Beauty of Magic Squares: 3240-Multiple Orders Bordered Magic Squares of Order 90

This paper presents a systematic construction of  multiple order bordered magic squares at orders 20, 30, 42, 56, 72 and 90. for a full structural summary. Unlike classical block-wise bordered magic squares, which are built as multiples of a single block size, the structures explored here incorporate successive borders drawn from magic squares of  different orders --- specifically orders 3 through 9 --- each contributing a distinct structural layer. The innermost core is a magic square of order~12 formed by different-sum magic squares of order~3.  Successive borders of orders 4, 5, 6, 7,  8 and 9 are then applied; even-order borders consist of equal-sum, while odd-order borders consist of  different-sum magic squares. In case of order 8, first three are of equal-sums and last three are of different sums. The combinatorial multiplicity of border types at each level yields a hierarchy of distinct configurations. The further study of multiple order bordered magic squares of orders 108, 110, 120, 132 and 144 are given in details in the reference list. The excel file of the work are also attached.

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