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MAGIC SQUARES

(Investigator 102, 2005 May)



Are "magic numbers" either lucky or magical? The following is a so-called "Magic Square":
 
 

4 3 8
9 5 1
2 7 6
 
A Magic Square is a square divided into equal smaller squares – 3x3, 4x4, 5x5, etc – with each small square or "cell" containing a number such that the sum of each column, and each row and each diagonal is equal. In the example above each adds up to 15.

The sum to which the rows, columns and diagonals each add up, is said to be the "magic number". In the example above the magic number is 15.

Magic squares originated in India or China thousands of years ago. Emperor Yu of China is said to have discovered the various patterns of the 3 x 3 square near BC 2200.

Magic squares reached Europe via the Muslims and were constructed in Europe by the 14th century.

The German painter and engraver Albrecht Durer (1471-1528 shows the following Magic Square in his painting titled Melancholy:
 
 

16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
 
The two middle cells of the bottom row give the date of the painting – 1514. In this case the "magic number" is 34. This particular Magic Square not only has equal sums, i.e. 34, for the columns, rows and diagonals but also for the four corner squares.

To work out the "magic number" quickly, use the formula

(N2 + N) / 2L

where N is the total number of "cells" and L is the number of cells in one row or column. (Note there is actually no 2 x 2 magic square although the formula gives a magic number "5".)

Long ago magic squares acquired the "magical" description and many people believe that "magic numbers" had magic powers. They, therefore, wore magic numbers or magic squares on a chain around their neck.

American statesman and scientist Benjamin Franklin (1706-1790) amused himself, when he became a clerk in 1736 for the Pennsylvania General Assembly, by making up magic squares. He later wrote:

In my younger days, having once had some leisure…I amused myself in making these kind of magic squares, and, at length, had acquired such a knack at it, that I could fill the cells of any magic square, of a reasonable size, with a series of numbers as fast as I could write them, disposed in such a manner, as that the sums of every row, horizontal, perpendicular or diagonal, should be equal… (Quoted by Stern 1980)
 
One of Franklin’s magic squares had 256 cells (16 x 16) containing the numbers from 1 to 256. Each column, row and diagonal totalled to 2,056. You can check this by applying the above formula, giving the calculation: (2562 + 256) / 2 x 16

To fully fill in a Magic Square requires more than just knowing the magic number, the sum of each row, column or diagonal.

There are, however, systematic procedures to work out what numbers go in what cells. For Franklin to fill in magic squares quickly he must have known about such procedures. Frenchman Antoine de la Loubere visited Siam in 1687-1688 and there learned a method for filling out magic squares that have an odd number of cells. It’s described in the book by David Stern and I won’t repeat it here.

Another method starts by working out all the ways or combinations that get us the "magic number". For the 3 x 3 square, where the magic number is 15, the different combinations are:
 

1 + 5 + 9 = 15
1 + 6 + 8 = 15
2 + 4 + 9 = 15
2 + 5 + 8 = 15
2 + 6 + 7 = 15
3 + 4 + 8 = 15
3 + 5 + 7 = 15
4 + 5 + 6 = 15


Note the number "5" occurs four times. Therefore "5" must be positioned in the only cell used on four occasions – i.e. the second row and second column and both diagonals. So, "5" goes in the centre.

Corner cells are used on three occasions that is in one column, one row and one diagonal. Therefore, in the four corners we put the numbers used on three occasions, these being "2", "4", "6" and "8". The rest is easy.

I won't go into larger magic squares. My point is merely that working them out is logical and mathematical and there is actually nothing "magical" about it. And if there is nothing "magical", then there is no reason to expect any of the numbers involved to be special or "lucky".
 

Reference:
Stern, D 1980 Math Squared, Teachers College Press, Columbia University.

(BS)
 
 

 

MAGIC SQUARES

Bob Potter

(Investigator 103, 2005 July)

As a child I spent many hours playing around with magic squares.  There were several formulae I used to know for constructing them.

There was one square I loved because it arrived at the number 78 (allegedly a Moorish Musselman's mystic number) in twenty three different combinations.

The square is four by four, and you must insert the numbers in the following order:
 
 

40 10 20 8
7 21 9 41
12 42 6 8
19 5 43 11

 

You always arrive at 78, namely rows, columns, diagonals, four corner cells, central square of four cells, the four corner squares of four cells, two sets of corresponding diagonal cells next to the corners, two sets of central cells on the top and bottom rows and on the outside columns.
 


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