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Benjamin Franklin, Magician?

Paul C. Pasles
Dept. of Mathematical Sciences
Villanova University
© 2000, 2003

[This short column, which appeared in the Fall 2000 Franklin Gazette, is intended for a general audience.]

In the final year of his life, Ben Franklin was living with his daughterís family on Market Street in Philadelphia. 1790 was also the year of the first U.S. Census. That census was conducted by order of Article I of the Constitution, a document whose passage owed much to Franklinís well-reasoned support. And as one of the worldís first demographers [1], he would have taken a keen interest in the census results.

In those days, the survey was not very detailed; it counted the number of people in a household and catalogued them by gender, nationality, slave or free status, etc., but it would be another half-century before the census asked for each personís occupational category [2]. Nevertheless, I wonder, what would Ben have listed as his occupation? Many titles and trades spring to mind: author, scientist, inventor, diplomat; not to mention postmaster, library founder, almanacist... Most likely he would go with "printer," of course. To this list I would like to add one other possible vocation: mathematician.

That Doctor Franklin was possessed of a wide variety of interests, both academic and otherwise, is well-known, but that is not so striking as the fact that he excelled in so many of those pursuits. Mathematics is a case in point. In the eighteenth century, America was still a century away from becoming a "world power" in mathematics; yet, in that vacuum stood Dr. Franklin, whom I have called the "reluctant colonial mathematician." [3]

Readers of the Autobiography may recall the passage wherein, bored with proceedings at the Assembly, Franklin says he drew "magic squares or circles" to occupy his mind. Van Dorenís Pulitzer-prize winning biography Benjamin Franklin goes into greater detail. It includes an account of visits to the home of James Logan, with whom Franklin perused several mathematical classics. The first of these volumes was Frénicle de Bessyís encyclopedic listing of the 4 by 4 magic squares (all 880 of them!) [4]. These are arrays of

Figure 1: A Frénicle Square.

numbers such as Figure 1 in which each row, each column, and each diagonal adds up to the same sum-- in this case, 34.  Such four-by-four configurations date back at least as far as the thirteenth century, when they were believed to possess mystical properties.

Such a design may appear at first glance a mere "trinket" or curiosity. In fact, magic squares are still a subject of research, despite their long history. For example, a current open problem  asks [5]: is there a 3 by 3 magic square whose entries are distinct perfect squares (like 1, 4, 9, ...)?  This last problem has stumped many great minds. Current efforts to solve it lead to the theory of elliptic curves (not to be confused with ellipses), the deep branch of mathematics which was used to settle Fermatís Last Theorem.

Thus, the study of the "magic square"ó which is purely theoretical and apparently useless aside from its role as an intellectual stimulantó is still a topic of interest in mathematics today. This only magnifies the accomplishment of a man who drew them as effortlessly as a doodle. (Ben once wrote apologetically that he could have spent his time better.)

Franklin preferred to measure Ďmagicí using a different kind of diagonal from his predecessor Frénicle, employing a shape which he called a "bent row." In his 8 by 8 square (Figure 2), each row and each column adds up to 260. Also, 52 + 3 + 5 + 54 + 43 + 28 + 30 + 45 = 260. Observe the placement of these terms in the diagram; they form a "bent row". So do their immediate downstairs neighbors: 14 + 60 + 59 + 10 + 23 + 38 + 37 + 19 = 260. Again, refer to the diagram: you can see that this set, too, is still worthy of the name "bent row". Moreover, those V-shapes can be taken sideways or even upside-down, and they will still sum to 260.

Figure 2: A Franklin Square.

Here is another, slightly weirder "bent row": 14 + 61 + 64 + 15 + 18 + 33 + 36 + 19 = 260. Get the idea? Try to find some others!

I think you will agree that Figure 2 satisfies properties which are more impressive than Frénicleís two-diagonal requirement. Franklin has arranged the integers 1 through 64 in such a way as to satisfy 8 row conditions, 8 column conditions, and 32 bent row conditions. You can convince yourself with a little thought that these 48 conditions are not all independent of one another, but itís an impressive feat nonetheless!

Now look at any 2 by 4 block and take its sum. What do you get?

There are other nifty properties you can look for which are a consequence of those already listed. If you find this square as fascinating as I do and you would like to learn more, a good place to start is Oystein Oreís Invitation to Number Theory, an inexpensive paperback whose cover displays one of Franklinís designs.

James Logan also introduced Franklin to the work of Michael Stifel (1487-1567), author of Arithmetica Integra. This German mathematician pioneered the use of negative exponents and algebraic symbols we take for granted today. Stifelís ideas would lead to the development of John Napierís logarithms. (Stifel and Napier also shared a tendency toward religious controversy, but thatís another story.) His book included Pascalís triangle, one of its first appearances in the Western world, well before Blaise Pascalís birth. He helped to popularize the radical sign still used to represent the square root.

What intrigued Logan and Franklin, however, was Stifelís gargantuan magic square composed of the numbers 1 through 256. Franklin was driven to create his own square of equal size, shown in Figure 3. Like the 8 by 8, it satisfies a sum condition (sum = 2056) on every row, column, and bent row. These 96 properties trump Stifelís 34 conditions by a mile!

Figure 3: Another Franklin Square.

Space does not permit us to examine the Franklin magic circle, an arrangement of the whole numbers from 12 to 75 in concentric circles wherein each radius and each ring adds up to 360.

I have spoken twice on Benjamin Franklinís mathematics, once before the Institute in the History of Mathematics and Its Use in Teaching, and also at the joint annual meeting of the Mathematical Association of America and the American Mathematical Society. Both times the following setup worked perfectly:

1. Show the three famous figures;
2. Describe some of their properties;
3. Ask the audience, "What are you all wondering?"

As one they responded: "Howíd he do it?"

Good question. Over a dozen scholarly papers were devoted to Benís squares in the past century alone. Some use mathematics not yet developed in his time. Some use ad hoc techniques which are hindsight-driven, and unlikely to be the true method. Not one of these authors has found a method which (a) actually works, and (b) could have been the one in use 250 years ago.

That there was a method, there can be no doubt. There are 64!/8 (or around 1.59 times ten to the eighty-eighth) different ways to arrange the integers from 1 to 64 in an 8 by 8 array. Franklin must have followed an algorithm, but no one yet has satisfactorily explained what it was. We can "see" the system by tracing the integers in order in Figure 1, but describing it is not the same as understanding the construction. How could I know in advance that this arrangement will work? Trial-and-error is an insufficient explanation when the cases to be considered number over a billion billion billion billion... well, you get the idea. Not bad for a fellow whose early experiences in arithmetick were unpleasant and unproductive.

Few contributions to mathematics are still remembered after two centuriesí time, but Ben made his mark. The three magic figures alone would be enough to guarantee him a small but secure place in math historyó but there is more! Alas, these discoveries will have to wait for a future issue.

As Ben himself remarked famously, the 16 by 16 square is "the most magically magical of any square ever made by any magician." [6]


1.  B. Franklin, Observations Concerning the Increase of Mankind and the Peopling of Countries (1751).

2.  United States Historical Census Browser, University of Virginia Library,

3.  P.C. Pasles, Abstracts of the American Mathematical Society 21:1 (2000), p. 3.

4.  B. Frénicle de Bessy, et al., Divers ouvrages de mathematique et de physique (1693).

5.  J. P. Robertson, "Magic Squares of Squares," Mathematics Magazine 69:4 (1996), pp. 289-293.

6.  Jared Sparks, ed., The Works of Benjamin Franklin Vol. VI (1856).

© 2000, 2003 Paul C. Pasles. All rights reserved. This page cannot be copied, published or redistributed in any form without the express written consent of the author.