From: "Josep M." <crull@ERASEtinet.fut.es>
Newsgroups: sci.math,rec.puzzles
Subject: De la Loubere. WasRe: 5 by 5 magic square solution?
Date: Fri, 23 Oct 1998 18:40:06 +0100

herkommer@my-dejanews.com wrote:
> 
> In article <70ktb4$c0q@panix3.panix.com>,
>   michael@panix.com (Chico Cherry) wrote:
> > Hi all,
> >
> >       I'm looking for a solution for a 5 by 5 magic square, using the
> > numbers 1 through 25, where the number 1 is placed in the very center
> > square. (Please forgive the fact that I'm an idiot and should be able to
> > work this out myself).
> 
> The de la Loubre method works fine with a minor modification. Here is a
> solution:
> 
>   10  12  19  21   3
>   11  18  25   2   9
>   17  24   1   8  15
>   23   5   7  14  16
>    4   6  13  20  22
> 

This is NOT a solution in the sense the original poster meant, for the
NE-SW diagonal does not add up to 65.
Actually, it can be proven that the sent program produces magic squares
(rows, colums + main diagonals) if and only if the central value
n(n+1)/2 is placed in the center (I mean the program after changing the
first position lines).

De la Loubere's method is equivalent to the following procedure:

let n be odd

let i range from 0 to n-1
let j range from 0 to n-1

let rows be numbered from 0 to n-1
let columns too

choose A,B,C,D,E,F

place i*n+j+1 at
row    A*i+B*j+C mod n
column D*i+E*j+F mod n

(in your program B=-1 E=1 A=2 D=-1 C=(n-1)/2=F  )

Then:
-all positions are filled iff n and A*E-B*D are relative prime
-all rows add up to the same iff A p.r. to n and B p.r. to n
-all columns ... iff D and E are p.r. to n
-if A,B,D,E satisfy all above then ( central value is placed in central
position iff the two main diagonals add up to the magic sum )
-ALL diagonals add up to magic sum iff A+D,B+E,A-D,B-E are all prime
relative to n

From  all this 
-a magic (r+c+d) square can be done by de la Loubere's method with any
value placed in central position iff
-there exist A,B,D,E such that A*E-B*D,A,B,D,E,A+D,B+E,A-D,B-E are all
prime relative to n iff
-n and 6 are prime relative

or 
-a magic (r+c+d) square can be done by de la Loubere's method iff
-there exist A,B,D,E such that A*E-B*D,A,B,D,E are all prime relative to
n iff
-n and 2 are prime relative

Let, for instance, A=2 B=-1 D=1 E=-2 . This method produces magic
squares with chess knight-like "jumps" for any odd n not multiple of 3.
These squares have the "all diagonals add up to the magic sum" property.

Greetings,

Josep M.