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Mathematical Treasure: Benjamin Franklin’s Magic Square of Squares

Author(s): 
Sidney J. Kolpas (Delaware County Community College)

The larger of the two magic squares most associated with Benjamin Franklin (1706–1790) was first published in James Ferguson's Tables and Tracts, Relative to Several Arts and Sciences (London, 1767) and reprinted one year later in volume 38 of London's Gentleman's Magazine, which was in operation from 1731 to 1922 and employed such famous figures as Samuel Johnson.

James Parton (1822–1891) discussed the origin of Benjamin Franklin’s 16X16 "Magic Square of Squares" in his biography, Life and Times of Benjamin Franklin (pp. 255–257):

The activity of Franklin's mind was shown in his trifling amusements. During the sessions of the Assembly, he had to endure many dull hours, perched in his seat as clerk, listening to debates in which he could take no part. His friend Logan showed him one day a French book of "Magical Squares," an idle game of the last century. Franklin, who had made these squares in his youth, now beguiled the tedium of the daily session by producing squares of extreme intricacy, surpassing all that had ever been done in that way. . . .

Nor was this the most wonderful of Franklin's magical squares. He made one of 16 cells in each row, which besides possessing the properties of the square given above (the amount, however added, being always 2056), had also this most remarkable peculiarity: a square hole being cut in a piece of paper of such a size as to take in and show through it just sixteen of the little squares, when laid on the greater square, the sum of the sixteen numbers, so appearing through the hole, wherever it was placed on the greater square, should likewise make 2056.

This square was executed in a single evening. It excited the boundless wonder of Mr. Logan, to whom Franklin sent it, and who styled it a "most stupendous piece." Franklin himself jocularly said it was the "most magically magical of any magic square ever made by any magician." Mr. Logan alludes to these squares in one of his letters to Peter Collinson of London: "Our Benjamin Franklin is certainly an extraordinary man, one of a singular good judgment, but of equal modesty. He is clerk of our Assembly, and there, for want of other employment, while he sat idle, he took it into his head to think of magical squares.

Cover of July 1768 Gentleman's Magazine.

Cover of The Gentleman's Magazine, July 1768, from the collection of Dr. Sid Kolpas.

Pages 312-313 from July 1768 Gentleman's Magazine.

"A Magic Square of Squares," Gentleman's Magazine (July 1768), pp. 312–313.

Transcription of page 313:

Surprizing Properties of Numbers placed in Dr. Franklin’s Magic Square of Squares. 

The great square is divided into 256 small squares, in which all the numbers from 1 to 256 are placed in 16 columns, which may be taken either horizontally or vertically.  The properties are as follows:

  1. The sum of the 16 numbers in each column, vertical or horizontal, is 2056.
  2. Every half column, vertical and horizontal, makes 1028, or half 2056.
  3. Half a diagonal ascending, added to half a diagonal descending, makes 2056; taking these half diagonals from the ends of any side of the square to the middle thereof, and so reckoning them either upward or downward; or sideways from left to right hand, or from right to left.
  4. The same with all the parallels to the half diagonals, as many as can be drawn in the great square: for any two of them being directed upward and downward, from where they began to where they end, their sums will make 2056.  The same downward and upward from where they begin to where they end; or the same if taken sidewise to the middle, and back to the same side again.
    N.B. One set of these half diagonals and their parallels, is drawn in the figure upward and downward.Another such set may be drawn from any of the other three sides.
  5. The four corner numbers in the great square added to the four central numbers, make 1028, equal to half the sum of any vertical or horizontal column, which contains 16 numbers, and equal to half a diagonal or its parallel.
  6. If a square hole (equal in breadth to four of the little squares) be cut in a paper, through which any of the 16 little squares in the great square may be seen, and the paper be laid on the great square; the sum of all the 16 numbers, seen through the hole, is equal to the sum of the 16 numbers in any horizontal or vertical column, viz, to 2056.

Transcription of Franklin's 16X16 magic square.

Transcription of Franklin's 16X16 magic square. Click this link to download an Excel file for the reader’s or students’ use.

For the Classroom

Have students verify properties 1 through 6 with the Excel file. For example:

Property 1: Excel function =SUM(B1:B16)  

Answer: 2056

Excel column needed for solving Property 1.

Property 2: Excel function =SUM(B1:B8)  

Answer: 1028

Excel column needed for solving Property 2.

Property 3: Excel Function =SUM(B1+C2+D3+E4+F5+G6+H7+I8+B16+C15+D14+E13+F12+G11+H10+I9)

Answer: 2056

Excel cells needed for solving Property 3.

Property 4: Excel Function =SUM(B15+C14+D13+E12+F11+G10+H9+I8+J8+K9+L10+M11+N12+O13+P14+Q15) 

Answer: 2056

Excel cells needed for solving Property 4.

Property 5: Excel Function =SUM(B1+Q1+B16+Q16+I8+J8+I9+J9)

Answer: 1028

Excel cells needed for solving Property 5.

Property 6: Excel Function =SUM(F1:F4)+SUM(G1:G4)+SUM(H1:H4)+SUM(I1:I4)

Answer: 2056

Excel cells needed for solving Property 6.

Students can also research how to make magic squares (there are algorithms) or learn more about Franklin’s work on magic squares; one of the best books on that topic is Benjamin Franklin’s Numbers: An Unsung Mathematical Odyssey by Paul C. Pasles. The American Mathematical Monthly has published numerous articles on Franklin's magic squares, including Pasles's 2001 "The Lost Squares of Dr. Franklin: Ben Franklin's Missing Squares and the Secret of the Magic Circle" and Christopher J. Henrich's "Magic Squares and Linear Algebra." Another article on magic squares in Convergence is P. G. Brown's "The Magic Squares of Manuel Moschopoulos."

References

"Surprizing Properties of Numbers placed in Dr. Franklin's Magic Square of Squares." The Gentleman’s Magazine 38 (July 1768): 312–313.

Parton, James.  Life and Times of Benjamin Franklin. Vol. 1. Boston: Ticknor and Fields, 1867.

Pasles, Paul C.  Benjamin Franklin’s Numbers: An Unsung Mathematical Odyssey. Princeton: Princeton University Press, 2008.

"The Gentleman’s Magazine." Wikipedia. https://en.wikipedia.org/wiki/The_Gentleman%27s_Magazine.

Index to Mathematical Treasures

Sidney J. Kolpas (Delaware County Community College), "Mathematical Treasure: Benjamin Franklin’s Magic Square of Squares," Convergence (November 2019)

Dummy View - NOT TO BE DELETED