# Bordered Magic Squares Multiples of 13

#### ByInder J. Taneja

Jul 16, 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. 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 and 11. 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.

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.

Below are examples of bordered magic squares multiples of order 13 up to order 52. Total we worked with 25 different types of magic squares of order 13. Up to order 52 pdf files of each order are attached for download. In the end there are excel files of whole are also attached.

This work can also be accessed at the following link:

Inder J. Taneja, Bordered Magic Squares Multiples of 13, Zenodo, July 25, pp. 1-32, 2023, https://doi.org/10.5281/zenodo.8178879.

#### Magic Squares of Order 13

Below are 25 magic square of order 13. We have used these magic squares to bring bordered magic squares up to order 130 as multiples of 13.

#### Block-Wise Bordered Magic Squares Multiples of 13Magic Squares of Order 39

Below are only few examples of magic square of order 39. The other examples are given in attached pdf file.

### Pdf file of Order 39

#### Block-Wise Bordered Magic Squares Multiples of 13Magic Squares of Order 52

Below are only few examples of magic square of order 52. The other examples are given in attached pdf file.

File 1:

File 2: