Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 144 – Part 1

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,

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 resulting in multiple order bordered magic squares of order 90. Considering again a border of order 9 we get multiple order bordered magic squares of order 108. Further, a border of order 18 is considered to bring the magic square of order 144. In this case we have used 6 different types of magic squares of order 18. Also these are used with different magic sums. Below are more details

See below the detailed structure of magic square of order 144. The changes are with different magic squarez and different styles and also different order borders. In some case they are of equal sums and sometimes different sums.

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 9
  • In this case we have considered 6 different types of magic squares of order 9.
    • The first is a single-digit bordered magic square of order 9 embedded with a double-digit bordered magic square of order 7. It contains 4 equal sums magic rectangles of order 2×3 with magic square of order 3 in the middle.
    • The second is again a single-digit bordered magic square of order 9 embedded with a cornered magic square of order 7. It contains again a cornered magic square of order 5 in the upper-left corner. The magic rectangles of order 2×3 and of order 2×5 are of equal sums in each case.
    • The third is again a single-digit bordered magic square of order 9 embedded with a cyclic double-digit bordered magic square of order 7. It contains 4 equal sums magic rectangles of order 2×5 with magic square of order 3 in the middle.
    • The forth is again a cornered magic square of order 9 having a single-digit bordered magic square of order 7 embedded with cornered magic square of order 5. The magic rectangles of orders 2×3 and 2×7 are of equal sums in each case.
    • The fifth is again a cornered magic square of order 9 having a double digit bordered magic square of order 7 in the upper-left corner. The four magic rectangles of orders 2×3 are of equal sums. Also the two magic rectangles of orders 2×7 are of equal sums.
    • The sixth is a corner-type magic square of order 9 formed by 2 equal sums magic recangles of orders 2×9 and 2 equal sums magic rectangles of order 2×5 having pandiagonal magic square of order 5 in upper-left corner. It is modified form of the fifth-type magic square.
• 9^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 a composed of 9 equal sums magic squares of order 6. Out of them five are single-layer magic square with pandigonal magic square of order 4 in the inner parts. The other four magic squares of order 6 are corner-type magic square with pandiagonal magic square of order 4 in the corner. There are total 9 equal-sums pandiagonal magic squares of order 4.
    • The second magic square of order 18 is composed of external border of order 4 and internal as magic square of order 10. The external border of order 4 is composed of 4 equal-sums pandiagonal sums magic squares of order 4 and 4 bordered magic rectangles of order 4×10. The inner part is a magic square of order 10 formed by single-layer magic square of order 10 with 4 equal-sums pandiagonal magic squares of order 4.
    • The third magic square is order 18 is composed of external border of order 3 formed different sums magic squares of order 3. The internal part is a magic square of order 12. It is again formed by bordered magic rectangle of order 10×12 and two single-rows of order 1×12.
    • The forth magic square of order 18 is composed of external border of order 4 and internal magic square of ordr 10. The external bordered is cyclic-type 4 equal sums magic rectangles of order 4×12. The internal part a single-layer bordered magic square of order 10. It has a pandiagonal magic square of order 4 in the middle.
    • The fifth is composed of double-layer cyclic-type 4 equal sums magic rectangles of order 2×16. as an external border. The internal part is a magic square of order 16 formed by two magic squares of orders 6 and 8 and two single-layer bordered magic rectanlges of order 6×10.
    • The sixth is a single-layer bordered magic square of order 18 with a pandigaonal magic square of order 12 in the middle.
      

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 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 9
  • 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 648*6=3888 multiple order bordered magic squares of order 108.
• 8^th Border:Different sums magic squares of order 18
  • In this case we have considered again 6 different types of magic squares of order 18. Combining with 3240*6=19440 magic squares of order 108. Instead of considering 19440 we have obtained only 3888 magic square of order 144. In this we have considered only 648 magic square of order 108. This lead us to 648*6=3888 multiple order bordered magic squares of order 144.

The complete work with excel files for download see the link below. For other orders see the reference list given at the end of this work.

• Inder J. Taneja, Beauty of Magic Squares: 3888-Multiple Orders Bordered Magic Squares of Order 144 – Part 1, Zenodo,

Magic Squares of Order 144

There are total 3240 magic squares of order 108 studied previously. This work is for the multiple order bordered 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 the details.

1. The first is a composed of 9 equal sums magic squares of order 6. Out of them five are single-layer magic square with pandigonal magic square of order 4 in the inner parts. The other four magic squares of order 6 are corner-type magic square with pandiagonal magic square of order 4 in the corner. There are total 9 equal-sums pandiagonal magic squares of order 4.
2. The second magic square of order 18 is composed of external border of order 4 and internal as magic square of order 10. The external border of order 4 is composed of 4 equal-sums pandiagonal sums magic squares of order 4 and 4 bordered magic rectangles of order 4×10. The inner part is a magic square of order 10 formed by single-layer magic square of order 10 with 4 equal-sums pandiagonal magic squares of order 4.
3. The third magic square is order 18 is composed of external border of order 3 formed different sums magic squares of order 3. The internal part is a magic square of order 12. It is again formed by bordered magic rectangle of order 10×12 and two single-rows of order 1×12.
4. The forth magic square of order 18 is composed of external border of order 4 and internal magic square of ordr 10. The external bordered is cyclic-type 4 equal sums magic rectangles of order 4×12. The internal part a single-layer bordered magic square of order 10. It has a pandiagonal magic square of order 4 in the middle.
5. The fifth is composed of double-layer cyclic-type 4 equal sums magic rectangles of order 2×16. as an external border. The internal part is a magic square of order 16 formed by two magic squares of orders 6 and 8 and two single-layer bordered magic rectanlges of order 6×10.
6. The sixth is a single-layer bordered magic square of order 18 with a pandigaonal magic square of order 12 in the middle.

Previouly, we worked with multiple order bordered magic square of order 108. There are total 3888 magic square of order 144 with different styles and orders of magic squares. Considering this 6 magic squares of order 18 in 6 different ways, we get 33240*6=19440 magic squares of order 108. Since this number is too high, we shall conder only 648 magic squares of order 108. This will give us 648*6= 3888 magic squares of order 144. See below few examples in figures (without numbers) in each case. The excel file with all magic squares of order 144 is given a work at Zenodo.

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.

Mixed Orders 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.