Recently, the author revised and improved the block-wise bordered magic squares multiples of even number blocks. This means, multiples of 4, 6, 8, etc. The work is up to order 20. These works can be accessed at the following links.
1. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4.
2. Block-Wise Bordered Magic Squares Multiples of Magic Squares of Order 6.
3. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 8 – Revised
4. Block-Wise Bordered Magic Squares Multiples of 10 – Revised.
5. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 12 – Revised.
6. Block-Wise Bordered Magic Squares Multiples of 14 – Revised.
7. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 16.
8. Block-Wise Bordered Magic Squares Multiples of 18.
9. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 20.
Author also worked with odd number blocks. The work is for multiples of order 3, 5, 7, 9, 11 and 13. See the links below:
1. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 3.
2. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 5.
3. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 7.
4. Block-Wise Bordered Magic Squares Multiples of 9 – Revised.
5. Block-Wise Bordered Magic Squares Multiples of Order 11- Revised.
6. Block-Wise Bordered Magic Squares Multiples of 13.
7. Block-Wise Bordered Magic Squares Multiples of 15.
8. Block-Wise Bordered Magic Squares Multiples of 17.
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.
We observe that the above study of even order blocks starting from 4. Recently, author studied the double digit or two digits borders resulting in interesting magic squares. These studies can be accessed at the following link:
1. Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24.
2. Two Digits Bordered Magic Squares of Orders 28 and 32.
3. Two digits Bordered Magic Squares of Order 36.
4. Two digits Bordered Magic Squares of Order 40.
The work is for the order of type 4k+2, where k>1, i.e., for the orders 10, 14, 18, 22, 26, 28, 30 and 32 can be access at the following links:
1. Two digits Bordered Magic Squares of Orders 10, 14, 18 and 22.
2. Two digits Bordered Magic Squares of Orders 26 and 30.
3. Two Digits Bordered Magic Squares of Orders 28 and 32.
Some studies towards cornered magic squares are also made. See the following links:
1. Cornered Magic Squares of Orders 5 to 10.
2. Cornered Magic Squares of Orders 11 to 13.
3. Cornered Magic Squares of Orders 14 to 24.
Below are examples of bordered magic squares multiples of order 19. These examples are only for the magic squares of orders 19 and 57. Total work is with 42 different types of magic squares of order 19. PDF files of magic squares of order 19 and 57 are also attached. In the end there are excel files of whole the up to order 133 multiples of 19.
This work can also be accessed at the following link:
Inder J. Taneja, Bordered Magic Squares Multiples of 19, Zenodo, July 25, pp. 1-31, 2023, https://doi.org/10.5281/zenodo.8180919.
Magic Squares of Order 19
Below are 42 magic square of order 19. We have used these magic squares to bring bordered magic squares up to order 133 as multiples of 19.
Pdf file of Magic Squares of Order 19
Block-Wise Bordered Magic Squares Multiples of 19
Magic Squares of Order 57
Below are few examples of magic square of order 57. The other examples are given in attached pdf file and in excel files.
Pdf file of Magic Squares of Order 57
Excel Files of Whole Work for Download
File 1:
File 2: