Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 144 – Part 2

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:

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, and 8 resulting in multiple order bordered magic squares of order 72. Considering again a border of magic squares of order 18, we get multiple order bordered magic squares of order 108. The even order borders are with magic squares, such as of orders 4, 6, 8 and 18 are with equal sums magic squares. The odd order borders of orders 5 and 7 are with different sums magic squares. In some cases the magic squares of order 18 are withdifferent sums magic squares. This lead us to a magic square of order 18. Further considereing again magic square of order 18, lead us to a magic square of order 144.

See below the details of abovemultiple order bordered magic squares of order 155:

The details of bordered magic squares are 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.
  • 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 18
  • In this case we have considered 6 different types of magic squares of order 18.
    • The first is composed of two types of 6 magic squares of order 6. One is single-digit bordered magic square of order 6 having a pandiagonal magic square of order 4 in the middle. The second type is a cornered magic square of order 6 with two equal sums magic rectangles of order 2×4 and a pandigonal magic square of order 4 in the upper-left corner. These are considered in four corners with four different directions resulting a beautiful symmetric magic square of order 18.
    • The second is single digit bordered magic square of order 18 embedded with four equal sums magic squares of order 8 having pandiagonal magic square of order 4 in the middle of 4.
    • The third is cyclic-type magic square of order 10 composed of four equal sums bordered magic rectangles of order 6×12 embedded with a single-digit bordered magic squares of order 6 having a pandiagonal magic square of order 4 in the middle,
    • The forth is triple-digit bordered magic square of order 18. The external boarder is composed of different sums magic squares of order 3. The internal part is a magic square of order 12 composed of two equal sums bordered magic rectangles of order 6×12.
    • The fifth is a double-digit bordered magic squares of order 18 bordered with four equal sums magic rectangles of order 2×16 embedded with a magic square of order 14. It is composed two equal sums single-digit bordered magic rectangles of order 6×8 and two equal sums single-digit bordered magic rectangles of order 4×6.
    • The sixth is again a double-digit bordered magic squares of order 18 bordered with four equal sums magic rectangles of order 2×16 embedded with a magic square of order 14. It is composed four equal sums single-digit bordered magic rectangles of order 4×10 embedded with single-digit magic square of order 6 having a pandiagonal magic square of order 4 in the middle.
• 7^th Border: Different sums magic squares of order 18
  • In this case we have considered 6 different types of magic squares of order 18.
    • The first magic square of order 18 is composed of four single-layer bordered magic rectangles of orders 6×18, and three 4×18. The three 4×18 are of equal sums.
    • The second magic square of 18 is composed of one single-layer bordered magic rectangles of order 6×18 and 9 equal sums magic rectangles of order 4×6.
    • The third magic square of order 18 is composed of 1 bordered magic rectangle of order 6×18 and one bordered magic rectangle of order 4×18. There is one more single-layer bordered magic rectangle of order 8×10 and one single-layer bordered magic square of order 8.
    • The forth magic rectangles of order 18 is composed of three single-layer magic rectangles. Two of them are of equal sums of order 4×18. The third one is also sinlge-layer bordred of order 10×18.
    • The fifth magic square of order 18 is composed of two single-layer bordered magic rectangles of order 4×18 and 2 of order 4×10. The internal part is a magic square of order 10. It is complsed of single-layer bordered magic square of order 10 with four equal sums pandiagonal magic squares of order 4.
    • The sixth magic square of order 18 is very much similar to third. The difference is that here we have two equal sums sinlge-layer magic rectangles of order 4×18. In the middle there is magic square of order 10×10 and a single-layer magic rectangle of order 8×10. The magic square of order 10 is also a single-layer bordered magic square of order 10.

Summarizing there are multiple order bordered magic square of order 144 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 18
  • In this case we have considered 6 different types of magic squares of order 18. Combining with 540 magic squares of order 72 we have 3240-different types of magic squares of order 108.
• 7^th Border:Different sums magic squares of order 18
  • In this case we have considered 6 different types of magic squares of order 18. Combining with 6400 magic squares of order 108 we have 3888-different types of magic squares of order 144.

This work is also available at the following link with excel files for download. For other others see the references in the end of this work.

• Inder J. Taneja, Beauty of Magic Squares: Multiple Orders Bordered Magic Squares of Order 144 – Part 2, Zenodo,

Magic Squares of Order 144

Let’s consider the following 6 magic squares of order 18:

Above there are 6 magic squares of order 18 with different styles. See below are the details.

1. The first magic square of order 18 is composed of four single-layer bordered magic rectangles of orders 6×18, and three 4×18. The three 4×18 are of equal sums.
2. The second magic square of 18 is composed of one single-layer bordered magic rectangles of order 6×18 and 9 equal sums magic rectangles of order 4×6.
3. The third magic square of order 18 is composed of 1 bordered magic rectangle of order 6×18 and one bordered magic rectangle of order 4×18. There is one more single-layer bordered magic rectangle of order 8×10 and one single-layer bordered magic square of order 8.
4. The forth magic rectangles of order 18 is composed of three single-layer magic rectangles. Two of them are of equal sums of order 4×18. The third one is also sinlge-layer bordred of order 10×18.
5. The fifth magic square of order 18 is composed of two single-layer bordered magic rectangles of order 4×18 and 2 of order 4×10. The internal part is a magic square of order 10. It is complsed of single-layer bordered magic square of order 10 with four equal sums pandiagonal magic squares of order 4.
6. The sixth magic square of order 18 is very much similar to third. The difference is that here we have two equal sums sinlge-layer magic rectangles of order 4×18. In the middle there is magic square of order 10×10 and a single-layer magic rectangle of order 8×10. The magic square of order 10 is also a single-layer bordered magic square of order 10.

Previouly, we worked with multiple order bordered magic square of order 1080. The borderes are of order 3,4,5,6,7, 18 resulting in 3888 magic squares of order 1080. Considering this 6 magic squares of order 18 in 6 different ways, we get 3888*6=23328 magic squares of order 108. Since this number is too high, we shall conder only 648 magic squares of order 108 to construct magic squares of order 144.. This will give us 648*6= 3888 multiple order bordered magic squares of order 144. See below few examples in figures (without numbers) in each case. The full excel file can be found in author’s Zenodo work given in references below.

In Figures or Designs

First-Type:

Second-Type:

Third-Type:

Forth-Type:

Fifth-Type:

Sixth-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, April 14, 2026, pp. 1-75, https://doi.org/10.5281/zenodo.19573409.
   • 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, April 14, 2026, pp. 1-50, https://doi.org/10.5281/zenodo.19571319.
   • 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, April 14, 2026, pp. 1-48, https://doi.org/10.5281/zenodo.19571287.
     • 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, April 14, 2026, pp. 1-52, https://doi.org/110.5281/zenodo.19571709
     • 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, April 14, 2026, pp. 1-48, https://doi.org/10.5281/zenodo.19571838
   • 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, April 14, 2026, pp. 1-54, https://doi.org/10.5281/zenodo.19571923.
   • 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, April 14, 2026, pp. 1-53, https://doi.org/10.5281/zenodo.19572065.
     • 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, April 14, 2026, pp. 1-49, https://doi.org/10.5281/zenodo.19572160.
     • 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, April 14, 2026, pp. 1-55, https://doi.org/110.5281/zenodo.19572664.
     • 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, April 20, 2026, pp. 1-43, https://doi.org/10.5281/zenodo.19572938.
     • Web-site Link: Beauty of Magic Squares: 3240 Multiple Order Bordered Magic Squares of Order 132 – Part 4

Total up to now we have constructed 32292 magic squares. It includes of orders 72, 90, 108, 110, 120 and 132. Still we have to wrok with order 144. Including 14256 of order 144, there shall be 46548 magic squares except striped. It is a seperate study.