Benjamin Franklin's Mathematics 

        Princeton University Press or Amazon. You can read the reviews here and the first chapter here. Interviews can be found at Radio Times and Technica, capsule reviews at and New Scientist , and the latest review at Convergence. Eminent Franklin biographer James Srodes weighs in for the Washington Times. Find out more at Science News or at my Franklin page.

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What is this page all about?

What is a magic square?

What does that have to do with Ben Franklin?

Articles by me

News articles about my research & related stuff

More items of interest (under development)


What is this page all about?

For several years now I have been devoted to educating the world about Benjamin Franklin's little-known role as a mathematician. He has been portrayed as a mathematical dilettante, but I can assure you, that is far from the truth! This page outlines some basics, but you can find much, much more in Benjamin Franklin's Numbers.

What is a magic square?

We will sooner see agreement on most other pressing issues of the day before the world sees eye-to-eye on a standard definition of a "magic square".

You draw a square array -- a square matrix, in the mathematical lingo-- and fill in the spaces with numbers. On that much, we can all agree. Oh, occasionally one sees letters used instead of numbers, especially in mystical contexts; some "New Age" books include this old example:

 S   A   T   O   R 
 A   R   E   P   O 
 T   E   N   E   T 
 O   P   E   R   A
 R   O   T   A   S 

which is supposed to ward off the evil eye. (I suppose it works... noone has successfully given me the evil eye since I started typing today.) But that superstitious nonsense is not so interesting mathematically, so let's stick with numbers.

Not only should the entries be numeric, they are usually presumed to be whole numbers. Franklin (and others) chose to use 1,2,3,... in some order. Thus the highest entry is n2, with n being the number of cells on a side. Here is one example with n = 3:

 2   9   4 
 7   5   3 
 6   1   8 
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and you can see more examples here. Every row and column adds up to the same number (in the figure above, the common sum is 15), and moreover the two diagonal sums also have that same "magic" sum.

What does any of this have to do with Ben Franklin?

However, Ben Franklin was not so impressed with this species of magic square, so he invented his own! 

The Autobiography acknowledges that Franklin drew "magic squares and circles". He provided no details there, but in other writings there are scattered examples of his creations. One example is shown at right.

A Franklin magic square has constant row and column sums, and while it may lack the diagonal properties (and usually does), it possesses other magical patterns instead. For example, if you join half of one diagonal to half of the other (see gray squares in the figure at right), you obtain what Ben called a "bent row". Its entries sum to 260, the same as a row or column sum in that figure. And his invention hides a lot more magic-- only some of which is traced out explicitly in the color-coded figure.

Articles by Paul Pasles

  1. "Benjamin Franklin, Magician?" Franklin Gazette, Fall 2000. While conducting preliminary research in 1999-2000, I communicated frequently with the dedicated folks at the American Philosophical Society. Several of them are members of the Friends of Franklin, and they encouraged me to write a preliminary piece for their newsletter. (This was the 2nd piece I wrote, but the first to be published.)
  2. "The Lost Squares of Dr. Franklin."  lead article American Mathematical Monthly, June-July 2001. I won't be posting this one on the web, since there is some overlap with a chapter in my forthcoming book on the subject. However, the abstract/summary is online, as are some addenda:
    1. Appendix 1: Franklin Squares. Here's one possible method BF might have used to build his most famous squares.
    2. Appendix 2: More Magic Squares. See some magic squares that figure prominently in Ben's story.
    3. Appendix 3: The Magic Circle. This is one of Ben's finest creations.
  3. "Benjamin Franklin." MacTutor entry, June 2001.
  4. "Digging For Squares." Math Horizons, April 2002. solutions to the exercises.
  5. "Franklin's Other 8-Square." Journal of Recreational Mathematics, 31:3, 2003.
  6. "A Bent for Magic." Mathematics Magazine, 79:1, 2006.
Articles on other subjects can be found here.


News Articles About Paul's Research + Book Mentions, Etc.

The "lost squares" research was reported in the journal Science, in "Random Samples" (4 May 2001). The same writeup was previewed at Postech's Science Now and Academic Press's world's most famous computer scientist wrote me a nice letter about "The Lost Squares". In response to my paper he is making a change in the new edition of his famous Selected Papers on Computer Science. The abstract section of November's Historia Mathematica listed "The Lost Squares". In December, Mathematics Magazine reviewed the Science and Monthly articles. Also, I was quoted in an article that appeared in the Franklin & Marshall Gazette.

The August 2002 issue of Pour la Science printed another of the 8-squares, plus a few statements by me on the subject. Historia abstracts the "Digging for Squares" piece. Cliff Pickover's book (2002) mentions my work on the "lost squares". Says he: "It is amazing that these new squares ... did not come to light until the twenty-first century" (pp. 382-3). The Monthly paper was reviewed in the usual places. Here's the Zentralblatt review. You can also read the MR review if your institution has a MathSciNet subscription.

More recently, my papers were cited in Polyhedral Cones of Magic Cubes and Squares by Ahmed, De Loera & Hemmecke; in The number of "magic" squares and hypercubes by Beck, et al.; and in How Many Squares Are There, Mr. Franklin?: Constructing and Enumerating Franklin Squares by Ahmed, which appeared in the May 2004 American Mathematical Monthly. My Franklin Squares page is recommended in the wonderful Everything Kids' Math Puzzles Book: Brain Teasers, Games, and Activities for Hours of Fun.

The Toronto Globe and Mail, Canada's "newspaper of record", ran a front-page story about Peter Loly's new result on Franklin squares (March 2006). I had the honor of being quoted. Read Peter's paper and the news article.

In January, Ed Rendell and I (together with 298 other people) lit the candles on a very big cake at the National Constitution Center. We were the " 300 Modern-Day Franklins"!


The most dangerous place to be is between a mathematician and a microphone.

In the summer of 1999 I spoke on "A Reluctant Colonial Mathematician" at the Institute for the History of Mathematics and Its Use in Teaching. A few months later, an updated version of this talk was delivered at the 2000 Joint Meetings of the AMS & MAA. I created a sequel for the 2001 Joint Meetings, and yet another for the Spring 2003 AMS Eastern Section Meeting. Here are the abstracts as they were printed in the Abstracts of Papers Presented to the American Mathematical Society:

Washington, DC meeting (January 2000). A Reluctant Colonial Mathematician.
Two centuries after their creation, Franklin semi-magic squares are unsurpassed in ingenuity. Two examples are quite well-known, two more are far less well-known, and a final pair seem to have been lost. I will discuss the circumstances surrounding the creation and subsequent fate of these half-dozen gems from a reluctant colonial mathematician.
AMS contributed paper session on Mathematics Education and History. Meeting 2074, abstract 950-01-98.
New Orleans, LA meeting (January 2001). The Lost Squares of Dr. Franklin.
B. Franklin's mathematical side extends beyond the two magic squares frequently seen in recreational mathematics books. This talk will cover his unpublished squares, almanac mathematics, and personal library. It will also describe his procedure for constructing "magic circles" similar to those seen in Smith & Mikami's "A History of Japanese Mathematics".
AMS special session on the History of Mathematics. Meeting 2025, abstract 962-01-113.

New York, NY meeting (April 2003). The Followers of Franklin.
For two hundred thirty-five years, recreational math enthusiasts have devoted themselves to decoding the methods used by a minor mathematician of major renown. Like the Founder himself, most of the contributors to this program were untrained amateurs working in isolation. This talk surveys the work and lives of those who bothered to leave a written record.
AMS special session on the History of Mathematics. Meeting 2094, abstract 986-01-44.

Ben had a hand in founding two great institutions of higher learning: the University of Pennsylvania and Franklin & Marshall College. In 2001 I was invited to preach the gospel at both schools. (Here's a press release.)

Other places where I've delivered different versions of this spiel are Bryn Mawr College, Villanova University, Adelphi University (at the Pohle Colloquium), Temple University, Cabrini College, Manhattan College, Rowan University and Haverford College.

In April 2004 I spoke at the NCTM (National Council of Teachers of Mathematics) meeting in Philadelphia, and in May I was keynote speaker at the CMC3 (California Mathematics Council) conference in Lake Tahoe. I led the Franklin-themed workshop at Mathfest 2005 in Albuquerque. In July 2006 I was a resident scholar at the NEH workshop for K-12 teachers, Benjamin Franklin and the Invention of America .

In 2009, I presented at the annual meeting of the History of Science Society in Phoenix and at Science & Math Night at Labyrinth Books in Princeton. In 2010, I spoke at Rutgers University.

More items of interest
(section under development)