Irregular panmagic squares of prime order | ||
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S = (LS(a) + Pad_{1}) * m + (LS(b) + Pad_{2}) + 1 = [LS(a) m + LS(b)] + [Pad_{1} m + Pad_{2}] + 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 Pad_{1} for the high order and Pad_{2} 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. |
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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 m^{2}-1, currently as an assumption I suppose da <= a and db <= b (verification left for future exploration)) |
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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 |