The Luoshu is a real magic square
The Luoshu arranges the digits 1 to 9 in a 3×3 grid so that every row, every column and both diagonals sum to 15:
| 4 | 9 | 2 |
| 3 | 5 | 7 |
| 8 | 1 | 6 |
verified
Check it yourself: 4+9+2 = 3+5+7 = 8+1+6 = 4+3+8 = 9+5+1 = 2+7+6 = 4+5+6 = 2+5+8 = 15. The arithmetic is not in dispute and never has been.
verified
It is also, up to rotation and reflection, the only 3×3 magic square using 1–9. There are eight arrangements and they are all the same square viewed differently. 5 must occupy the centre, and the even numbers must occupy the corners — both facts follow from the arithmetic and neither is a choice.
That uniqueness is why this object recurs. Anyone who sets themselves the puzzle finds this square, because there is no other to find. It appears in Islamic letter-magic and in Agrippa's planetary squares as the square of Saturn, with the same sum of 15.
Yu the Great and the turtle
verified
The received legend has the pattern revealed on the shell of a turtle emerging from the river Luo, in the time of Yu the Great, and the Hetu diagram carried by a dragon-horse from the Yellow River. The story is a long-standing part of Chinese tradition.
verified
The phrase is genuinely early: the Analects and the Book of Documents refer to the He tu, and the Xici commentary to the Yijing mentions both. Early texts knew the names.
disputed
What those early texts do not do is show the diagrams. The surviving visual forms — the arrangement of black and white dots that everyone now reproduces — are attested from the Song dynasty (11th century), in the work of scholars such as Liu Mu and Shao Yong. Between the early textual references and the Song diagrams there is a gap of well over a millennium, and reconstruction across it is inference.
This gap is the most important fact on this page, and it is routinely omitted. “An ancient Chinese diagram” is true of the name and misleading about the picture.
The nine palaces
verified
The jiugong (nine palaces) scheme maps the nine cells onto directions, trigrams, seasons and phases, and is used in fengshui, in ritual movement patterns and in divination. Its use is well documented.
Notice the direction of derivation, which is the same as in Mesopotamia: the arrangement exists because of arithmetic, and the cosmological readings are attached to it afterwards. A 3×3 grid with a forced centre and forced corners is an unusually good armature for any ninefold classification, and Chinese cosmology had several ready to hand.
The same square elsewhere
verified
The 3×3 square appears in Islamic magic-square literature, in Agrippa as the square of Saturn, and in Indian and European magic-square traditions. Larger magic squares were studied seriously as mathematics in the Islamic world and later in Europe.
exploratory
Whether any of these appearances represents transmission from China, or independent discovery of a uniquely determined object, is not settled and probably cannot be settled from the surviving evidence. Given that there is exactly one such square to find, independent discovery is entirely plausible — and the ordinary rule applies: a shared result is only evidence of contact if the result was hard to reach independently. This one is not.
Hetu
verified
The Hetu is the companion diagram, arranging 1–10 in a cross pattern with the odd (yang) and even (yin) numbers paired by position, and is used to expound the generative relationships of the five phases.
disputed
The Hetu is not a magic square and has no equivalent arithmetic constraint; its arrangement is conventional rather than forced. It has the same Song-dynasty attestation problem as the Luoshu, and rather less to anchor it, since there is no mathematical property that would make it uniquely reconstructible.
I Ching & Daoism
The Yijing, whose commentaries these diagrams are attached to.
Confucian Numerics
The classical corpus that transmitted the references.
Western Esotericism
The same 3×3 square as Agrippa’s square of Saturn.
Numeral Systems
How Chinese tradition encodes number more generally.
Cite this page
Formatted references for this page. The evidence grade is part of the claim — when quoting a graded statement, carry the grade with it.