The Magic Encyclopedia ™

Irregular panmagic squares of prime order
{note: investigative article}
(by Aale de Winkel)

Recently I discovered the page of M. Suzuki on irregular squares, a few days later <George Chen> allerted me that he intended to investigate the order 13 irregular squares which seemed to be of the same type. An investigation into the matter lead me to conclude, that these squares use partly the regular panmagic square theory with a pandiagonal add-on on both components of the regular part. Below I define the framework I work with handling these squares.

Irregular panmagic squares of prime order
S = (LS(a) + Pad₁) * m + (LS(b) + Pad₂) + 1 = [LS(a) m + LS(b)] + [Pad₁ m + Pad₂] + 1
The above formula depicts the situation for prime orders, for a description of LS(a) and LS(b) see the
regular theory for prime order panmagic squares. The add onns I currently denoted by Pad₁ for the
high order and Pad₂ for the low order pandiagonal add-onn. Currently I believe that these two form
a sort "choreography" for a number dance rearanging the numbers of the regular square into an irregular square
thereby moving the square outside the squares obtainable by the regular theory, while the regular number range
is maintained.

Pandiagonal
add-onn An pandiagonal order p square with numbers ranging fro -(m-1) to m-1 with sum 0
All numbers have a unique decomposition in m as N = a m + b, henche the deviations of numbers
N = (a + da) m + (b + db) = (a m + b) + (da m + db) need to cancel each other out in each line
so both the add-onn need to be zero summing pandiagonal squares. The range of numbers for these
add-onns are in the range -(m-1) .. (m-1) (perhaps +/-m is possible as long as in the balance
the numbers are between 0 and m²-1, currently as an assumption I suppose
da <= a and db <= b (verification left for future exploration))

distance An measure on the pandiagonal add-onns
Experience show that the irregular squares differ from the regular veriety by add-onns, however
changing the regular part merely change the add-onns, so some kind of measure might be defined
to define which of the possibilities is the best. Currently I'm trying to work with the sum of
the positive numbers in the add-onns. Summing both these distances the lowest of the
possibilities given two orthogonal LS's I'll try to work with. Currently however the "digit
changing' and the rearangements are beyond my Excell programming capabilities

Irregular panmagic squares of prime order
S = (LS(a) + Pad₁) * m + (LS(b) + Pad₂) + 1 = [LS(a) m + LS(b)] + [Pad₁ m + Pad₂] + 1
The above formula depicts the situation for prime orders, for a description of LS(a) and LS(b) see the regular theory for prime order panmagic squares. The add onns I currently denoted by Pad₁ for the high order and Pad₂ for the low order pandiagonal add-onn. Currently I believe that these two form a sort "choreography" for a number dance rearanging the numbers of the regular square into an irregular square thereby moving the square outside the squares obtainable by the regular theory, while the regular number range is maintained.
Pandiagonal add-onn	An pandiagonal order p square with numbers ranging fro -(m-1) to m-1 with sum 0
Pandiagonal add-onn	All numbers have a unique decomposition in m as N = a m + b, henche the deviations of numbers N = (a + da) m + (b + db) = (a m + b) + (da m + db) need to cancel each other out in each line so both the add-onn need to be zero summing pandiagonal squares. The range of numbers for these add-onns are in the range -(m-1) .. (m-1) (perhaps +/-m is possible as long as in the balance the numbers are between 0 and m²-1, currently as an assumption I suppose da <= a and db <= b (verification left for future exploration))
distance	An measure on the pandiagonal add-onns
distance	Experience show that the irregular squares differ from the regular veriety by add-onns, however changing the regular part merely change the add-onns, so some kind of measure might be defined to define which of the possibilities is the best. Currently I'm trying to work with the sum of the positive numbers in the add-onns. Summing both these distances the lowest of the possibilities given two orthogonal LS's I'll try to work with. Currently however the "digit changing' and the rearangements are beyond my Excell programming capabilities