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 120 and 110. Based on these two big magic squares, the inner order magic squares multiples of 10 are studied. By inner orders we understand as the magic squares of orders 100, 90, 80, etc. Instead of working in decreasing order, we worked with increasing orders, such as, orders 10, 20, 30, etc. The construction of the block-wise bordered magic squares multiples of 10 is based on equal sum blocks of magic squares of order 10. It is done in three different ways.
One as normal magic squares of order 10
- Normal magic squares of order 10;
- Block-bordered magic squares of order 10, where the inner magic square of order 8 is pandiagonal formed by four equal sums pandiagonal magic squares of order 4;
- Bordered magic square of order 10. In this case, the inner blocks are magic squares of orders 8, 6 and 4. The magic square of order 4 is pandiagonal.
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 10. This work is for multiples of order 10. For multiples of orders 4, 6 and 8 see the following links:
- Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4
- Block-Wise Bordered Magic Squares Multiples of Magic and Bordered Magic Squares of Order 6
- Block-Wise Bordered Magic Squares Multiples of 8
For the multiples of 12 and 14, see the following links:
For this work the examples below are only up to order 70. 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:
- Inder J. Taneja, Block-Wise Bordered Magic Squares Multiples of 10, Zenodo, September 17, pp. 1-170, https://doi.org/10.5281/zenodo.5514398
- Excel file for download:
Below are some examples studied in the work. The work is up to order 120. The examples below are only up to order 70. As written above, in each case, there are three examples: