Franklin Squares
Part 1


My Construction Method

Original 8x8 Square B
3
4
3
4
3
4
3
4
5
2
5
2
5
2
5
2
4
3
4
3
4
3
4
3
2
5
2
5
2
5
2
5
6
1
6
1
6
1
6
1
0
7
0
7
0
7
0
7
1
6
1
6
1
6
1
6
7
0
7
0
7
0
7
0

 

Consider square B of Franklin’s original 8x8. Because it is a destruction product of his square it has all the features in that square including those not yet identified. It also has features not in his square because they aren’t in square A.

Square B is a pandiagonal magic square with both full diagonals and all broken diagonals totaling row totals. It has imbedded squares. All four corner 4x4s are pandiagonal magic squares. It has complimentary symmetry. A center line drawn between columns four and five becomes an axis of symmetry. All pairs of cells symmetrical about that axis are complimentary pairs.

The placement of complimentary pairs in column one assures a complete run of consecutive numbers in the combined magic square. Any sequence can be used in row one of square A as long as all numbers in the number set are included and the square is completed by complimentary construction. That assures a match up of all numbers in B with all numbers in A.

Square B construction can be extended to all orders that are multiples of eight. Column one can be divided into blocks of four rows. There will always be an even number of blocks as long as the order number is a multiple of eight with half the blocks above the center line and the other half below. Each block of four cells can be filled with two complimentary pairs, one pair in the odd rows and the other in the even rows. All the pairs will be used when the column is filled. Complimentary construction fills the square. That constructs half columns with half row totals as well as all other Franklin square features.

This construction will be used for all squares B going forward for multiples of eight.
The challenge is in producing row one square A to achieve a desired feature without losing any other features.

 

Some materials on this site are Copyright © Donald Morris 2005 all rights reserved