Order 8 Franklin Squares

Description

There are 90 principal magic and 180 principal semimagic Franklin squares. These transform into 368,640 magic and 737,280 semimagic Franklin squares.

From Reversible

The order 8 principal Franklin squares can be made by transformations of the principal reversible squares.

Magic

There are 3 transforms that make first rows and first columns for principal Franklin magic squares from each of the 10 principal reversible squares. The row and column transforms are the same, and each row 1 transform applies with each column 1 transform. This completely determines the 3×3×10 = 90 principal Franklin magic squares.

When the square numbers are 0..n2-1, the remainder of the transformed square X is filled as:

Xrow,col = Xrow,0 + X0,col

Complementing alternate cells then completes the Franklin square.

Transforms

   row 1    reversible square          col 1    reversible square
 transform     column order          transform      row order
 ---------   ---------------         ---------   ---------------
      I      1 3 8 6 2 4 7 5              I      1 3 8 6 2 4 7 5 
     II      1 2 8 7 3 4 6 5             II      1 2 8 7 3 4 6 5
    III      1 2 8 7 4 3 5 6            III      1 2 8 7 4 3 5 6

Example

Row transform I and column transform I of the first principal reversible square:

Semimagic

Note: Here semimagic squares are exclusive of magic squares.

There are 2 sets of 3 transforms that make first rows and first columns for principal Franklin semimagic squares from each of the 10 principal reversible squares. Set A row transforms are the same as set B column transforms and vice versa. For each set, each row 1 transform applies with each column 1 transform. This completely determines the 2×3×3×10 = 180 principal Franklin semimagic squares.

Transforms

Note 1: Set B row, (set A column), transforms are the same as for magic squares.
Note 2: Set A row, (set B column), transforms make every principal semimagic an adjacent side paired square.

      row 1    reversible square          col 1    reversible square
    transform     column order          transform      row order
    ---------   ---------------         ---------   ---------------
   Set A:
         I      1 1 4 4 2 2 3 3              I      1 3 8 6 2 4 7 5 
        II      1 1 6 6 2 2 5 5             II      1 2 8 7 3 4 6 5
       III      1 1 7 7 3 3 5 5            III      1 2 8 7 4 3 5 6

   Set B:
         I      1 3 8 6 2 4 7 5              I      1 1 4 4 2 2 3 3 
        II      1 2 8 7 3 4 6 5             II      1 1 6 6 2 2 5 5 
       III      1 2 8 7 4 3 5 6            III      1 1 7 7 3 3 5 5

REFERENCES

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

Schindel, Daniel, Matthew Rempel and Peter Loly "Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS"
http://www.physics.umanitoba.ca/news/loly_paper.pdf.