https:\/\/doi.org\/10.5281\/zenodo.8019330<\/a>.<\/li>\n<\/ul>\n\n\n\n<\/p>\n\n\n\n
Striped Magic Square of Order 12<\/h3>\n\n\n\n <\/p>\n\n\n\n
Below is a striped <\/strong>magic square of order 12. <\/p>\n\n\n\nLet’s consider first the following striped magic square of order 4:<\/p>\n\n\n
\n
<\/p>\n\n\n\n
It is magic square of order 4 consisting to two equal sums magic rectangles of order 2×4. Based on this magic square, below is a striped magic square of order 12. <\/p>\n\n\n
\n
<\/p>\n\n\n\n
Striped Magic Squares of Order 24<\/h3>\n\n\n\n We observe that the above magic square of order 12 is divisible by 6. Based this idea we shall bring magic square of order 24 considering the futher border of order 6. It is based on the following two variations of striped magic squares of order 6.<\/p>\n\n\n\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
These are two striped<\/strong> magic squares of order 6, written in different forms. As we have seen that the magic square of order 12 is constructed based on 9 equal sums striped<\/strong> magic squares of order 4. But in this case, the situation is different. In this the striped<\/strong> magic squares of order 24 are constructed with external border of differendt sums striped<\/strong> magic squares of order 6. See below:<\/p>\n\n\n\n<\/p>\n\n\n\n
Striped Magic Squares of Order 24.<\/h4>\n\n\n\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
By considering a little different way of writing striped<\/strong> magic square of order 12, we have the same striped<\/strong> magic squares of order 24 given as below:<\/p>\n\n\n\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
For further work, we shall consider only the first two striped<\/strong> magic squares of order 24. In future also, we can construct striped<\/strong> magic square of order 24 using 9 equal sums striped<\/strong> magic squares of order 8.<\/p>\n\n\n\n<\/p>\n\n\n\n
Striped Magic Squares of Order 40<\/h3>\n\n\n\n <\/p>\n\n\n\n
We observe that the above magic squares of order 24 are divisible by 8. Based this idea we shall bring magic square of order 40 considering the futher border of order 8. It is based on the following four variations of striped<\/strong> magic squares of order 8.<\/p>\n\n\n\n
<\/p>\n\n\n
\n
These are four striped<\/strong> magic squares of order 8, written in different forms. There are much more striped magic squares of order 8 given in the reference list above, but we have considered only four just to have an idea. Let’s see below the 8 striped magic squares of order 40 based on an external border of order 8 over the two striped magic squares of order 24. In this case, not all the magic squares for border 8 are of equal sums. <\/p>\n\n\n\n<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
Magic Squares of Order 60<\/h3>\n\n\n\n <\/p>\n\n\n\n
We observe that the above magic squares of order 40 are divisible by 5, 8 and 10. Based this idea we shall bring magic square of order 60 considering the futher border of order 10. It is based on the following 6 striped<\/strong> magic squares of order 10.<\/p>\n\n\n\n<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
These are six striped<\/strong> magic squares of order 10, written in different forms. There are much more striped magic squares of order 10 given in the reference list above, but we have considered only six just to have an idea. Let’s see below the 48 striped magic squares of order 60 based on external borders of order 10 over the 8 striped magic squares of order 40. Since the order 60 is a big, don’t fit properly in the screen, we shall represent these only with figure: <\/p>\n\n\n\nIn Figures:<\/h4>\n\n\n\n <\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
Above there are only few examples. More examples with excel file can be seen in a work given above in a link<\/p>\n\n\n\n
<\/p>\n\n\n\n
Magic Squares of Order 84<\/h3>\n\n\n\n <\/p>\n\n\n\n
Let’s consider the following 5 magic squares of order 7:<\/p>\n\n\n\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
Above there are five magic squares of order 7. See below the details<\/p>\n\n\n\n
\nThe first one is a pandiagonal<\/strong> magic square of order 7.<\/li>\n\n\n\nThe second one is double-digit bordered<\/strong> 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. <\/li>\n\n\n\nThe third one is a cornered<\/strong> 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.<\/li>\n\n\n\nThe forth one is also a double-digit bordered<\/strong> 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<\/strong> magic square of order 7.<\/li>\n\n\n\nThe fifth one is single-digit bordered magic square of order 7, where there is a magic square of order 3 in the middle. <\/li>\n<\/ol>\n\n\n\nAbove there are 18 magic squares of order 42 and combining with these five possibilities, we have total 90 magic squares of order 56. Below there are only few possibilities are given in figures. The complete 90 magic squares of order 56 in numbers is given in excel file attached at the end of this work:<\/p>\n\n\n\n
<\/p>\n\n\n\n
In Figures or Designs:<\/h4>\n\n\n\nFirst-Type<\/h5>\n\n\n\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
Second-Type<\/h5>\n\n\n\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
Third-Type<\/h5>\n\n\n\n <\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
Forth-Type<\/h5>\n\n\n\n <\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
Fifth-Type<\/h5>\n\n\n\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
\nSee in the end of this work excel file<\/strong> of above magic squares in numbers.<\/em><\/li>\n<\/ul>\n\n\n\n<\/p>\n\n\n\n
Magic Squares of Order 72<\/h3>\n\n\n\n Let’s consider the following 5 magic squares of order 8:<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
About there are 6 magic squares of order 8. See below the details:<\/p>\n\n\n\n
\nThe first one is a pandiagonal<\/strong> magic square of order 8 formed by four equal sums pandiagonal <\/strong>magic squares of order 4.<\/li>\n\n\n\nThe second one is a cornered<\/strong> 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. <\/li>\n\n\n\nThe third one is a double-digit bordered<\/strong> 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. <\/li>\n\n\n\nThe forth one is also a double-digit bordered <\/strong>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.<\/strong><\/li>\n\n\n\nThe 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.<\/strong><\/li>\n\n\n\nThe sixth one is single-digit bordered magic square of order 8 having a pandiagonal <\/strong>magic square of order 4 in the middle.<\/li>\n<\/ol>\n\n\n\n<\/p>\n\n\n\n
Since there are 90 magic squares of order 56. Making combinations with 6 external border of order 8 we have toal 540 magic squares of order 72. Below are few examples in figures. The total 540 magic squares of order 72 are given in the end as an excel file for download.<\/p>\n\n\n\n
In Figures or Designs:<\/h4>\n\n\n\nFirst-Type<\/h5>\n\n\n\n <\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
Second-Type<\/h5>\n\n\n\n <\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
Third-Type<\/h5>\n\n\n\n <\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n\n
Forth-Type<\/h5>\n\n\n\n <\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
<\/p>\n\n\n
\n
![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-4x4-1.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-12x12-1.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-6x6-1-2.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-24x24-1.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-24x24-2.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-24x24-3.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-24x24-4.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-8x8-1-2.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-8x8-3-4.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-40x40-1-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-40x40-2-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-40x40-3-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-40x40-4-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-40x40-5-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-40x40-6-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-40x40-7-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-40x40-8-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-10x10-1-2.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-10x10-3-4.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/ms-10x10-5-6.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-1-2-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-3-4-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-5-6-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-7-8-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-9-10-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-11-12-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-13-14-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-15-16-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-17-18-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-19-20-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-21-22-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2026\/05\/f-60x60-23-24-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/ms-7x7-1.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/ms-7x7-2.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-1-2-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-5-6-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-17-18-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-21-22-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-25-26-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-31-32-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-35-36-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-39-40-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-43-44-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-47-48-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-53-54-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-55-56-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-61-62-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-67-68-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-71-72-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-73-74-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-79-80-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-79-80-1-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-83-84-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-56x56-87-88-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/ms-8x8-1.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/ms-8x8-2.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-1-2-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-5-6-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-9-10-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-13-14-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-17-18-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-105-106-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-109-110-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-113-114-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-117-118-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-121-122-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-209-210-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-213-214-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-217-218-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-221-222-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-313-314-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-317-318-scaled.png\")
<\/figure>\n<\/div>\n\n\n![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/06\/f-72x72-321-322-scaled.png\")
<\/figure>\n<\/div>\n\n\n