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.

In previous work there are also 3240 multiple order bordered magic squares of order 108. It is done by considering double border of magic squares of order 9 ovedr magic square of order 72. In this work instead of double border of order 9, we have considered a border of magic squares of order 18 obrt the magic squares of order 72. See the figure above where the upper border is of magic square of order 18.

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, 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.

See below the details of above multiple order bordered magic squares of order 108:

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 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.

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 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.

For the results on multiple bordered magic squares of orders 12-70, 90, 108 and 110 refer to following 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 540 magic squares of order 72 studied previously. This work is for the multiple order bordered magic squares of order 108. Let’s consider the following 6 magic squares of order 10:

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

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.

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.