# Even Order Magic Squares Using Bordered Magic Rectangles

#### ByInder J. Taneja

Dec 2, 2022

For the first time we are introducing a new idea of magic squares by figures. This work brings magic squares of even orders from 6 to 40. From order 14 onwards, construction idea is explained. It is based on external borders made by cornered magic squares of orders 4, 6, 8, etc. and in middle bordered magic rectangles. The inner blocks are previously known magic squares. The inicial work was up to order 30. Then it is extended up to order 40. Further orders shall be written in futures. The lack of numbers in figures is due to the fact that the readers can also try to complete them in their own ways. Otherwise, the readers can see the results in the author’s other works. Up to order 20 the results can be seen here. In some cases, the PDF files with numbers are embedded for downloads.

The bordered magic squares and the bordered magic rectangles are with the property that if we remove external borders in each case, still we are left with bordered magic squares and bordered magic rectangles respectively with sequential number entries. The whole work is in figures instead of numbers. In the same magic square, the small equal figures represent equal sums magic squares and/or equal sums bordered magic rectangles. The whole work is done manually except the bordered magic squares and bordered magic rectangles. This construction is based on the software due to H. While. Later, these bordered magic rectangles are readopted manually according to distribution of each magic square. The distribution of magic squares or bordered magic rectangles is based on half-sequential numbers. By half-sequential numbers we understand that the total numbers in each case are divided in two equal parts. The first part is one sequence, and the second part is another sequence. They are considered according to order of magic squares.

Below is a list of work of even order magic squares from orders 6 to 40. The readers can download the work from the links or can see it at their respective site links. Moreover, at the end of each site, there is a PDF file of figured magic squares for download. In some cases, these files are with numbers. These are from order 18 onwards. These files are divided in parts. First part brings magic squares of orders 6 to 30. The second part brings magic squares of orders 32 to 40. In this part, the files of magic squares are given in figures as well as in numbers. The third part from orders 42 to 50 is in construction.

## Part 1: Orders 6 to 30

1. Inder J. Taneja, Figured Magic Squares of Orders 6, 10, 12, 14 and 16 Using Bordered Magic Rectangles, Zenodo, November 29, 2022, pp. 1-31, https://doi.org/10.5281/zenodo.7377674.
2. Inder J. Taneja, Figured Magic Squares of Orders 18 and 20 Using Bordered Magic Rectangles, Zenodo, November 29, 2022, pp. 1-87, https://doi.org/10.5281/zenodo.7377689.
3. Inder J. Taneja, Figured Magic Squares of Order 22 Using Bordered Magic Rectangles, Zenodo, November 29, 2022, pp. 1-61, https://doi.org/10.5281/zenodo.7377706
4. Inder J. Taneja, Figured Magic Squares of Order 24 Using Bordered Magic Rectangles, Zenodo, November 29, 2022, pp. 1-104, https://doi.org/10.5281/zenodo.7377779
5. Inder J. Taneja, Figured Magic Squares of Order 26 Using Bordered Magic Rectangles, Zenodo, November 29, 2022, pp. 1-88, https://doi.org/10.5281/zenodo.7377794
6. Inder J. Taneja, Figured Magic Squares of Order 28 Using Bordered Magic Rectangles, Zenodo, December 02, 2022, pp. 1-179, https://doi.org/10.5281/zenodo.7390666.
7. Inder J. Taneja, Figured Magic Squares of Order 30 Using Bordered Magic Rectangles, Zenodo, December 02, 2022, pp. 1-179, https://doi.org/10.5281/zenodo.7390705.

## Part 2: Orders 32 to 40

1. Inder J. Taneja, Figured Magic Squares of Order 32 Using Bordered Magic Rectangles: A Systematic Procedure, Zenodo, December 22, 2022, pp. 1-310, https://doi.org/10.5281/zenodo.7472891
2. Inder J. Taneja, Figured Magic Squares of Order 34 Using Bordered Magic Rectangles: A Systematic Procedure, Zenodo, December 27, 2022, pp. 1-193, https://doi.org/10.5281/zenodo.7486540.
3. Inder J. Taneja, Figured Magic Squares of Order 36 Using Bordered Magic Rectangles: A Systematic Procedure, Zenodo, December 27, 2022, pp. 1-140, https://doi.org/10.5281/zenodo.7486548.
4. Inder J. Taneja, Figured Magic Squares of Order 38 Using Bordered Magic Rectangles: A Systematic Procedure, Zenodo, January 03, 2023, pp. 1-133, https://doi.org/10.5281/zenodo.7500188.
5. Inder J. Taneja, Figured Magic Squares of Order 40 Using Bordered Magic Rectangles: A Systematic Procedure, Zenodo, January 03, 2023, pp. 1-157, https://doi.org/10.5281/zenodo.7500192.

## Crossed Bordered Magic Rectangles and Magic Squares

1. Inder J. Taneja, Single Crossed Bordered Magic Rectangles and Magic Squares of Order 40, Zenodo, January 24, 2023, pp. 1-76, https://doi.org/10.5281/zenodo.7565946.
2. Inder J. Taneja, Double Crossed Bordered Magic Rectangles and Magic Squares of Order 40, Zenodo, January 30, 2023, pp. 1-102, https://doi.org/10.5281/zenodo.7585787. Also see at: Double Crossed Bordered Magic Rectangles and Magic Squares of Order 40
3. Inder J. Taneja, Single-Cross Bordered Magic Rectangles and Magic Squares of Order 42, Zenodo, March 03, 2023, pp. 1-69, https://doi.org/10.5281/zenodo.7695939. Also see at:
4. Inder J. Taneja, Double-Cross Bordered Magic Rectangles and Magic Squares of Order 42, Zenodo, March 03, 2023, pp. 1-59, https://doi.org/10.5281/zenodo.7696070. Also see at:
5. Inder J. Taneja, Closed Double-Cross Bordered Magic Rectangles and Magic Squares of Order 42, Zenodo, March 03, 2023, pp. 1-28, https://doi.org/10.5281/zenodo.7696181. Also see at:

## Part 3: Orders 42 to 50

1. Inder J. Taneja, Magic Squares of Order 42 Using Bordered Magic Rectangles: A Systematic Procedure, Zenodo, March 03, 2023, pp. 1-92, https://doi.org/10.5281/zenodo.7695834. Also see at:

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