Franklin squares of order n have these requirements:

- The sum of each half row, (half column), is 1/2 the magic constant.
- The sum of each bent diagonal, (pan-4-way), is the magic constant.
- The sum of each 2x2 block, (including wrap-around), is 4/n of the magic constant.

As such, the squares are semi-magic, (the sum of each main diagonal is
**not** required to be the magic constant).
There are Franklin squares for all doubly even orders except 4 and 12.

And, for orders 8k, k = 1,2,3,... there are Franklin magic squares, i.e.,
for which the sum of each main diagonal **is** the magic constant.
FranklinMagicSquares makes
these squares.

Franklin8 makes all the order 8 Franklin squares. The construction is based on the method given by Hurkens.

There are no Franklin magic squares of order 8k + 4, k = 0,1,2,3,... .

When the below proof was announced, (January, 2012), Miguel Angel Amela |

From the properties of a Franklin square plus the main diagonal sums, it can be shown that for a Franklin magic square:

- The square is pandiagonal.
- Each quarter of the square is pandiagonal.

See derivation.

Such a quarter square cannot exist when **its** order is singly even.

We can apply
Planck's proof to a quarter of the square, to show that it
cannot be pandiagonal. Let the order of this quarter square be **2m**.
Its magic constant is half that of the full square
= **Σ/2** = **m(16m ^{2}+1)**.
Then the sum of each of Planck's four species of cells is

Hurkens analysis shows that Franklin square properties are preserved under these transformations:

. permutation of n/4: odd rows 1 .. n/2-1 even rows 2 .. n/2 odd rows n/2+1 .. n-1 even rows n/2+2 .. n odd columns 1 .. n/2-1 even columns 2 .. n/2 odd columns n/2+1 .. n-1 even columns n/2+2 .. n . exchanging all n/4: odd rows 1 .. n/2-1 with odd rows n/2+1 .. n-1 even rows 2 .. n/2 with even rows n/2+2 .. n odd columns 1 .. n/2-1 with odd columns n/2+1 .. n-1 even columns 2 .. n/2 with even columns n/2+2 .. n

So, each Franklin square can be transformed into (n/4)!^{8}2^{4} or:

**16(n/4)! ^{8}**

squares. For order 8, that is 4096; for order 16, it is 1,761,205,026,816.

Franklin square properties are also preserved by complementing the square. However, this will sometimes not double the number because some Franklin squares are associative and some are side-to-side symmetric, and squares of these types are self-complementary.

It is useful to define a principal or standard form for all the Franklin squares that
can be transformed into each other. The form outlined by
Hurkens in the search for order
12 Franklin squares is defined by first row and first column tuples
**x = x _{0}**, ... ,

Program Hurkens puts Franklin squares in principal form. Program Frenicle can then be used on the output file to remove duplicates. For order 8 Franklin squares, this reduces the 368,640 magic to 90 and the 737,280 semimagic to 180. Note that Frénicle form may reverse property 8 above.

There are also Franklin non-square rectangles. Here is a 24x16 by
Phan Văn Khái.

.txt
See description: .pdf or
.docx.

Amela, Miguel Angel "Structured 8 x 8 Franklin Squares" http://www.region.com.ar/amela/franklinsquares/

"Franklin Magic Square" http://taliscope.com/franklin_en.html

"Franklin's Magic Squares" https://mathpages.com/home/kmath155.htm

Heinz, Harvey "Franklin Squares" http://recmath.org/Magic%20Squares/franklin.htm

Hurkens, C.A.J. "Plenty of Franklin Magic Squares, but none of order 12"

http://www.win.tue.nl/bs/spor/2007-06.pdf

Hurkens, Cor "Constructing Franklin Magic Squares" http://www.win.tue.nl/~wscor/Magic/mag.pdf

Phan Văn Khái 133 Le Loi, Quang Ngai City, Vietnam khaiphanvan65@gmail.com

Planck, C. "PANDIAGONAL MAGICS OF ORDERS 6 AND 10 WITH MINIMAL NUMBERS."

http://www.archive.org/stream/monistquart29hegeuoft#page/306/mode/2up