To: alt.magick
From: milmoe@symcom.math.uiuc.edu (Andrew Milmoe)
Subj: Magick Squares Formula (0000.msqfmla.am)
Date: unknown
Quoting: |unknown
|...Andrew, are you willing to check the magic squares I have against
|a math text on them? (Magic squares, having specific mathematical rules,
|can actually be checked for accuracy, unlike most other things in magic.)
Just an off-the-cuff analysis: for an NxN magic square, there are N^2
consecutive integers (our variables) to fill the squares with, and there
are 2N+2 equations to satisfy (N rows, N columns, and two diagonals that each
add up to (N^4 + N^2)/2N ). If we were dealing with real numbers here
and not just integers, I'd say that there are N^2-(2N+2) degrees of freedom
(that's the number of variables that we can assign any value to and still
be able to solve the equations). Since we're dealing with consecutive
integers, we're a bit more constrained - there may be equations to describe
the system and account for this that I haven't thought of yet.
Anyway, for Saturn's 3x3 square, we have 9 consecutive integers to deal
with, and 8 equations, yielding 1 degree of freedom - I feel that only
reflects the fact that it's possible to rotate the square and still
satisfy all the conditions (barring having the same magickal significance).
As I said, I'll have to check on what sort of equation would express the
constraint that we're dealing with consecutive integers, and see if it
has any effect (barring a rotation of the square) on the solutions.
Whew! Did you really need to hear all that? ;)
Andrew Milmoe
milmoe@symcom.math.uiuc.edu