The Magic Encyclopedia ™

Dynamic Numbering.
(by Aale de Winkel)

I but recently got to understand what <Gil Lamb> calls Dynamic Numbering. His spreadsheets merely arrowed one number toward another showing me the underlying "latin-structures". What he really wanted to show is where the succesive numbers of the "Generating Squares" are placed, changing "Generating Squares" and placing the succesive numbers the same way another square is obtained. After I got this picture clear it dawned me that this was the same thing as using a given magic square as an index into the "Generating Square", a view which makes "Dynamic Numbering" real easy to implement, since one merely needs to move the "Magic Squares" to the "analitic numberrange" and transform each number into coordinates onto the Generating square.
The below formalises this point of view. Currently concentrated around the square, but the method generalises obviously onto hypercubes of any dimension. After implementation of this I'll augment this (and related) texts accordingly

Dynamic Numbering
The following formalises "Dynamic Numbering"
Generating Squares Generating squares can be formulated as products of "Normal Rectangles"
which allows the formula
(i=0n piNqi)[x,y] = i=0n k=0i pkqk {
([x % k=0i pk] \ k=0i-1 pk) + pn ([y % k=0i qk] \ k=0i-1 qk) };
i=0n pi = i=0n qi = m
Examples up till 4 rectangles:
(pNq * rNs)[x,y] = (x%r) + r (y%s) + rs [(x\r) + p (y\s)]
(tNu * pNq * rNs)[x,y] = (x%r) + r (y%s) + rs [((x%pr)\r) + p ((y%qs)\s)] +
pqrs [(x\pr) + t (y\qs)]
(vNw * tNu * pNq * rNs)[x,y] = (x%r) + r (y%s) + rs [((x%pr)\r) + p ((y%qs)\s)] +
pqrs [((x%prt)\pr) + t ((y%qsu)\qs)] + pqrstu [(x\prt + v (y\qsu)]
Coordinate Squares A coordinate square is a (magic) square in analitic numberrange
where the numbers are transformed into coordinates
MS[x,y] => [ MS[x,y] % m, MS[x,y] \ m ].
Generated Squares Using the coordinate Square into the Generating Square Generates a new square
S[x,y] => GS[ MS[x,y] % m, MS[x,y] \ m ].

With the above point of view "Dynamic Numbering" moves into the realm of the more basic isomorphisms on a magic square. As another method which uses "Generating Squares" is my own Pan-Transform for the double even orders, as it turns out: Pan(GS) = GS(Pan(Nm)) meaning that the pan-transform applied on a generating square gives the same result as applying "Dynamic Numbering" with the same "Generating Square" on the pan-transformed "Normal Square" Nm.

It is my current understanding that any construction method for a (Magic) square one obtains say Con(Nm), applying dynamic numbering to this square (with GS) is the same as doing the construction itself on GS ie:
{GS} Con(Nm) = Con(GS)
Application of "Dynamic Numbering" however is not limited to using only "Generating Squares" but can be done with any square. Doing this for order 8 on my "Pan(N8)" introduced me to a 26 {compact complete} order 8 squares {GS}k Pan(N8) of which I only reckognized the 10 (k=1) squares (note {GS}k S, means the application of Dynamic Numbering k times with the same GS). A lot more squares can be obtained by application of different GS's one after another

Dynamic Numbering can of course also be applied to Generating Squares, these squares seem to share the {pandiagonal} feature, but don't have there monagonal sums in incremental order.

Dynamic Numbering forms a powerfull tool combining a lot of squares to each other. The three {compact complete} square families I already knew as:
Pan(N4), Pan(N2 N2) and Pan(2N1 2N4)
with dynamic numbering also can be called:
Pan(N4), {N2 N2}Pan(N4) and {2N1 2N4} Pan(N4)
which dispenses with 2 counts of performing the construction, Dynamic Numbering merely changes the number source.