Benjamin Franklin is well known for such practical inventions as bifocals and the lightning rod. Often overlooked are his purely mathematical creations. Researchers have just discovered several new demonstrations of Franklin's love for numbers.
| Math mastery. Not only do the numbers in each row and column add up to 260, but so do the numbers in each of the color-coded "bent columns." The color coding can be slid left or right, and the same magic occurs. |
CREDIT: B. FRANKLIN/B. CIPRA
The ever-busy Franklin spent idle time concocting "magic" squares and circles: geometric arrangements of numbers with many ways of getting the same sum. He once told a friend he could fill in the numbers "as fast as I could write them." But he published just three examples, two squares and a circle, which have long been in the mathematical literature.
Now Paul Pasles, a number theorist at Villanova University in Philadelphia, has brought to light four more. He found three squares, with 4, 6, and 8 numbers to a side, in the 35-volume Papers of Benjamin Franklin, which began publication in 1959. And he uncovered a 16 x 16 monster in the library of the Royal Society in London, in a facsimile of a letter Franklin wrote in 1765. "They didn't know they had it!" he says. It will appear for the first time in the June/July issue of the American Mathematical Monthly.
The expanded oeuvre shows that Franklin "had at least four different methods" for constructing magic squares, Pasles says. Roughly speaking, there are three mathematical challenges: inventing algorithms to generate the squares; classifying them (there are, for example, 880 possible 4 x 4 squares); and inventing squares with extra magic properties, such as "bent" rows and columns with the same sum (see illustration).
Could there be other squares still buried in Franklin's papers? "No comment," says Pasles. "If there are, I want to be the one to find them."
Paul Pasles's home page