## Franklin Square: Construction Algorithm

### by Neil Abrahams

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#### Construction Algorithm

The first step in constructing a Franklin square, after drawing its grid, is to seed it. Begin by placing n/2 in the top row of the right seed column. (The n/4th and (n/4 + 1)th columns are the seed columns.) Next, while alternating between this and the other seed column, place successively lower numbers in the cells until you reach n/4, instead of which write n/2 + 1. Now increase the value of the next seed cells by 1 until you reach n + 1, instead of which write n/4 and return to your original pattern. Seeding is complete when you reach 1.

Now create a list of root pairs, one for each row. Begin by writing only their first components These are one less than twice the value of the seed cell in the associated row. For example, the first component of the first row's root pair is 2(n/2) - 1 = n - 1.

Complete your list by filling in the second component for all of the root pairs; these are the differences between the first ones and 2n.

You can now complete your square row-by-row by alternately adding the two components of the root pair for each row to a cell (starting with the seed cell and the second component of the root pair) to get the value of the next (in the direction of the seed cell with respect to both seed columns) until all of the cells have been filled. When you have finished, you will have a neo-magic square with several unusual properties.

Notes:
1. The magic constant is (n/2) * (n2 + 1).
2. All of the observations to follow (except those referring to top, bottom, or center rows) use a pan-diagonal viewpoint.
3. Congruent pairs of cells or columns are pairs of cells or columns that are symmetric about the same vertical line (e.g. column pairs 6 and 7, and 2 and 3 are congruent since both pairs of columns are symmetric about the lines between the two columns of each pair. View the square pan-diagonally to see this).
4. Congruent quadruples of cells are sets of four cells each of which forms the corners of imaginary concentric even-ordered squares. (See gridsheet for two examples.)
5. The value of a set of one or more cells is the number in it or the sum of the numbers in them. Cells or sets of cells of equal value are equivalent. Sets of cells whose value is the magic constant are magical. Those whose value is a fraction - (x/y) - of the magic constant are (x/y)-magical.
6. Rows in which the numbers, beginning with the seed cell, increase from left to right are called plus rows; the others are minus rows. The direction of increasing cell values in a row is called its plus direction; that of decreasing cell values, its minus direction.
7. The columns containing the end and seed cells are collectively referred to as the seed column pair.

#### Next Page: Observations on Rows

Questions? Comments? Write to Neil Abrahams