Magic Squares

A "magic square" is a rectangular array of numbers, usually from 1 to n2,  so that each column, row, and both diagonals have the same sum.  Other types of magic squares, with other shapes or special properties are common in recreational math.  You can find a very great number of examples of different squares and special features at the web site of Harvey Heinz with additional information on magic stars.

The history of magic squares dates back to at least 1000 BC in China.  A Chinese book called Lo Shu (book of the River Lo) relates the story of how a magic square on the back of a turtle saved the city. The image at right is from the web page of Karen Verschooren and Tine Uytdenhouwen
By the 2nd Century BC there were 4x4 magic squres appearing, often in connection with religious practice.  The Islamic/Arab mathematicians probably were introduced to the magic square from India, but they quickly developed squares of higher order.  One of the most famous illustrations of a magic square is in the famous Albrecht Durer woodcut, Melancholia.  The illustration and a blow up of the square can be seen at these links to the St Andrews University site.  Durer was a major contributor to the mathematics of art and is often credited with being the founder of descriptive geometry, yet he is probably known to most students, if at all, for this single woodcut. There are 880 different solutions to the 4x4 magic square, including the one in the Durer painting.

Here is a quote from David Singmaster on the history of Magic Squares,

MAGIC SQUARES. These are certainly Chinese in origin. Needham, Ho Peng Yoke, Camman and Lam cite a number of books which are not available in translations. These are the following.
Hsu Yo (Xu Yue). Shushu Jiyi.
I Wei Ch'ien Tso Tu. c1C.
Ta Tai Li Chi. c80. Chap 67: Ming Thang. Or: Chap. 8, p. 43 of Szu-pu ts'ung-k'an edition, Shanghai, 1919-1922.
Ts'ai Yuan-Ting. c1160. (He is cited by an author, quoting Needham, but I can't find him in Needham.)

I would be interested in seeing these and any similar early Chinese material.
Li Yan & Du Shiran's Chinese Mathematics - A Concise History, section 5.6 (and other books) describes 6 x 6 magic squares on iron tablets, using Arabic numerals, found at Xian in 1956. Can anyone give more details on these? These are apparently in the new museum in Xian.

I later found an image, shown below, referring to these metal plates at Xian in "A History of Chinese Mathematics by Jean-Claude Martzloff, J. Gernet and J Dhombres.

the "Mongol Period", it seems, is around the 13th century.

Here are some additional notes on the history of magic squares from the on-line magazine Convergence.

The idea of the magic square was transmitted to the Arabs from the Chinese, probably through India, in the eighth century and is discussed by Thabit ibn Qurra (known for his formula for amicable numbers) in the early ninth. A list of squares of all orders from 3 to 9 are displayed in the Encyclopaedia, (the Rasa`il), compiled about 990 by a group of Arabic scholars known as the "brethren of purity" (the Ikhwan al- safa), (see [11], Vol. 1, pp. 660f.). Despite all this, no general constructive methods appeared until slightly later. In 1225, Ahmed al-Buni showed how to construct magic squares using a simple bordering technique, but he may not have discovered the method himself. Biggs ([3], p.120), referring to a paper by Camman ([5] ), suggests that the methods explained by Moschopoulos may have been of Persian origin and be linked to those expounded by al-Buni. Camman indeed claims that the two methods given by Moschopoulos for constructing odd magic squares were known to the Persians, citing an anonymous Persian manuscript (Garrett Collection no. 1057, Princeton University). Even so, this document contains examples and not explicit methods.

It appears that magic squares were introduced to Europe through Spain. Indeed, Abraham ben Meir ibn Ezra (c. 1090-1167), an Hispano-Jewish philosopher and astrologer, translated many Arabic works into Hebrew and had a deep interest in magic squares and numerology in general. He travelled widely throughout Italy and beyond, and may have been the one of the people responsible for the introduction of magic squares into Europe.

The magic square at the right comes from pg 104 of The Man Who Counted, a fable about mathematics by Malba Tahan (which I belive is the pseudonym of Júlio César de Mello, a Brazilian Mathematics Professor). The square is called a diabolical magic square. Supposedly you can find the sum of 34 in 86 different ways. For example, the four numbers in the four corners, any quadrant of the square, or the four nearest the center all add up to 34.

The photo below shows a different kind of magic square. This one is on the Passion facade of the Sagrada Familia, the unfinished cathedral in Barcelona designed by Antoni Gaudi. Each row and column add up to 33, the supposed age of Christ at his death. In fact there are supposed to be exactly 33 differnt four number groupings that add up to 33; can you find them all?

Maurice Weitman was kind enough to send his image of this square and the facade

In October of 2004 Ms. Sheila Knight sent me a note indicating that she and her Year 7 students at The Grays School Media Arts College had found a total of 42 four number groupings. She added, "Perhaps Gaudi was a fan of “The Hitch-hiker’s Guide to the Galaxy” where 42 is “the answer to life , the universe and everything” – this works when you consider it is on a church!". She was kind enough to send a PDF file of her results.
In Dec of 2006, Ahto Truu sent me an email in which he expanded her list to 54 different sums. He also reasoned that probably Gaudi had no knowledge of "Hitch-hiker’s Guide". Here is his post, clipped a bit

In fact, there are 48 of them if we don't allow the two 10's or the two 14's to be used at once, and 54 if we do.
In the list below, all the possible sums are listed in a systematic order, with the entries not on Ms. Knight's list marked:
1 3 14 15
1 4 13 15
1 5 13 14
1 6 11 15
1 7 10 15
1 7 11 14
1 8 9 15
*1 8 10 14
1 8 11 13
1 9 10 13
2 3 13 15
2 4 13 14
2 5 11 15
2 6 10 15
2 6 11 14
2 7 9 15
2 7 10 14
2 7 11 13
2 8 9 14
2 8 10 13
3 4 11 15
3 5 10 15
3 5 11 14
3 6 9 15
3 6 10 14
3 6 11 13
3 7 8 15
3 7 9 14
3 7 10 13
3 8 9 13
3 9 10 11
4 5 9 15
4 5 10 14
4 5 11 13
4 6 8 15
4 6 9 14
4 6 10 13
*4 7 8 14
*4 7 9 13
*4 8 10 11
5 6 7 15
5 6 8 14
5 6 9 13
5 7 8 13
*5 7 10 11
5 8 9 11
6 7 9 11
*6 8 9 10

*1 4 14 14
*2 3 14 14
*2 10 10 11
*4 9 10 10
*5 8 10 10
*6 7 10 10

Also, I find it quite unlikely that Antoni Gaudi could have been a fan of "The Hitch-Hikers Guide to the Galaxy", as he died in 1926, more than 50 years before the Guide appeared (the first episode of the radio show was aired in 1978, and the first book was published in 1979 :)

A couple of years later, in 2009, I received a note from Christopher Mata that pointed out that the wall may have been influenced by the Hitch-Hikers Guide after all...
"I'd just like to note that the Passion façade was sculpted by Josep Maria Subirachs when he started work on the Sagrada Familia temple in 1987, not by Gaudí, and by then the hitchhiker's guide to the galaxy was in full circulation. Not that it is of any relevance to the matter, but for historical accuracy it isn't so much Gaudí's magic square as it is Subirach's, and was acquired bu rotating Dürer's Melancholia square and subtracting 1 from 15, 11, 10 and 16."
n More information about the design and its transformation from the Durer Square can be found at this web page

I'm not sure if there is any symbolism intended in the location of the square, but it is located on the wall behind the sculpture depicting Judas betrayal of Christ with a kiss. You can see a photo of the setting here.

One of the famous historical documents about Magic Squares is Euler's De quadratis magicis. It can be found here, and an English translation is here.

A different type of Sculptured Magic Square is in the garden of the Eaton Fine Art Gallery in West Palm Beach, Florida. The image at right shows a sculpture by Patrick Ireland in which the number of blocks form a magic square with a sum of... oh count them yourself...

Teachers interested in lesson plans related to magic squares, and links to lots more pages about them should visit Suzanne Alejandre's web page.   She includes graphic links to the turtle from the story of Lo Shu, and links to many cross-curricular themes with each unit.

For more on Magic Squares, you can download a report by Helen Porter that she has gratiously consented for me to archive.