The Magic Encyclopedia ™

Franklin Squares
(by Aale de Winkel)

<Donald Morris> gives on his site bestfranklinsquares an excellent analysis of a construction method Benjamin Franklin might have used in constructing what now is called a Franklin Square. A type of square that does not necessarily has its diagonals summing but has summing bentdiagonals either shaped as "/\" "\/" ">" and "<" as seen from the topleft corner denoted as "V" and ">".
As Morris shows in a construction method he formulated himself, component are formed by repeating a rectangle with equal summing diagonals and equally summing columns. The off row-sums are compensated by pasting the horizontal mirror into the other halve of the rows. Basically the low component is a transposed version of the highcomponent, but as is usual with components any orthogonal pair can be combined into a single square.
The above described is the general notion Morris describes, carefully chosen twists in the components might expose features in the resulting square.
In time this page will show the types of twists I'll implement in my own program which will be obtainable elsewhere on this site at some future date. The morris construction will be added to the XmlHypercubes description as it will become reconstructable from the given data. ne of the ingredient of the Morris-Construction is "Tweaking", something that is difficult to program. Yet the procedure can be implemented and tweaks can be added in due time. I have still not decided the format of the tweaker, so the table below holds the implemented Morris-Construction uptill now.

The Morris Construction
The Morris construction is like the "Latin Prescription" and the "KnightJump Procedure" a parametrizable method
Basically a line of digits is copied throughout the square either normally or complemetairy.
Below the morris construction thus far implemented!
perm(0..m-1)l denotes a permutation of l of the m digits (complementairy exclusive)
MC({perm(0..m-1)m},{perm(0..m-1)m}) The basic format of the construction
the first permutation is placed in the high components first row
and complemented into the second row
these two lines are repaeted to fill up the high component.
simularly the second permatation is placed in the low components first column
complemented into the second column
and repeated to fill up the entire low component.
MC({perm(0..m-1)m/2},{perm(0..m-1)m/2}) The second format of the construction
Half the digits are mentioned and their complements are copied into the second half
this form merely shortens the basic form the basic format follows from:
perm(0..m-1)m[i] = perm(0..m-1)m/2[i] i = 0 .. m/2 - 1
perm(0..m-1)m[m/2 + i] = m - 1 - perm(0..m-1)m/2[i] i = 0 .. m/2 - 1
Note: the two permutation forms can be combined


<Peter D Loly>'s iteration of all order 8 Franklin Square allowed us to investigate these. To me it turns out these are best viewed as having three radix 4 components in stead of two radix 8 components, the spreadsheet Order8Franklin.xlsx exploits these and using excell 2007 conditional formatting shows features of selected squares. The spreadsheet holds a full listing of 576 squares with all component quaters solid order 4 latin squares, a selection of the 1440 "magic" as well as all 4320 Franklin squares Perter Loly found in a normalized listing. Component permutation over the three components show squares that are and that are not in the squares listed by Peter Loly note that this is the first time I have used three components used in squares, the conditions resulting in Loly's listing I haven't analysed with respect too this novel type oof decomposition.