Beauty of Magic Squares: 3240 Multiple Order Bordered Magic Squares of Order 132 – Part 4

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.

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

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 further borders are magic squares of orders 4, 5, 6, 7, 8, 9 and 15 resulting in multiple order bordered magic squares of order 120. Considering one more border of order 6, we get multiple order bordered magic square of order 132,. Similar kind of results are done for orders 108, 110 and 132 see the reference list given in the end. The even order borders are with magic squares, such as of orders 4, 6, 8 are with equal sums magic squares. The odd order borders are with magic squares, such as of orders 5, 7 and 15 are with different sums of magic squares.

See below the details of above multiple order bordered magic squares:

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

• 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.
    • The first one is pandiagonal magic square of order 4.
    • The second one 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.
  • In this case, we have considered 6 different types of 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.
• 6^th Border: Different sums magic squares of order 9
  • In this case we have considered 6 different types of magic squares of order 9.
    • First one is composed of 9 semi-magic squares of order 3 resulting in a pandiagonal magic square of order 9.
    • The second one is a cornered magic square, where magic squares of orders 7 and 5 are also cornered magic squares at the upper-left corner having magic square of order 3.
    • The third one is double-digit bordered magic square of order 8 having cornered magic squaRE of order 5 in the middle. Two magic rectangles of orders 2×3 and two magic rectangles of orders 2×5 are of equal sums in each case.
    • The forth one is again a double-digit bordered with single-digit bordered magic square of order 5 at the middle. The external border is composed of 2 magic rectangles of order 2×9 and two magic rectangles of order 2×5. Both are of equal sums in each case. In the middle there is a magic square of order 3.
    • The fifth one is again a double digit-digit bordered magic square of order 9 with external bordered as 4 equal sums magic rectangles of order 2×7. In the middle there is a pandiagonal magic square of order 5. This kind of magic square we call as cyclic-type double digit bordered magic square of order 9
    • The sixth is a normal single-digit bordered magic square of order 9 having magic square of order 3 in the middle.
• 7^th Border: Different sums magic squares of order 15
  • In this case we have considered 6 different types of magic squares of order 15.
    • The first is a double-digit bordered magic square of order 15 embedded with double-digit bordered magic square of orders 11 and 7 with magic square of order 3 in the middle. It contains 4 equal sums magic rectangles of order 2×11, 2×7 and 2×3 in each case.
    • The second is also a double-digit bordered magic square of order 15 with double-digit bordered magic square of orders 11 and 7 with magic square of order 3 in the middle. The four magic rectangles of order 2×13, 2×9 and 2×5 equal sums in each case. Sometimes, this kind of magic squares we call as cyclic-type double-digit bordered magic squares.
    • The third is also a double-digit bordered magic square of order 15 embedded with double-digit bordered magic square of order 11 and 7. It contains 2 magic rectangles of orders 2×15, and 4 or orders 2×11, and four of oders 2×7 and 2 of orders 2×3. In each case they are equal sums. Also there is a magic square of order 3 in the middle. This kind of magic square sometimes we call as flat-type-double-digit bordered magic square..
    • The forth is a cornered magic square of order 15 having again cornered magic squares orders 13, 11, 9, 7 and 5 at the upper-left corner with magic square of order 3 at the upper-left corner. The two magic rectangles of orders 2×3, 2×5, 2×7, 2×9 and 2×11 are of equal sums in each case.
    • The fifth is again a composed of 9 different sums single-digit bordered magic squares of order 5 having magic squares of order 3 in the middle of each.
    • The sixth is a single-digit bordered magic square of order 15 with magic square of order 3 in the middle.
• 8^th Border: Different sums magic squares of order 6 and Bordered Magic Rectangle of Order 6×24
  • In this case we have considered 4 different types of magic squares of order 6 and one borderd magic rectangle of order 6×26.
    • The first is a magic square is a normal magic square of order 6 without any special properties.
    • The second is a cornered magic square of order 6, where there is a pandiagonal magic square of order 4 at the upper left corner. The two magic rectangles of order 2×4 are of equal sums.
    • The third is a striped magic square of order order 6. It composed of a magic rectangle of order 2×6. There are also three equal sums magic rectangles of order 2×4. When a magic square composed of only magic rectangles of equal width and different or same lengths, we call these kind of magic squares as striped magic squares.
    • The forth is a single-layer bordered magic square of order 6 embedded with a pandiagonal magic square of order 4 in the middle.
    • The last one a bordered magic rectangle of order 6×24.

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.
• 6^th Border:Different sums magic squares of order 9
  • In this case we have considered 6 different types of magic squares of order 9. Combining with 540 magic squares of order 72 we have 3240-different types of magic squares of order 90.
• 7^th Border:Different sums magic squares of order 15
  • In this case we have considered again 6 different types of magic squares of order 9. Combining with 3240*6=19440 magic squares of order 90. Instead of 19440 we have obtained only 3888 magic square of order 90. In this we have considered only 540 magic square of order 90. This lead us to 540*6=3240 multiple order bordered magic squares of order 120.
• 8^th Equal and different sums magic squares of order 6 and bordered magic rectangle of order 6×24
  • In this case we have considered again 4 different types of magic squares of order 6 and one bordered magic rectangle of order 6×24. Combining with 3240*6=19440 magic squares of order 132. Instead of 19440 we have obtained only 3240 magic square of order 132. In this we have considered only 648 magic square of order 120. This lead us to 648*5=3240 multiple order bordered magic squares of order 132.

For the previous results on multiple bordered magic squares of orders 20, 30, 42, 56, 72 and 90 refer to the link:

• Inder J. Taneja, Multiple Orders Bordered Magic Squares, Zenodo,
  • Site link: Beauty of Magic Squares: 3240 Multiple Order Bordered Magic Squares of Order 90.

Magic Squares of Order 120

There are total 3240 magic squares of order 120. This work is for the multiple order bordered magic squares of order 132. Let’s consider the following 4 magic squares of order 6 and one bordered magic rectangle of order 6×24:

Above there are 4 6 magic squares of order 6 and a magic rectangle of order 6×26. Below are the details

1. The first is a magic square is a normal magic square of order 6 without any special properties.
2. The second is a cornered magic square of order 6, where there is a pandiagonal magic square of order 4 at the upper left corner. The two magic rectangles of order 2×4 are of equal sums.
3. The third is a striped magic square of order order 6. It composed of a magic rectangle of order 2×6. There are also three equal sums magic rectangles of order 2×4. When a magic square composed of only magic rectangles of equal width and different or same lengths, we call these kind of magic squares as striped magic squares.
4. The forth is a single-layer bordered magic square of order 6 embedded with a pandiagonal magic square of order 4 in the middle.
5. The last one a bordered magic rectangle of order 6×24.

Previouly, we worked with multiple order bordered magic square of order 120. The borderes are of order 3,4,5,6,7, 9 and 15 resulting in 3240 magic squares of order 120. Considering this 4 magic squares of order 6 and a bordered magic rectangle of order 2×24 in different ways, we get 3240*6=19440 magic squares of order 1320. Since this number is too high, we shall conder only 648 magic squares of order 120. This will give us 648*5= 3240 a multiple order bordered magic squares of order 132. See below few examples in figures (without numbers) in each case.

In Figures or Designs

First-Type:

Second-Type:

Third-Type:

Forth-Type:

Fifth-Type:

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.

Multiple Orders Bordered Magic Squares

1. Inder J. Taneja, Beauty of Magic Squares: 540 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: 7128 Multiple Order Bordered Magic Squares of Order 108
   • 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
   • 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
4. 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.
5. 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.
6. Inder J. Taneja, Beauty of Magic Squares: 14256 Multiple Order Bordered Magic Squares of Order 132
   • 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
   • 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
   • 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
   • Beauty of Magic Squares: Multiple Orders Bordered Magic Squares of Order 132 – Part 4, Zenodo,
     • Web-site Link: Beauty of Magic Squares: 3240 Multiple Order Bordered Magic Squares of Order 132 – Part 4
7. Total till up to here: 32292: It includes of orders 72, 90, 108, 110, 120 and 132. Including 14256 of order 144, there wil total of 46548