Beauty of Magic Squares: 3888 Multiple Order Bordered Magic Squares of Order 110

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, 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:

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 there are two ways. One is pandiagonal magic squares of order 4. The second one is a magic square of order 4 composed of two equal sums magic rectangles of order 2×4.
2^nd Border: Different sums magic squares of order 5.
- In this case there are two ways. One is pandiagonal magic squares of order 5. The second is bordered magic square of order 3 with inner part as a magic square of order 3. In both the cases the magic sums of order 5 are different.
3^rd Border: Equal sums magic squares of order 6.
- In this case there are two ways. One is magic squares of order 6. The second is bordered magic square of order 6 with inner part as a pandiagonal magic square of order 4.
4^th Border: Different sums magic squares of order 7.
- In this case also there are two ways. One is pandiagonal magic squares of order 7. The second one is bordered magic square of order 7 with inner parts as a magic square of orders 5 and 3. In both the cases the magic sums of order 7 are different.
5^th Border: Equal sums magic squares of order 8.
- In this case there are three ways. One is pandiagonal magic squares of order 8 with equal sums pandiagonal magic squares of order 4. The second one is flat-type magic square of order 8 with inner part as a magic square of order 4, and the magic rectangles are of orders 2×8 and 2×4. The third one is cyclic-type double-digit magic squares of order 8 with inner part as a magic square of order 4 with four magic rectangles of orders 2×6 are of equal sums.
6^th Border: Different sums magic squares of order 9.
- In this case also there are three ways. One is equal sum semi-magic squares of 3 resulting in magic square of order 9. The second one is bordered magic squares of order 9 having magic squares of orders 7, 5 and 3 in the middle as magic square of order 3, while the orders 7 and 5 again bordered magic squares.
7^th Border: Equal sums magic squares of order 10.
- In this case also there are three ways. One is equal sum semi-magic squares of 3 resulting in magic square of order 9. The second one is bordered magic squares of order 9 having magic squares of orders 7, 5 and 3 in the middle as magic square of order 3, while the orders 7 and 5 again bordered magic squares.

For the previous results on multiple bordered magic squares of orders 20, 30, 42, 56 and 72 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: Multiple Order Bordered Magic Squares of Orders 20, 30, 42, 56 and 72

Magic Squares of Order 110

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.

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.

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 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. See below few examples in figures (without numbers) in each case. The complete list of 3888 magic squares of order 110 is attached at the end of this work.