Structured 8 x 8 Franklin Squares

by: Miguel Angel Amela

Definition of 8 x 8 Franklin Square

                       F1                                   F2

The well-known square F1, as well as the less familiar F2, were constructed by Benjamin Franklin.

In a letter to Peter Collinson he describes four properties of the 8 × 8 square F1 as follows:

     1. The entries of every row and column add to a common sum called the magic sum.

     2. In every half-row and half-column the entries add to half the magic sum.

     3. The entries of the main bent diagonals and all the bent diagonals parallel to it add to the magic sum.

     4. The four corner entries together with the four middle entries add to the magic sum.

According to Maya Mohsin Ahmed (1) of the Department of Mathematics, University of California, Davis, in this work is defined 8 × 8 Franklin squares to be squares with nonnegative integer entries that have the properties (1)–(4) listed by Benjamin Franklin and the additional property that every 2 × 2 subsquare adds to one-half the magic sum. The 8 × 8 squares constructed by Franklin have this additional property (this might be one of the unstated curious properties to which Franklin was alluding in his letter). It is worth noticing that the fourth property listed by Benjamin Franklin becomes redundant with the assumption of this additional property.

In (2), Paul Pasles of the Department of Mathematical Sciences, Villanova University, Pennsylvania says: "... readers of the Autobiography may recall the passage wherein, bored with proceedings at the Assembly, Franklin says he drew "magic squares or circles" to occupy his mind. Van Doren’s Pulitzer-prize winning biography Benjamin Franklin goes into greater detail. It includes an account of visits to the home of James Logan, with whom Franklin perused several mathematical classics. The first of these volumes was Frenicle de Bessy’s encyclopedic listing of the 4 by 4 magic squares (all 880 of them!) ..."

Is of highlighting that the additional property confers to the 8 x 8 Franklin squares the structure a (alpha) of Frenicle for the 4 x 4 magic squares (3).-

Structured 8 x 8 Franklin Square

By analogy to the concept of structured magic square defined by Yuri Chebrakov of the Department of Mathematics, Technical University, St. Petersburg, for the 4 x 4 magic squares (4); we shall say that the structure of a natural 8 x 8 Franklin Square  is given  by  the  disposition  of  the thirty two pairs of complementary numbers of the arithmetic progression: 1–64;  2–63;  3–62; ... 32–33.-

For obtaining the structural pattern of a natural 8 x 8 Franklin Square, is sufficient to connect by lines each pair of complementary numbers, forming this structure directly in the square.-

Structures, examples, number of solutions and symmetry operations

General computer program:

             *  Name:         F-total.exe

             *   Language:    BASIC

             *  Compiler:    Microsoft QuickBasic Extended v7.1 for DOS

             *  Size:            62.7 KB

             *  Strategy:      Nine independent variables

             *  Result:         1105920 Franklin Squares

             *  Run time:     38 minutes in a PC with microprocessor Pentium of 2.26 GHz ( program not optimized )

Is of highlighting, that it was not programmed in function of the bent diagonals and with surprise, it was found that they are not restrictive, therefore the program not check neither; in other words, the thirty two bent diagonals are not an added complication to solve the problem.-

Orientation of the squares:

n1                   n8

  n57                  n64

n64  >  n1  <  n8  <  n57 

Then n1, is the smaller number of the corner numbers

            n1       Number of solutions

     1                  69120

     2                  68160

     3                  66624

     4                  65664

     5                  63648

     6                  62688

     7                  61152

     8                  60192

     9                  54624

   10                  53664

   11                  52128

   12                  51168

   13                  49152

   14                  48192

   15                  46656

   16                  45696

   17                  23424

   18                  22464

   19                  20928

   20                  19968

   21                  17952

   22                  16992

   23                  15456

   24                  14496

   25                    8928

   26                    7968

   27                    6432

   28                    5472

   29                    3456

   30                    2496

   31                      960

    Total             1105920 (*)

(*) The result is the same than Daniel Schindel and collaborators of the Department of Physics and Astronomy, of the University of Manitoba in Winnipeg, Canada; obtained in about 15 hours with advanced techniques of computing, coding in C++ and running under Linux on a 2 GHz Pentium 4 machine.(5).-

According to the chosen orientation of the squares, in the 1105920 solutions were identified fifteen structures as follows:

Structure I

Structure II

Structure III

Structure IV

Structure V

Structure VI

Structure VII

Structure VIII

Structure IX

Structure X

Structure XI

Structure XII

Structure XIII

Structure XIV

Structure XV

Number of structural solutions

The run time of each one of the fifteen programs ( partially optimized ) of the structural solutions they were as follows:

Programs             Number of solutions            Run time (*)  

                     F-I          368640          227    

                    F-II           92160           36    

                   F-III           92160           36    

                    F-IV           53760           20    

                     F-V           53760           20    

                    F-VI           53760           20    

                   F-VII           53760           20    

                  F-VIII           53760           16    

                    F-IX           53760           16    

                     F-X           38400           14    

                    F-XI           38400           14    

                   F-XII           38400           14    

                  F-XIII           38400           14    

                   F-XIV           38400           14    

                    F-XV           38400           14    

                                  ------------                        -----           

                                 1105920          495    

(*) In seconds ( approximately the 40% of the total time, is to show the solutions in screen ).-

The fifteen structural patterns are symmetry operations, in consequence any Franklin Square can be transformed in other Franklin Square by means of those operations. Here some examples as follows:

The transformation of Franklin Squares with the operation III, it produces rotated isomorphic squares:

Reduced program

Coded the symmetries described by Maya Mohsin Ahmed (1), result of interchanging the alternate columns and or alternate rows: 1«3; 2«4; 5«7; 6«8 this reduced the run time to 9 seconds, producing a set of 4320 squares (1105920 / 256); the same result obtained by Daniel Schindel and collaborators in 10 minutes (5).-

The number of structural solutions of the 4320 squares is as follows:

   I   1440

 III    720

   V    540

  VI    300

VIII    420

  XI    420

XIII    180

  XV    300

       ----

       4320

Observation

If not is considered the orientation of the squares, then there are eight structures:

1) I                     

2) II                    

3) III                   

4) IV = X                

5) V = XI                

6) VI = VII = XII = XIII 

7) VIII = XIV            

8) IX = XV               

Miguel Angel Amela

[email protected]

General Pico - La Pampa- Argentina

( May - 2006 )

References:

(1) Ahmed, M. M.  How many squares are there, Mr Franklin? Constructing  and  Enumerating Franklin Squares. American

(2) Pasles P.  Benjamin Franklin Magician ? Franklin Gazette, Fall 2000.-

(3) Frenicle de Bessy. Mémoires de l’Académie  Royale des Sciences,  Depuis 1666, jusqu’à 1699, Tome V. Des Quarrez

     de Quatre, 1729, 348-354.-

(4) Chebrakov, Y.V.  Analytical  Formulae  and  Algorithms  for  Constructing  Magic  Squares  from an Arbitrary set of 16

      numbers. Smarandache Notions, Book 8, 1997, 29-46.-

(5) Schindel D., Rempel M., and Loly P. Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS. Proceedings

     of the Royal Society A, 2006, January.-

My unpublished works:

(1)  El cuadrado mágico 3 x 3 (Método de las repeticiones de los ni ); 1987.-

(2)  Total de cuadrados mágicos 4 x 4 [880] (Método de las diagonales potencialmente mágicas); 1991.- (**)

(3)  Cuadrados mágicos y simetría (370 geometrías simétricas diferentes del cuadrado mágico 4x4); 1996.-

(4)  La ley de los cuadrados mágicos (Las constantes concéntricas); 1998.-

(5)  Total de cuadrados mágicos 5 x 5 [275305224] (Método de las constantes concéntricas); 2003.-

(6)  Momento de inercia de cuadrados y cubos mágicos, suma de distancias al cuadrado y números poliédricos; 2005.-

(7)  Frenicle se equivocó; 2005.-

(8)  Total de cuadrados mágicos 4 x 4 con un conjunto arbitrario de números sin usar la suma mágica; 2005.-

---  Redefinición de cuadrado de Franklin 8 x 8; ( en preparación ).-

(**) Published in internet (http://amela.vr9.com); 2000.- The server vr9.com closed in the year 2001.-

© 2006 Miguel Angel Amela. All rights reserved. This page cannot be copied, published or redistributed in any form without the express written consent of the author.