Block-Wise Bordered Magic Squares Multiples of 12

During past years author worked with block-wisebordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 120 and 108. Based on these two big magic squares, the inner order magic squares multiples of 12 are studied. By inner orders we understand as the magic squares of orders 96, 84, 72, etc. Instead of working in decreasing order, we worked with increasing orders, such as, orders 12, 24, 36, etc. The construction of the block-wise bordered magic squares multiples of 12 is based on equal sum blocks of magic squares of order 12. It is done in six different ways. First three ways are such that each magic square of order 12 is composed by blocks of orders 3, 4 and 6. The fourth way is bordered magic squares. Two blocks of order 12 composed with small blocks of order 3 and 4 are pandiagonal. This led us to write all orders multiples of 12 as pandiagonal magic squares. The only difference is that the pandiagonal magic squares multiples of 12 are no more block-wise bordered magic squares. Moreover, the magic squares from orders 36 onwards are block-wise bordered magic squares. Below are details of these 6 types:

  1. Blocks of Magic squares of order 12 with blocks of order 3;
  2. Blocks of Magic squares of order 12 with blocks of order 4;
  3. Blocks of Magic squares of order 12 with blocks of order 6;
  4. Blocks of Bordered Magic squares of order 12;
  5. Blocks of Pandiagonal Magic squares of order 12 with blocks of order 3;
  6. Blocks of Pandiagonal Magic squares of order 12 with blocks of order 4;

The advantage in studying block-wise bordered magic squares is that when we remove external borders, still we left with magic squares with sequential entries. The bordered magic squares also have the same property. The difference is that instead of numbers here we have blocks of equal sum magic squares multiples of 12.

For multiples of orders 4, 6, 8 and 10, see the following links:

  1. https://inderjtaneja.com/2021/08/31/block-wise-bordered-and-pandiagonal-magic-squares-multiples-of-4/
  2. https://inderjtaneja.com/2021/09/19/block-wise-bordered-magic-squares-multiples-of-magic-and-bordered-magic-squares-of-order-6/
  3. https://inderjtaneja.com/2022/02/13/block-wise-bordered-magic-squares-multiples-of-8/
  4. https://inderjtaneja.com/2022/02/13/block-wise-bordered-magic-squares-multiples-of-10/

The advantage in studying block-wise bordered magic squares is that when we remove external borders, still we left with magic squares with sequential entries. It is the same property of bordered magic squares. The difference is that instead of numbers here we have blocks of equal sum magic squares of order 12.

For this work the examples below are only up to order 72. Higher order examples can be seen in Excel file attached with the work. The total work is up to order 120. Below are links for the download of work:

  1. Inder J. Taneja, Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 12, Zenodo, September 23, pp. 1-170, https://doi.org/10.5281/zenodo.5523608
  2. Excel file for download: 120-108-12×12-onlineDownload

Below are some examples studied in the work. The work is up to order 120 but the examples below are only up to order 72. As written above, in each case, there are four examples:

Magic and Bordered Magic Squares of Order 12

Magic and Bordered Magic Squares of Order 24

Magic and Bordered Magic Squares of Order 36

Magic and Bordered Magic Squares of Order 48

Magic and Bordered Magic Squares of Order 60

Magic and Bordered Magic Squares of Order 72

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