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 and Pandiagonal Magic Squares Multiples of 16.
14. Block-Wise Bordered Magic Squares Multiples of 18.
15. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 20.

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:

• 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 pandiagnonal magic squares of order 4.
• 2^nd Border: Different sums magic squares of order 5.
  • In this case there are two ways. One is pandiagonal magic squares of order 5. The second is bordered magic square of order 3 with inner part as a magic square of order 3. In both the cases the magic sums of order 5 are different.
• 3^rd Border: Equal sums magic squares of order 6.
  • In this case there are two ways. One is magic squares of order 6. The second is bordered magic square of order 6 with inner part as a pandiagonal magic square of order 4.
• 4^th Border: Different sums magic squares of order 7.
  • In this case also there are two ways. One is pandiagonal magic squares of order 7. The second one is bordered magic square of order 7 with inner parts as a magic square of orders 5 and 3. In both the cases the magic sums of order 7 are different.
• 5^th Border: Equal sums magic squares of order 8.
  • In this case there are three ways. One is pandiagonal magic squares of order 8 with equal sums pandiagonal magic squares of order 4. The second one is flat-type magic square of order 8 with inner part as a magic square of order 4, and the magic rectangles are of orders 2×8 and 2×4. The third one is cyclic-type double-digit magic squares of order 8 with inner part as a magic square of order 4 with four magic rectangles of orders 2×6 are of equal sums. This lead us to a multiple order bordered magic square 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.

Magic Square of Order 20

Below in a magic square of order 20. It is with blocks of order 3 and 4. Blocks of order 3 are forming a magic square of order 12. The border with magic squares of order 4 give us a magic square of order 20. The block of order 4 are equal magic sums pandiagonal magic squares. The magic squares of orders 3 and 4 used are as follows:

Below is a bordered magic square of order 20.

In Figure:

Magic Squares of Order 30

Below are two different types of magic squares of order 30. These are formed by external border of order 5 in two different ways. One with pandiagonal magic squares of order 5. The second with bordered magic squares of order 5. The blocks are of order 20 are the same as given above formed by blocks of orders 3 and 4. In both the examples, the magic squares of order 5 are with different magic sums. The magic squares of order 5 are as follows:

Based on these two magic square of order 5 below are two bordered magic square of order 30.

In Figures:

Magic Squares of Order 42

Below are four different types of magic squares of order 42. These are formed by external border of order 6 in two different ways. One with magic square of order 6 and another with bordered magic squares of order 6. These external borders are ove magic squares of order 30 given above formed by blocks of orders 3, 4 and 5. In both the situations the magic squares of orders 6 are of equal magic sums.

The magic squares of order 6 are as follows:

Based on these two magic square of order 6 below are two bordered magic square of order 42.

In Figures:

Magic Squares of Order 56

Below are eight different types of magic squares of order 56. These are formed by external border of order 7 in two different ways. One with pandiagonal magic square of order 7 and another with bordered magic squares of order 7. These external borders to magic squares of order 56. Since there are four magic squares of order 42. It gives 8 magic squares of order 56. Thus we have 8 magic squares order 56 formed by blocks of order 3, 4, 5, 6 and 7.

The magic squares of orders 3, 5 and 7 are with different magic sums and the magic squares of orders 4 and 6 are with equal sums magic squares. The magic squares of order 7 are as follows:

Based on these two magic square of order 7 below are eight bordered magic square of order 42 (figures). The excel file with with eight magic squares of order 56 is given at the end for download.

In Figures or Designs:

Figures: 1-2

Figures: 3-4

Figures: 5-6

Figures: 7-8

Magic Squares of Order 72

There are 32 different types of magic squares of order 72. These are formed by external border of order 8 in four different ways.

1. A pandiagonal magic square of order 8 formed by four pandiagonal magic squares of order 4.
2. It is formed by two magic rectangles of orders 2×8 and two magic rectangles of orders 2×4. In the middle there is a pandiagonal magic square of order 4.
3. It is formed four equal sums magic rectangles of orders 2×6 with pandigonal magic square of order 4 in the middle part.
4. It is a single-digit bordered magic square of order 8.

Since there are 8 magic squares of order 56. Along with this four magic squarer of order 8 we have 32 magic squares of order 72. Thus, we have 32 magic squares order 72 formed by blocks of orders 3, 4, 5, 6, 7 and 8. Below are only examples in figures. The complete excel file is given at the end for download.

In Figures or Designs:

Figures: 1-2

Figure: 3-4

Figures: 5-6

Figures: 7-8

Figures: 9-10

Figures: 11-12

Figures: 13-14

Figures: 15-16

Figures: 17-18

Figures: 19-20

Figures: 21-22

Figures: 23-24

Figures: 25-26

Figures 27-28

Figures: 29-30

Figures: 31-32

Excel files 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: Multiple Order Bordered Magic Squares of Order 90 – Recreating Numbers and Magic Squares

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

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.
2. Inder J. Taneja, Bordered Magic Squares Multiples of 6, Zenodo, July 25, 2023, pp. 1-32, https://doi.org/10.5281/zenodo.8184983.
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.
4. Inder J. Taneja, Bordered Magic Squares Multiples of 10, Zenodo, July 26, pp. 1-40, https://doi.org/10.5281/zenodo.8187888.
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.
6. Inder J. Taneja, Bordered Magic Squares Multiples of 14, Zenodo, July 27, pp. 1-33, https://doi.org/10.5281/zenodo.8188395.
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.
8. Inder J. Taneja, Bordered Magic Squares Multiples of 18, Zenodo, July 28, pp. 1-31, https://doi.org/10.5281/zenodo.8191223.
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.

Odd Orders Magic Squares

1. Inder J. Taneja, Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 3, Zenodo, May 05, pp. 1-29, 2023, https://doi.org/10.5281/zenodo.7898383.
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.
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.
4. Inder J. Taneja, Bordered Magic Squares Multiples of 9, Zenodo, July 23, 2023, pp. 1-28, https://doi.org/10.5281/zenodo.8176357.
5. Inder J. Taneja, Bordered Magic Squares Multiples of 11, Zenodo, July 24, pp. 1-34, 2023, https://doi.org/10.5281/zenodo.8176475.
6. Inder J. Taneja, Bordered Magic Squares Multiples of 13, Zenodo, July 24, pp. 1-32, 2023, https://doi.org/10.5281/zenodo.8178879.
7. Inder J. Taneja, Bordered Magic Squares Multiples of 15, Zenodo, July 24, pp. 1-35, 2023, https://doi.org/10.5281/zenodo.8178935.
8. Inder J. Taneja, Bordered Magic Squares Multiples of 17, Zenodo, July 25, pp. 1-26, 2023, https://doi.org/10.5281/zenodo.8180706.
9. Inder J. Taneja, Bordered Magic Squares Multiples of 19, Zenodo, July 25, pp. 1-31, 2023, https://doi.org/10.5281/zenodo.8180919.

Mixed Orders Magic Squares

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