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 140 and 126. Based on these two big magic squares, the inner order magic squares multiples of 14 are studied. By inner orders we understand as the magic squares of orders 112, 98, 84, etc. Instead of working in decreasing order, we worked with increasing orders, such as, orders 14, 28, 42, etc. The construction of the block-wise bordered magic squares multiples of 14 is based on equal sum blocks of magic squares of order 14. It is done in six different ways. First way is a general magic square of order 14. The next three ways are block-bordered magic squares of order 14 with blocks of orders 3, 4 and 6 forming magic square of order 12. The fifth way is bordered magic squares of order 14. The last way is the block-bordered magic squares of order 14 with equal sums bordered magic squares of order 6. See below these six ways:
- Magic Squares of Order 14;
- Block-Bordered Magic square order 14 with inner magic square of order 12 having blocks of order 3;
- Block-Bordered Magic square order 14 with inner magic square of order 12 having blocks of order 4;
- Block-Bordered Magic square order 14 with inner magic square of order 12 having blocks of order 6;
- Bordered Magic square order 14;
- Block-Bordered Magic square order 14 with inner magic square of order 12 having bordered magic squares of order 6.
Similar kind of work for the multiples of orders 4, 6, 8, 10 and 12 is already done by the author. See the following link:
For multiples of orders 4, 6, 8 and 10, see the following links:
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 14.
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 140. Below are links for the download of work:
- Inder J. Taneja, Block-Wise Bordered Magic Squares Multiples of 14 (Version 2). Zenodo, February 16, 2022, pp. 1-233, https://doi.org/10.5281/zenodo.6107943.
- Excel file for download: 140-126-14×14-revisedDownload
Below are some examples studied in the work. The work is up to order 120 but the examples below are only up to order 70. As written above, in each case, there are six examples: