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# Conclusion

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 ) 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=4m+2) perfect magic square does not exist also (see page 150 of  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.   Next: Appendix Up: 2D number-tiling Previous: Relation between fundamental tilings

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