Beauty of Magic Squares: 540-Multiple Order Bordered Magic Squares of Orders 20, 30, 42, 56 and 72

This paper presents a systematic construction of multiple order bordered magic squares at orders 20, 30, 42, 56, and 72  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 orders --- specifically orders 3 through 8 --- 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, and 8 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 90, 108, 110, 120, 132 and 144 are given in details in the reference list.

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