https:\/\/doi.org\/10.5281\/zenodo.8019330<\/a>.<\/p>\n\n\n\n<\/p>\n\n\n\n
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 upper borders are magic squares of orders 4, 5, 6, 7, 8 and 9. The even order borders are with magic squares, such as of orders 4, 6 and 8 are with equal sums magic squares<\/strong>. The odd order borders are with magic squares, such as of orders 5, 7 and 9 are with different sums magic squares. <\/strong>See below the details of each order:<\/p>\n\n\n\n<\/p>\n\n\n
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See below the details of above multiple order bordered magic square. <\/p>\n\n\n\n
\n0 Border:<\/strong> Different sums magic squares of order 3.<\/strong>\n\nInitially, we have a magic square of order 12 formed by different sums<\/strong> magic squares of order 3.<\/li>\n<\/ul>\n<\/li>\n\n\n\n1st<\/sup> Border: Equal sums magic squares of order 4. <\/strong>\n\nHere have considered two types of magic squares of order 4. One is pandiagonal <\/strong>magic square of order 4 and another is formed by two equal sums magic sums magic rectangles<\/strong> of order 2×4 resulting in a magic square of order 4.<\/li>\n<\/ul>\n<\/li>\n\n\n\n2nd<\/sup> Border: Different sums magic squares of order 5.<\/strong>\n\nIn 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 <\/strong>of order 2×3 and a magic square of order 3 at the upper-left corner. The third one is single-digit bordered<\/strong> magic square with magic square of order 3 in the middle.<\/li>\n<\/ul>\n<\/li>\n\n\n\n3rd<\/sup> Border: Equal sums magic squares of order 6.<\/strong>\n\nIn 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<\/strong> 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.<\/li>\n<\/ul>\n<\/li>\n\n\n\n4th<\/sup> Border:<\/strong> Different sums magic squares of order 7.<\/strong>\n\nIn this case, we have considered 5 different ways of magic squares of order 7. The first one is a pandiagonal<\/strong> magic square of order 7. The 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. The 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.The 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. The 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<\/ul>\n<\/li>\n\n\n\n5th<\/sup> Border:<\/strong> Equal sums magic squares of order 8.<\/strong>\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. The 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. The 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. The 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>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.<\/strong>The sixth one is single-digit bordered magic square of order 8 having a pandiagonal <\/strong>magic square of order 4 in the middle. This lead us to a multiple order bordered magic square of order 72.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<\/p>\n\n\n\n
Summarizing there are multiple order bordered magic square of order 72 as given below<\/p>\n\n\n\n
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\n0 Border:<\/strong> Different sums magic squares of order 3.<\/strong>\n\nInitially, we have a magic square of order 12 formed by different sums<\/strong> magic squares of order 3.<\/li>\n<\/ul>\n<\/li>\n\n\n\n1st<\/sup> Border: Equal sums magic squares of order 4. <\/strong>\n\nIn 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<\/strong> of magic squares of order 20.<\/li>\n<\/ul>\n<\/li>\n\n\n\n2nd<\/sup> Border: Different sums magic squares of order 5.<\/strong>\n\nIn 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<\/strong> of magic squares of order 20.<\/li>\n<\/ul>\n<\/li>\n\n\n\n3rd<\/sup> Border: Equal sums magic squares of order 6.<\/strong>\n\nIn 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<\/strong> of magic squares of order 42.<\/li>\n<\/ul>\n<\/li>\n\n\n\n4th<\/sup> Border:<\/strong> Different sums magic squares of order 7.<\/strong>\n\nIn 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 <\/strong>of magic squares of order 56.<\/li>\n<\/ul>\n<\/li>\n\n\n\n5th<\/sup> Border:<\/strong> Equal sums magic squares of order 8.<\/strong>\n\nIn this case we have considered 6 different types of magic squares. Combining with 18 magic squares of order 42 we have 540-different types <\/strong>of magic squares of order 72.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<\/p>\n\n\n\n
Magic Square of Order 12<\/h3>\n\n\n\n <\/p>\n\n\n\n
Below is a magic square of order 12. It is with different magic sums magics squares of order 3.<\/p>\n\n\n\n
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Let’s consider the following magic square of order 3:<\/p>\n\n\n
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Based on the above magic square of order 3, we have contructed the following magic square of order 12:<\/p>\n\n\n\n
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Magic Square of Order 20<\/h3>\n\n\n\n Let’s consider the following magic squares of orders 3 and 4:<\/p>\n\n\n\n
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Above there are two different magic squares of order 4. One is pandiagonal<\/strong> magic square and aonther is with two magic rectangles of orders 2×4. There is also a magic square of order 3. Based on above three magic squares there are two different magic square of order 20. The only difference is in magic square of order 4. While the magic square of order 3 remains the same in both the magic squares. See below in figures:<\/p>\n\n\n\n<\/p>\n\n\n\n
In Figures:<\/h4>\n\n\n\n
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\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 30<\/h3>\n\n\n\n <\/p>\n\n\n\n
Let’s consider following three magic squares of order 3:<\/p>\n\n\n
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Above there are three magic squares of order 5. See the details below:<\/p>\n\n\n\n
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\nThe first one is pandiagonal <\/strong>magic square of order 5. <\/li>\n\n\n\nThe second one is cornered magic square of order 5, where there is a magic square of order 3 at the upper-left corner and there are two equal sums magic rectangles of order 2×3. <\/li>\n\n\n\n The third one is traditional bordered magic square of order 5 with magic square of order 3 in the middle. Based on these three magic squares of order 5 there are six magic square of order 30, where the upper borders are magic squares of order 5 formed by above three magic squares of order 5. The inner magic squares of order 20 are the two as given above resulting in 6 magic squares of order 30. See below in figures:<\/li>\n<\/ol>\n\n\n\nIn Figures:<\/h4>\n\n\n\n
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\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 42<\/h3>\n\n\n\n <\/p>\n\n\n\n
Let’s consider the following magic squares of order 6:<\/p>\n\n\n\n
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Above there are three magic squares of order 6. See the details below:<\/p>\n\n\n\n
\nThe first one isa nornal magic square of order 6. <\/li>\n\n\n\n The second one is cornered magic square of order 6, where there is a pandiagonal<\/strong> magic square of order 4 at the upper-left corner with two equal sums magic rectangles of order 2×4. <\/li>\n\n\n\nThe third one is traditional bordered magic square of order 6 with pandiagonal magic square of order 4 in the middle. <\/li>\n<\/ol>\n\n\n\n<\/p>\n\n\n\n
Above there are 6 magic squares of order 30 and combining with these three possibilities, we have total 18 magic squares of order 42. See below all these 18 possibilities. <\/p>\n\n\n\n
In Figures:<\/h4>\n\n\n\n <\/p>\n\n\n
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\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 56<\/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
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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
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In Figures or Designs:<\/h4>\n\n\n\nFirst-Type<\/h5>\n\n\n\n
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Second-Type<\/h5>\n\n\n\n
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Third-Type<\/h5>\n\n\n\n <\/p>\n\n\n
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Forth-Type<\/h5>\n\n\n\n <\/p>\n\n\n
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Fifth-Type<\/h5>\n\n\n
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