Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 132

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 multiple order 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 upper borders are magic squares of orders 4, 5, 6, 7, 8, 9 and 10. 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 figure and details:

The above figure is up to order 110. See below the details of above multiple order bordered magic squares of order 110.

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 10
- In this case we have considered 6 different types of magic squares of order 10.
  - First one is a tradicional square of order 10.
  - The second one is a cornered magic square of order 10, where magic squares of orders 8 and 6 are also cornered magic squares at the upper-left corner having magic square of order 4.
  - The third one is double-digit bordered magic square of order 10 having single-digit magic square of order 6 in the middle. Four magic rectangles of order 2×8 are of equal sums magic sums. This kind of magic square we call as cyclic-type double digit bordered magic square of order 10.
  - The forth one is again a double-digit bordered with cornered magic squares of order 6 in the middle. It contains magic square of order 4 at the upper-left corner. The external border is composed of 2 magic rectangles of order 2×10 and two magic rectangles of order 2×6. Both are of equal sums in each case.
  - The fifth one is again a single-digit bordered magic square of order 10 embedded with a striped magic square of order 8 composed 4 equal sums magic squares of order 4 or 8 equal sums magic rectangles of order 2×4.
  - The sixth is a normal single-digit bordered magic square of order 10 having magic square of order 4 in the middle.
8^th Border: Different Sums Magic Squares of Order 11
- In this case we have considered 6 different types of magic squares of order 10.
  - First one is a double-layer bordered magic square of order 11 with magic square of order 3 is in the middle. The 4 magic rectangles of orders 2×3 and 4 magic rectangles of order 2×7 are of equal magic sums in each case.
  - The second one is a cornered magic square of order 11, where magic squares of orders 9, 7 and 5 are also cornered at the upper-left corner with magic square of order 3. The 2 magic rectangles of orders 2×3, 2 of order 2×5, 2 of order 2×7 and 2 of order 2×9 are of equal sums in each case.
  - The third one is again a double-layer bordered magic squares of orders 11 and 7 having magic square of order 3 in the middle. In this case the magic rectangles 2 of orders 2×11, 4 of order 2×7 and 2 of order 2×3 are of equal magic sums in each case. This types of magic squares sometimes we call as flat-double-layer bordered magic squares.
  - The forth is a cyclic-type-ouble-layer bordered with magic square of order 3 in the middle. The 4 magic rectangles of order 2×9 and 4 of order 2×5 are of equal sums in each case.
  - The fifth is a single-layer bordered magic square of order 11 embedded with a magic square of order 9 composed of 9 equal sums magic squares of order 3.
  - The sixth is a single-layer bordered magic square of order 11 having magic square of order 3 in the middle.

Summarizing there are multiple order bordered magic square of order 110 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 10
- In this case we have considered 6 different types of magic squares of order 10. Considering this 6 magic squares of order 10 in 6 different ways, we get 3240*6=19440 magic squares of order 110. 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 110 of multiple order bordered magic squares of order 110.
8^th Border: Different Sums Magic Squares of Order 11
- In this case we have considered 6 different types of magic squares of order 11. Considering this 6 magic squares of order 10 in 6 different ways, we get 3888*6=23328 magic squares of order 110. Since this number is too high, we shall conder only 648 magic squares of order 110. This will give us 648*6= 3888 magic squares of order 132 of multiple order bordered magic squares of order 132.

For the previous results on multiple bordered magic squares of orders 20, 30, 42, 56, 72, 90 and 110 refer to the link:

Inder J. Taneja, Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 110, Zenodo,
- Web-site Link: Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 110.

Magic Squares of Order 132

There are total 3880 magic squares of order 110 studied previously. This work is for the multiple order bordered magic squares of order 132 based on magic squares of order 11. Let’s consider the following 6 magic squares of order 11

Above there are 6 magic squares of order 11 with different styles. See below the details.

First one is a double-layer bordered magic square of order 11 with magic square of order 3 is in the middle. The 4 magic rectangles of orders 2×3 and 4 magic rectangles of order 2×7 are of equal magic sums in each case.
The second one is a cornered magic square of order 11, where magic squares of orders 9, 7 and 5 are also cornered at the upper-left corner with magic square of order 3. The 2 magic rectangles of orders 2×3, 2 of order 2×5, 2 of order 2×7 and 2 of order 2×9 are of equal sums in each case.
The third one is again a double-layer bordered magic squares of orders 11 and 7 having magic square of order 3 in the middle. In this case the magic rectangles 2 of orders 2×11, 4 of order 2×7 and 2 of order 2×3 are of equal magic sums in each case. This types of magic squares sometimes we call as flat-double-layer bordered magic squares.
The forth is a cyclic-type-ouble-layer bordered with magic square of order 3 in the middle. The 4 magic rectangles of order 2×9 and 4 of order 2×5 are of equal sums in each case.
The fifth is a single-layer bordered magic square of order 11 embedded with a magic square of order 9 composed of 9 equal sums magic squares of order 3.
The sixth is a single-layer bordered magic square of order 11 having magic square of order 3 in the middle.

Previouly, we worked with multiple order bordered magic square of order 110. The borderes are of order 3,4,5,6,7, 9 and 10 resulting in 3880 magic squares of order 110. Considering this 6 magic squares of order 11 in 6 different ways, we get 3888*6=23328 magic squares of order 132. Since this number is too high, we shall conder only 648 magic squares of order 110. This will give us 648*6= 3888 magic squares of order 132 of multiple order bordered magic squares of order 132. See below few examples in figures (without numbers) in each case. The complete list of 3888 magic squares of order 132 is given in a work appearing zenodo. See the link above