a square divided into nine, and above it the numbers 1 through 9.
“The game,” she says, “is to put all nine numbers in the grid so that every row, column, and diagonal adds up to the same thing.”
The class erupts. Bottle caps, torn calendar corners. This is the Luo Shu (洛书), China’s legendary “Luo Writing” — the oldest magic square on Earth, a pattern China once carried out of a river on the back of a turtle, and the children are about to build it one sticky desk at a time.
A boy named Chuanlin puts 9 in the center and is refuted within a minute. A girl called Nannan works with her lips moving, slides 5 into the middle square first, then builds outward: 4 9 2 across the top. 3 5 7 through the waist. 8 1 6 along the bottom. Ms. Lin walks the rows like an auditor. Rows: fifteen, fifteen, fifteen. Columns the same. Both diagonals too.
Eight sums, all the same. For a moment Room 4-2 is quiet — even the radiator seems to listen — then someone demands a harder one. Nannan shrugs, the way children shrug at small miracles, and asks if they can play again tomorrow.
Ms. Lin looks out the window. Rain on the glass, the gray city, and — she knows, because she grew up here — under that same rain, a few kilometers north, the Luo River is running. She could tell them. She doesn’t, yet. The game has no name on the worksheet. It should. It has been sacred longer than Luoyang has existed.
The Question: Whose Numbers Are These?
Here is the puzzle behind the grid. Nine numbers, one square, a perfect balance around a center — this could have stayed a children’s game, a coincidence of arithmetic. Instead, for most of Chinese history, these particular numbers were treated as a message from the rivers themselves: carried out of the Yellow River by a dragon-horse, out of the Luo by a sacred turtle, before the first dynasty was founded. Emperors built halls to them. Geomancers read them in compasses. Mathematicians, a thousand years before Europe knew such squares existed, listed them alongside the equations of surveying.
The same nine digits have lived three lives — myth, cosmology, mathematics — without changing a single position. Which raises the only interesting question: when a culture finds a perfect pattern, what does it do with it?
Chapter I: Two Rivers, Two Gifts
The myth comes in two panels, like a pair of scrolls.
In the first, the world is still young. Pangu has long since become the mountains and Nüwa’s patched sky is holding (their creations are told here), and along the Yellow River a culture-hero named Fu Xi (伏羲) is trying to think. Out of the water rises a dragon-horse (龙马), the old texts insist — horse in body, dragon in scale and breath — with dots glowing on its flanks. Fu Xi reads the dots, counts their clusters, and from them derives the Eight Trigrams (八卦), the broken and unbroken lines that would become the atoms of the Yijing (易经), the Book of Changes.
In the second panel, generations later, the world is drowning. Yu the Great (大禹) has been walking the flooded land for years, channeling rivers, dragging order out of mud. One day, from the Luo River — the river that runs through modern Luoyang — a turtle surfaces and climbs the bank at his feet. On its shell: the same kind of pattern. Nine clusters of marks, one to a plate. Yu reads them, the legend says, and derives from them the Nine Provinces (九州) — the shape of a governable China — and the Nine Categories of the Hong Fan (洪范), the “Great Plan” the Book of Documents (尚书) says Heaven bestowed for taming the flood.
The Yijing’s appendix, the Xici (系辞), folds both gifts into one sentence of scripture-grade authority: “The River brought forth the Chart; the Luo brought forth the Writing; the sages took them as their model.” A chart from the Yellow River — the He Tu (河图). A writing from the Luo — the Luo Shu. When Confucius, in the Analects, despairs that his age has fallen out of heaven’s favor, his proof is a river: “The phoenix does not come, and the River gives forth no Chart.”
Chapter II: Reading the Turtle’s Back
Put the turtle on the table and count. This is what Yu saw — or what generations of ritual masters, counting turtle shells by lamplight, agreed he must have seen.
The pattern survives as a little verse, memorized by feng shui students the way medical students memorize bones: Nine at the head, one at the tail; three on the left, seven on the right; two and four as shoulders, six and eight as feet; five enthroned in the center. Lay it out and there stands Nannan’s grid — 4 9 2 / 3 5 7 / 8 1 6 — the turtle’s spine running through the five.
Look at what the numbers are doing. Every line through the center sums to fifteen: three rows, three columns, two diagonals — eight balances for eight directions, with five sitting at the still point. The odd numbers, yang by classical convention, hold the cross of the cardinal directions; the even numbers, yin, hold the corners. Every opposite pair around the rim — 4 and 6, 9 and 1, 3 and 7, 2 and 8 — sums to ten, which the Han cosmologists read as a completed cycle of generation. The square is balanced eight ways at once — and balanced in a way that maps onto space: the four directions and the four corners between.
And here is the fact that makes the classroom game worth playing past the bell: there is exactly one answer. Of the 362,880 ways to arrange nine digits in nine squares, only eight give a true 3×3 magic square, and those eight are one solution turning in place: four rotations and their four mirror images. Nannan did not choose her pattern. She found the only one there is.
Why fifteen? Arithmetic shrugs: with 1 through 9, the total is 45, one third per row, 15. But China noticed what else fifteen is. Fifteen days from new moon to full. Fifteen days in each of the twenty-four solar terms — the calendar’s unit for one thing becoming another. A perfect balance on a cosmic rhythm — coincidence, certainly. But it is the kind of coincidence that religions are built on.
Chapter III: From Shrine to Study
For most of the Han dynasty (206 BCE – 220 CE), the Luo Shu lived as architecture and ritual. The ritual hall of the kings, the Mingtang (明堂), was described as a square of nine rooms — and the Da Dai Liji (大戴礼记) records the room-numbers in one laconic line, “two-nine-four, seven-five-three, six-one-eight”: Nannan’s grid read aloud like a roll call. Heaven’s pattern as a floor plan. The medical classic Lingshu (灵枢) gave the nine palaces their winds, and diviners built boards to track the god Taiyi walking its rooms.
One of those boards survived. In 1977, archaeologists opened a Western Han tomb at Shuangguchui, near Fuyang in Anhui province — the tomb of the marquis Xiahou Zao, sealed around 165 BCE — and lifted out a lacquered divination board, the Taiyi Nine-Palaces board, its nine fields numbered exactly as the Luo Shu numbers them. It is the oldest physical magic square on Earth — two thousand years underground before Dürer was born.
One honesty the record deserves: the dot-diagrams as pictures first appear in the Song dynasty, credited to the recluse Chen Tuan (陈抟) and canonized by Zhu Xi (朱熹) on the frontispiece of his Zhouyi Benyi (周易本义, 1177); in 1706 the scholar Hu Wei (胡渭) demonstrated that the pictures were younger than the ideas they pictured. The myth is old; the diagram is medieval; the arithmetic is eternal. All three claims are true at once — which is how myth usually works.
But the square was never just venerated — it was studied. In 1275 the mathematician Yang Hui (杨辉) published his Xugu Zhaiqi Suanfa and treated these grids as a genre he called zongheng tu (纵横图), “vertical-horizontal diagrams”: he gave the method for building them and pushed on to squares of order ten. Europe’s first printed magic squares — Agrippa in the 1530s, Dürer’s Melencolia I leaning over its 4×4 square dated 1514 — arrive nearly three centuries after Yang Hui, two millennia after the marquis’s tomb was sealed. The puzzle page in your newspaper descends from Euler, not from the turtle. But the shiver of it — nine small numbers refusing to add up wrong — is the same shiver in Luoyang and in Nuremberg.
Meanwhile the square kept its day job. The Luo Shu became the Nine Palaces (九宫) of geomancy: the luopan compass carries its ring, the “flying star” school of feng shui flies its numbers through nine chambers year by year, and the Yijing tradition held that all of this — chart, writing, trigrams, the five phases themselves (traced here) — flowed from those two riverine gifts. Not arithmetic, exactly — China’s working image of a cosmos that stays whole: a still center, everything around it paired.
What the Nine Numbers Ask of You
Tonight, if you want to know how Yu the Great felt, you need no river. Take nine coins, number them, and try the game on the kitchen table. Most people need twenty minutes. The moment of finding it — the sudden stillness when every line agrees — is three thousand years old and completely fresh: the world, briefly, balancing.
None of this is destiny. The Luo Shu will not tell you your future; the ancients who mapped its winds would have said the same of your horoscope. It is something quieter and sturdier: pure order that human beings found — or invented, the distinction barely matters — and refused to put down. For three millennia they painted it on temple beams, built it into halls, marched it across compasses, proved theorems about it. A culture, given one perfect pattern, spent its whole history saying thank you.
So carry the question instead of the answer. What is the five at your center — the thing that does not move while everything else turns? Look at the square again: 4 9 2 / 3 5 7 / 8 1 6. A girl found it in a math class last Friday. A turtle carried it out of a flood. Both sentences are true, and that — not magic — is what a culture is: every generation finding the same balance for itself.
A Moment to Keep
Nine numbers, one center, eight directions agreeing. The ancients put it on a turtle’s back because some balances deserve a body. Mark yours — the afternoon the pattern finally held, on paper, on a napkin, in a name you give the day — because a moment that adds up in every direction is not repeated. It is kept.
Frequently Asked Questions
What are the He Tu and Luo Shu in Chinese mythology? The He Tu (河图, River Chart) and Luo Shu (洛书, Luo Writing) are two legendary number-patterns: a dragon-horse rose from the Yellow River bearing the Chart for the culture-hero Fu Xi, and a sacred turtle climbed from the Luo River bearing the Writing for Yu the Great. The Yijing says “the River brought forth the Chart, the Luo brought forth the Writing; the sages took them as their model.”
Is the Luo Shu really the world’s oldest magic square? The Luo Shu is the classic 3×3 magic square — the digits 1 to 9 arranged so every row, column, and diagonal sums to 15. In China the pattern is described in Han-dynasty texts, and the earliest surviving physical magic square is a lacquered divination board from a tomb sealed around 165 BCE — centuries before magic squares appear anywhere else.
What does the Luo Shu have to do with feng shui? The square’s nine positions became the Nine Palaces (九宫): eight directions plus a center. Feng shui compasses carry the Luoshu ring, and the “flying star” schools track numbers migrating through the nine palaces over time. It is read as a symbolic map of space and change — cultural interpretation, not prediction.
How do you solve the 3×3 magic square? The key is the center: since each of the four lines through the middle accounts for all three lines summing to the total, 5 must sit in the center, and the pairs around it must total 10. Place evens at the corners, odds at the cardinal points — 4 9 2 / 3 5 7 / 8 1 6 — and every line sums to 15. The solution is unique up to rotation and reflection.
Did the dragon-horse and the turtle really exist? The bearers are myth; the pattern is real arithmetic. Historians trace the dot-diagrams themselves to Song-era transmission, while the numbers and their lore are far older — a reminder that myths grow backward onto genuine wonders, the way a temple (the Dragon-Horse Temple still stands near the Yellow River) grows around a story.
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