[Caution, the quotes in this file have been edited for research purposes.
 Tyagi]
_Nagasiva_: A Compendium of Chess, I Ching and the Magick Square of Mercury
Research by Tyagi Nagasiva for the Purposes of Assisting my King in Bringing the
Law of Thelema to Promulgation More Swiftly and Successfully
Section I
Research on the Symbology of Chess and the Possibility of Using it
for the Purposes of Divination
a) Nigel Pennick
"Chess, the best known and most widespread board game in modern times,
is of divinatory origin. Some Chess historians have connected it with
the Chinese game of *Siang k'i*, whilst others deny the Chinese connection.
Whether or not it was a forerunner of Chess, *Siang k'i* has strong
connections with divination. *Siang k'i* is translated usually as 'The
Astronomical Game' or 'The Figure Game'. Here, there is a close affinity
between the gameboard and the diviner's boards which gave rise to the
magnetic compass used today in *Feng Shui*, the Chinese art of placement
[geomancy]. This game was played with the aid of dice.
"The earliest known instructions for the canonical form of cities are in
the 'Arthasastra' of Kautilya, who was prime minister of the ancient
Indian ruler Candragupta Maurya. This gives the grid of nine lines by nine
lines as the canonical forms, making the city have a layout of eight by
eight square blocks. The legendary city of Krishna, Dvarati... was
believed to have been laid out with eight streets crossing at right angles.
Similarly, the *K'aokung Chi, a classical Chinese text, describes the
layout of the royal Chao, capital city, thus:
'The carpenters demarcated the capital as a square with sides of
nine *Li*, each side having three gateways. Within the capital,
there were nine meridional and nine latitudinal streets, each of
the former being nine chariottracks wide.'
This description shows us that the capital was a 'chessboard' grid of 64
squares. The administrative centre was the central quarter area of the grid,
four by four.
"In Vedic India, the eight by eight chequerboard was called *Ashtapada*.
The earliest known mention of board games in literature is in the *BrahmaJala
Sutra*, which contains words attributed to the Buddha Gautama. There, he
describes the trivial things that occupy the thoughts of the unenlightened,
among them playing *Ashtapada* and *Dasapada*. The *Mahabhashya* (second
century CE) defines *Ashtapada* as 'a board in which each line has eight
squares', and the word *Ashtapada* was used to describe a grid employed in
the land survey [Feng Shui?]. The French
Section III
Research on the 8 x 8 'Magic(k) Square'
(including the writings of the evil Frater Nigris)
a) Andrew Milmoe, who writes:
In article <29folb$61e@vixen.cso.uiuc.edu>
ba@mrcnext.cso.uiuc.edu (B.A. DavisHowe) writes:
>...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 offthecuff 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.
Andrew Milmoe
milmoe@symcom.math.uiuc.edu
===================================================================
Then Keith Ramsay:
In <1993Apr11.4711.7901@dosgate> "james gillen"
writes:
Tyagi: What does (a+b)^2 have to do with the I Ching?

James : Should have been (a^2 + b^2)^3 which I believe has 64
 terms in its expansion  I remember a diagram showing
 a geometric representation of two squares moving in a
 cube.
Try (a+b)^6 = ((a+b)^2)^3, which can be expanded as aaaaaa+aaaaab+...
+bbbbbb with 64 terms in an obvious way (two ways of choosing each
factor). (Of course in commutative algebra one usually combines them
as a^6+6a^5b+15a^4b^2+20a^3b^3+15a^2b^4+6ab^5+b^6 to get just 7 terms,
but this is beside the point here.)
Keith Ramsay
ramsay@unixg.ubc.ca
===================================================================
From: geversti@oasys.dt.navy.mil (Gordon Everstine)
Subject: Re: More than one 8 x 8 Magick Square?
To: Tagi@cup.portal.com (Tagi Mordred Nagashiva)
There are several algorithms for magic squares.
Odd and even order squares must be constructed by different approaches.
For odd orders, the most common algorithm is a recursive scheme devised
by de la Loubere about 300 years ago. For even orders, one procedure
is the Devedec algorithm, which treats even orders not divisible by
4 slightly differently from those which are divisible by 4 (doubly even).
These algorithms and others are described in, for example, "Magic
Squares and Cubes" by W.S. Andrews (The Open Court Publishing Co.,
Chicago, 1908; may also be available from Dover) and "Mathematical
Recreations" by M. Kraitchik (Dover, 1953).
++++++++++++++++++++++++++++++++++++++++
In a previous article, Tagi@cup.portal.com (Tagi Mordred Nagashiva) says:
>Is there more than one major solution (aside from rotation) for the
>8 x 8 Magick Square? What are the other possibilities? Is there
>a method to generate evennumbered Squares aside from the 4 x 4?
>
>Below is the solution I know of for the 8 x 8. Alternatives would
>be welcomed via email or post. Thanks!
To any magic square you can do the following:
Scale each element by a constant
Exchange two rows
Exchange two columns
and the square will still work.
Now suppose you have a magic square AxA and an algorythm to construct a
magic square of BxB with an arbitrary sum.
To construct a square (A*B) on a side, construct A*A squares BxB, with
sums that correspond to each square in A. Then arrange them (Replace
17 with the square that totals 17, etc) into one larger square.
There seems to be an algorythm for both odd x odd squares, and
multiple_of_four x multiple_of_four squares, but I don't know what it
is...
I would be interested in a post of C source code that could do the
job... (Or Pascal, I actually prefer that language.)
>
>Tagi
>
>
>
>01 56 48 25 33 24 16 57
>63 10 18 39 31 42 50 07
>62 11 19 38 30 43 51 06
>04 53 45 28 36 21 13 60
>05 52 44 29 37 20 12 61
>59 14 22 35 27 46 54 03
>58 15 23 34 26 47 55 02
>08 49 41 32 40 17 09 64
>
>The one I know about.
>
>
>

Jeff Epler Additions Welcome c(8 ;) >{8) 
 :) (=( =] (= Celebrating the variety of faces => :^) {= ) (: 
 Lincoln, Nebraska
+++++++++++++++++++++++++++++++++++++++++++++++
From: goddard@nextwork.rosehulman.edu
To: Tagi@cup.portal.com (Tagi Mordred Nagashiva)
Subject: Re: More than one 8 x 8 Magick Square?
In rec.puzzles article <64230@cup.portal.com> you wrote:
> Is there more than one major solution (aside from rotation) for the
> 8 x 8 Magick Square? What are the other possibilities? Is there
An exercise in Ken Rosen's Elementary Numer Theory says:
To make an n x n magic square, choose integers c,d,e,and f relatively
prime to n. Then put the integer k from the set 0,1,...,n^21 into
the ith row and jth column, where
i \equiv a+ck+e[k/n] (mod n)
j \equiv b+dk+f[k/n] (mod n).
\equiv means congruent to
[x] is the greatest integer function
So different choices of c, d, e, and f should give different magic
squares.....
A followup problem states that if in addition you have
c+d, cd, e+f, and ef also relatively prime to n, then the sum along
any diagonal (with its "complement") will also be the same.
Bart
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
In article <1992Aug12.094525.12675@phillip.edu.au>
x01233@phillip.edu.au writes:
> I once used, years ago, an algorithm for creating magic squares (i.e.
a square
> full of numbers, all rows and all columns add up to the same number).
> The algorithm stated where to put each and every number in the
matrix.
>
> Sadly I have lost this algorithm... Does anyone know what it is?
>
There is an algorithm for oddorder magic squares:
The general idea is to write the numbers in order, always moving
diagonally upright. The snag is that one crashes into the sides of
the squares or into squares that already have numbers in them, and one
needs rules to handle these cases.
1. If the next box would cross the right side, go to the leftmost box
of the row above (which will always be empty!)
2. If the next box would cross the top side, go to the lowest box of
the column to the right.
3. If the next box is filled, go to the box directly below the CURRENT
box (not the box below the "next" box).
4. When you hit the upper right corner, go down one (apply rule 3.)
5. Start with 1 (or any integer for that matter) in the middle box in
the top row.
Confused?
Here's the 5x5:
17 24 1 8 15
23 5 7 14 16
4 6 13 20 22
10 12 19 21 3
11 18 25 2 9
I put 1 in the center top (rule 5). I'd LIKE to put 2 in the box
upright, but there isn't one, so I apply rule 2, and put it in the
lowest box in the next column. The box upright from the 2 is empty,
so the 3 goes there.
But now I've hit the right side, so the 4 goes in the leftmost box of
the row above (rule 1). The 5 goes in the upright box from the 4.
Now I've hit the box that has the 1 in it (rule 3), so the 6 goes in
below the 5. 7 and 8 follow nicely. I've hit the top, so the 9 goes
in the bottom of the next column. Now I've hit the side, so the 10 goes
up and way to the left. The next box has the 6 in it already, so the
11 goes below the 10. 1215 go in nicely, and then I use rule 4 and put
16 below the 15. I think that's an example of everything that can
happen.
The only one I know for evenorder squares is for the 4x4 case, where
the numbers are written in order and the diagonals flipped:
1 2 3 4 16 2 3 13
5 6 7 8 5 11 10 8
9 10 11 12 > 9 7 6 12
13 14 15 16 4 14 15 1
Further, if one constructs a magic square of magic squares, one gets a
magic square. So I could construct a 12x12 magic square by the
following method:
Make a 3x3 magic square using the algorithm above and the numbers 19,
and call this MS1. Make a second 3x3 using the numbers 1018. This is
MS2.
MS3 uses 1927 and so on, until MS16. Then arrange these 16 3x3
according to the 4x4 pattern above (MSj goes in the box with j in it.)
In elementary school I won a local math contest by constructing an
81x81 magic square which was really a 3x3 of 27x27's which in turn were
3x3's of
9x9's which were 3x3's of 3x3's. (The tedium wasn't worth the 5 bucks
I got for first prize.)
Hope this helps.
Bart Goddard From: andyb@europa.coat.com (Andy Behrens)
Date: Sun, 23 Aug 1992 10:39:44 0400
To: Tagi@cup.portal.com
Subject: More than one 8 x 8 Magick Square?
Here is another 8x8 magic square. It can be generated from your square
by interchanging and reversing some of the rows and columns.
1 63 3 61 60 6 58 8
16 10 54 52 13 51 15 49
41 23 19 45 44 22 42 24
40 26 38 28 29 35 31 33
32 39 30 36 37 27 34 25
17 47 43 21 20 46 18 48
56 50 11 12 53 14 55 9
57 2 62 5 4 59 7 64
Following is a program which generates magic squares of both even and odd
orders. It is written in Algol 60. If you are familiar with Pascal,
you should at least be able to understand the algorithms.
I apologize for the programming style  I wrote the program 25 years
ago, and was just learning about programming.
 Andy
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