Recently, author worked with cornered Magic Squares based on two-by-two divisions. These works can be accessed at the following links:

1. Inder Jeet Taneja, Cornered Magic Squares of Orders 5 to 10.

2. Inder Jeet Taneja, Cornered Magic Squares of Orders 11 to 13.

3. Inder Jeet Taneja, Cornered Magic Squares of Orders 14 to 24.

Also, the author worked on **even order** **double digits bordered magic squares**. This works starts from order 8 and goes up to order 40. See below the links:

The work for order of type 4k, k=1, 2, 3, … i.e., for orders 8, 12, 16, 20, 24, 28, 32, 36 and 40:

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, k=1, 2, 3, …, i.e., for the orders 10, 14, 18, 22, 26, 30, 34 and 38:

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 Order 34.

4. Two digits Bordered Magic Squares of Order 38.

This work is for magic square of order 31. It includes five type of magic squares:**1. Traditional centered magic squares.2. Magic Squares with blocks of order 93. Bordered with magic squares of order 9.4. Bordered with magic squares of order 3.5. Double digits bordered magic squares.**

**5.1 Embedded Lower Order Magic Squares.**

**5.2. Embedded Cornered Magic Squares.**

Double digits magic squares can also be constructed using software, the reader can follow H. White’s site: Magic Squares.

This work can also be accessed online at:

Inder J. Taneja, Magic Squares of Order 31, Zenodo, August 06, 2023, pp. 1-35, https://doi.org/10.5281/zenodo.8219053.

For the previous work on

**odd orders from 3 to 29**see the following links:.

For the previous work on **odd orders from 3 to 31** see the following links:

- Inder J. Taneja, Odd Order Magic Squares: Orders 3 to 15, Zenodo, June 15, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.8043030.

Site Link: Odd Order Magic Squares: Orders 3 to 15 - Inder J. Taneja, Magic Squares of Orders 17 and 19, Zenodo, June 15, 2023, pp. 1-38, https://doi.org/10.5281/zenodo.8043105.

Site Link: Magic Squares of Orders 17 and 19 - Inder J. Taneja, Magic Squares of Orders 21 and 23, Zenodo, June 15, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.8043198.

Site Links: Magic Squares of Order 21. - Inder J. Taneja, Magic Squares of Orders 21 and 23, Zenodo, June 15, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.8043198.

Site Links: Magic Squares of Order 23. - Inder J. Taneja, Magic Squares of Order 25, Zenodo, June 15, 2023, pp. 1-27, https://doi.org/10.5281/zenodo.8043228.

Site Links: Magic Squares of Order 25. - Inder J. Taneja, Magic Squares of Order 27, Zenodo, August 06, 2023, pp. 1-32, https://doi.org/10.5281/zenodo.8218291.

Site Links: Magic Squares of Order 27. - Inder J. Taneja, Magic Squares of Order 29, Zenodo, August 06, 2023, pp. 1-30, https://doi.org/10.5281/zenodo.8218771.

Site Links: Magic Squares of Order 29.

To have better idea, **pdf file** of total 640 magic squares of order 31 is attached for download.

### Magic Squares of Order 31

As explained above, the construction of magic squares of order 31 are of five types. Below are examples of each case.

### 1. Traditional Centered Magic Squares

### 2. Magic Squares with Blocks of Order 9

### 3. Bordered with Magic Squares of Order 9

### 3. **Bordered magic squares with magic squares of order 3**

### 5. Double Digits Magic Squares:

5.1 Embedded with Lower Order Magic Squares

### 5. Double Digits Magic Squares

5.2. Embedded Cornered Magic Squares with Lower Orders

### Pdf file of Magic Squares of Order 31

There are total 640 magic squares of order 31. These are embedded as **pdf file.**