Revised on July 21, 2023
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. Bordered and Pandiagonal Magic Squares Multiples of 4.
2. Bordered Magic Squares Multiples of Magic Squares of Order 6.
3. Bordered and Pandiagonal Magic Squares Multiples of 8 – Revised
4. Bordered Magic Squares Multiples of 10 – Revised.
5. Bordered and Pandiagonal Magic Squares Multiples of 12 – Revised.
6. Bordered Magic Squares Multiples of 14 – Revised.
7. Bordered and Pandiagonal Magic Squares Multiples of 16.
8. 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. Bordered and Pandiagonal Magic Squares Multiples of 3.
2. Bordered and Pandiagonal Magic Squares Multiples of 5.
3. Bordered and Pandiagonal Magic Squares Multiples of 7 – Revised.
4. Bordered Magic Squares Multiples of 9 – Revised.
5. Bordered Magic Squares Multiples of Order 11- Revised.
6. Bordered Magic Squares Multiples of 13.
7. Bordered Magic Squares Multiples of 15.
8. Bordered Magic Squares Multiples of 17.
9. Bordered Magic Squares Multiples of 19.
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 5. These examples are only for the magic squares of orders 5, 15, 20, 25, 30, 35 and 40. Total work is with 3 different types of magic squares of order 5. In the end there an excel file of whole the work up to order 150 multiples of 5 is attached for download. Some pandiagonal magic squares with excel files multiples of 5 are also given at the end.
Whole can also be accessed at the the following link:
Inder J. Taneja, Bordered and Pandiagonal Magic Squares Multiples of 5, Zenodo, July 23, 2023, pp. 1-36, https://doi.org/10.5281/zenodo.8175759.
Magic Squares of Order 5
Below are 3 examples of magic square of order 5. These are used to bring bordered magic squares multiples of 5 up to order 150.
Magic Squares of Order 15 Multiples of 5
Magic Squares of Order 20 Multiples of 5
Magic Squares of Order 25 Multiples of 5
Magic Squares of Order 30 Multiples of 5
Magic Squares of Order 35 Multiples of 5
Magic Squares of Order 40 Multiples of 5
Excel file of Bordered Magic Squares Multiples of 5
Pandigonal Magic Squares Multiples of 5
Equal Sums of Order 5
Below examples of equal sums pandiagonal magic squares multiples of 5. These examples are of orders 15, 25, 35 and 45. The magic square of order 25 is also bimagic. Non equal sums pandiagonal magic squares shall be dealt elsewhere.
An excel file at the end of equal sums magic squares is also attached for download.
