Franklin Magic Squares

Description

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.

Order 8k+4

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
advised that he had already proved this in 2008. Here is Miguel's paper,
No Magic Franklin Square of Order n = 8k + 4.

Proof

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(16m²+1). Then the sum of each of Planck's four species of cells is (m/2)(Σ/2) = 8m⁴ + m²/2. If m is odd this cannot be, because m²/2 is fractional.

Transformations

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)!⁸2⁴ or:

16(n/4)!⁸

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

Complementing

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.

Hurkens Principal Form

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 = x₀, ... , x_n-1 and y = y₀, ... , y_n-1. For convenience, every second element, (starting with x₁ and y₁), is the complement of the actual number. The tuples then have these properties:

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.

Franklin Magic Rectangles

There are also Franklin non-square rectangles. Here is a 24x16 by Phan Văn Khái.
.txt See description: .pdf or .docx.

REFERENCES

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