More than Magic Squares

"In my younger days, having once some leisure (which I still think I might have employed more usefully), I had amused myself in making . . . magic squares."

Benjamin Franklin, who made this comment in a letter written more than 200 years ago, was certainly not the first to experience the fascination of magic squares. People have been toying with these number patterns for more than 2,000 years.

Typically, a magic square consists of a set of integers arranged in the form of a square so that the sum of the numbers in each row, each column, and each diagonal add up to the same total. If the integers are consecutive numbers from 1 to n^2, the square is said to be of nth order.

Here's an example of a magic square of the fourth order, made up of the first 16 integers. The sum of the numbers in each row, column, and diagonal is 34.

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

There are 880 possible magic squares of the fourth order, not counting reflections or rotations of each pattern. One of the most remarkable of these squares is one that dates back to India in the eleventh or twelfth century.

7 12 1 14

2 13 8 11

16 3 10 5

9 6 15 4

Notice that not only the rows, columns, and diagonals add up to 34 but also the corner 2 x 2 subsquares. And there's more! The four corner numbers add up to 34, as do the four numbers in the center. Other subsquares (such as 3 + 10 + 6 + 15) give the same result. It's also possible to find "split" subsquares and "split" diagonals that work: 7 + 2 + 14 + 11, and so on. In fact, there is an astonishing number of different ways to get the sum 34 out of this particular magic square.

Victor E. Hill IV, a math professor at Williams College, used this example to make an interesting point during a lecture he presented at last summer's Seattle Mathfest. His topic was mathematical aspects of the music of Bach.

Hill wondered whether the discoverer of the magic square actually set out to find an example that added up correctly in so many ways. Or did that individual serendipitously stumble upon the patterns when examining different squares? Was the discoverer even aware of all the different ways in which the sum 34 arises, or did others point them out later? In sum, how much was accident and how much intent?

Hill asked the same, essentially unresolvable question of Bach's music. How much of the mathematical order that we find in his compositions did Bach deliberately insert? How much simply reflects the perspectives that we bring to our analysis of his music or the patterns inherent in music itself? There is evidence favoring each of these possibilities.

Just as readers or listeners bring their own insights and perceptions to any novel, poem, song, or symphony, mathematicians bring their own experiences into play in studying theorems and mathematical structures. Mathematical discoveries often have more to say than their discoverers had ever intended or anticipated.

References:

Ball, W.W. Rouse, and H.S.M. Coxeter. 1974. Mathematical Recreations & Essays. Toronto: University of Toronto Press.

Gardner, Martin. 1988. Time Travel and Other Mathematical Bewilderments. New York: W.H. Freeman.

______. 1988. Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles & Games. Chicago: University of Chicago Press.

______. 1961. The 2nd Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon & Schuster.

Hofstadter, Douglas R. 1979. Godel, Escher, Bach: An Eternal Golden Braid. New York. Basic Books.

Rothstein, Edward. 1995. Emblems of Mind: The Inner Life of Music and Mathematics. A good starting point for exploring magic squares on the web is the Math Forum site at http://forum.swarthmore.edu/alejandre/magic.square.html.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.