Talk:Magic square

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[edit] M+N, M*N, kM

I believe that if two matrices M, N are magic squares, then M+N, M*N, kM are also magic squares. If I'm right, could someone add this?

Also, out of interest -- let M be the set of all n x n magic squares, for some n. Let V be the real vector space (M,+), where + is matrix addition. Does V exist? How many dimensions has V? Can you give a basis for V? (I am asking this question for all n, but would appreciate answers even for a particular n > 1). -- SJK

Addition of magic squares produces magic squares (ignoring that the coefficients are not consecutive integers). Multiplication of magic squares, however, does not: 8163574922 is
91 67 67
67 91 67
67 67 91. -phma

[edit] To SJK:

Provided your definition of a magic square is just that the columns, rows and long diagonals have the same sum, then the magic squares with real entries do form a vector space over R.

For n = 2, the dimension is 1, because all the entries must be equal.

For n = 3, the dimension is 3, because you can choose the first row freely, but this then determines all the remaining entries. The following three matrices form a basis:

/ 3  0  0 \      / 0  3  0 \      / 0  0  3 \
|-2  1  4 |      | 1  1  1 |      | 4  1 -2 |
\ 2  2 -1 /      \ 2 -1  2 /      \-1  2  2 /

I don't know how to determine the dimension in general, although it must be at least n2-2n-1 and no more than n2. --Zundark 14:51 Sep 13, 2002 (UTC)

The dimension is well known to be n2-2n-1. Zaslav 03:17, 23 January 2006 (UTC)

[edit] Loh-Shu magic square

Isn't there a legend that the Loh-Shu magic square was first seen written on the shell of a turtle? -- Tarquin 21:34 Feb 1, 2003 (UTC)

See [1] Cheers Chas zzz brown 07:59 Feb 2, 2003 (UTC)
See Lo Shu Square. —Herbee 17:53, 20 Mar 2005 (UTC)

The following section is not very enlightening, so I'm taking it out until someone can elaborate.

Euler showed how to derive magic squares from Latin squares.

—Herbee 18:01, 20 Mar 2005 (UTC)


Yes, It is a legend. So that I personally believe that it is not a truth.--Meavel 06:05, 8 July 2007 (UTC)

[edit] Definition

  • Halló! I have some problems with the definition: Both according to Mathworld and to Harvey Heinz (this link should be included at #External links) the numbers start with 1 and end with n2.
  • I would suggest that Wikipedia should use the same definition. To my understanding the "other" squares are somehow related or just simple "magic patterns".
  • For analysis purposes more equivalent representations can be given: numbers from 0 to n2-1; odd numbers from -n2+1 to n2-1 and probably others.
  • I developed a set of templates to be used to ilustrate many properties of 4x4 type magic squares at meta:Category:4x4 type square. Please let me know if you have some time to work on this subject and to help to make a wikibook at en: in other languages. We should make some documenation first and a list of items to work on. Thanks in advance! Best regards Gangleri | Th | T 22:40, 28 July 2005 (UTC)

The term "magic square" covers many different though related ideas. They all have in common that the rows and columns have the same "magic sum". The diagonals should be included, but research mathematicians have often been very sloppy about that and other requirements; properly, a square that is magic for rows and columns, ignoring the diagonals, is "semimagic", not "magic". Recreational mathematicians (a different group from research mathematicians) have always included the diagonal sums in the requirements. I also have never heard of any magic squares whose entries are not integers, except in the vector-space generalization alluded to in another comment. Historically, squares with negative entries seem not to have appeared, but squares with 0 have appeared (centuries ago) and squares with entries that are consecutive integers but not starting with 1 have also appeared for centuries (usually, I think, as tools to make larger squares out of smaller ones). Squares with entries that are nonconsecutive integers have been studied in the twentieth century and possibly the nineteenth. Thus, your two references are excessively narrow and not historically justified.

Subtracting or adding any constant to the entries of a magic square, e.g., 1, will always give a magic square. This is so obvious that it seems silly to discuss it at length. (It might be worth a mention in the appropriate subsection, if there is one.)

Zaslav 03:28, 23 January 2006 (UTC)

[edit] Yang Hui's square and "table of Jupiter"

[edit] Constructing a magic square of doubly even order

It is necessary write the numbers from right to left? (Keyword: symmetry)

[edit] Western occult

  • Durer's Melancholia square, noted in the article, is reportedly based on the magic squares of his contemporary, occultist Heinrich Cornelius Agrippa. [2] Agrippa apparently assigned magic squares of the orders of 3 through 9 to the planets (and astrological/alchemical figures) Saturn, Jupiter, Mars, Sol (the sun), Venus, Mercury, and Luna (the moon) respectively, hence the "table of Jupiter" noted above. (See Agrippa's text for more.) I believe his "tables" would be inscribed on magic talismans, in hopes of harnessing the various virtues and powers of each particular planet.

[edit] Historic image available

If anyone's interested, I just uploaded Image:16th century arabic magic square.jpg. Couldn't see an obvious place to add it, so I thought I'd just let you know. — Laura Scudder ☎ 01:40, 10 February 2006 (UTC)

[edit] Ben Franklin and others

Mr. Franklin was a great fan of magic squares, and created quite a few HUGE (16x16, 24x24) magic squares with a number of interesting attributes.

I also recall a magic square that had to do with the 365 days in our year. After a brief web search, I found on this site: http://www.jainmathemagics.com/page/1/default.asp that it says "This 27 x 27 Magic Square Calendar has, as its central cell, the number 365 which is the number of days in a solar year. It has 364 dark cells which represent the number of nights, and 365 white cells which represent the number of days. The Magic Sum of the inner and central 3x3 square is 1,095 being the number of days in a 3 year period. The Magic Sum of the 9x9 square is 3,285 being the number of days in a 9 year period. The Magic Sum of the whole 27x27 square is 9,855 being the number of days in a 27 year period."

I also recall a few other crazy interesting magic squares (http://mathworld.wolfram.com/PrimeMagicSquare.html is neat), and know that there have been some famous people (besides Franklin and Durer) that dabbled with/used magic squares.

If anyone wants to run with this idea to add a section on a list of famous people who played with magic squares or who wants to add a section talking about the many connections with various occultist/whatever things ... go ahead. Email me at cht13er a t gmaildotcom if you like :-) cheater 14:03, 15 March 2006 (UTC)

[edit] Magic Squares of odd order

Perhaps I have missed something, but aren't the first and second paragraphs of this section saying the same thing? Bio 18:08, 22 November 2006 (UTC)

[edit] Source of info

I saw this site on smagic quares and I think the construction of magic squares there is easier. Can someone investigate this? --Blonkm 02:38, 17 December 2006 (UTC)

[edit] Islam

Islam is a religion, I think the part titled "Islam" should be renamed. It is about arabic uses of magic squares. It should be called "Arabs".

What about mid-eastern? Yongke 17:09, 11 March 2007 (UTC)


[edit] More "See also"

I moved 2 cube links to end, then added 3 hypercube links.Harvey Heinz 17:11, 12 September 2007 (UTC)