Franklin Square: Construction Algorithm
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Construction AlgorithmThe 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.
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