Magic Rectangles in Construction of Block-Wise Pan Magic Squares

The idea of magic rectangles is used to bring pan magic squares of orders 15, 21, 27 and 33, where 3×3 blocks are with equal sums entries,  and are semi-magic squares of order 3 (in rows and columns). The magic squares of orders 12, 18, 24, 30 and 36 are calculated, with the property that 3×3 blocks are magic squares of order 3 with different magic sums. All the magic squares constructed are pan diagonal except the orders 18 and 30. Exceptionally, the pan magic square of order 35 is of type 5×7 or 7×5. It is constructed in two different approaches. One with 49 blocks of equal sums magic squares of order 5 and second with 25 blocks of equal sums magic squares of order 7.

Links for download the complete work: Block-Wise Magic Squares

Below are examples of work divided in three Groups:

Group 1: Pan magic squares of order 9, 15, 21, 27 and 33 – 3×3 blocks are with equal sums entries, and are semi-magic squares of order 3 (in rows and columns) with equal values;

Group 2: Magic squares of order 12, 18, 24, 30 and 36 with the property that each block of order 3×3 is magic square of order 3 with diferente magic sums resulting in another magic square. In this case, the magic squares are pan magic squares except the orders 18 and 30.

Group 3: Pan magic square of order 35 in two diferente forms: one of type 5×7 (49 pan magic square of order 5 with equal magic sums), and second of type 7×5 (25 pan magic squares of order 7 with equal magic sums).

  • Examples of Group 1: Orders 9, 15, 21, 27 and 33

B3k1B3k3B3k5B3k7

  • Group 2: Orders 12, 18, 24, 30 and 36

B3k2B3k4B3k6B3k8B3k10

  • Group 3: Order 35 – Two ways (5×7 and 7×5)

B3k11B3k12

Links for download the complete work: Block-Wise Magic Squares

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