Monday, July 5, 2010

de Morgan's magic square

In The Elements of Arithmetic (1830), Augustus De Morgan (noted in Wikipedia for his peculiarities) has a little fun after explaining how to add multi-digit numbers. On page 20, De Morgan presents this exercise:

Not the most gentle exercise for someone who just learned how to add.

If you divide all the entries by 36, you'll see that this is a magic square of order 11 (magic constant 671).

The Wolfram Mathworld article on magic squares explains various ways of generating magic squares. It looks like  De Morgan used what came to be known as the "Siamese method" starting in the cell (6,7) with an order vector of (1,1) (shown in blue below) and a break vector of (0,2) (shown in red below).