{"id":17179,"date":"2025-12-07T01:53:01","date_gmt":"2025-12-07T04:53:01","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=17179"},"modified":"2026-01-17T02:11:17","modified_gmt":"2026-01-17T05:11:17","slug":"algebraic-double-digit-and-cornered-magic-squares-of-odd-orders-from-5-to-19","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=17179","title":{"rendered":"Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19"},"content":{"rendered":"\n

This work brings double-digit<\/strong> or double-layer algebraic<\/strong> magic squares of odd orders from 5 to 19 for reduced entries<\/strong>. This study include three types of algebraic magic squares, i.e., cyclic-type, flat-type<\/strong> and corner-type<\/strong>. Cyclic-type<\/strong> and Flat-type<\/strong> are two different ways of writing as double-digit<\/strong> magic squares. Sometimes, these algebraic<\/strong> magic squares, we call as self-made<\/strong>, because they are complete in themselves. Just choose the entries and magic sum, we always get a magic square. In this work we use always magic rectangles of width 2 except in the middle or corner, where there is a magic square of order 3 or 5. The idea of double-digit<\/strong> and cornered<\/strong> magic squares for sequential entries is already studied by the authors. For details see the reference list given at the end.<\/p>\n\n\n\n

We know that magic sum of a magic square of order n<\/em><\/strong> having 1<\/em><\/strong> to n2<\/sup><\/em><\/strong> number of entries is given by<\/p>\n\n\n\n

Snxn<\/sub>:= n*(1+n2<\/sup>)\/2<\/em><\/strong><\/p>\n\n\n\n

By algebraic magic squares<\/strong> we understand that the entries are vaiables<\/strong> and their combinations. Thus, instead of squential entries, we have non-sequential<\/strong> entries. These can be positive, negative <\/strong>or decimal numbers<\/strong>.<\/p>\n\n\n\n

<\/p>\n\n\n\n

Whole work is also available at the following link:

Inder J. Taneja<\/strong>, Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo<\/strong>, December 08, 2025, pp. 1-46, https:\/\/doi.org\/10.5281\/zenodo.17859037<\/a>.<\/p>\n\n\n\n

Below are few examples of magic squares of odd orders fro 5 to 19<\/p>\n\n\n\n

<\/p>\n\n\n\n

Algebraic Magic Square of Order 5<\/h4>\n\n\n\n

<\/p>\n\n\n\n

Result 1: Algebraic Magic Square of Order 5<\/h4>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic <\/em>cornered magic square of order 5<\/strong>, where there is a magic square of order 3 at the upper-left corner. There are two magic rectangles of order 2×5 and 2×3. The magic sum of order 5 is given as S5×5<\/sub> := 5*S\/3<\/strong>, where S<\/strong> is the magic sum of order 3. In order to avoid decimal entries the the magic sum of order 3,i.e., S should be multiple of 3. m:=2*S\/3<\/strong> is the width of the magic rectangle. See below few examples:<\/em><\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub> :=30<\/strong>, S5×5<\/sub> :=45<\/strong> and m:=18<\/strong>.
In the second example<\/strong> the magic sums are S3×3<\/sub> :=3<\/strong>3, S5×5<\/sub> :=55<\/strong> and m:=22<\/strong>.<\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Algebraic Magic Squares of Order 7<\/h4>\n\n\n\n

Below are three types of striped algebraic magic squares of order 7.<\/p>\n\n\n\n

Result 2: Algebraic Cyclic-Type Magic Square of Order 7<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic magic square of order 7<\/strong> composed of four equal sums magic rectangles of orders 2×5 and embedded with a magic square of order 3. Since the external four strips are of equal sums, we can name it is as an algebraic cyclic-type magic square of order 7<\/strong>. In this case the magic sums are S3×3<\/sub>:=S <\/strong>and S7×7<\/sub>:=7*S\/3<\/strong>, where S<\/strong> is the magic sum of magic square of order 3. In order to avoid decimal entries the the magic sum of order 3 should be multiple of 3. The width of magic rectangles is given as m:=2*S\/3<\/strong>. See below few examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub> :=21<\/strong>, S7×7<\/sub> :=49<\/strong> and m:=14<\/strong>.
In the second example<\/strong> the magic sums are S3×3<\/sub> :=24<\/strong>, S7×7<\/sub> :=56<\/strong> and m:=16<\/strong>.<\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Result 3: Algebraic Flat-Type Magic Square of Order 7<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic magic square of order 7<\/strong> composed of twor equal sums magic rectangles of orders 2×7 and two equal sum magic rectangles of order 2×3 embedded with a magic square of order 3. For simplicity, this types of magic sequares we call as flat-type. Thus we have an algebraic flat-type magic square of order 7<\/strong><\/em> In this case the magic sums are S3×3<\/sub>:=S <\/strong>and S7×7<\/sub>:=7*S\/3<\/strong>, where S<\/strong> is the magic sum of magic square of order 3. In order to avoid decimal entries the the magic sum of order 3 should be multiple of 3. The width of magic rectangles is given as m:=2*S\/3<\/strong>. See below few examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub> :=57<\/strong>, S7×7<\/sub> :=133<\/strong> and m:=38<\/strong>.
In the second example<\/strong> the magic sums are S3×3<\/sub> :=60<\/strong>, S7×7<\/sub> :=140<\/strong> and m:=40<\/strong>.<\/em><\/em><\/p>\n\n\n\n

Result 4: Algebraic Cornered Magic Square of Order 7<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic cornered magic square of order 7<\/strong>, where the magic squares of orders 3 and 5 are at the upper-left corner. In this case the magic sums are S3×3<\/sub>:= S<\/strong>, S5×5<\/sub>:=5*S\/3<\/strong> and S7×7<\/sub>:=7*S\/3<\/strong>, where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. The magic rectanges are of orders 2×3, 2×5 and 2×7. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub> :=51<\/strong>, S5×5<\/sub> :=85<\/strong><\/em><\/em>, S7×7<\/sub> :=119<\/strong> and m:=34<\/strong>.
In the second example<\/strong> the magic sums are S3×3<\/sub> :=63<\/strong>, S5×5<\/sub> :=105<\/strong><\/em><\/em>, S7×7<\/sub> :=147<\/strong> and m:=42<\/strong>.<\/em><\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Algebraic Magic Squares of Order 9<\/h4>\n\n\n\n

Below are three types of algebraic striped magic squares of order 9.<\/p>\n\n\n\n

<\/p>\n\n\n\n

Result 5: Algebraic Cyclic-Type Magic Square of Order 9<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic cyclic-type magic square of order 9<\/strong> composed of four equal sums magic rectangles of orders 2×7 embedded with a magic square of order 5. In this case the magic sums are S5×5<\/sub>:=S<\/strong> and S9×9<\/sub>:=9*S\/5<\/strong>, where S<\/strong> is the magic sum of order 5. In this case, m:=2*S\/5<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 5 should be multiple of 5. See below two examples:<\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S5×5<\/sub> :=45<\/strong><\/em><\/em>, S9×9<\/sub> :=81<\/strong> and m:=18<\/strong>.
In the second example<\/strong> the magic sums are S5×5<\/sub> :=50<\/strong><\/em><\/em>, S9×9<\/sub> :=90<\/strong> and m:=20<\/strong>.<\/em><\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Result 6: Algebraic Flat-Type Magic Square of Order 9<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic flat-type magic square of order 9<\/strong> composed of two equal sums magic rectangles of order 2×9 and two equal sums magic rectangles of order 2×5 <\/em>embedded with a magic square of order 5. In this case the magic sums are S5×5<\/sub>:=S<\/strong> and S9×9<\/sub>:=9*S\/5<\/strong>, where S<\/strong> is the magic sum of order 5. In this case, m:=2*S\/5<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 5 should be multiple of 5. See below two examples:<\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S5×5<\/sub> :=85<\/strong><\/em><\/em>, S9×9<\/sub> :=153<\/strong> and m:=34<\/strong>.
In the second example<\/strong> the magic sums are S5×5<\/sub> :=105<\/strong><\/em><\/em>, S9×9<\/sub> :=189<\/strong> and m:=42<\/strong>.<\/em><\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Result 7: Algebraic Cornered Magic Square of Order 9<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic cornered striped magic square of order 9<\/strong>, where the magic squares of order 3, 5 and 7 are at the upper-left corner. The magic squares of orders 5 and 7 are also an algebraic cornered magic squares<\/strong> In this case the magic sums are S3×3<\/sub>:=S<\/strong>, S5×5<\/sub>:=5*S\/3<\/strong>, S7×7<\/sub>:=7*S\/3<\/strong><\/strong> and S9×9<\/sub>:=3*S,<\/strong> where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub> := 39<\/strong>, S5×5<\/sub> := 65<\/strong>, S<\/b>7×7<\/sub><\/span>:=<\/b>91<\/span>. <\/strong>S9×9<\/sub><\/strong> :=117<\/strong> and <\/i>m:=26<\/strong>.<\/i>
In the second example<\/strong> the magic sums are S3×3<\/sub> := 42<\/strong>, S5×5<\/sub> := 70<\/strong>, S<\/b>7×7<\/sub><\/span>:=<\/b><\/span>98, <\/strong>S9×9<\/sub><\/strong> :=126<\/strong> and <\/i>m:=28<\/strong>.<\/i><\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Algebraic Magic Squares of Order 11<\/h4>\n\n\n\n

Below are three types of algebraic striped magic squares of order 11.<\/p>\n\n\n\n

<\/p>\n\n\n\n

Result 8: Algebraic Cyclic-Type Magic Square of Order 11<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic cyclic-type magic square of order 11<\/strong> composed of four equal sums magic rectangles of orders 2×9 embedded with a magic square of order 7. It is again composed of four equal sums magic rectangles of order 2×5 having a magic square of order 3 in the middle. In this case the magic sums are S3×3<\/sub>:=S<\/strong>, S7×7<\/sub>:=7*S\/3<\/strong> and S11×11<\/sub>:=11*S\/3,<\/strong> where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub> := 33<\/strong>, S<\/b>7×7<\/sub><\/span>:=<\/b>77<\/span>. <\/strong>S11×11<\/sub><\/strong> :=121<\/strong> and <\/i>m:=22.<\/strong>
In the second example<\/strong> the magic sums are S3×3<\/sub> := 39<\/strong>, S<\/b>7×7<\/sub><\/span>:=<\/b><\/span>91, <\/strong>S11×11<\/sub><\/strong> :=143<\/strong> and <\/i>m:=26<\/strong>.<\/i><\/em><\/em><\/p>\n\n\n\n

Result 9: Algebraic Flat-Type Magic Square of Order 11<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic flat-type magic square of order 11<\/strong> composed of two equal sums magic rectangles of orders 2×11 and two equal sums magic rectangles of order 2×7 embedded again with a flat-type <\/strong>magic square of order 7. In this case the magic sums are S3×3<\/sub>:=S<\/strong><\/em><\/em>, S7×7<\/sub>:=7*S\/3<\/strong> and S11×11<\/sub>:=11*S\/3,<\/strong> where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub> := 48<\/strong>, S<\/b>7×7<\/sub><\/span>:=<\/b>112<\/span>. <\/strong>S11×11<\/sub><\/strong> :=176<\/strong> and <\/i>m:=32.<\/strong>
In the second example<\/strong> the magic sums are S3×3<\/sub> := 51<\/strong>, S<\/b>7×7<\/sub><\/span>:=<\/b><\/span>119, <\/strong>S11×11<\/sub><\/strong> :=187<\/strong> and <\/i>m:=34<\/strong>.<\/i><\/em><\/em><\/p>\n\n\n\n

Result 10: Algebraic Cornred Magic Square of Order 11<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic cornered striped magic square of order 11<\/strong>, where the magic squares of order 3, 5, 7 and 9 are at the upper-left corner. The magic squares of orders 5, 7 and 9 are also algebraic cornered magic squares<\/strong> In this case the magic sums are S3×3<\/sub>:=S<\/strong>, S5×5<\/sub>:=5*S\/3<\/strong>, S7×7<\/sub>:=7*S\/3<\/strong><\/strong>, S9×9<\/sub>:=3*S and S11×11<\/sub>:=11*S<\/em>\/3 <\/strong>where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub>:=54<\/strong>, S5×5<\/sub>:= 90<\/strong>, S7×7<\/sub>:=126<\/strong>, S9×9<\/sub>:=162, S11×11<\/sub>:=198<\/em><\/strong><\/em><\/em> and <\/i>m:=36.<\/strong>
In the second example<\/strong> the magic sums are S3×3<\/sub>:=57<\/strong>, S5×5<\/sub>:= 9<\/strong>5 S7×7<\/sub>:=133<\/strong>, S9×9<\/sub>:=171, S11×11<\/sub>:=209<\/em><\/strong><\/em> and <\/i>m:=38.<\/strong><\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Algebraic Magic Squares of Order 13<\/h4>\n\n\n\n

Below are three types of algebraic striped magic squares of order 13.<\/p>\n\n\n\n

Result 11: Algebraic Cyclic-Type Magic Square of Order 13<\/h4>\n\n\n
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It is an algebraic cyclic-type magic square of order 13<\/strong> composed of four equal sums magic rectangles of orders 2×11 embedded with a magic square of order 9. It is again composed of four equal sums magic rectangles of order 2×7 having a magic square of order 5 in the middle. In this case the magic sums are S5×5<\/sub>:=S<\/strong>, S9×9<\/sub>:=9*S\/5<\/strong><\/em><\/strong> and S13×13<\/sub>:=13*S\/5,<\/strong> where S<\/strong> is the magic sum of order 5. In this case, m:=2*S\/5<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 5 should be multiple of 5. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S5×5<\/sub>:=65<\/strong>, S9×9<\/sub>:=117<\/strong>, S13×13<\/sub>:=169<\/strong><\/em> and <\/i>m:=26.<\/strong>
In the second example<\/strong> the magic sums are S3×3<\/sub> := 70<\/strong>, S<\/b>9×9<\/sub><\/span>:=<\/b><\/span>126, <\/strong>S13×13<\/sub><\/strong> :=182<\/strong> and <\/i>m:=28<\/strong>.<\/i><\/em><\/em><\/p>\n\n\n\n

Result 12: Algebraic Flat-Type Magic Square of Order 13<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic flat-type magic square of order 13<\/strong> composed of two equal sums magic rectangles of orders 2×13 and two equal sums magic rectangles of order 2×9 embedded again with a flat-type <\/strong>magic square of order 9. It is avgain a algebraic flat-type magic square of order 9 <\/strong>embedded with a magic square of order 5. <\/em><\/em> In this case the magic sums are S5×5<\/sub>:=S<\/strong>, S9×9<\/sub>:=9*S\/5<\/strong><\/em><\/strong> and S13×13<\/sub>:=13*S\/5,<\/strong> where S<\/strong> is the magic sum of order 5. In this case, m:=2*S\/5<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 5 should be multiple of 5. See below two examples:<\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S5×5<\/sub>:=75<\/strong>, S9×9<\/sub>:=135<\/strong>, S13×13<\/sub>:=195<\/strong><\/em> and <\/i>m:=30.<\/strong>
In the second example<\/strong> the magic sums are S5×5<\/sub> := 85<\/strong>, S<\/b>9×9<\/sub><\/span>:=<\/b><\/span>153 <\/strong>S13×13<\/sub><\/strong> :=221<\/strong> and <\/i>m:=34<\/strong>.<\/i><\/em><\/em><\/p>\n\n\n\n

Result 13: Algebraic Cornered Magic Square of Order 13<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic cornered striped magic square of order 13<\/strong>, where the magic squares of order 3, 5, 7, 9 and 11 are at the upper-left corner. The magic squares of orders 5, 7, 9 and 11 are also algebraic cornered magic squares<\/strong> In this case the magic sums are S3×3<\/sub>:=S<\/strong>, S5×5<\/sub>:=5*S\/3<\/strong>, S7×7<\/sub>:=7*S\/3<\/strong><\/strong>, S9×9<\/sub>:=3*S, S11×11<\/sub>:=11*S<\/em>\/3<\/strong><\/em><\/strong><\/em> and S13×13<\/sub>:=13*S<\/em>\/3, <\/strong>where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub>:=72<\/strong>, S5×5<\/sub>:= 120<\/strong>, S7×7<\/sub>:=168<\/strong>, S9×9<\/sub>:=216, S11×11<\/sub>:=264, S13×13<\/sub>:=312 <\/em><\/strong><\/em>and <\/i>m:=48.<\/strong>
In the second example<\/strong> the magic sums are S3×3<\/sub>:=63<\/strong>, S5×5<\/sub>:= 10<\/strong>5 S7×7<\/sub>:=147<\/strong>, S9×9<\/sub>:=189, S11×11<\/sub>:=231, S13×13<\/sub>:=273<\/em><\/strong><\/em><\/em><\/em> <\/em><\/strong><\/em>and <\/i>m:=42.<\/strong><\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Algebraic Magic Squares of Order 15<\/h4>\n\n\n\n

Below are three types of algebraic striped magic squares of order 15.<\/p>\n\n\n\n

Result 14: Algebraic Cyclic-Typec Magic Squares of Order 15<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic cyclic-type magic square of order 15<\/strong> composed of four equal sums magic rectangles of orders 2×13 embedded with a magic square of order 11. It is again composed of four equal sums magic rectangles of order 2×9 and so on having a magic square of order 3 in the middle. In this case the magic sums are S3×3<\/sub>:=S<\/strong>, S7×7<\/sub>:=7*S\/3<\/strong>, S11×11<\/sub>:=11*S\/3 and S15×15<\/sub>:=5*S<\/strong>,<\/strong> where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub> := 63<\/strong>, S<\/b>7×7<\/sub><\/span>:=<\/b>147<\/span>. <\/strong>S11×11<\/sub><\/strong> :=231<\/strong>, S15×15<\/sub><\/strong> :=315<\/strong> <\/i><\/em>and <\/i>m:=42.<\/strong>
In the second example<\/strong> the magic sums are S3×3<\/sub> := 72<\/strong>, S<\/b>7×7<\/sub><\/span>:=<\/b>168<\/span>. <\/strong>S11×11<\/sub><\/strong> :=264<\/strong>, S15×15<\/sub><\/strong> :=360<\/strong> <\/i><\/em>and <\/i>m:=48.<\/strong><\/em><\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Result 15: Algebraic Flat-Type Flat Magic Square of Order 15<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic flat-type magic square of order 15<\/strong> composed of two equal sums magic rectangles of orders 2×13 and two equal sums magic rectangles of order 2×9 embedded again with a flat-type <\/strong>magic square of order 11 and so on. In this case the magic sums are S3×3<\/sub>:=S<\/strong><\/em><\/em>, S7×7<\/sub>:=7*S\/3<\/strong>, S11×11<\/sub>:=11*S\/3<\/strong> and S15×15<\/sub>:=5*S<\/strong><\/em>,<\/strong> where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub> := 93<\/strong>, S<\/b>7×7<\/sub><\/span>:=<\/b>217<\/span>. <\/strong>S11×11<\/sub><\/strong> :=341<\/strong>, S15×15<\/sub><\/strong> :=465<\/strong> <\/i><\/em>and <\/i>m:=62.<\/strong>
In the second example<\/strong> the magic sums are S3×3<\/sub> := 102<\/strong>, S<\/b>7×7<\/sub><\/span>:=<\/b>238<\/span>. <\/strong>S11×11<\/sub><\/strong> :=374<\/strong>, S15×15<\/sub><\/strong> :=510<\/strong> <\/i><\/em>and <\/i>m:=68.<\/strong><\/em><\/em><\/em><\/p>\n\n\n\n

Result 16: Algebraic Cornered Magic Square of Order 15<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic cornered striped magic square of order 15<\/strong>, where the magic squares of order 3, 5, 7, 9, 11 and 13 are at the upper-left corner. The magic squares of orders 5, 7, 9, 11 and 13 are also algebraic cornered magic squares<\/strong> In this case the magic sums are S3×3<\/sub>:=S<\/strong>, S5×5<\/sub>:=5*S\/3<\/strong>, S7×7<\/sub>:=7*S\/3<\/strong><\/strong>, S9×9<\/sub>:=3*S, S11×11<\/sub>:=11*S<\/em>\/3<\/strong><\/em><\/strong><\/em>, S13×13<\/sub>:=13*S<\/em>\/3 <\/strong>and S15×15<\/sub>:=5*S<\/em><\/strong><\/em><\/strong><\/em>, <\/strong>where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In the first example<\/strong> the magic sums are S3×3<\/sub>:=81<\/strong>, S5×5<\/sub>:= 135<\/strong>, S7×7<\/sub>:=189<\/strong>, S9×9<\/sub>:=243, S11×11<\/sub>:=297, S13×13<\/sub>:=351<\/em><\/strong><\/em>, S15×15<\/sub>:=405<\/em><\/strong><\/em><\/em> and <\/i>m:=54.<\/strong>
In the second exemplem<\/strong> the magic sums are S3×3<\/sub>:=90<\/strong>, S5×5<\/sub>:= 150<\/strong>, S7×7<\/sub>:=210<\/strong>, S9×9<\/sub>:=270, S11×11<\/sub>:=330, S13×13<\/sub>:=390<\/em><\/strong><\/em>, S15×15<\/sub>:=450<\/em><\/strong><\/em><\/em> and <\/i>m:=60.<\/strong><\/em><\/em><\/em><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Algebraic Magic Squares of Order 17<\/h4>\n\n\n\n

Below are three types of algebraic striped magic squares of order 17.<\/p>\n\n\n\n

Result 17: Algebraic Cyclic-Type Magic Square of Order 17<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

It is an algebraic cyclic-type magic square of order 17<\/strong> composed of four equal sums magic rectangles of orders 2×15 embedded with a magic square of order 13. It is again composed of four equal sums magic rectangles of order 2×11 and so on having a magic square of order 5 in the middle. In this case the magic sums are S5×5<\/sub>:=S<\/strong>, S9×9<\/sub>:=3*S<\/strong>, S13×13<\/sub>:=13*S\/3 <\/strong>and S17×17<\/sub>:=17*S\/3<\/strong><\/em>,<\/strong> where S<\/strong> is the magic sum of order 5. In this case, m:=2*S\/5<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 5 should be multiple of 5. See below two examples: <\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S5×5<\/sub>:=85<\/strong>, S9×9<\/sub>:=153<\/strong>, S13×13<\/sub>:=221<\/strong><\/em>, S17×17<\/sub>:=289<\/strong><\/em><\/em> and <\/i>m:=34.<\/strong><\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S5×5<\/sub>:=105<\/strong>, S9×9<\/sub>:=189<\/strong>, S13×13<\/sub>:=273<\/strong><\/em>, S17×17<\/sub>:=357<\/strong><\/em><\/em> and <\/i>m:=42.<\/strong><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Result 18: Algebraic Flat-Type Magic Square of Order 17<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic flat-type magic square of order 17<\/strong> composed of two equal sums magic rectangles of orders 2×17 and two equal sums magic rectangles of order 2×13 embedded again with a flat-type <\/strong>magic square of order 13. It is again a algebraic flat-type magic square of order 13<\/strong> and so on having a magic square of order 5 in the middle. <\/em>In this case the magic sums are S5×5<\/sub>:=S<\/strong>, S9×9<\/sub>:=3*S<\/strong>, S13×13<\/sub>:=13*S\/3 <\/strong>and S17×17<\/sub>:=17*S\/3<\/strong><\/em>,<\/strong> where S<\/strong> is the magic sum of order 5. In this case, m:=2*S\/5<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 5 should be multiple of 5. See below two examples: <\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S5×5<\/sub>:=95<\/strong>, S9×9<\/sub>:=171<\/strong>, S13×13<\/sub>:=247<\/strong><\/em>, S17×17<\/sub>:=323<\/strong><\/em><\/em> and <\/i>m:=38.<\/strong><\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S5×5<\/sub>:=125<\/strong>, S9×9<\/sub>:=225<\/strong>, S13×13<\/sub>:=325<\/strong><\/em>, S17×17<\/sub>:=425<\/strong><\/em><\/em> and <\/i>m:=50.<\/strong><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Result 19: Algebraic Cornered Magic Square of Order 17<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic cornered striped magic square of order 17<\/strong>, where the magic squares of order 3, 5, 7, 9, 11, 13 and 15 are at the upper-left corner. The magic squares of orders 5, 7, 9, 11, 13 and 15 are also algebraic cornered magic squares<\/strong>. In this case the magic sums are S3×3<\/sub>:=S<\/strong>, S5×5<\/sub>:=5*S\/3<\/strong>, S7×7<\/sub>:=7*S\/3<\/strong><\/strong>, S9×9<\/sub>:=3*S, S11×11<\/sub>:=11*S<\/em>\/3<\/strong><\/em><\/strong><\/em>, S13×13<\/sub>:=13*S<\/em>\/3, S15×15<\/sub>:=5*S<\/em><\/strong><\/em><\/strong><\/em> <\/strong>and S17×17<\/sub>:=17*S\/3<\/em><\/strong><\/em><\/strong><\/em>, <\/strong>where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S3×3<\/sub>:=99<\/strong>, S5×5<\/sub>:= 165<\/strong>, S7×7<\/sub>:=231, <\/strong> S9×9<\/sub>:=297, S11×11<\/sub>:=363, S13×13<\/sub>:=429<\/em><\/strong><\/em>, S15×15<\/sub>:=495<\/em><\/strong><\/em><\/em>, S17×17<\/sub>:=561<\/em><\/strong><\/em><\/em><\/i><\/em> and <\/i>m:=78.<\/strong><\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S3×3<\/sub>:=117<\/strong>, S5×5<\/sub>:= 195<\/strong>, S7×7<\/sub>:=273, <\/strong> S9×9<\/sub>:=351, S11×11<\/sub>:=429, S13×13<\/sub>:=507<\/em><\/strong><\/em>, S15×15<\/sub>:=585<\/em><\/strong><\/em><\/em>, S17×17<\/sub>:=663<\/em><\/strong><\/em><\/em><\/i><\/em> and <\/i>m:=66.<\/strong><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Algebraic Magic Squares of Order 19<\/h4>\n\n\n\n

Below are three types of algebraic striped magic squares of order 19.<\/p>\n\n\n\n

Result 20: Algebraic Cyclic-Type Magic Square of Order 19<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic cyclic-type magic square of order 19<\/strong> composed of four equal sums magic rectangles of orders 2×17 embedded with a magic square of order 15. It is again composed of four equal sums magic rectangles of order 2×13 and so on having a magic square of order 3 in the middle. In this case the magic sums are S3×3<\/sub>:=S<\/strong>, S7×7<\/sub>:=7*S\/3<\/strong>, S11×11<\/sub>:=11*S\/3, S15×15<\/sub>:=5*S<\/strong><\/strong><\/em> and S19×19<\/sub>:=19*S\/3<\/strong>,<\/strong> where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S3×3<\/sub>:=195,<\/strong> S7×7<\/sub>:=455, S11×11<\/sub>:=715, <\/em><\/strong><\/em>S15×15<\/sub>:=975<\/em><\/strong><\/em><\/em>, S19×19<\/sub>:=1235 <\/em><\/strong><\/em><\/em><\/i><\/em>and <\/i>m:=130.<\/strong><\/em><\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S3×3<\/sub>:=198,<\/strong> S7×7<\/sub>:=462, S11×11<\/sub>:=726, <\/em><\/strong><\/em>S15×15<\/sub>:=990<\/em><\/strong><\/em><\/em>, S19×19<\/sub>:=1254 <\/em><\/strong><\/em><\/em><\/i><\/em>and <\/i>m:=132.<\/strong><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Result 21: Algebraic Flat-Type Magic Square of Order 19<\/h4>\n\n\n
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\"\"<\/figure>\n<\/div>\n\n\n

It is an algebraic flat-type magic square of order 19<\/strong> composed of two equal sums magic rectangles of orders 2×19 and two equal sums magic rectangles of order 2×15 embedded again with a flat-type <\/strong>magic square of order 15 and so on. In this case the magic sums are S3×3<\/sub>:=S<\/strong><\/em><\/em>, S7×7<\/sub>:=7*S\/3<\/strong>, S11×11<\/sub>:=11*S\/3<\/strong>, S15×15<\/sub>:=5*S<\/strong><\/em> and S19×19<\/sub>:=19*S\/3<\/strong><\/em>,<\/strong> where S<\/strong> is the magic sum of order 3. <\/em><\/strong><\/em>In this case, m:=2*S\/3<\/strong> is the width of magic rectangles, where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S3×3<\/sub>:=123,<\/strong> S7×7<\/sub>:=287, S11×11<\/sub>:=451, <\/em><\/strong><\/em>S15×15<\/sub>:=615<\/em><\/strong><\/em><\/em>, S19×19<\/sub>:=779 <\/em><\/strong><\/em><\/em><\/i><\/em>and <\/i>m:=82.<\/strong><\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S3×3<\/sub>:=231,<\/strong> S7×7<\/sub>:=539, S11×11<\/sub>:=847, <\/em><\/strong><\/em>S15×15<\/sub>:=1155<\/em><\/strong><\/em><\/em>, S19×19<\/sub>:=1463 <\/em><\/strong><\/em><\/em><\/i><\/em>and <\/i>m:=154.<\/strong><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

Result 22: Algebraic Cornered Magic Square of Order 19<\/h4>\n\n\n
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It is an algebraic cornered striped magic square of order 19<\/strong>, where the magic squares of order 3, 5, 7, 9, 11, 13, 15 and 17 are at the upper-left corner. The magic squares of orders 5, 7, 9, 11, 13, 15 and 17 are also algebraic cornered magic squares<\/strong>. In this case the magic sums are S3×3<\/sub>:=S<\/strong>, S5×5<\/sub>:=5*S\/3<\/strong>, S7×7<\/sub>:=7*S\/3<\/strong><\/strong>, S9×9<\/sub>:=3*S, S11×11<\/sub>:=11*S<\/em>\/3<\/strong><\/em><\/strong><\/em>, S13×13<\/sub>:=13*S<\/em>\/3, S15×15<\/sub>:=5*S<\/em><\/strong><\/em><\/strong><\/em>, S17×17<\/sub>:=17*S\/3<\/em><\/strong><\/em> <\/strong>and S19×19<\/sub>:=19*S\/3<\/em><\/strong><\/em><\/strong><\/em><\/em>, where S<\/strong> is the magic sum of order 3. In this case, m:=2*S\/3<\/strong> is the width of magic rectangles. To avoid decimal entries the magic sums of order 3 should be multiple of 3. See below two examples:<\/em><\/p>\n\n\n\n

<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S3×3<\/sub>:=147<\/strong> S5×5<\/sub>:= 245<\/strong>, S7×7<\/sub>:=343,<\/strong> S9×9<\/sub>:=441, S11×11<\/sub>:=539, S13×13<\/sub>:=637<\/em><\/strong><\/em>, S15×15<\/sub>:=735<\/em><\/strong><\/em><\/em>, S17×17<\/sub>:=833<\/em><\/strong><\/em><\/em><\/i><\/em>, S19×19<\/sub>:=931<\/em><\/strong><\/em><\/em><\/i><\/em><\/i><\/em> and <\/i>m:=98.<\/strong><\/em><\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

<\/p>\n\n\n\n

In this example<\/strong> the magic sums are S3×3<\/sub>:=132,<\/strong> S5×5<\/sub>:= 220<\/strong>, S7×7<\/sub>:=308,<\/strong> S9×9<\/sub>:=396, S11×11<\/sub>:=484, S13×13<\/sub>:=572<\/em><\/strong><\/em>, S15×15<\/sub>:=660<\/em><\/strong><\/em><\/em>, S17×17<\/sub>:=748<\/em><\/strong><\/em><\/em><\/i><\/em>, S19×19<\/sub>:=836<\/em><\/strong><\/em><\/em><\/i><\/em><\/i><\/em> and <\/i>m:=88.<\/strong><\/em><\/p>\n\n\n\n

<\/p>\n\n\n\n

References<\/h4>\n\n\n\n

<\/p>\n\n\n\n

Part 1: Day and Dates of the Year – 2025 in Terms of Magic Squares<\/mark><\/h4>\n\n\n\n