Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 144 – Part 3

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.

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, 8, 9 and 15 resulting in multiple order bordered magic squares of order 120. Consdering again a magic square of order 12 as a border, this lead us to magic square of order 144. Below are some details of magic squares of order 144.

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

See below the details of above multiple order bordered magic squares. The bordered applied 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 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 15
- In this case we have considered 6 different types of magic squares of order 15.
  - The first is a double-digit bordered magic square of order 15 embedded with double-digit bordered magic square of orders 11 and 7 with magic square of order 3 in the middle. It contains 4 equal sums magic rectangles of order 2×11, 2×7 and 2×3 in each case.
  - The second is also a double-digit bordered magic square of order 15 with double-digit bordered magic square of orders 11 and 7 with magic square of order 3 in the middle. The four magic rectangles of order 2×13, 2×9 and 2×5 equal sums in each case. Sometimes, this kind of magic squares we call as cyclic-type double-digit bordered magic squares.
  - The third is also a double-digit bordered magic square of order 15 embedded with double-digit bordered magic square of order 11 and 7. It contains 2 magic rectangles of orders 2×15, and 4 or orders 2×11, and four of oders 2×7 and 2 of orders 2×3. In each case they are equal sums. Also there is a magic square of order 3 in the middle. This kind of magic square sometimes we call as flat-type-double-digit bordered magic square..
  - The forth is a cornered magic square of order 15 having again cornered magic squares orders 13, 11, 9, 7 and 5 at the upper-left corner with magic square of order 3 at the upper-left corner. The two magic rectangles of orders 2×3, 2×5, 2×7, 2×9 and 2×11 are of equal sums in each case.
  - The fifth is again a composed of 9 different sums single-digit bordered magic squares of order 5 having magic squares of order 3 in the middle of each.
  - The sixth is a single-digit bordered magic square of order 15 with magic square of order 3 in the middle.
8^th Border: Different sums magic squares of order 12
- In this case we have considered 6 different types of magic squares of order 12
  - The first magic square of order 12 is composed 4 equal sums magic squares of order 6. The are corner-type magic squares of order 6 have a pandiagonal magic square of order 4 in the corner.
  - The second magic square of order 12 is a cornered 2 magic rectangles of order 2×12 and 2 of order 2×8. The innner part is a magic square of order 8.
  - The third magic square of order 12 is similar to previous one. The only change is in the internal magic square of order 12. It is different from the one given in second type.
  - The forth magic square of order 12 is composed of 2 equal sums single-layer magic rectangles of order 6×12.
  - The fifth magic square of order 12 composed of 2 equal sums magic rectangles of order 6×8 and 2 magic rectangles of order 4×6.
  - The sixth magic square of order 12 is composed of single-layer magic square of order 12 embedded with a magic square of order 10. This magic square of order 10 is again composed two single-layer magic rectangles of order 4×10 and one magic rectangle of order 2×10.

Summarizing there are multiple order bordered magic square of order 132 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 15
- 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 540*6=3240 multiple order bordered magic squares of order 120.
8^th Border: Different sums magic squares of order 12
- In this case we have considered again 6 different types of magic squares of order 12 3888*6=23328 magic squares of order 144. Instead of 23328 we have obtained only 3888 magic square of order 144. In this we have considered only 648 magic square of order 120. This lead us to 648*6=3888 multiple order bordered magic squares of order 1442.

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,
- Site link: Beauty of Magic Squares: 3240 Multiple Order Bordered Magic Squares of Order 90.

Magic Squares of Order 144

This work is for the multiple order bordered magic squares of order 144. Let’s consider the following 6 magic squares of order 12.

Above there are 6 magic squares of order 12. The details are as follows:

The first magic square of order 12 is composed 4 equal sums magic squares of order 6. The are corner-type magic squares of order 6 have a pandiagonal magic square of order 4 in the corner.
The second magic square of order 12 is a cornered 2 magic rectangles of order 2×12 and 2 of order 2×8. The innner part is a magic square of order 8.
The third magic square of order 12 is similar to previous one. The only change is in the internal magic square of order 12. It is different from the one given in second type.
The forth magic square of order 12 is composed of 2 equal sums single-layer magic rectangles of order 6×12.
The fifth magic square of order 12 composed of 2 equal sums magic rectangles of order 6×8 and 2 magic rectangles of order 4×6.
The sixth magic square of order 12 is composed of single-layer magic square of order 12 embedded with a magic square of order 10. This magic square of order 10 is again composed two single-layer magic rectangles of order 4×10 and one magic rectangle of order 2×10.

Below are few examples of magic squares of order 144. These are only figures (without numbers). The complete excel files are given author’s work in Zenodo. Reference is given below in reference list.