{"id":17009,"date":"2025-11-18T18:44:05","date_gmt":"2025-11-18T21:44:05","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=17009"},"modified":"2025-12-01T09:58:07","modified_gmt":"2025-12-01T12:58:07","slug":"double-digit-cyclic-type-bordered-algebraic-magic-squares-of-orders-7-to-20-for-reduced-entries","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=17009","title":{"rendered":"Double-Digit Cyclic-Type Bordered Algebraic Magic Squares of Orders 7 to 20 for Reduced Entries"},"content":{"rendered":"\n

This work brings double-digit<\/strong> or double-layer algebraic<\/strong> magic squares of orders 7 to 20 for reduced entries<\/strong>. Sometimes, these types of magic squares, we call as self-made<\/strong>, beacause they are complete in themselves. Just choose the entries and magic sum, we always get a magic square.

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

In this work the entries are written as vaiables<\/strong> and their combinations. The work is based on four equal sums magic rectangles in each layer or border. The width is always 2. The size of length magic rectangle depends on the orders of the magic squares. For simplicity, these types of magic squares we call as cyclic. Similar kind of study for the sequential entries can be seen at the following work.<\/p>\n\n\n\n

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

Whole work can be downloaded at the following link:

Inder J. Taneja<\/strong>, Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo<\/strong>, November 21, 2025, pp. 1-37, https:\/\/doi.org\/10.5281\/zenodo.17675032<\/a>.<\/p>\n\n\n\n

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

Double-Digit Bordered Algebric Magic Square of Order 7<\/h4>\n\n\n\n

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

Result 1: Double-Digit Bordered Algebric Magic Square of Order 7<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles of order 2×5 embedded with a magic square of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we have decimal entries, then the magic sum of order 7 is also multiple of 3. The magic sum of order 7 is given as S7×7<\/sub> :=7*S\/3<\/strong><\/em>, where S<\/strong> is the magic sum of order 3. <\/strong>See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

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

Double-Digit Bordered Algebric Magic Square of Order 8<\/h4>\n\n\n\n

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

Result 2: Double-Digit Bordered Algebric Magic Square of Order 8<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles of order 2×6 embedded with a magic square of order 4. Since the magic square of order 4 requires magic sum as multiple of 2, otherwise we have decimal entries, then the magic sum of order 8 is also multiple of 2. The magic sum of order 8 is given as <\/em>S8×8<\/sub> :=2*S, <\/em><\/strong>where S <\/strong>is the magic sum of order 4.<\/em> See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

In the first example the magic sums are S4×4<\/sub> :=26<\/em><\/strong> and S8×8<\/sub> :=52<\/em><\/strong>.
In the second example the magic sums are S4×4<\/sub> :=34<\/em><\/strong> and S8×8<\/sub> :=68<\/em><\/strong>.<\/p>\n\n\n\n

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

Double-Digit Bordered Algebric Magic Square of Order 9<\/h4>\n\n\n\n

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

Result 3: Double-Digit Bordered Algebric Magic Square of Order 9<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles<\/strong> of order 2×7 embedded with a pandiagonal <\/strong>magic square of order 5. Here the magic sum of order 5 don’t have any condititon, but the magic sum of order 9 depends on number 5, i.e., S9×9<\/sub> :=9*S\/5<\/strong><\/em><\/em><\/strong><\/em>, where S <\/strong>is the magic sum of order 5. This requires the magic sum of order 5 should be multiple of 5, otherwise we may have decimal entries. <\/em>See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

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

Double-Digit Bordered Algebric Magic Square of Order 10<\/h4>\n\n\n\n

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

Result 4: Double-Digit Bordered Algebric Magic Square of Order 10<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles of order 2×8 embedded with a magic square of order 6. This magic square of order 6 is again composed of four equal sums magic squares of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we have decimal entries, then the magic sum of order 10 is also multiple of 3. The magic sum of order 10 is given as S10×10<\/sub> :=10*S\/3<\/strong><\/em>, where S<\/strong> is the magic sum of order 3. <\/strong>In this case the magic sum of order 6 is given as S6×6<\/sub> :=2*S.<\/strong><\/em> See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

In the first example the magic sums are S3×3<\/sub> :=45<\/em><\/strong>, S6×6<\/sub> :=90<\/em><\/strong> and S10×10<\/sub> := 150<\/em><\/strong>.
In the second example the magic sums are S3×3<\/sub> :=48<\/em><\/strong>, S6×6<\/sub> :=96<\/em><\/strong> and S10×10<\/sub> := 160<\/em><\/strong>.<\/p>\n\n\n\n

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

Double-Digit Bordered Algebric Magic Square of Order 11<\/h4>\n\n\n\n

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

Result 5: Double-Digit Bordered Algebric Magic Square of Order 11<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles<\/strong> of orders 2×5 and 2×9 (equality in each case) embedded with a magic square of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we have decimal entries, then the magic sums of orders 7 and 11 are also multiples of 3. The magic sum of orders 7 and 11 are given as S11×11<\/sub> :=7*S\/3<\/strong><\/em><\/strong><\/em> and S11×11<\/sub> :=11*S\/3<\/strong><\/em>, where S<\/strong> is the magic sum of order 3. <\/strong>See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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In the first example the magic sums are S3×3<\/sub> :=33<\/em><\/strong>, S7×7<\/sub> :=77<\/em><\/strong> and S11×11<\/sub> := 121<\/em><\/strong>.
In the second example the magic sums are S3×3<\/sub> :=39<\/em><\/strong>, S7×7<\/sub> :=91<\/em><\/strong> and S11×11<\/sub> := 143<\/em><\/strong>.<\/p>\n\n\n\n

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

Double-Digit Bordered Algebric Magic Square of Order 12<\/h4>\n\n\n\n

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

Result 6: Double-Digit Bordered Algebric Magic Square of Order 12<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles<\/strong> of orders 2×6 and 2×10 (equality in each case) embedded with a magic square of order 4. Since the magic square of order 4 requires magic sum as multiple of 2, otherwise we have decimal entries, then the magic sum of order 12 is also multiple of 2. The magic sum of orders 12 and 8 ares given as S12×12<\/sub> :=3*S<\/em><\/strong><\/em> and S8×8<\/sub> :=2*S, <\/em><\/strong>where S <\/strong>is the magic sum of order 4.<\/em> See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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In the first example the magic sums are S4×4<\/sub> :=46<\/em><\/strong>, S8×8<\/sub> :=92<\/em><\/strong> and S12×12<\/sub> := 138<\/em><\/strong>.
In the second example the magic sums are S4×4<\/sub> :=52<\/em><\/strong>, S8×8<\/sub> :=104<\/em><\/strong> and S12×12<\/sub> := 156<\/em><\/strong>.<\/p>\n\n\n\n

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

Double-Digit Bordered Algebric Magic Square of Order 13<\/h4>\n\n\n\n

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

Result 7: Double-Digit Bordered Algebric Magic Square of Order 13<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles<\/strong> of orders 2×7 and 2×11 (equality in each case) embedded with a pandiagonal magic square of order 5. The magic square of order 5 don’t requires any condition but the magic sums of orders 13 and 9 depends on five, we mucht have these magic sums as multiples of 5 to avoid decimal entries. The magic sum of orders 13 and 9 ares given as <\/em>S13×13<\/sub> :=13*S<\/em><\/em>\/5 <\/strong>and S9×9<\/sub> :=9*S\/5, <\/em><\/strong>where S <\/strong>is the magic sum of order 5.<\/em> See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

In the first example the magic sums are S5×5<\/sub> :=65<\/em><\/strong>, S9×9<\/sub> :=117<\/em><\/strong> and S13×13<\/sub> := 169<\/em><\/strong>.
In the second example the magic sums are S5×5<\/sub> :=70<\/em><\/strong>, S9×9<\/sub> :=126<\/em><\/strong> and S13×13<\/sub> := 182<\/em><\/strong>.<\/p>\n\n\n\n

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

Double-Digit Bordered Algebric Magic Square of Order 14<\/h4>\n\n\n\n

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

Result 8: Double-Digit Bordered Algebric Magic Square of Order 14<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles of orders 2×8 and 2×12 (equal in each case) embedded with a magic square of order 6. This magic square of order 6 is again composed of four equal sums magic squares of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we may have decimal entries. The magic sums of orders 10 and 14 are also multiple of 3. The magic sum of orders 10 and 14 are given as S10×10<\/sub> :=10*S\/3<\/strong><\/em> and S10×10<\/sub> :=14*S\/3<\/strong><\/em>, where S<\/strong> is the magic sum of order 3. <\/strong>In this case the magic sum of order 6 is given as S6×6<\/sub> :=2*S.<\/strong><\/em> See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

In the first example the magic sums are S3×3<\/sub> :=66<\/em><\/strong>, S6×6<\/sub> :=132<\/em><\/strong>, S10×10<\/sub> :=220<\/em><\/strong> and S14×14<\/sub> := 308<\/em><\/strong>.
In the second example the magic sums are S3×3<\/sub> :=75<\/em><\/strong>, S6×6<\/sub> :=150<\/em><\/strong>, S10×10<\/sub> :=250<\/em><\/strong> and S14×14<\/sub> := 350<\/em><\/strong>.<\/p>\n\n\n\n

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

Double-Digit Bordered Algebric Magic Square of Order 15<\/h4>\n\n\n\n

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

Result 9: Double-Digit Bordered Algebric Magic Square of Order 15<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles<\/strong> of orders 2×5, 2×9 and 2×13 (equality in each case) embedded with a magic square of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we have decimal entries, then the magic sums of order 11 and 15 are also multiples of 3. The magic sum of orders 11 and 15 are given as S11×11<\/sub> :=11*S\/3<\/strong><\/em> and S15×15<\/sub> :=15*S\/3=5*S<\/strong><\/em>, where S<\/strong> is the magic sum of order 3. <\/strong>See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

In the first example the magic sums are S3×3<\/sub> :=63<\/em><\/strong>, S7×7<\/sub> :=147<\/em><\/strong>, S11×11<\/sub> :=231<\/em><\/strong> and S15×15<\/sub> := 315<\/em><\/strong>.
In the second example the magic sums are S3×3<\/sub> :=72<\/em><\/strong>, S7×7<\/sub> :=168<\/em><\/strong>, S11×11<\/sub> :=264<\/em><\/strong> and S15×15<\/sub> := 360<\/em><\/strong>.<\/p>\n\n\n\n

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

Double-Digit Bordered Algebric Magic Square of Order 16<\/h4>\n\n\n\n

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

Result 10: Double-Digit Bordered Algebric Magic Square of Order 16<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles<\/strong> of orders 2×6, 2×10 and 2×14 (equality in each case) embedded with a magic square of order 4. Since the magic square of order 4 requires magic sum as multiple of 2, otherwise we have decimal entries, then the magic sum of orders, 8, 12 and 16 are also multiple of 2. The magic sum of orders 16, 12 and 8 ares given as S16×16<\/sub> :=4*S<\/em><\/strong><\/em><\/em>, S12×12<\/sub> :=3*S<\/em><\/strong><\/em> and S8×8<\/sub> :=2*S, <\/em><\/strong>where S <\/strong>is the magic sum of order 4.<\/em> See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

In the first example the magic sums are S4×4<\/sub> :=102<\/em><\/strong>, S8×8<\/sub> :=204<\/em><\/strong>, S12×12<\/sub> := 312<\/em><\/strong> and S16×16<\/sub> := 408<\/em><\/strong>
In the second example the magic sums are S4×4<\/sub> :=110<\/em><\/strong>, S8×8<\/sub> :=220<\/em><\/strong>, S12×12<\/sub> := 330<\/em><\/strong> and S16×16<\/sub> := 440<\/em><\/strong><\/p>\n\n\n\n

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

Double-Digit Bordered Algebric Magic Square of Order 17<\/h4>\n\n\n\n

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

Result 11: Double-Digit Bordered Algebric Magic Square of Order 17<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles<\/strong> of orders 2×7, 2×11 and 2×15 (equality in each case) embedded with a pandiagonal <\/strong>magic square of order 5. The magic square of order 5 don’t requires any condition but the magic sums of orders 17, 13 and 9 depends on five. Thus, we must have these magic sums as multiples of 5 to avoid decimal entries. The magic sum of orders 17, 13 and 9 ares given as S17×17<\/sub> :=17*S<\/em><\/em>\/5<\/strong><\/em>, S13×13<\/sub> :=13*S<\/em><\/em>\/5 <\/strong>and S9×9<\/sub> :=9*S\/5, <\/em><\/strong>where S <\/strong>is the magic sum of order 5.<\/em> See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

<\/p>\n\n\n

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

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

Double-Digit Bordered Algebric Magic Square of Order 18<\/h4>\n\n\n\n

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

Result 12: Double-Digit Bordered Algebric Magic Square of Order 18<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles of orders 2×8, 2×12 and 2×16 (equal in each case) embedded with a magic square of order 6. This magic square of order 6 is again composed of four equal sums magic squares of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we may have decimal entries. The magic sums of orders 10, 14 and 18 are also multiple of 3. The magic sum of orders 10, 14 and 18 are given as S10×10<\/sub> :=10*S\/3<\/strong><\/em>, S14×14<\/sub> :=14*S\/3<\/strong><\/em> and S18×18<\/sub> :=18*S\/3 = 6*S<\/strong><\/em>, where S<\/strong> is the magic sum of order 3. <\/strong>In this case the magic sum of order 6 is given as S6×6<\/sub> :=2*S.<\/strong><\/em> See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

In this example the magic sums are S3×3<\/sub> :=81<\/em><\/strong>, S6×6<\/sub>:=162<\/em><\/strong>, S10×10<\/sub> :=270<\/em><\/strong>, S14×14<\/sub> :=378<\/em><\/strong> and S18×18<\/sub> := 486<\/em><\/strong>.<\/p>\n\n\n\n

<\/p>\n\n\n

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

In this example the magic sums are S3×3<\/sub> :=99<\/em><\/strong>, S6×6<\/sub>:=198<\/em><\/strong>, S10×10<\/sub> :=330<\/em><\/strong>, S14×14<\/sub> :=462<\/em><\/strong> and S18×18<\/sub> := 594<\/em><\/strong>.<\/p>\n\n\n\n

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

Double-Digit Bordered Algebric Magic Square of Order 19<\/h4>\n\n\n\n

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

Result 13: Double-Digit Bordered Algebric Magic Square of Order 19<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles<\/strong> of orders 2×5, 2×9, 2×13 and 2×17 (equality in each case) embedded with a magic square of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we have decimal entries, then the magic sums of orders 7, 11, 15 and 19 are also multiples of 3. The magic sum of orders 7, 11, 15 and 19 are given as S11×11<\/sub> :=7*S\/3<\/strong><\/em><\/em>, S11×11<\/sub> :=11*S\/3<\/strong><\/em>, S15×15<\/sub> :=15*S\/3=5*S<\/strong><\/em> and S19×19<\/sub> :=19*S\/3<\/strong><\/em> where S<\/strong> is the magic sum of order 3. <\/strong>See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

<\/p>\n\n\n

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

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

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

Double-Digit Bordered Algebric Magic Square of Order 20<\/h4>\n\n\n\n

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

Result 14: Double-Digit Bordered Algebric Magic Square of Order 20<\/h4>\n\n\n
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It is a composed of four equal sums magic rectangles<\/strong> of orders 2×6, 2×10, 2×14 and 2×18 (equality in each case) embedded with a magic square of order 4. Since the magic square of order 4 requires magic sum as multiple of 2, otherwise we have decimal entries, then the magic sum of orders, 8, 12 and 16 are also multiple of 2. The magic sum of orders 20, 16, 12 and 8 ares given as S16×16<\/sub> :=5*S<\/em><\/strong><\/em><\/em>,<\/em> S16×16<\/sub> :=4*S<\/em><\/strong><\/em><\/em>, S12×12<\/sub> :=3*S<\/em><\/strong><\/em> and S8×8<\/sub> :=2*S, <\/em><\/strong>where S <\/strong>is the magic sum of order 4.<\/em> See below two examples:<\/p>\n\n\n\n

<\/p>\n\n\n

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

In this example the magic sums are S4×4<\/sub> :=102<\/em><\/strong>, S8×8<\/sub> :=204<\/em><\/strong>, S12×12<\/sub> := 312<\/em><\/strong>, S16×16<\/sub> := 408<\/em><\/strong> and S20×20<\/sub> := 510<\/em><\/strong><\/p>\n\n\n\n

<\/p>\n\n\n

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

In this example the magic sums are S4×4<\/sub> :=152<\/em><\/strong>, S8×8<\/sub> :=304<\/em><\/strong>, S12×12<\/sub> := 456<\/em><\/strong>, S16×16<\/sub> := 608<\/em><\/strong> and S20×20<\/sub> := 760<\/em><\/strong><\/p>\n\n\n\n

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

References: Reduced Entries Magic Squares<\/mark><\/h3>\n\n\n\n

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