Introduction to Magic Squares

A Magic Square is a square grid arrangement, where each square is filled with a number such that each column, each row and both diagonals sum to the same number.  In the case of the magic square shown above each row, each column and each diagonal sums to 15.  The grids do not have to have consecutive numbers (like the example above) but they do have to have different numbers in each square.  The square grids can be of different sizes: 3x3, 4x4, 5x5, etc.

The Lo Shu magic square is the oldest know magic square, discovered in Chinese literature date to about 650 B.C.  It is basically the same magic square as the one shown above.

Many magic squares have been created that have additional properties.  They may be created with just prime numbers, or us multiplication instead of addition, or have multiple magic squares inside the main magic squares.

Magic squares were popular evening entertainment years ago (before radio, TV and the internet).  Benjamin Franklin (the $20 bill guy) created many magic squares with additional properties.  In fact there is a recently published book about Benjamin Franklin and his magic squares: “Benjamin Franklin's Numbers” by Paul C. Pasles.  (This would be a good place of an Amazon.com add to make a little change.)

The largest magic square that I have found was a 3559 x 3559 magic square created by Peter Weber and Tassilo Herbig in Germany in 2012.  (I won’t reprint it here.  That magic square requires 12,666,481 different numbers!)  The largest magic square that I am aware of that was created by hand was a 1111 x 1111 magic square created by Norbert Behnke, also from Germany, in 1990.  This magic square only requires 1,234,321 different numbers.

My personal favorite magic squares is a 13 x 13 magic square that contains 12 other magic squares inside it (well almost – I think one diagonal did not add up to the correct number in one of the smaller magic squares, and a 3 x 3 alphamagic square the was created by the MBIOM faculty, staff, and student that we believe is the first alphamagic square to use Pig Latin (I will explain some of the newer special versions soon in other posts.

REFERENCES:

http://en.wikipedia.org/wiki/Magic_square

http://en.wikipedia.org/wiki/Lo_Shu_Square

David