Magic Squares

Magic Squares are so named because each of the rows, columns and diagonals add to the same sum. BUT does it stop there.

The earliest known magic square is Chinese, recorded around 2800 B.C. Fuh-Hi described the "Loh-Shu", or "scroll of the river Loh". It is a typical 3x3 magic square except that the numbers were represented by patterns not numerals

Magic Squares seem to crop up in some unfamiliar places. 
     Perhaps the most famous of them all is Durer’s magic square in the back of a picture he drew in 1514. 

    

More recently there is a Magic Square with a difference on the doors of Familia Sagrada, Gaudi’s Cathedral in Barcelona, which is still under construction.  (photo by Len)


Can you see how this Magic Square is different to most others?




Primary Students are often asked to use the digits 1-9 and place them on a 3x3 grid so that each row, column and diagonal add to the same total.

Beside trial and error, or thinking laterally-“put the middle number in the middle square and then use pairs of numbers like 1 and 9 2 and 8, there is a ‘cheat’s way’ especially for teachers!
1. Draw a 3x3 grid

 2. Now put a square next to the middle square on each side

3. Take any consecutive set of nine numbers and put numbers in each square working diagonally upwards from the middle square on the left.

4. Now take the numbers in the outside squares and move them across to the empty square opposite

5. Each row, column and diagonal adds to 30!
6. Can you find a way of easily creating a 4x4 Magic Square like Durer's?

Durer’s Magic Square has the 'magic' sum of 34.  This total is easily found in the two diagonals.  Set you students the task of seeing how many sets of 4 numbers add to 34 throughout the whole Magic Square.  (I found in excess of 25!)

And here is one to get you started

Finally, senior students may wish to explore this page about Magic Squares:
http://plus.maths.org/content/maths-magic-squares-0?nl=0