MAGIC SQUARES

 

MADE BY : Karen Verschooren and Tine Uytdenhouwen

E-mail addresses :
karenverschooren@ hotmail.com
tineuytdenhouwen @hotmail.com

  

  1. What is a "Magic square"

A Magic square is a square in which the following numbers (1,2,3,...,n2) are ordered so that the sums of the elements on each line, on each column and on both diagonals are the same. In each Magic square we can also use this formula : (n3+n)/2 to find the sum.

16

 3 2

13

5

10

11

8
9

6

7

12

4

15 14

1

n2 is the highest number of the square ( here 16 )

n2 = 16 Ž n = 4

(n3 + n)/2 = (64 + 4)/2 = 34

  1. History

The square of Lo Shu

We don't know if this story is true, but the legend tells us that in the 26th century before Christ, Chinese people brought sacrifices to their rivergod from the river Lo, because the river almost flooded.

After every sacrifice a turtle came out of the water and crept around the sacrifice and disappeared back into the water. The rivergod didn't accepted the sacrifices until a child noticed a special figure on the back of the animal.

 

 

On the back of the turtle you can discover the following Magic square:

4

9

2

3

5

7

8

1

6

 

The sum of this Magic square is 15. Having noticed this Magic square, the people knew that 15 sacrifices were needed to please the rivergod.

This Magic square is the oldest known Magic square. And a lot of research followed during the next centuries...

The magic square of Dürer

In 1514, the German artist Albrecht Dürer made his most famous engraving : Melancholia. It contains a Magic square. The two numbers in the middle of the fourth line represent the year in which he made the engraving.

  1. Special Magic squares

Complete Magic squares

These are Magic squares in which the sum of the numbers on the pan-diagonals is also the same as the sum of the numbers on each line, column or diagonal. Pan-diagonals are lines parallel to the diagonal.

For example:

In a Magic square of order 4, the sum of the pan-diagonals of 3 and 1 or of the pan-diagonals of 2 and 2 has to be the same as the magic sum (in the following example : 34).

The complete Magic square has also some supplementary qualities. For a complete Magic square of order 4, the sum of the numbers indicated with a +-sign is also equal to the magic sum.

For a complete Magic square of order 5, we can find the sum (in the following example : 65) like this :

A Latin square is a sort of a Magic square, but with this difference that each number is only used once on each line. Between such a square and a complete Magic square, there is a remarkable connection, namely that you can find a complete Magic square with the sum of a few orthogonal Latin squares.

For example :

 we find the following complete Magic square:

 Symmetrical magic squares

If we substitute each number x of a Magic square of order n by n2 + 1 - x (the x from the formula signifies the number in the original square), then it can be proved that we get another Magic square. We call them joined squares. Some Magic squares are equal(after some reflections) to their joined squares. Those squares are named self-joined or symmetrical.

For example:

 

n2 + 1 - x = 16 + 1 - 3 = 14

 Ultra-super Magic squares

An ultra-super Magic square is a Magic square that is complete and symmetrical. Ultra-super Magic squares of order 4 don' t exist. There are only 16 different ultra-super Magic squares of order 5.

For example :

1

15

22

18

9

23

19

6

5

12

10

2

13

24

16

14

21

20

7

3

17

8

4

11

25

 

Semi-Magic squares

A Semi-Magic square is a square in which the sums of the numbers on each line and column are the same.

!!! Not on the diagonals !!!

For example :

1

2

15

16

6

11

7

10

13

12

4

5

14

9

8

3

  

   This Semi-Magic square (not normal) is hidden in the following fragment from Goethe's Faust.

Du musst verstehen!

Aus Eins mach Zehn,
Und Zwei lass gehen
,
Und Drei mach gleich 
So bist du reich.
Verlier die Vier!
Aus Fünf und Sechs,
So sagt die Hex,
Mach Sieben und Acht,
So ist's vollbracht:
Und Neun ist Eins,
Und Zehn ist keins.

Das ist das Hexen-Einmaleins!

 
  1. Constructions

The problem of composing a Magic square can be separated into two different categories.

Unpair order: method of De La Loubičre

The unpair order means that the quantity of the compartments is also unpair.

1. Write number 1 in the middle of the first line.

2. Write the following numbers in the compartment right above the previous one. Pretend that the first and last line and the left and right side are stuck together.

   

1

   
 

5

     

4

       
       

3

     

2

 

 

3. If the compartment right above is already filled, you write the number in the compartment beneath.

   

1

   
 

5

7

   

4

6

     
       

3

     

2

 

 

4. Keep using this process and you will have made your own Magic square !!!

17

24

1

8

15

23

5

7

14

16

4

6

13

20

22

10

12

19

21

3

11

18

25

2

9

 

 Quadruple order

To compose a Magic square of quadruple order (= order 4), you can follow these rules.

1. Make a little square inside your Magic square, and put a sign in all the corners of your square, like this:

 

+

   

+

 

+

+

 
 

+

+

 

+

   

+

 

2. Start counting, and put in each +-compartment the corresponding number.

1

   

4

 

6

7

 
 

10

11

 

13

   

16

 

3. When you have done this, you start counting from the bottom in the opposite way, and fill the empty compartments.

1

15

14

4

12

6

7

9

8

10

11

5

3

3

2

16


4. Now you have made another Magic square !!!

  1. Applications

Filling in the compartments

For this method, you have to transpose each number of your square into a binary number. We divide every compartment into 4 triangles and colour them according to the binary number like this:


So you will come to this combination:


For this Magic square,

8

5

2

15

6

11

12

1

13

0

7

10

3

14

9

4


you will get this:

 

And if you put a lot of these squares together you will become this strange picture:

 

 

You can make other variations by using other Magic squares and of course by using colours. For this patron, we used the same Magic square, but a colour for the pair numbers.

 

We found a lot of information in : " Magische vierkanten en andere matrices" (Bart Windels, University of Antwerp) 
and on the the internet.

If you want to put your name for ever on the internet, you can go to the following address:
http://www.amersol.edu.pe/highschool/magic.cgi

There you can participate to a game in which you'll have to construct a little Magic square.
But with this information, it can't be a problem anymore!!!

Good luck!

 

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