Franklin's Magic Squares

Here's an amusing passage from Benjamin Franklin's autobiography:

"Being one day in the country at the house of our common friend,
the late learned Mr. Logan, he showed me a folio French book
filled with magic squares, wrote, if I forget not, by one M.
Frenicle [Bernard Frenicle de Bessy], in which, he said, the
author had discovered great ingenuity and dexterity in the
management of numbers; and, though several other foreigners had
distinguished themselves in the same way, he did not recollect
that any one Englishman had done anything of the kind remarkable.
I said it was perhaps a mark of the good sense of our English
mathematicians that they would not spend their time in things
that were merely 'difficiles nugae', incapable of any useful
application."

Logan disagreed, pointing out that many of the math questions
publically posed and answered in England were equally trifling
and useless.  After some further discussion about how things of
this sort might perhaps be useful for sharpening the mind,
Franklin says

"I then confessed to him that in my younger days, having once
some leisure which I still think I might have employed more
usefully, I had amused myself in making these kind of magic
squares..."

Franklin then described an 8x8 magic square he had devised in his
youth, and the special properties it possessed.  Here is the square:

            52 61  4 13 20 29 36 45
            14  3 62 51 46 35 30 19
            53 60  5 12 21 28 37 44
            11  6 59 54 43 38 27 22
            55 58  7 10 23 26 39 42
             9  8 57 56 41 40 25 24
            50 63  2 15 18 31 34 47
            16  1 64 49 48 33 32 17

As explained by Franklin, each row and column of the square have the
common sum 260.  Also, he noted that half of each row or column sums
to half of 260.  In addition, each of the "bent rows" (as Franklin
called them) have the sum 260.  The "bent rows" are patterns of 8
numbers with any of the shapes and orientations shown below

 # - - - - - - -   # - - - - - - #
 - # - - - - - -   - # - - - - # -
 - - # - - - - -   - - # - - # - -
 - - - # - - - -   - - - # # - - -
 - - - # - - - -   - - - - - - - -
 - - # - - - - -   - - - - - - - -
 - # - - - - - -   - - - - - - - -
 # - - - - - - -   - - - - - - - -

 - - - - - - - -   - - - - - - - #
 - - - - - - - -   - - - - - - # -
 - - - - - - - -   - - - - - # - -
 - - - - - - - -   - - - - # - - -
 - - - # # - - -   - - - - # - - -
 - - # - - # - -   - - - - - # - -
 - # - - - - # -   - - - - - - # -
 # - - - - - - #   - - - - - - - #


It isn't clear from his verbal description whether Franklin was
claiming just the five parallel patterns of each of these types
that fall strictly within the square, or if he was claiming all
eight, counting those that "wrap around".  In any case, his square
does possess this property.  For example, if we shift the first
"bent row" to the left, wrapping the ends around, we have the
patterns

- - - - - - - #    - - - - - - # -    - - - - - # - -
# - - - - - - -    - - - - - - - #    - - - - - - # -
- # - - - - - -    # - - - - - - -    - - - - - - - #
- - # - - - - -    - # - - - - - -    # - - - - - - -
- - # - - - - -    - # - - - - - -    # - - - - - - -
- # - - - - - -    # - - - - - - -    - - - - - - - #
# - - - - - - -    - - - - - - - #    - - - - - - # -
- - - - - - - #    - - - - - - # -    - - - - - # - -


In addition, Franklin noted that the "shortened bent rows" plus
the "corners" also sum to 260.  An example of this pattern is
shown below:

    # - # - - # - #
    - # - - - - # -
    # - - - - - - #
    - - - - - - - -
    - - - - - - - -
    - - - - - - - -
    - - - - - - - -
    - - - - - - - -

As with the previous patterns, this template can be rotated in
any of the four directions, and shifted parallel into any of
the eight positions (with wrap-around), and the sum of the
highlighted numbers is always 260.

Finally, Franklin noted that the following two sets of eight
numbers also sum to 260

- # - - - - # -    # - - - - - - #
# - - - - - - #    - - - - - - - -
- - - - - - - -    - - - - - - - -
- - - - - - - -    - - - # # - - -
- - - - - - - -    - - - # # - - -
- - - - - - - -    - - - - - - - -
# - - - - - - #    - - - - - - - -
- # - - - - # -    # - - - - - - #

He doesn't explicitly mention it, but these patterns can also
be translated (with wrap-around), and since they are symmetrical
between horizontal and vertical, they can be translated in either
direction.

After showing off this 8x8 square to his friend, Franklin continued:

"Mr Logan then showed me an old arithmetical book in quarto,
wrote, I think, by one [Michel] Stifelius, which contained a
square of 16x16 that he said he should imagine must have been
a work of great labor; but if I forget not, it had only the
common properties of making the same sum, viz 2056, in every
row, horizontal, vertical, and diagonal.  Not willing to be
outdone by Mr. Stifelius, even in the size of my square, I
went home and made that evening the following magical square
of 16, which, besides having all the [special] properties of
[his earlier 8x8 square], had this added: that a four-square
hole being cut in a piece of paper of such a size as to take
in and show through it just 16 of the little squares, when
laid on the greater square, the sum of the 16 numbers so
appearing through the hole, wherever it was placed on the
greater square, should likewise make 2056.

This I sent to our friend the next morning, who, after some
days, sent it back in a letter with these words: 'I return
to thee thy astonishing or most stupendous piece of the
magical square, in which' - but the compliment is too
extravagant, and therefore, for his sake as well as my own,
I ought not to repeat it.  Nor is it necessary; for I make
no question but you will readily allow this square of 16 to
be the most magically magical of any magic square ever made
by any magician."


               

    

More info

Document on history of magic squares