*Never Seen Before*

This work brings magic squares in very different way. These are based on three different types of **magic rectangles**:**1. Bordered Magic Rectangles:2. Double Digits Magic Rectangles:3. Cornered magic rectangles. **

First of let’s understand these magic rectangles one by one. Removing external borders in each case still we left with lower order

**magic rectangles**. See below few examples in each case.

### 1. Bordered Magic Rectangles

Below are two examples of bordered magic rectangles

**Example 1. Bordered Magic Rectangle of Order 12×18**

The entries according to colors are as follows:

#### Example 2. Bordered Magic Rectangle of Order 6×16

The entries according to colors are as follows:

More details are given in author’s recent work:

**Inder J. Taneja,**Different Types of Magic Rectangles,**Zenodo**, September 04, 2023, pp. 1-26, https://doi.org/10.5281/zenodo.8316719.

### 2. Double Digits Magic Rectangles

Below are two examples of bordered magic rectangles

**Example 1. Double Digits Magic Rectangle of Order 14×20**

Except the corners, the entries **entries sums** are understood as

**Example 2. Double Digits Magic Rectangle of Order 12×18**

Except the corners, the entries **entries sums** are understood as

After reorganizing the **internal magic rectangle of order 4×16**, we have the following **double digits magic rectangle**:

More details are given in author’s recent work:

**Inder J. Taneja,**Different Types of Magic Rectangles,**Zenodo**, September 04, 2023, pp. 1-26, https://doi.org/10.5281/zenodo.8316719.

### 3. Cornered Magic Rectangles

Below are two examples of cornered magic rectangles

**Example 1. Cornered Magic Rectangle of Order 8×12**

Let’s consider a cornered magic rectangle of order **8×12** formed by 96 sequencial entries, i.e., from 1 to 96:

Distributions in colors as follows:

**Example 2. Cornered Magic Rectangle of Order 8×12**

Let’s consider a cornered magic rectangle of order **10×24** formed by 240 sequencial entries, i.e., from 1 to 240:

Distributions in colors as follows:

More details are given in author’s recent work:

**Inder J. Taneja,**Different Types of Magic Rectangles,**Zenodo**, September 04, 2023, pp. 1-26, https://doi.org/10.5281/zenodo.8316719.

Based on ideas given above, we shall construct magic squares of order 14 and 18. These are constructed with **four magic rectangles**. As explained above these are of are of **equal sums in pairs except in corners.** These are made with the help of script by H. White (Downloads (budshaw.ca) – *NestedCornerRectangles*)). These kinds of magic squares are never seen in the history. There are of different styles of this magic squares of order 14. See below:

### Magic Squares of Order 14

Initially below are two **double digits** and **cornered magic squares** of order 14:

### 1. Bordered Magic Rectangles and Magic Squares of Order 14

### 2. Cornered Magic Rectangles and Magic Squares of Order 14: First Type

### 2. Cornered Magic Rectangles and Magic Squares of Order 14: Second Type

### Magic Squares of Order 18

Initially below are two **double digits** and **cornered magic squares** of order 18:

### 1. Bordered Magic Rectangles and Magic Squares of Order 18

Below are examples of magic **squares of order 18** centered in magic squares of order 6 and 10 respectively. These are constructed with four equal sums **bordered magic rectangles **of orders **6×12** and **4×14**

### 2. Double Digits Magic Rectangles and Magic Squares of Order 18

Below is a **magic square of order 18** centered in magic squares of order 6. It is constructed with four equal sums **double digits magic rectangles **of order **6×12.**

In this case we don’t have magic square of order 18 centered in magic square of order 10 having four equal sums double digits magic rectangles.

### 3. Cornered Magic Rectangles and

Magic Squares of Order 18: First Type

Below are examples of magic **squares of order 18** centered in magic squares of order 6 and 10 respectively. These are constructed with four equal sums **cornered magic rectangles **of orders **6×12** and **4×14**

### 3. Cornered Magic Rectangles and

Magic Squares of Order 18: Second Type

Below is an example of **magic square of order 18** centered in magic squares of order 6. It is constructed with four equal sums **cornered magic rectangles **of orders **6×12**

### 3. Cornered Magic Rectangles and

Magic Squares of Order 18: Third Type

Below are examples of magic **squares of order 18** centered in magic squares of order 6 and 10 respectively. These are constructed with four equal sums **cornered magic rectangles **of orders **6×12** and **4×14**

### PDF File of Magic Squares of Order 18

Below is a pdf file of 61 magic squares of order 18 for download. These are constructed with 4 **equal sums magic rectangles of three types** as explained above. Magic squares number 1 and 2 are basic.

## References

**Inder J. Taneja**, Double Digits Even and Odd Orders Bordered Magic Squares.

Also see: New Concepts in Magic Squares: Double Digits Bordered Magic Squares of Orders 7 to 108, pp. 1-30, August 09, 2023,**Zenodo**. https://doi.org/10.5281/zenodo.8230214.**Inder J. Taneja**, New Concepts in Magic Squares: Cornered Magic Squares of Orders 5 to 81, pp. 1-27, August 09, 2023,**Zenodo**. https://doi.org/10.5281/zenodo.8231157.**Inder J. Taneja**, Different Types of Magic Rectangles,**Zenodo**, September 04, 2023, pp. 1-26, https://doi.org/10.5281/zenodo.8316719.**Inder J. Taneja**, Different Types of Magic Rectangles in Construction of Magic Squares of Orders 14 and 18,**Zenodo**, September 10, 2023, pp. 1-32, https://doi.org/10.5281/zenodo.8331709.**Inder J. Taneja**, Different Types of Magic Rectangles in Construction of Magic Squares of Order 22,**Zenodo**, September 10, 2023, pp. 1-36, https://doi.org/10.5281/zenodo.8331743.**Inder J. Taneja**, Different Types of Magic Rectangles in Construction of Magic Squares of Order 26,**Zenodo**, September 10, 2023, pp. 1-39, https://doi.org/10.5281/zenodo.8331750.**Inder J. Taneja**, Different Types of Magic Rectangles in Construction of Magic Squares of Order 30,**Zenodo**, September 10, 2023, pp. 1-44, https://doi.org/10.5281/zenodo.8331755.**Inder J. Taneja**, Different Types of Magic Rectangles in Construction of Magic Squares of Order 34,**Zenodo**, September 10, 2023, pp. 1-49, https://doi.org/10.5281/zenodo.8331759.**Inder J. Taneja**, Cornered Magic Squares in Construction of Magic Squares of Orders 16, 20, 24 and 28,**Zenodo**, September 10, 2023, pp. 1-35, https://doi.org/10.5281/zenodo.8332156.