Beauty of Magic Squares: Multiple Order Bordered Magic Squares of Orders 20, 30, 42, 56 and 72

During past years author worked with block-wise bordered magic squares multiples of even and odd number blocks. This means, multiples of 3, 4, 5, 6, etc. These works can be accessed at the following links.

1. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 3
2. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4.
3. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 5.
4. Block-Wise Bordered Magic Squares Multiples of Magic and Bordered Magic Squares of Order 6.
5. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 7.
6. Block-Wise Bordered Magic Squares Multiples of 8.
7. Block-Wise Bordered Magic Squares Multiples of 9.
8. Block-Wise Bordered Magic Squares Multiples of 10.
9. Block-Wise Bordered Magic Squares Multiples of 11.
10. Block-Wise Bordered Magic Squares Multiples of 12.
11. Block-Wise Bordered Magic Squares Multiples of 13.
12. Block-Wise Bordered Magic Squares Multiples of 14.
13. Block-Wise Bordered Magic Squares Multiples of 15.
14. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 16.
15. Bordered Magic Squares Multiples of 17.
16. Block-Wise Bordered Magic Squares Multiples of 18.
17. Bordered Magic Squares Multiples of 19.
18. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 20.
19. Beauty of Magic Squares: Multiple Order Bordered Magic Squares of Orders 20, 30, 42, 56 and 72

The advantage in studying block-wise bordered magic squares is that when we remove external borders, still we are left with magic squares with sequential entries. The bordered magic squares also have the same property. The difference is that instead of numbers here we have blocks of magic squares.

This work bring magic squares, based on multiple order magic squares in the same magic squares. This means same magic square contains borders of order 3, 4, 5, etc. It can be accessed at the following link:

Inder J. Taneja– Multiple Orders Bordered Magic Squares, Zenodo, Jun 9, 2023, pp. 1-43
https://doi.org/10.5281/zenodo.8019330.

This work brings brodered magic squares in such a way that in the beginning there is magic square of order 12 with different sums magic squares of order 3. The upper borders are magic squares of orders 4, 5, 6, 7, 8 and 9. The even order borders are with magic squares, such as of orders 4, 6 and 8 are with equal sums magic squares. The odd order borders are with magic squares, such as of orders 5, 7 and 9 are with different sums magic squares. See below the details of each order:

See below the details of above multiple order bordered magic square.

• 0 Border: Different sums magic squares of order 3.
  • Initially, we have a magic square of order 12 formed by different sums magic squares of order 3.
• 1^st Border: Equal sums magic squares of order 4.
  • Here have considered two types of magic squares of order 4. One is pandiagonal magic square of order 4 and another is formed by two equal sums magic sums magic rectangles of order 2×4 resulting in a magic square of order 4.
• 2^nd Border: Different sums magic squares of order 5.
  • In this case, we have considered three different types of magic squares of order 5. One is pandiagonal magic squares of order 5. The second is cornered magic square of order 5, where there are two equal sums magic rectangles of order 2×3 and a magic square of order 3 at the upper-left corner. The third one is single-digit bordered magic square with magic square of order 3 in the middle.
• 3^rd Border: Equal sums magic squares of order 6.
  • In this case also we have considered three different types of magic squares of order 6.The first one isa nornal magic square of order 6. The second one is cornered magic square of order 6, where there is a pandiagonal magic square of order 4 at the upper-left corner with two equal sums magic rectangles of order 2×4. The third one is traditional bordered magic square of order 6 with pandiagonal magic square of order 4 in the middle.
• 4^th Border: Different sums magic squares of order 7.
  • In this case, we have considered 5 different ways of magic squares of order 7. The first one is a pandiagonal magic square of order 7. The second one is double-digit bordered magic square of order 7, where there is a magic square of order 3 in the middle and 4 equal sums magic squares of orders 2×3. The third one is a cornered magic square of order 7 composed of one cornered magic square of order 3 and magic squares of order 3 at the upper-left corner. Two equal sums magic rectangles of order 2×3 and two equal sums magic rectangles of order 2×5.The forth one is also a double-digit bordered magic square of order 7 with magic square of order 3 in the middle. Also there are four equal sums magic rectangles of order 2×5. This type of magic squares we call as cyclic-double-digits magic square of order 7. The fifth one is single-digit bordered magic square of order 7, where there is a magic square of order 3 in the middle.
• 5^th Border: Equal sums magic squares of order 8.
  • The first one is a pandiagonal magic square of order 8 formed by four equal sums pandiagonal magic squares of order 4. The second one is a cornered magic square of order 8, where there is cornered magic square of order 6 with pandiagonal magic square of order 4 at the upper-left corner. The magic rectangles of orders 2×4 and 2×6 are of equal sums in each case. The third one is a double-digit bordered magic square with pandiagonal magic square of order 4 in the middle having 2 equal sums magic rectangles of order 2×8 and two equal sums magic rectangles of order 2×4. The forth one is also a double-digit bordered magic square of order 8 with magic square of order 4 in the middle and 4 equals sums magic rectangles of order 2×6. Since it is formed by only magic rectangles of equal width it is known as striped magic square.The fifth one is four equal sums magic squares of order 4, where each magic square of order 4 is formed by two equal sums magic rectangles of order 2×4. Since it is formed by only magic rectangles of equal width it is known as striped magic square.The sixth one is single-digit bordered magic square of order 8 having a pandiagonal magic square of order 4 in the middle. This lead us to a multiple order bordered magic square of order 72.

Summarizing there are multiple order bordered magic square of order 72 as given below

• 0 Border: Different sums magic squares of order 3.
  • Initially, we have a magic square of order 12 formed by different sums magic squares of order 3.
• 1^st Border: Equal sums magic squares of order 4.
  • In this case we have considered two different types of magic squares of order 4. Combining with magic square of order 12 we have 2-different types of magic squares of order 20.
• 2^nd Border: Different sums magic squares of order 5.
  • In this case, we have considered 3-different types of magic squares of order 5. Combining with 3 magic squares of order 20 we have 6-different types of magic squares of order 20.
• 3^rd Border: Equal sums magic squares of order 6.
  • In this case also we have considered 3-different types of magic squares of order 6. Combining with 6 magic squares of order 30 we have 18-different types of magic squares of order 42.
• 4^th Border: Different sums magic squares of order 7.
  • In this case, we have considered 5-different ways of magic squares of order 7. Combining with 18 magic squares of order 42 we have 90-different types of magic squares of order 56.
• 5^th Border: Equal sums magic squares of order 8.
  • In this case we have considered 6 different types of magic squares. Combining with 18 magic squares of order 42 we have 540-different types of magic squares of order 72.

Magic Square of Order 12

Below is a magic square of order 12. It is with different magic sums magics squares of order 3.

Let’s consider the following magic square of order 3:

Based on the above magic square of order 3, we have contructed the following magic square of order 12:

Magic Square of Order 20

Let’s consider the following magic squares of orders 3 and 4:

Above there are two different magic squares of order 4. One is pandiagonal magic square and aonther is with two magic rectangles of orders 2×4. There is also a magic square of order 3. Based on above three magic squares there are two different magic square of order 20. The only difference is in magic square of order 4. While the magic square of order 3 remains the same in both the magic squares. See below in figures:

In Figures:

• See in the end of this work excel file of above magic squares in numbers.

Magic Squares of Order 30

Let’s consider following three magic squares of order 3:

Above there are three magic squares of order 5. See the details below:

1. The first one is pandiagonal magic square of order 5.
2. The second one is cornered magic square of order 5, where there is a magic square of order 3 at the upper-left corner and there are two equal sums magic rectangles of order 2×3.
3. The third one is traditional bordered magic square of order 5 with magic square of order 3 in the middle. Based on these three magic squares of order 5 there are six magic square of order 30, where the upper borders are magic squares of order 5 formed by above three magic squares of order 5. The inner magic squares of order 20 are the two as given above resulting in 6 magic squares of order 30. See below in figures:

In Figures:

• See in the end of this work excel file of above magic squares in numbers.

Magic Squares of Order 42

Let’s consider the following magic squares of order 6:

Above there are three magic squares of order 6. See the details below:

1. The first one isa nornal magic square of order 6.
2. The second one is cornered magic square of order 6, where there is a pandiagonal magic square of order 4 at the upper-left corner with two equal sums magic rectangles of order 2×4.
3. The third one is traditional bordered magic square of order 6 with pandiagonal magic square of order 4 in the middle.

Above there are 6 magic squares of order 30 and combining with these three possibilities, we have total 18 magic squares of order 42. See below all these 18 possibilities.

In Figures:

• See in the end of this work excel file of above magic squares in numbers.

Magic Squares of Order 56

Let’s consider the following 5 magic squares of order 7:

Above there are five magic squares of order 7. See below the details

1. The first one is a pandiagonal magic square of order 7.
2. The second one is double-digit bordered magic square of order 7, where there is a magic square of order 3 in the middle and 4 equal sums magic squares of orders 2×3.
3. The third one is a cornered magic square of order 7 composed of one cornered magic square of order 3 and magic squares of order 3 at the upper-left corner. Two equal sums magic rectangles of order 2×3 and two equal sums magic rectangles of order 2×5.
4. The forth one is also a double-digit bordered magic square of order 7 with magic square of order 3 in the middle. Also there are four equal sums magic rectangles of order 2×5. This type of magic squares we call as cyclic-double-digits magic square of order 7.
5. The fifth one is single-digit bordered magic square of order 7, where there is a magic square of order 3 in the middle.

Above there are 18 magic squares of order 42 and combining with these five possibilities, we have total 90 magic squares of order 56. Below there are only few possibilities are given in figures. The complete 90 magic squares of order 56 in numbers is given in excel file attached at the end of this work:

In Figures or Designs:

First-Type

Second-Type

Third-Type

Forth-Type

Fifth-Type

• See in the end of this work excel file of above magic squares in numbers.

Magic Squares of Order 72

Let’s consider the following 5 magic squares of order 8:

About there are 6 magic squares of order 8. See below the details:

1. The first one is a pandiagonal magic square of order 8 formed by four equal sums pandiagonal magic squares of order 4.
2. The second one is a cornered magic square of order 8, where there is cornered magic square of order 6 with pandiagonal magic square of order 4 at the upper-left corner. The magic rectangles of orders 2×4 and 2×6 are of equal sums in each case.
3. The third one is a double-digit bordered magic square with pandiagonal magic square of order 4 in the middle having 2 equal sums magic rectangles of order 2×8 and two equal sums magic rectangles of order 2×4.
4. The forth one is also a double-digit bordered magic square of order 8 with magic square of order 4 in the middle and 4 equals sums magic rectangles of order 2×6. Since it is formed by only magic rectangles of equal width it is known as striped magic square.
5. The fifth one is four equal sums magic squares of order 4, where each magic square of order 4 is formed by two equal sums magic rectangles of order 2×4. Since it is formed by only magic rectangles of equal width it is known as striped magic square.
6. The sixth one is single-digit bordered magic square of order 8 having a pandiagonal magic square of order 4 in the middle.

Since there are 90 magic squares of order 56. Making combinations with 6 external border of order 8 we have toal 540 magic squares of order 72. Below are few examples in figures. The total 540 magic squares of order 72 are given in the end as an excel file for download.

In Figures or Designs:

First-Type

Second-Type

Third-Type

Forth-Type

Fifth-Type

Sixth-Type

Magic Squares of Orders 12, 20, 30, 42, 56 and 72
Excel file for download

This file contains the multiple order bordered magic squares of orders 12, 20, 30, 42, 56 and 72.

Magic Squares of Order 90

There are 160 different types of magic squares of order 90. These are formed by external border of order 9 in five ways. These forms an external border to magic squares of order 90. Thus, we have 160 magic squares order 90 formed by blocks of order 3, 4, 5, 6, 7, 8 and 9. For this order 90 refer the link below:

• Beauty of Magic Squares: 3240 Multiple Order Bordered Magic Squares of Order 90

In the above link the figures along with excel file for download is also enclosed.

Magic Squares of Order 108

There are 3888 different types of magic squares of order 108. These are formed by external border of order 9 in double over the magic squares of order 72.

Magic Squares of Order 110

here are 144 different types of magic squares of order 110. These are formed by external border of order 10 in three ways. One with magic square of order 10. The second with block border magic squares of order 10 with four magic squares of order 4. The third as bordered magic squares of order 10. These give external borders to magic squares of order 110. Thus, we have 144 magic squares order 110 formed by blocks of order 3, 4, 5, 6, 7, 8, 9 and 10. Below are only three examples as figures without numbers. Since there are lot of examples, the excel file contains few of them.

This shall be given in another work.

Magic Squares of Order 132

There are 288 different types of magic squares of order 132. These are formed by external border of order 11 in two ways. One with magic square of order 11. The second as bordered magic squares of order 11. These give external borders to magic squares of order 132. Thus, we have 288 magic squares order 132 formed by blocks of order 3, 4, 5, 6, 7, 8, 9, 10 and 11. Below are only three examples as figures without numbers. The full work with numbers can be seen in excel file attached with the work. Since there are lot of examples, the excel file contains few of them.

This shall be given in another work.

References

Even Orders Magic Squares

1. Inder J. Taneja, Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4, Zenodo, August 31, 2021, pp. 1-148, https://doi.org/10.5281/zenodo.5347897.
   Web-site Link: Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4.
2. Inder J. Taneja, Bordered Magic Squares Multiples of 6, Zenodo, July 25, 2023, pp. 1-32, https://doi.org/10.5281/zenodo.8184983.
   Web-site Link: Block-Wise Bordered Magic Squares Multiples of Magic and Bordered Magic Squares of Order 6.
3. Inder J. Taneja, Bordered and Pandiagonal Magic Squares Multiples of 8, Zenodo, July 26, 2023, pp. 1-58, https://doi.org/10.5281/zenodo.8187791.
   Web-site Link: Block-Wise Bordered Magic Squares Multiples of 8.
4. Inder J. Taneja, Bordered Magic Squares Multiples of 10, Zenodo, July 26, pp. 1-40, https://doi.org/10.5281/zenodo.8187888.
   Web-site Link: Block-Wise Bordered Magic Squares Multiples of 10.
5. Inder J. Taneja, Bordered and Pandiagonal Magic Squares Multiples of 12, Zenodo, July 27, 2023, pp. 1-31, https://doi.org/10.5281/zenodo.8188293.
   Web-site Link: Block-Wise Bordered Magic Squares Multiples of 12.
6. Inder J. Taneja, Bordered Magic Squares Multiples of 14, Zenodo, July 27, pp. 1-33, https://doi.org/10.5281/zenodo.8188395.
   Web-site Link: Block-Wise Bordered Magic Squares Multiples of 14.
7. Inder J. Taneja, Bordered and Pandiagonal Magic Squares Multiples of 16, Zenodo, July 27, pp. 1-30, https://doi.org/10.5281/zenodo.8190884.
   Web-site Link: Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 16.
8. Inder J. Taneja, Bordered Magic Squares Multiples of 18, Zenodo, July 28, pp. 1-31, https://doi.org/10.5281/zenodo.8191223.
   Web-site Link: Block-Wise Bordered Magic Squares Multiples of 18.
9. Inder J. Taneja, Bordered and Pandiagonal Magic Squares Multiples of 20, Zenodo, July 28, pp. 1-45, https://doi.org/10.5281/zenodo.8191426.
   Web-site Link: Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 20.

Odd Orders Magic Squares

1. Inder J. Taneja, Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 3, Zenodo, May 5, pp. 1-29, 2023, https://doi.org/10.5281/zenodo.7898383.
   Web-Site Link: Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 3.
2. Inder J. Taneja, Bordered and Pandiagonal Magic Squares Multiples of 5, Zenodo, July 23, 2023, pp. 1-36, https://doi.org/10.5281/zenodo.8175759.
   Web-site Link: Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 5.
3. Inder J. Taneja, Bordered and Pandiagonal Magic Squares Multiples of 7, Zenodo, July 23, pp. 1-34, 2023, https://doi.org/10.5281/zenodo.8176061.
   Web-site Link: Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 7.
4. Inder J. Taneja, Bordered Magic Squares Multiples of 9, Zenodo, July 23, 2023, pp. 1-28, https://doi.org/10.5281/zenodo.8176357.
   Web-site Link: Block-Wise Bordered Magic Squares Multiples of 9.
5. Inder J. Taneja, Bordered Magic Squares Multiples of 11, Zenodo, July 24, pp. 1-34, 2023, https://doi.org/10.5281/zenodo.8176475.
   Web-site Link: Block-Wise Bordered Magic Squares Multiples of 11.
6. Inder J. Taneja, Bordered Magic Squares Multiples of 13, Zenodo, July 24, pp. 1-32, 2023, https://doi.org/10.5281/zenodo.8178879.
   Web-site Link: Bordered Magic Squares Multiples of 13.
7. Inder J. Taneja, Bordered Magic Squares Multiples of 15, Zenodo, July 24, pp. 1-35, 2023, https://doi.org/10.5281/zenodo.8178935.
   Web-site Link: Block-Wise Bordered Magic Squares Multiples of 15.
8. Inder J. Taneja, Bordered Magic Squares Multiples of 17, Zenodo, July 25, pp. 1-26, 2023, https://doi.org/10.5281/zenodo.8180706.
   Web-site Link: Bordered Magic Squares Multiples of 17.
9. Inder J. Taneja, Bordered Magic Squares Multiples of 19, Zenodo, July 25, pp. 1-31, 2023, https://doi.org/10.5281/zenodo.8180919.
   Web-site Link: Bordered Magic Squares Multiples of 19.

Mixed Orders Magic Squares

1. Inder J. Taneja, Beauty of Magic Squares: Multiple Order Bordered Magic Squares of Orders 20, 30, 42, 56 and 72, Zenodo,
   • Web-site Link: Beauty of Magic Squares: Multiple Order Bordered Magic Squares of Orders 20, 30, 42, 56 and 72.
2. Inder J. Taneja, Beauty of Magic Squares: 3240 Multiple Orders Bordered Magic Squares of Order 90, Zenodo,
   • Web-site Link: Beauty of Magic Squares: 3240 Multiple Order Bordered Magic Squares of Order 90.
3. Inder J. Taneja, Beauty of Magic Squares: 3888 Multiple Orders Bordered Magic Squares of Order 108 – Part 1, Zenodo,
   • Web-site Link: Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 108 – Part 1
4. Inder J. Taneja, Beauty of Magic Squares: 3240 Multiple Orders Bordered Magic Squares of Order 108 – Part 2, Zenodo,
   • Web-site Link: Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 108-Part 2
5. Inder J. Taneja, Beauty of Magic Squares: 3888 Multiple Orders Bordered Magic Squares of Order 110, Zenodo,
   • Web-site Link: Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 110.
6. Inder J. Taneja, Beauty of Magic Squares: 3240 Multiple Orders Bordered Magic Squares of Orders 120, Zenodo,
   • Web-site Link: Beauty of Magic Squares: 3240 Multiple Order Bordered Magic Squares of Order 120
7. Inder J. Taneja, Beauty of Magic Squares: 3888 Multiple Orders Bordered Magic Squares of Order 132 – Part 1, Zenodo,
   • Web-site Link: Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 132- Part 1
8. Inder J. Taneja, Beauty of Magic Squares: 3888 Multiple Orders Bordered Magic Squares of Order 132 – Part 2, Zenodo,
   • Web-site Link: Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 132-Part 2
9. Inder J. Taneja, Beauty of Magic Squares: Multiple Orders Bordered Magic Squares of Order 132 – Part 3, Zenodo,
   • Web-site Link: Beauty of Magic Squares: 3240 Multiple Order Bordered Magic Squares of Order 132-Part 3