Quick Answer: What Is the Luoshu Magic Square?
The Luoshu (洛书, Luòshū) is associated with a three-by-three square using the numbers 1 through 9 once each. In that square, every row, column and main diagonal adds to 15. Chinese traditions also connect the nine positions with the Luo River story and the nine palaces (九宫, jiǔgōng).[1][5]
- The center of the standard arrangement is 5.
- Opposite numbers across the center add to 10.
- Yang Hui later discussed this size and larger numerical squares.[4]
The sums can be checked directly. The story attached to the square has its own historical sources.
Read the three numbers in any line through the middle of the Luoshu square. The outer numbers add to 10, and the 5 in the center brings the line to 15.
How to Check the Square
One traditional orientation reads 4, 9, 2 across the top, 3, 5, 7 across the middle and 8, 1, 6 across the bottom. The two side columns are 4 + 3 + 8 and 2 + 7 + 6. Each gives 15. The diagonals are 4 + 5 + 6 and 2 + 5 + 8, and both give 15 too.[1][5]
A rotation or reflection gives another orientation of the same arrangement. Reflecting this grid left to right changes the top row to 2, 9, 4 and preserves every sum.
Why the Sum Has to Be 15
The numbers from 1 through 9 add to 45. The three rows together contain all nine numbers. If each row has the same sum, that sum must be 45 divided by 3, or 15. Columns and diagonals still need separate checks, because a square with equal row totals alone does not necessarily meet the full condition.
In this arrangement, the four opposite pairs are 1 and 9, 2 and 8, 3 and 7, and 4 and 6. Each sums to 10. Adding the center explains all four lines that pass through it.
The Luo River Story and the Texts
One legend says a turtle emerged from the Luo River bearing the design. Textual references, descriptions of number positions and later illustrated forms need to be dated on their own evidence.[1][5]
A passage associated with the Book of Changes refers to writing from the Luo without giving nine numbers in that passage. The diagram became part of later interpretive traditions. It can appear alongside the eight trigrams, which form a different system of symbols.[2][3]
What Yang Hui Studied
The thirteenth-century mathematician Yang Hui presented numerical diagrams of several sizes. The three-by-three example belongs with larger squares in that work. He called them number diagrams, a useful reminder that the English term magic square describes a mathematical arrangement in this context.[4]
For other parts of Chinese mathematical history, compare counting rods and the problems in The Nine Chapters.
FAQs About the Luoshu Magic Square
Why are there several pictures of the Luoshu?
A rotation or reflection changes where a number appears on the page while preserving the row, column and diagonal sums.
Does the turtle story establish when the square was invented?
It is a legend associated with the diagram. The age of a given numerical arrangement depends on dated textual or material evidence.[1][3]
Did Yang Hui work with larger squares?
Yes. His number diagrams include squares of several orders, from the three-by-three example to order ten.[4]
Источники
[1] Mathematical Association of America. Legacy of the Luoshu
[2] Chinese Text Project. Book of Changes, Appended Statements
[3] The Mathematical Gazette. Magic cubes
[4] MacTutor, University of St Andrews. Yang Hui
[5] National Museum of Ethnology, Japan. Study in the history of magic squares before the sixteenth century