Magic Squares
Introduction
A magic square is a table of numbers that, if you add all the numbers in each row and column and in each of its two diagonals, adds up to the same sum and called magic constant. A magic square is also the below.
As we can see, in this magic square, all the rows, columns and diagonals add up to 111!
43+1+67=111, 61+37+13=111, 7+73+31=111,
43+61+7=111, 1+37+73=111, 67+13+31=111,
43+37+31=111, 67+37+7=111
In the below magic square of order 3, (The magic square 4 by 4 is of order 4, the magic square 5 by 5 is of order 5, the magic square 6 by 6 is of order 6, etc. ), it is interesting that all the numbers in the table are singular.
The first record of a magic square
The first record of a magic square, (of order 3), appears in Ancient China in the 5th century B.C. A legend says that this magic square of the numbers 1 to 9 was a gift to the Chinese emperor Yu from a turtle of the Lo River, and this magic square is still used today as an amulet.
The magic square that was on the turtle is the below. This magic square has a sum of 15, in all the numbers in each row and in each column and in each of its two diagonals.
Magic squares, Part 3: The Albrecht Dürer magic square.
Albrecht Dürer, ( Albrecht Dürer, born 21 May 1471, died 6 April 1528), was a German painter, engraver and mathematician, and creator of the magic square of 34 of order 4.
This magic square makes its appearance, in his own engraving of 1514, “Melancholy”.
We can see that it is a magic square that all the lines add up to 34:
16+3+2+13=34, 5+10+11+8=34, 9+6+7+12=34, 4+15+14+1=34.
We see that all the columns also make 34:
16+5+9+4=34, 3+10+6+15=34, 2+11+7+14=34, 13+8+12+1=34.
We see that the diagonal rows also add up to 34:
16+10+7+1=34 and 13+11+6+4=34.
So far, it has the properties that all magic squares have… but the surprises of this square don’t stop there!!!
We see that if we divide the square into four, each piece comes out to 34:
16+3+5+10=34, 2+13+11+8=34, 9+6+4+15=;34, 7+12+14+1=34.
We see that the four end numbers add up to 34: 16+13+4+1=34. The four central squares add up to 34: 10+11+6+7=34. We see that the middle numbers of the first and last row, yield 34: 3+2+15+14=34. Also the middle numbers of the first and last columns, yields 34: 5+9+8+12=34.
We see that the first four squares clockwise, after the corners, make 34: 3+8+14+9=34. Also, the first four squares counterclockwise, after the corners make 34: 2+12+15+5=34.
We see that the squares enclosing the opposite corners make 34:
5+3+14+12=34 and 9+15+2+8=34.
The sum 34, appears in other formations on the square as well…
Finally, on the last line we see the numbers: 4,15,14,1.
Albrecht Dürer created the magic square in 1514.
His surname in German starts with the 4th letter of the alphabet, while his name starts with the 1st letter!
Why would the sum be 34?
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16=136,
136/4=34, so the minimum magic square 4*4, must add up to 34…
Albrecht Dürer was very proud of his creation, and as we have seen, he was not wrong!!!
