MAGIC SQUARES

Lies De Sutter

An De Brandt

Katrien De Bruycker

Vicky De Marteau

Why this subject ?

When we think about Mathematics we think most of the time of long lessons but there is more…

One day our teacher of Mathematics talked to us about Magic Squares.

The magic immediately caught us in it's mysterious arms, so we wanted to know more about this subject.

It is merely coincidence or is there another unknown power behind it?

Is it the same fascinating magic as we can also find in fractals?

Does nature have a secret that it will never reveal totally for mankind?

Is the entire universe with it's living creatures build up by a complicated (or simple ?!) pattern?

We wanted an answer on all these questions but actually … there are no answers at this moment.

So we decided to start with basic research of these intriguing constructions : Magic Squares.

What are they, do they have a use, how were they discovered, what will they learn us in the future ?

Let us take you on a journey through this magical world.

What Is A Magic Square?

A magic square is a simple mathematical game developed during the 1500s. You prepare a single square which is divided into the same number of rows and columns. Then, start filling each little square with the number from 1 to x where x equals the number of rows multiplied by the number of columns. You can only use a number once. Fill each square so that the sum of each row is the same as the sum of each column. In the example shown here, the sum of each row is 34, and the sum of each column is also 34. Note that the numbers from 1 through 16 is used only once. This is called a magic square.

Some of The Greatest Magic Squares

Take a look at this magic square. The sum of each row is 260, and so is the sum of each column. The numbers from 1 through 64 is used only once, so this is a valid magic square. But that's not all. When you add the first four numbers of any rows or columns, you get 130 which is the half of the sum of each row and column. But that's not the point! The greatest thing is that you can trace all the numbers, from 1 through 64, by "knight jumps". (A knight jump is a movement that a knight can perform in a chess game.) Try it. This amazing magic square was created at the beginning of the 18th century.

How about this magic square? Just like the previous example, the sum of each row is 260, and so is the sum of each column. The numbers from 1 through 64 is used only once, so this is a valid magic square. Like the previous example, when you add the first four numbers of any rows or colums, you get 130 which is the the half of the sum of each row and column. But from here is the different point than the previous one. Try adding any four little squares whose distances from the origin are the same. Also, take any squares consist of 4 little squares, and add those 4 numbers. In both cases, you get 130.

Review of Magic Cubes

"The construction of magic squares is an amusement of great antiquity; we hear of magic squares in India and China before the Christian era, while they appear to have been introduced to Europe by Moscopulus who flourished Constantinople early in the fifteenth century.

"However, what was at first merely a practice of magicians and talisman makers has now for a long time become a serious study for mathematicians. Not that they have imagined that they would lead them to anything of solid advantage, but because the theory was seen to be fraught with difficulty, and it was considered possible that some new properties of numbers might be discovered which mathematicians could turn to account. This has in fact proved to be the case, for from a certain point of view the subject has been found to be algebraical rather than arithmetical and to be intimately connected with great departments of science such as the 'infinitesimal calculus,' the 'calculus of operations', and the theory of groups.

"No person living knows in how many ways it is possible to form a magic square of any order exceeding 4x4. The fact is that before we can attempt to enumerate magic squares we must see our way to solve problems of far more simple character.

"To say and to establish that problems of the general nature of the magic square are intimately connected with the infinitesimal calculus and the calculus of finite differences is to sum the matter up."

Background

Magic cubes are extensions of some of the principles of magic squares. Magic squares are square arrays of numbers, in which the sum of each row column, and diagonal is identical. The sum of each row column and diagonal is called the magic number. The simplest magic square is a 3x3. There are countless ways to construct magic squares of even and odd order.

The characteristics of numeric magic cubes, odd or even, are that all straight columns, whether running from the top of the cube to the bottom, from the front to the back, or from one side to the other, should add up to the same number, and the four triagonals which unite of the cube also add up to the same number.

Magic squares and cubes are more than just number games; they use the principles of mathematics in their construction. Magic squares/cubes are included in the areas of: set theory, matrices, number theory, and combinatorics.

One might assume that some general formula exists such that all magic cubes and squares can be automatically produced by the computer, but there are countless and varying methods of construction. This field of study has continued to expand as computers are used to study the implications of number theory. Harold Stark of MIT included a chapter on "The Uniform Step Theory" (A method of constructing magic squares) in An Introduction to the Theory of Numbers, published in 1994. Because number theory is concerned with the properties of numbers, particularly integers, he considered the topic of theoretical interest.

The Rubiks Cube is sometimes called a magic cube. It has some similar properties; but it is constructed with colors rather than numbers. These magic cubes are not the topic of this project.

Magic cubes have several interesting properties. They are classified as either odd or even. In odd magic cubes, the sum of any two numbers diametrically opposite to each other should be equal to the sum of the first and the last series used. In these points the cube is close to perfection, but the corner diagonals of the 6 outside squares consist of various sums besides 42. Only one magic cube has been discovered with perfect characteristics, an 8x8x8. All others have perfect diagonals, sometimes called pandiagonals, or triagonals, in which the sums equal the magic number. On these cubes the diagonals on the faces of the cubes do not have the sum of the magic number. The smallest magic cube is 3x3x3. There are 27 numbers in this square. The magic number is 42.

Where do they come from?

Yin King is the name of a very old Chinese book. No one knows who has written it! In this book there is a story about a great turtle that appeared out of the Yellow river one day. On this great turtle's back were strange marks. These marks were dots that stood for the numbers 1 to 9. They were arranged in such way that no matter which direction you added the number up the answer was always 15. It was a magic square.

A brief history of magic square in Japan

Japan was isolated from western society in the Edo erra (1603-1867). During those days Japanese mathematician created their own mathematical world. Many difficult problems were presented and solved. Most of the answers were dedicated to temples or shrines as beautiful pannels which were called "San-Gaku". Many of them were lost during the tide of modernization after the Meiji revolution. About nine hundred San-Gaku, howver, are seen nowadays in rural area of Japan.
Many books on the Japanese mathematics (called "Wa-San") were also published during the Tokugawa-Shogun's period.
These mathematics are called "Wa-San"(old Japanese Math) in order to distinguish them from western mathematics.
Beauty of the Wa-San was recently introduced to the western society by "Japanese Temple Geometry Problems" written by Mr. H. Fukagawa, published from Charles Babbage Research Centre (1989).
Mr. Fukagawa wrote another book on Wa-San (not translated into English yet). Following short comments on the magic squares in Japan were seen in the book.

  2  1  3
  3  2  1
  1  3  2

Above magic square using 1 , 2 and 3 was seen in a book publised in 1743.

  6  1  8
  7  5  3
  2  9  4

This well known square was seen in another book published in 1840.

           19

           16       

17         13          18
    14             15
       11       12
            1
        9       10
     6              7
 3          8           4 

            5
 
            2

This magic circle was reported in 1660.

World Records from 1975 to now

1975 105 x 105 Richard Suntag (Pomona, USA)

1979 501 x 501 Gerolf Lenz ( Wuppertal, Germany)

1987 897 x 897 Frank Tast and Uli Schmidt ( Pforzheim, Germany)

1988 1000 x 1000 Christian Schaller ( Munich, Germany)

1989 2001 x 2001 Sven Paulus and Ralph Bülling and Jörg Sutter

( Pforzheim, Germany)

1991 2121 x 2121 Ralf Laue (Leipzig, Germany)

1994 3001 x 3001 Louis Caya ( Sainte-Foy, Canada)

Largest magic square written by hand

1990 1111 x 1111 Norbert Behnke ( Krefeld, Germany)

A very, very magic square

The following 25x25 matrix is a magic square.It has the following properties:

• Then entries of the square are the numbers 1,2,...,625=25x25.
• All its column sums, row sums and both diagonal sums are all equal to 7825=25x(1+625)/2.
• If the square is divided up in 25 5x5 squares, then all those little squares are magic too: they all have row, column and diagonal sums equal 1565=5x(1+625)/2.

If one squares all entries in in the square, the square remains magic: all row,column and diagonal sums are equal to 3263025

A very, very magic square

Ultra Super Magic Squares of 5 x 5

What is ultra super magic square ?

These squares are complete- and self-similar square and have special constellation patterns all at a time !

(1) complete property;
In the squares,not only the major but olso the minor diagonals add up to the same number 65.
For examples;

  A * * * *
  * B * * *
  * * C * *
  * * * D *    A + B + C + D + E = 65    (Major diagonal)
  * * * * E

  * A * * *  
  * * B * * 
  * * * C * 
  * * * * D    A + B + C + D + E = 65  
  E * * * *

  * * A * *
  * * * B *
  * * * * C
  D * * * *    A + B + C + D + E = 65
  * E * * *

...
...   and so on.

(2) Self complement;
The square is invariant for the complemental transform. If you change all the number "n" of the square by "26-n" , you get the rotated same square.
You can say that it is axisymmetry or self-simillar, in another word.

(3) five-star constellation patterns;
Sum of the five numbers of all small squares and the centers sum to the same number 65.

  1  *  *  *  9    1  * 22  *  *    * 15  * 18  *    *  * 22  *  9   
  *  *  *  *  *    * 19  *  *  *    *  *  6  *  *    *  *  *  5  *
  *  * 13  *  *   10  * 13  *  *    *  2  * 24  *    *  * 13  * 16
  *  *  *  *  *    *  *  *  *  *    *  *  *  *  *    *  *  *  *  *
 17  *  *  * 25    *  *  *  *  *    *  *  *  *  *    *  *  *  *  * 
  ....
  ....
                                                      *  *  *  *  *
                                                      *  *  *  *  *
                                                      *  * 13  * 16
     ....      ....     ....     ....    ....         *  *  *  7  *
                                                      *  *  4  * 25
Rhonbohedral patterns;
Furthermore, five numbers of all small rhonbohedrons add up to
the same number 65.

  * 15  *  *  *    *  * 22  *  *
 23 19  6  *  *    * 19  6  5  *
  *  2  *  *  *    *  * 13  *  * 
  *  *  *  *  *    *  *  *  *  *  
  *  *  *  *  *    *  *  *  *  *   ......

  ......
  ......

  *  *  *  *  *    *  * 22  *  *
  *  *  *  *  *    *  *  *  *  * 
  *  *  * 24  *   10  * 13  * 16 
  *  * 20  7  3    *  *  *  *  *
  *  *  * 11  *    *  *  4  *  *  

Sixteen Ultra Super Magic Squares

Note :

We found most of the information on Magic Squares on the Internet.

Next are the internet URL's (Uniform Resource Locator) where you can find more info.

http://www.indiaheritage.com/magicsq.htm

http://www.pse.che.tohoku.ac.jp/~msuzuki/MagicSquare.5x5.selfsim.html

http://www.amersol.edu.pe/highschool/magic.cgi

http://www.inetworld.net/~houlton/

http://www.imn.htwk-leipzig.de/~saxonia/records/magic.html

http://www.math.unibas.ch/~hderksen/magic.html

http://forum.swarthmore.edu/alejandre/magic.square/ben.html

http://users.powernet.co.uk/higgs/f3_sun.html

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