# Bordered Magic Squares Multiples of 9 – Revised

#### ByInder J. Taneja

May 24, 2023

Revised on July 13, 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. 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.
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.

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 and 30 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.

This work revises the previous work on bordered magic squares multiple of magic squares of order 9. This work can also be accessed at the following link:
Inder J. Taneja, Block-Wise Bordered Magic Squares Multiples of 9, Zenodo, July 23, pp. 1-28, 2023,  https://doi.org/10.5281/zenodo.8176357.

Below are few examples of bordered magic squares multiples of order 9. These are up to order 45. Excel files of whole work are attached at the end.

#### Block-Wise Bordered Magic Squares Multiples of 9Magic Squares of Order 9

Below are 11 examples of magic squares of order 9. We shall work these magic squares of order 9 to bring bordered magic squares multiples of 9.

#### Block-Wise Bordered Magic Squares Multiples of 9Magic Squares of Order 27

Above there are only three examples. The other examples can be seen at pdf file given below or in excel file attached at the end of the work.

### Pdf file of Magic Squares of Order 27

#### Block-Wise Bordered Magic Squares Multiples of 9Magic Squares of Order 36

Above there are only three examples. The other examples can be seen at pdf file given below or in excel file attached at the end of the work.

### Pdf file of Magic Squares of Order 36

#### Block-Wise Bordered Magic Squares Multiples of 9Magic Squares of Order 45

Above there are only three examples. The other examples can be seen at pdf file given below or in excel file attached at the end of the work.