Franklin finally was successful. This square is a pandiagonal magic square with bent diagonals, half rows and half columns, and 2x2s with the proper totals. For the first time he used duplication in a starting column. He built column one in square A with pairs of numbers. He used a consecutive sequence from zero to seven alternating back and forth between columns one and two. This produced half columns with proper totals. The rest of the square was finished with complimentary construction. Every pair of cells starting in an odd row down to the right or down to the left is a complimentary pair. All diagonals of all types have row total. Square B can be broken down into 4x4s, as in both of unknown author’s component squares. Constructing the square in the upper left corner was the challenge. He placed two complimentary pairs in row one. He finished the 4x4 with complimentary construction. That achieved everything as in unknown author’s squares but the diagonal totals were flawed. In rows five to eight he reversed the pairs of columns in rows one to four placing column one into column two and column two into column one, a technique he learned in earlier squares. He did the same with columns three and four. That reversed the flaw in rows one to four and created bent and broken diagonals with row total. He duplicated the four column block in rows one to eight in rows nine to 16. He then filled the 16x16 square by substituting the other six complimentary pairs in four column blocks using the same form as in columns one to four. Note the sequence of complimentary pairs in the four column blocks: zero and one, two and three, four and five, and six and seven. It’s possible that Franklin and unknown author worked together on this problem. These two squares have much in common with Franklin's techniques in unknown author’s square and unknown author’s techniques in Franklin’s square. Franklin and James Logan worked together on mathematical problems. The first problem they addressed was Frenicle de Bessy’s book in which he catalogued all 7040 order four magic squares by grouping them into 880 groups of eight. It’s possible unknown author was Logan but that’s just a guess. Without more documentation there is no proof possible that a given construction technique attributed to Franklin was the one he used. Every proposal must be measured against every other proposal. The one that is simplest, most elegant, most logical, uses the math of the time and, above all, works is the one to be chosen. I feel the comparison between Franklin’s three 8x8s showing an evolution of structure to cure the diagonal totals and the overall regularity and order in the structure of all the component squares of all the squares discussed, especially his improved 16x16, are definitive.
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Franklin’s Construction Method Part 5 |