A Math Forum Web Unit
Multiplying Magic Squares
About Allan Adler
Suzanne Alejandre's Magic Squares || Multiplying Magic Squares: Contents
I first became interested in magic squares when I was about 10 years old, when my mother found a copy of Jerome S. Meyer's book Fun with Mathematics and bought it for me. She knew I liked math and generally looked for ways to bring me intellectual stimulation that I might not get in school. If she met someone who knew a lot about something interesting, she would invite him or her over for dinner so we could talk. It was on such an occasion that I was first introduced to Latin, a language that has continued to be of interest to me and which now plays an important role in some of my studies of mathematics.
Meyer's book started me thinking about magic
squares at age 10, and I tried to solve some problems naturally
suggested by the material in the book. I was unable to solve them and forgot about them for a while, but retained an interest in magic squares as I went on to learn other mathematics, partly from school, partly from tutors my father hired to teach me algebra and calculus when I was in 7th and 8th grade, partly from browsing in the excellent math library at my high school, and eventually from being a math major in college and specializing in mathematics in grad school.
In graduate school I realized that a puzzle that had been given to me in high school by a math club lecturer (one which I had not been able to solve at the time) was closely related to the problems I had considered in connection with Meyer's book when I was 10 years old. By exploring this unexpected connection, I eventually arrived at the some of the results I published in the American Mathematical Monthly (see article (1) in the bibliography below). Returning to the ideas of that paper from time to time and making further improvements and innovations, I eventually arrived at other publishable results.
Thus, even when very young and inexperienced, one can try to do original work in mathematics and other subjects. One doesn't have to wait until one has gone to college and graduate school, although it might take that long to figure out the answers, and one doesn't need anyone's permission to begin.
I have published the following papers on magic squares and their higher dimensional analogues:
- "Magic N-Cubes and Prouhet Sequences," American Mathematical Monthly, vol.84, 1978, pp. 618-627 (with S.-Y. Robert Li).
- "Magic cubes and the 3-adic zeta function," Mathematical Intelligencer, vol.14, 1992, pp.14-23.
- "p-adic L-functions and Higher Dimensional Magic Cubes,"Journal of Number Theory, vol. 52, 1995, pp.179-197 (with Lawrence C. Washington).
- "Magic N-cubes Form a Free Monoid," accepted for publication by the European Journal of Combinatorics .
I also work in other areas of mathematics such as algebraic geometry, group theory, representation theory, invariant theory, and aspects of the history of mathematics. Most professional mathematicians consider "magic squares" to be purely recreational mathematics and unworthy of their attention. I believe that, partly as a result of the above publications, this attitude may change in the future. The reason is twofold: first, as shown by article (4), it is really possible to prove general theorems about magic squares; second, the techniques of constructing magic squares are beginning to involve more and more sophisticated mathematical objects, the study of which is of independent interest to professional mathematicians, as shown in articles (1), (2), and (3).
I became involved in producing this Web unit on magic squares after seeing a posting by Eric Sasson on the USENET newsgroup sci.math asking for contributions. I wrote to him of my work on magic squares and he and Sarah Seastone put me in touch with Suzanne Alejandre. What you are about to read is the result of our discussions.
Basically, what is presented here is an account of the results of my papers, to the extent that they can be presented in an elementary manner. In the professional journals in which they appear, the results are presented in a fairly general setting, using concepts and terminology normally beyond the background of people who have not spent a few years studying mathematics in college or graduate school. On the other hand, it has been my own experience that the basic ideas can be presented in a very concrete and visual manner to practically anybody, regardless of their background. This experience, gathered from explaining magic squares to people I have met casually at parties, on trains, and elsewhere, has persuaded me that my results on magic squares ought to be suitable for inclusion in this project.
In addition to the intrinsic interest of the results, I believe that the topics presented here offer a number of pedagogical advantages and can serve to convey something that is often difficult to communicate. Apart from the novelty of some of the results, notice that:
Although Suzanne Alejandre and Sarah Seastone and I hope in the near future to make versions of these Web pages that will discuss material from all of my articles on magic squares, so far we have only covered the results of article (4) and only in the case of magic squares (as opposed to magic cubes, etc.). We also plan to add a Zoo of magic squares and their higher dimensional cousins.
- Magic squares can be used to illustrate fundamental concepts of arithmetic, such as associativity and factorization into primes, and using them in this way serves to shed greater light on the way these notions pertain to ordinary numbers.
- The results presented here are the work of a living mathematician: I arrived at them through luck and experimentation, while drawing on concepts I learned from standard curricula as well as from extra-curricular sources such as Jerome Meyer's book, guest lectures at the math club in my high school, and conversations with people who liked discussing mathematics. I can trace the precise evolution of these ideas and results, but what is more important for students and others who might enjoy these pages is that, unlike the ancient methods and results for magic squares, which one finds repeated over and over again down the centuries, what one finds here is alive and growing. It also serves to show that anyone can follow the path to original discoveries by following his or her curiosity wherever it leads, and this is a lesson that needs very much to be communicated.
Apart from my interest in helping out with this Web page, I have had in mind for some time the idea of writing a book on magic squares in which these original discoveries could be presented along with their connections to a number of diverse areas of mathematics. I hope that my work on these magic squares Web pages will be a first step towards writing that book.