Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 108

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.

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 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. Instead order 9 if we consider a border of order 10, we get multiple order bordered magic squares of order 110. This is done in another link. See the references below. The even order borders are with magic squares, such as of orders 4, 6, 8 and 10 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 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 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 cyclic double-digit bordered magic square of order 9 having magic square of order 5 in the middle and 4 equal sums magic rectangles of order 2×7.
  - The forth is again double-digit bordered magic square of order 9 having a pandiagonal magic square of order 5 in the middle. The external border is formed by two equal sums magic rectangles of order 2×9 and another two equal sums magic rectangles of order 2×5. Sometimes we call it as flat-double-digit bordered magic square of order 9.
  - 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.

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 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 648 magic square of order 90. This lead us to 648*9=3888 multiple order bordered magic squares of order 108.

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, Jun 9, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.8019330.
- Site link: Beauty of Magic Squares: 3240 Multiple Order Bordered Magic Squares of Order 90

Magic Squares of Order 108

There are total 3240 magic squares of order 90 studied previously. This work is for the multiple order bordered magic squares of order 110. Let’s consider the following 6 magic squares of order 10:

Above there are 6 magic squares of order 9 with different styles. See below the details.

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 cyclic double-digit bordered magic square of order 9 having magic square of order 5 in the middle and 4 equal sums magic rectangles of order 2×7.
The forth is again double-digit bordered magic square of order 9 having a pandiagonal magic square of order 5 in the middle. The external border is formed by two equal sums magic rectangles of order 2×9 and another two equal sums magic rectangles of order 2×5. Sometimes we call it as flat-double-digit bordered magic square of order 9.
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.

Previouly, we worked with multiple order bordered magic square of order 90. The borderes are of order 3,4,5,6,7, 9 resulting in 3240 magic squares of order 90. Considering this 6 magic squares of order 9 in 6 different ways, we get 3240*6=19440 magic squares of order 108. Since this number is too high, we shall conder only 648 magic squares of order 90. This will give us 648*6= 3888 magic squares of order 108 of multiple order bordered magic squares of order 108. See below few examples in figures (without numbers) in each case.