in the case of the regular magic figures the generating function is implicitely:

f(i) = i; i = 1..n

Since a "magic object" consists of a distinct set of numbers one can either put 'step-functions' into f or simply limit the 'domain of the function' to a distinct set of numbers. Also since the position of the actual numbers is arbitrary, one can put the uniqueness requirement in functional terms as: df/dx > 0.

The uniqueness requirement is however not necessary (see the 'matrix' vector spaces), but in the context of regular magic figures the most interesting.

When the domain however consists of a 'grid' of evenly spread numbers, (ie: x

The above suagests that one only gets new thing iff the domain of the number generating function consists of irregularly spread numbers, and/or non lineair number generating functions. Currently I only know the case of the p-MultiMagic squares mentioned on MultiMagic Squares which has

f(i) = i

the references there given to this subject there given are:

Ball, W.W.R and Coxeter, H.S.M. Mathemetical recreations and Essays, 13th ed.

New York: Dover pp 212-213, 1987.

Kraitchik, M, "Multimagic Squares." &7.10 in Mathematical Recreations,

New York: W.W. Norton, pp. 176-178, 1942

Probably these squares will also appear on my magic squares an cubes pages, as soon as I know how I can construct them.

I don't know whether someone has 'conjured up' simular concepts as the one discussed on these pages The reader who knows anything simular, I appreciate email (preferably with url included, I don't have access to a (mathematical) library, Bookreferences I'll put in on my pages but can't consult myself) Also the reader is invited to investigate the subject, open a internet page on the subject, and email me the corresponding URL.

One has pe. (for squares)

the "Regular Magic Condition":

all horizontal, vertical and both diagonals sum up to the same number

the "semi-Magic Condition":

Regular condition but with off diagonals

the "semi Magic Condition": (sub-constant d)

The diagonals sum up to the same number (say s),

the verticals sum up blockwise to s +/- d;

the horizontals sum up blockwise to s +/- 2 d;

However one can also define some irregular magic conditions pe some variations on the p-MultiMagic squares, regularly defined as:

where 'sums' stands for the sums in the sense of the regular magic condition over the mentioned factors

the second function might be said as equivalent to the first but defines the multimagic property explicitely as a property of the first (ie: k = 1) square.

(the second formulation therefore might be better then the first)

The third form seems equivalent, but is a property of all magic squares it therefore is a rather useless formulation.

for the mathematical inclined reader: sorry for the 'loose formulation' of the above, I currently have none better, as soon as I have a better formulation I'll correct it.