During past years author worked with block-wise, bordered 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 108 and 104. Based on these two big magic squares inner order magic squares multiples of 4 are studied. By inner order we understand that magic squares of orders 100, 96, 92, etc. Instead of working in decreasing order, we worked with increasing orders, such as, orders 4, 8, 12, etc. The construction of the block-wise bordered magic squares multiples of 4 is based on equal sum blocks of pandiagonal magic squares of order 4. The block-wise bordered magic squares studied are not pandiagonal. Redistributing the same blocks in each case, we get pandiagonal magic squares of order 4, 8, 12, etc. This work is only for the multiples of order 4. The further multiples, such as multiples, 6, 8, 10, etc. shall be done in another works. Examples are given only up to order 48. Higher orders examples can be seen in Excel file attached with the work. The total work is up to order 108.
Below are the links of the work for download including Excel file:
- Inder J. Taneja, Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4. Zenodo, August 31, 2021, pp. 1-148, https://doi.org/10.5281/zenodo.5347897.
Below are some examples studied in the work. The work is upto order 108. The example below are from order 68 downwords in decreasing orders. The block-wise magic squares multiples of order 4 reduces by order 8. The examples in three blocks.
- First Block: Orders 68, 60, 52, etc.
- Second Block: Orders 64, 56, 48, etc.
- Pandiagonal Magic Squares: Orders 68, 64, 60, ….
- Higher orders are already in attached excel file.
First Block: Block-Borderd Magic Squares
- Block-Wise Bordered Magic Square of Order 68
- Block-Wise Bordered Magic Square of Order 60
- Block-Wise Bordered Magic Square of Order 52
- Block-Wise Bordered Magic Square of Order 44
- Block-Wise Bordered Magic Square of Order 36
- Block-Wise Bordered Magic Square of Order 28
- Block-Wise Bordered Magic Square of Order 20
- Block-Wise Bordered Magic Square of Order 12
- Magic Square of Order 4
Second Block: Block-Borderd Magic Squares
- Block-Wise Bordered Magic Square of Order 64
- Block-Wise Bordered Magic Square of Order 56
- Block-Wise Bordered Magic Square of Order 48
- Block-Wise Bordered Magic Square of Order 40
- Block-Wise Bordered Magic Square of Order 32
- Block-Wise Bordered Magic Square of Order 24
- Block-Wise Bordered Magic Square of Order 16
- Magic Square of Order 8
Third Block: Pandiagonal Magic Squares
- Pandiagonal Magic Square of Order 68
- Pandiagonal Magic Square of Order 64
- Pandiagonal Magic Square of Order 60
- Pandiagonal Magic Square of Order 56
- Pandiagonal Magic Square of Order 52
- Pandiagonal Magic Square of Order 48
- Pandiagonal Magic Square of Order 44
- Pandiagonal Magic Square of Order 40
- Pandiagonal Magic Square of Order 36
- Pandiagonal Magic Square of Order 32
- Pandiagonal Magic Square of Order 28
- Pandiagonal Magic Square of Order 24
- Pandiagonal Magic Square of Order 20
- Pandiagonal Magic Square of Order 16
- Pandiagonal Magic Square of Order 12
- Pandiagonal Magic Square of Order 8
- Pandiagonal Magic Square of Order 8
- Pandiagonal Magic Square of Order 4
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