# Figured Magic Squares of Order 40: A Systematic Procedure

#### ByInder J. Taneja

Jan 3, 2023

For the first time we are introducing the idea of magic squares by figures. This work brings magic squares of order 40. In some cases, the idea of construction is explained. It is based on external borders made by magic squares or bordered magic rectangles. Then inner blocks are previously known magic squares of smaller orders. The lack of numbers in figures is due to the fact that the readers can also try to complete them in their own way. Otherwise, the readers can see the results in the author’s other works. Up to order 20 the results can be seen here. Below is a list of work of even order magic squares from orders 6 to 40. The readers can download from the links

### Block 1: Even Order Magic Squares from 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.

### Block 2: Even Order Magic Squares from 32 to 40

1. Inder J. Taneja, Figured Magic Squares of Order 32 Using Bordered Magic Rectangles, 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, 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, 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, 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, Zenodo, January 03, 2023, pp. 1-157, https://doi.org/10.5281/zenodo.7500192.

## Magic Squares of Order 40

Below are a few Examples of Figured Magic Squares of Order 40 using bordered magic rectangles. Below are only 144 magic squares of order 40, but there are many more possibilities. This shall be presented later on. In the end a PDF file of work in numbers is embedded for download. The examples are divided in following parts:

• Part 1: Block-Wise with Orders 4 and 5
• Part 2: Equal Sums of Order 8
• Part 3: Equal Sums of Order 10
• Part 4: Only with BMRs
• Part 5: Cornered Magic Squares of Order 4
• First Type
• Second Type
• Part 6: Closed Border of Order 4
• Part 7: Cornered Magic Squares of Order 6
• Part 8: Cornered Magic Squares of Order 8
• Part 9: Cornered Magic Squares of Order 10
• Part 10: Cornered Magic Squares of Order 12
• Part 11: Closed Border of Order 12
• Part 12: Cornered Magic Squares of Order 14
• Part 13: Cornered Magic Squares of Order 16
• Part 14: Cornered Magic Squares of Order 18
• Part 15: Equal Sums Magic Squares of Order 20
• Part 16: Consecutive BMRs of Orders 18×20 and 20×22