The Magic Encyclopedia ™

Panmagic squares of compound order
{note: investigative article}
(by Aale de Winkel)

The compound order panmagic square is the most illusive, it is proven that doubly odd order panmagic square don't even exist. An investigation of an order 15 panmagic square showed me that it is even possible to have non-panmagic components within a panmagic square, below I present preliminairy generalisations of things found, future investigation might change the position taken below.
Because of certain (yet) uncertain factors no counting arguments can be given, the prime number and square orders investigation certain certain numbers which can be used to predict some lower bounds given the qualitative description below, orders with no primefactor 3 might even lead to more exact numbers since the prime number investigation is complete (according to numbers predicted, which confirms those achieved by others). Future upload might have some counting arguments in, based on current evidence it would be highly speculative, since the proposed construction method is speculative itself in nature, and awaits programming into a real program.

compound order panmagic squares (orders: m = i=0npi)
Latin component squares
LSC(Pi)
a m by m square of squares with digits radix pi (ie.: {0 .. pi-1})
when pi is a prime > 3 the seperate squares are easily made panmagic by the
formulae given in my prime order page and so easily put together in a resulting order m
panmagic component square. The order 15 evidence suggest it also possible when pi = 3
horewver evidence shows that components need not be panmagic themselves to make up a panmagic
square, off sums in one component can be compensated by off sums in an other component.
Latin squares
LC(....)

parametrizable perhaps
(suggested be square orders)
LC(....) = i=0n j=i+1n pj LSC(Pi)
The somewhat strange generalisation of regular decomposition with a single number as a base
The product factor is thus that enough space is given to the lower factors just as in the
regular number decomposition. So in principle given the right LSC one ought to be able to
obtain a regular panmagic square.
prime digital equations Si,j = (a i + b j + c) mod p; a,b,c,i,j ε [0 .. m-1]; p prime factor of m
The current evidence I have suggest it possible to have a kind of p-digital equations to
obtain squares with radix p digits, given these squares it is then a matter of combining
these squares, just like in all other cases (note: this is a new field of investigation!)
Colorisation patterns an m/p by m/p square of digit changing permutations to be
applied on the p by p subsquares having radix p digits
Ci,j = (=[perm(p)])i,j; i,j ε [0 .. m/p - 1]
Colorisation patterns are needed to be applied on the squares obtained by means of
forementioned prime digital equatations, depending on future investigation the above
might be quantized further.
LSCi,j = Si,j(Ci,j) i,j ε [0 .. m-1]
Carpet Colorisation Carpet colorisation (see order 9 investigation) might also here be applied to obtain
LSC's not obtainable by proposed prime digital scheme
The method discussed above notwithstanding, notice the prime order investigation, according to the obtained numbers,
which matches amounts obtained by others, the outlined parameter method results in all possible LSC's for prime order p,
thus obtained LSC's pasted onto a m/p by m/p square of p by p cells, thus the needed panmagic m by m LSC for order m is
obtained. Of course carpet colorisation is applicable independently on each p by p subblock.
When p = 3 things are a bit trickier, however the obtained order 9 LSC's (see database) which can be bordered to obtain
order m LSC's for primefactor 3. Thus obtained LSC's need simply be combined and checked for regularity.
Pandiagonal bordering In case the trivial border (border with only '1') is aplicable (as in case of the primefactor 3 lSC's)
one can place in 1/8 of the border a random selection of patterns, and distribute these patterns over
the rest of the border compatible with the "pandiagonal add-on", thus obtained a pandiagonal LSC.
note: Aside from using pandiagonal LSC's it is possible to have non-pandiagonal LSC's as components for
latin squares, one prime factor LSC's off sums can be compensated by higher prime factors LSC's
multiplied by this prime factor. Evidence for this I have however the obtained Latin Square is also
obtainable by regular pandiagonal LSC's, further investigation is needed to study this peculiar fact
probably one need to decompose latin squares by highest prime factor to the lowest, to avoid
nonpandiagonal LSC's altogether (as said this needs further study to be certain)

NOTE: The above are preliminairy statements based upon current evidence this author has, more involved production schemes than above mentioned might give other LSC's not obtainable by methods above, furher investigation might reveil that
An order 15 spreadsheet might be uploaded once I get more insight onto the forementioned principles. LSC's accomplished utilising "prime-digital equations", the colorisation scheme might be tricky to implement by spreadsheet means (currently beyond my understanding of Excell programming).