# Bordered and Pandiagonal Magic Squares Multiples of 7 – Revised

#### ByInder J. Taneja

May 6, 2023

Revised on July 20, 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, 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 7. These examples are only for the magic squares of orders 7, 21 and 28. Total work is with 7 different types of magic squares of order 7. In the end there an excel file of whole the work up to order 140 multiples of 7 is attached for download. Some pandiagonal magic squares with excel files multiples of 7 are also given at the end.

Whole the work can also be accessed at the following link:
Inder J. Taneja, Bordered and Pandiagonal Magic Squares Multiples of 7, Zenodo, July 23, 2023, pp. 1-34, https://doi.org/10.5281/zenodo.8176061.

#### Magic Squares of Order 7

Below are 7 magic square of order 7. We have used these magic squares to bring bordered magic squares up to order 140 as multiples of 7.

#### Block-Wise Bordered Magic Squares Multiples of 21

Below are 7 magic square of order 21.

#### Magic Squares of Order 28 Multiples of 7

Below are 7 magic square of order 28.

#### Pandigonal Magic Squares Multiples of 7

The magic square of order 49 is also a bimagic square.