The fundamental tilings which are identical to its rotation and
transposition
are given in particular orders ( ). The numbers of the
fundamental tiling are 1, 1, 2, 2 for *N*=2,3, 4, 5 respectively.
The perfect magic squares of order 4
are algorithmically given by the fundamental tilings of order 2.
It is conjectured that the perfect magic squares in order *N*
are written by
the fundamental tilings in the orders of *N*'s elementary (prime) factors
using (14).
The number of 4x4 fundamental perfect magic squares are 3.
The earliest report of the number of 4x4- and 5x5- fundamental perfect magic squares
appears in Oomori(1992)(see page 120 of
[2])
with its proof as long as I know.
Recently, Grogono successfully presents
the 4x4 fundamental magic squares with more elegant derivation
and the proof of the
number 3 in his web-page.
The fundamental tiling scheme presented in the previous sections gives
us their generalization.
If the conjecture 1 was proved in arbitrary
order *N*, we can obtain the numbers of perfect magic squares
by the numbers of fundamental tilings in the order of the divisors of *N*.
When we calculate the number, we need to remind that the association low
is broken in the product which includes transposition operator *T*.
However, how to find all of the fundamental tiling of arbitrary prime order is
still under investigations.
Though we need to recall the proof that half-even (*N*=4*m*+2) perfect magic
square does not exist also (see page 150 of
[2] for the proof),
it does not mean non-existence of fundamental tilings.
The explicit representations of symmetrical *N*-let
are obtained by thumb rule in each order in
the present study.
In order to obtain general formulae for the
symmetrical *N*-let
pattern, we need to wait for the future studies.