{"id":16149,"date":"2025-07-24T02:17:29","date_gmt":"2025-07-24T05:17:29","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=16149"},"modified":"2025-07-24T20:41:48","modified_gmt":"2025-07-24T23:41:48","slug":"reduced-entries-algebraic-magic-and-panmagic-squares-of-order-12","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=16149","title":{"rendered":"Reduced Entries Algebraic Magic and Panmagic Squares of Order 12"},"content":{"rendered":"\n

This work brings magic<\/strong>, panmagic<\/strong> and semi-magic<\/strong> squares of order 12 for the reduced entries. By reduced<\/strong> or less entries<\/strong> we understand that instead of considering 144 entries in a sequential way, we are using non-sequential<\/strong> entries in less numbers . These non-sequential<\/strong> entries may be positive<\/strong> and\/or negative<\/strong> numbers. In some cases, these may be decimal<\/strong> or fractional values<\/strong>. It depends on the type of magic squares. Initially, the work is written in terms of letters<\/strong> instead of numbers and then are followed by examples<\/strong>. These kind of magic squares sometimes we call as algebraic magic squares<\/strong>. The complete work is composed of 57 different types of magic squares of order 12. Out of them 28 are magic<\/strong>, 4 are panmagic<\/strong> or pandiagonal<\/strong> and 25 are semi-magic<\/strong> squares. By panmagic we understand that the magic squares are pandiagonal<\/strong>. We have divided this work in two parts. This part is composed of magic <\/strong>and panmagic<\/strong> squares. The second part is with semi-magic<\/strong> squares of order 12. In case of semi-magic<\/strong> squares some conditions are also explained to change them in magic squares. These are based on four types of magic squares, i.e., pandiagonal<\/strong>, cornered<\/strong>, single-digit bordered<\/strong> and double-digit bordered<\/strong> magic squares. This work also include the idea of magic rectangles<\/strong>. In each magic square, the magic rectanges of same order are equal<\/strong> in width<\/strong> and length<\/strong>. For similar kind of work for the orders 3 to 10 see below the reference list. Previously, the author also brought similar kind of work for the orders 3 to 12 for the dates and days of the year 2025, where the dates<\/strong> are few entries<\/strong> and days<\/strong> are the sums<\/strong> of magic squares. For this kind of work also see the reference list. This is extended and enlarged version of author\u2019s previous works. This is the first part only on magic<\/strong> and panmagic<\/strong> squares. The second part of this work on semi-magic<\/strong> squares is given separately. <\/p>\n\n\n\n

Below are the links to download both the works:

Inder J. Taneja<\/strong>, Reduced Entries Algebraic Magic and PanMagic Squares of Order 12, Zenodo<\/strong>, July 23, 2025, pp. 1-74, https:\/\/doi.org\/10.5281\/zenodo.16370556<\/a>.

Inder J. Taneja<\/strong>, Reduced Entries Algebraic Semi-Magic Squares of Order 12, Zenodo, July 23, 2025, pp. 1-60, https:\/\/doi.org\/10.5281\/zenodo.15692014<\/a>.<\/p>\n\n\n\n

Below are details the first part. As specified above this work is based on first work.
According to nature of magic squares, these are divided in sub-groups<\/strong>:
1. Algebraic Pandiagonal Magic Squares of Order 12 (4 results).
2. Algebraic Double-Digit Bordered Magic Squares Order 12 (3 results).
3. Algebraic Cornered Magic Squares Order 12 (6 results).
4. Algebraic Cornered with Double-Digit Bordered Magic Squares Order 12 (5 results).
5. Algberaic Cornered with Single-Digit Bordered Magic Squares Order 12 (14 results).
<\/p>\n\n\n\n

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

Algebraic Pandiagonal Magic Squares of Order 12<\/h3>\n\n\n\n

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

Result 1: Algebraic Pandiagonal Magic Square of Order 12<\/h4>\n\n\n
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It is a pandiagonal<\/strong> magic square of order 12 with 16 equal sums<\/strong> magic squares of order 3. The letter M represents the magic sum of order 3. In this case, R=4*M is the sum of the magic square of order 12. The letter M and R represents the magic squares of orders 3 and 12. Below are two examples.<\/p>\n\n\n

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

Result 2: Algebraic Pandiagonal Magic Square of Order 12<\/h4>\n\n\n
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It is also a pandiagonal<\/strong> magic square of order 12 with 12 equal sums<\/strong> magic squares of order 4. The letter M represents the magic sum of order 4. In this case, R=3*M is the sum of the magic square of order 4. The letters M and R represents the magic sums of orders 4 and 12 respectively. The magic squares of order 4 are also pandiagonals<\/strong>. In order to avoid decimal entries the magic sums of orders should be even<\/strong> numbers. Below are two examples.<\/p>\n\n\n

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The second example is with decimal entries. It because to have odd<\/strong> number magic sum of order 12, we must have magic sum of order 4 also odd numbers. <\/p>\n\n\n\n

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

Result 3: Algebraic Pandiagonal Magic Square of Order 12<\/h4>\n\n\n
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It is also a pandiagonal<\/strong> magic square of order 12 with 18 equal sums magic rectangles<\/strong> of order 2×4<\/strong>. The letter m<\/strong> represents the width of magic rectangle. In this case, R=6*m<\/strong> is the sum of the magic square of order 12, where m is the width<\/strong> of each magic rectangle<\/strong>. These types of magic squares we call as striped<\/strong> magic squares. For more details on striped<\/strong> magic squares refer author’s work<\/a>. Below are two examples.<\/p>\n\n\n

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

Result 4: Algebraic Pandiagonal Magic Square of Order 12<\/h4>\n\n\n
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It is a double-digit bordered<\/strong> magic square of order 12 embedded with four equal sums pandiagonal<\/strong> magic squares of order 4. It is almost a pandiagonal<\/strong> magic square, except at two cross-pandiagonals<\/strong>. In order to bring it as a pandiagonal<\/strong> we must have a condition, i.e., R=3*S, where the letters S, T and R are magic sums of orders 4, 8 and 12. Also T=2*S. The magic squares of order 4 and 8 are pandiagonal<\/strong>. See the examples below.<\/p>\n\n\n

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The magic sums of first example are R12×12<\/sub> := 240<\/strong> and M4×4<\/sub> := 90<\/strong>, and the second example are R12×12<\/sub> := 180<\/strong> and M4×4<\/sub> := 60<\/strong>. The second example is a pandiagonal<\/strong> magic squares. It satisfies the condition R=3*S, i.e. 180=3*60. In both the examples the magic squares of order 4 and 8 are pandiagonals<\/strong>.<\/p>\n\n\n\n

Algebraic Double-Digit Bordered Magic Squares Order 12<\/h3>\n\n\n\n

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

Result 5: Algebraic Double-Digit Bordered Magic Square of Order 12<\/h4>\n\n\n
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It is a double-digit bordered<\/strong> magic square of order 12 embedded with a double-digit bordered<\/strong> magic square of order 8. The internal magic square is of order 4. The magic rectangles of orders 2×4 and 2×8 are of equal width<\/strong> and length<\/strong> in each case. The letters M, T and R represents the magic squares of orders 4, 8 and 12. In order to avoid decimal entries, the magic sums T and R should be of same type, i.e., either even<\/strong> or odd<\/strong>. See below two examples:<\/p>\n\n\n

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

Result 6: Algebraic Double-Digit Bordered Magic Square of Order 12<\/h4>\n\n\n
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It is a double-digit bordered<\/strong> magic square of order 12 embedded with a pandiagonal<\/strong> magic square off order 8. The magic square of order 8 is composed with four equal sums pandiagonal<\/strong> magic squares of order 4. The letters S, T and R represents the magic sums for the magic squares of orders 4, 8 and 12 respectively, where T=2* S. In this case, the magic sum of order 12 is always an even number. To get magic sum as odd numbers, some of the entries may be decimal numbers. See below two examples:<\/p>\n\n\n

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

Result 7: Algebraic Double-Digit Bordered Magic Square of Order 12<\/h4>\n\n\n
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It is a double-digit bordered<\/strong> magic square of order 12 embedded with a pandiagonal<\/strong> magic square off order 8. The magic square of order 8 is composed with equal sum magic rectangles<\/strong> of order 2×4. The letters T<\/strong> and R<\/strong> represents the magic sums for the magic squares of orders 8 and 12 respectively. The letter m<\/strong> represents the width of each magic rectangle<\/strong> of order 2×4. In this case T=4*m. The magic sum of order 12 is always an even number. To get magic sum as odd numbers, some of the entries may be decimal numbers. See below two examples:<\/p>\n\n\n

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Algebraic Cornered Magic Squares Order 12<\/h3>\n\n\n\n

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Result 8: Algebraic Cornered Magic Square of Order 12<\/h4>\n\n\n
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It is an algebraic<\/strong> cornered<\/strong> magic square of order 12, where the magic squares of orders 4, 6, 8 and 10 are in the upper-left corner. The magic rectangles of orders 2×4, 2×6, 2×8 and 2×10 are of equal width and length seperately for each case. The letters M, S, T, L and R represents the magic squares of orders 4, 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums S, T, L and R should be of same type, i.e., either even<\/strong> or odd<\/strong> numbers. See below two examples with even<\/strong> and odd <\/strong>magic sums: <\/p>\n\n\n

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

Result 9: Algebraic Cornered Magic Square of Order 12<\/h4>\n\n\n
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It is also a cornered<\/strong> magic square of order 12 with magic square of order 6 at the upper-left corner. The blocks of orders 8 and 10 are also cornered<\/strong> magic squares of orders 8 and 10. The magic rectangles of orders 2×6, 2×8 and 2×10 are of equal width and length in each case. The letters S, T, L and R represents the magic sums for the magic squares of orders 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums in pairs (S,T), (T, L) and (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

Result 10: Algebraic Cornered Magic Square of Order 12<\/h4>\n\n\n
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It is also a cornered<\/strong> magic square of order 12 with magic square of order 6 at the upper-left corner. This magic square of order 6 is composed of four equal sums<\/strong> magic squares of order 3. The blocks of orders 8 and 10 are also cornered<\/strong> magic squares. The magic rectangles of orders 2×6, 2×8 and 2×10 are of equal width and length in each case. The letters S, T, L and R represents the magic sums for the magic squares of orders 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums in pairs (S,T), (T, L) and (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums. <\/p>\n\n\n

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Since the magic sum of order 6 is always even number, then in order to get odd number magic sum of order 12, then in the pair (S,T) , S is even number and T is odd number. This results in few decimal entries. It given some entries with decimal numbers.<\/p>\n\n\n\n

Result 11: Algebraic Cornered Magic Square Order 12<\/h4>\n\n\n
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It is also a cornered<\/strong> magic square of order 12 with a pandiagonal<\/strong> magic square of order 8 at the upper-left corner containing four equal sums pandiagonal<\/strong> magic square of order 4. The block of order 10 is also a cornered<\/strong> magic square of order 10. The magic rectangles of orders 2×8 and 2×10 are of equal width and length in each case. The letters S, T, L and R represents the magic sums for the magic squares of orders 4, 8, 10 and 12 respectively. In this case T=2*S. To avoid decimal entries, the magic sums in pairs (L, R) and (T,L) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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Since the magic sum of order 8 is always an even number, then in order to get odd number magic sum of order 12, we consider the pair (R, L) with even and odd magic sums resulting in odd number magic sum of order 12. It given some entries with decimal numbers.<\/p>\n\n\n\n

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

Result 12: Algebraic Cornered Magic Square Order 12<\/h4>\n\n\n
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It is also a cornered<\/strong> magic square of order 12 with pandiagonal<\/strong> magic square of order 8 at the upper-left corner containing 8 equal sums magic rectangles of order 2×4. The block of order 10 is also a cornered<\/strong> magic square of order 10. The magic rectangles of orders 2×8 and 2×10 are of equal width and length in each case. The letters T, L and R represents the magic sums for the magic squares of orders 8, 10 and 12 respectively. The letter m<\/em><\/strong> represents the width of magic rectangles appearing in magic squares of order 8. In order to avoid decimal entries, the magic sums in pairs (L, R) and (T,L) should be of same type, i.e., either even<\/strong> or odd<\/strong> numbers. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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Since the magic sum of order 8 is always an even number, then in order to get odd number magic sum of order 12, we considered the pair (R, L) with even and odd magic sums resulting in odd number<\/strong> magic sum of order 12. It given some entries with decimal numbers.<\/p>\n\n\n\n

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

Result 13: Algebraic Cornered Magic Square Order 12<\/h4>\n\n\n
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It is also a cornered<\/strong> magic square of order 12 with pandiagonal<\/strong> magic square of order 10 at the upperr-left corner. It is composed of 4 equal sum pandiagonal<\/strong> magic squares of order 5. The magic rectangles of orders 2×10 are of equal width and length The letters S, L and R represents the magic sums for the magic squares of orders 5, 10 and 12 respectively. Here L=2*S. In order to avoid decimal entries, the magic sums in pair (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong> numbers. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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Since the magic sum of order 10 is always an even number, then in order to get odd number magic sum of order 12, we considered the pair (R, L) with even and odd magic sums resulting in odd number<\/strong> magic sum of order 12. It given some entries with decimal numbers.<\/p>\n\n\n\n

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

Algebraic Cornered with Double-Digit Bordered Magic Squares Order 12<\/h3>\n\n\n\n

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

Result 14: Algebraic Cornered with Double-Digit Bordered Magic Square Order 12<\/h4>\n\n\n
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It is a cornered<\/strong> magic square of order 12 with double-digit bordered<\/strong> magic square of order 10 at the upper-left corner. This magic square of order 10 is having magic square of order 6 in the middle. The magic rectangles of orders 2×6, and 2×10 are of equal width and length in each case. The letters S, L and R represents the magic sums for the magic squares of orders 6, 10 and 12 respectively. In order to avoid decimal entries, the magic sums in pairs (S, L) and (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

Result 15: Algebraic Cornered with Double-Digit Bordered Magic Square Order 12<\/h4>\n\n\n
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It is also a cornered<\/strong> magic square of order 12 with double-digit bordered<\/strong> magic square of order 10 at the upper-left corner. The magic square of order 10 contains a pandiagonal<\/strong> magic square of order 6 in the middle. This magic square of order 6 is again formed by four equal sums magic squares of order 3. The magic rectangles of orders 2×6 and 2×10 are of equal width and length in each case. The letters M, S, L and R represents the magic sums for the magic squares of orders 3, 6, 10 and 12 respectively. In order to avoid decimal entries, the magic sums in pairs (S,L) and (L, R) should be of same type, i.e., either even or odd. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

Since the magic sum of order 6 is always an even number, then in order to get odd number magic sum of order 12, we considered the pair (R, L) with even<\/strong> and odd<\/strong> numbers magic sums resulting in odd number<\/strong> magic sum of order 12. It given some entries with decimal numbers.<\/p>\n\n\n\n

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

Result 16: Algebraic Cornered with Double-Digit Bordered Magic Square Order 12<\/h4>\n\n\n
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It is also a cornered<\/strong> magic square of order 12 with double-digit bordered<\/strong> magic square of order 10 at the upper-left corner embedded with a cornered<\/strong> magic square of order 6 containing magic square of order 4. The magic rectangles of orders 2×4, 2×6 and $2×10 are of equal width and length in each case. The letters M, S, L and R represents the magic squares of orders 4, 6, 10 and 12 respectively. In order to avoid decimal entries, the magic sums S, L and R should be of same type in pairs, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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Result 17: Algebraic Cornered with Double-Digit Bordered Magic Square Order 12<\/h4>\n\n\n
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It is also a cornered<\/strong> magic square of order 12 with double-digit bordered<\/strong> magic square of order 10 at the upper left corner. The inner part is a single-digit bordered <\/strong>magic square of order 6 embedded with magic square of order 4. The magic rectangles of orders 2×6 and 2×10 are of equal width and length in each case. The letters M, S, L and R represents the magic sums for the magic squares of orders 4, 6, 10 and 12 respectively. In order to avoid decimal entries, the of magic sums M, S, L and R should be of same type in pairs, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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Result 18: Algebraic Cornered with Double-Digit Bordered Magic Square Order 12<\/h4>\n\n\n
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It is a cornered<\/strong> magic square of orders 10 and 12 with double-digit bordered<\/strong> magic square of order 8 at the upper left corner embedded with a magic square of order 4. The four magic rectangles of order 2×4 are of equal width and length. Also the magic rectangles of orders 2×8 and 2×10 are of equal width and length in each case. The letters S, T, L and R represents the magic squares of orders 4, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums T, L and R should be of same type in pairs, i.e., either even<\/strong> or odd<\/strong> numbers. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h3>\n\n\n\n

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

Result 19: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
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It is a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 6 at the upper left corner embedded with magic square of order 4. Magic square of orders 8 and 10 are also parts of cornered magic square. Even though the magic square of order 6 may be semi-magic<\/strong> but still we get a magic square of order 12. The magic rectangles of orders 2×6, 2×8 and 2×10 are of equal width and length in each case. The letters M, S, T, L and R represents magic sums for the magic squares of orders 4, 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the of magic sums S, T, L and R should be of same type in pairs, i.e., either even<\/strong> or odd<\/strong> numbers. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

Result 20: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
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It is also a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 8 at the upper left corner again embedded with cornered magic square of order 6. Magic square of order 10 is also a part of cornered<\/strong> magic square. Even though the magic square of order 8 may be semi-magic<\/strong> but still we get a magic square of order 12. The magic rectangles of orders 2×4, 2×8 and 2×10 are of equal width and length in each case. The letters M, S, T, L and R represents magic sums for the magic squares of orders 4, 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums T, L and R should be of same type in pairs , i.e., either even<\/strong> or odd<\/strong> numbers. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

Result 21: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
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\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is also a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 8 at the upper left corner. Magic square of order 10 is also a part of cornered<\/strong> magic square. Even though the magic square of order 8 may be semi-magic<\/strong> but still we get a magic square of order 12. The magic rectangles of orders 2×8 and 2×10 are of equal width and length in each case. The letters M, S, T, L and R represents magic sums for the magic squares of orders 4, 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums T, L and R should be of same type in pairs, i.e., either even<\/strong> or odd<\/strong> numbers. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

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

Result 22: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
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\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is also a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper left corner containing double-digit bordered<\/strong> magic square of order 8. It contains in the middle a magic square of order 4. Even though the magic square of order 10 may be semi-magic<\/strong> but still we get a magic square of order 12. The magic rectangles of orders 2×4 and 2×10 are of equal width and length in each case. The letters S, T, L and R represents magic sums for the magic squares of orders 4, 8, 10 and 12 respectively. In order to avoid decimal entries, the pair of magic sums (S,T) and (L,R) should be of same type, i.e., either even<\/strong> or odd<\/strong> numbers. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

Result 23: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
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\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is also a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper-left corner containing cornered<\/strong> magic squares of orders 6 and 8. This block of order 10 is a reduced entries semi-magic<\/strong> square, but the way it is constructed lead us to a magic square of order 12. The magic rectangles of orders 2×4, 2×6 and 2×10 are of equal width and length in each case. The letters M, S, T, L and R represents magic sums for the magic squares of orders 4, 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums M, S, T, L and R should be of same type in pairs , i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

Result 24: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper-left corner. Order 10 magic square is embedded with a pandiagonal<\/strong> magic square of order 8 formed by equal sum magic rectangles<\/strong> of order 2×4. The block of order 10 may be a semi-magic<\/strong> square but still we can get order 12 as a magic square. The magic rectangles of orders 2×4 and 2×10 are of equal width and length in each case. The letters T, L and R represents the magic sums for the magic squares of orders 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums of the pair (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong> numbers. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

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

Result 25: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper-left corner. Order 10 magic square is embedded with a pandiagonal <\/strong>magic square of order 8 formed by four equal sum pandiagonal<\/strong> magic squares of order 4. The block of order 10 may be a semi-magic<\/strong> square but still we can get order 12 as a magic square. The magic rectangles of order 2×10 are of equal width and length. The letters T, L and R represents the magic sums for the magic squares of orders 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums of the pair (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

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

Result 26: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper-left corner. Order 10 magic square is embedded with a cornered magic square of order 8 having order 6 magic square at the upper-left corner. The block of order 10 may be a semi-magic<\/strong> square but still we can get order 12 as a magic square. The magic rectangles of orders 2×6 and 2×10 are of equal width and length in each case. The letters S, T, L and R represents the magic sums for the magic squares of orders 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums of the pairs (S, T) and (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

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

Result 27: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper-left corner. Order 10 magic square is embedded again with a cornered<\/strong> magic square of order 8 having order 6 pandiagonal<\/strong> magic square at the middle. This order 6 magic square is again composed of four equal sums magic squares of order 3. The block of order 10 may be a semi-magic<\/strong> square but still we can get order 12 as a magic square. The magic rectangles of orders 2×6 and 2×10 are of equal width and length of the same type. The letters M, S, T, L and R represents the magic sums for the magic squares of orders 3, 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums of the pair (S, T) and (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

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

Result 28: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is also a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper left corner containing cornered<\/strong> magic square of order 8. This cornered<\/strong> magic square of order 8 again contains a single-digit bordered<\/strong> magic square of order 6 with a magic square of order 4. The blocks of orders 6 and 10 are reduced entries semi-magic <\/strong>squares, but the way they are constructed lead us to a magic square of order 12. The magic rectangles of orders 2×6 and 2×10 are of equal width and length in each case. The letters M, S, T, L and R represents magic sums for the magic squares of orders 4, 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the pair of magic sums (S, T) and (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

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

Result 29: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper-left corner. Order 10 magic square is embedded again with a single-digit bordered<\/strong> magic square of order 8 having order 6 magic square at the middle. The blocks of orders 8 and 10 may be a semi-magic<\/strong> squares but still we can get order 12 as a magic square. The magic rectangles of orders 2×10 are of equal width and length. The letters S, T, L and R represents the magic sums for the magic squares of orders 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums of the pair (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

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

Result 30: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper-left corner. Order 10 magic square is embedded again with a single-digit bordered<\/strong> magic square of order 8 having order 6 magic square at the middle. This magic square of order 8 is again composed with four equal sums magic squares of order 3. The blocks of orders 8 and 10 may be a semi-magic<\/strong> squares but still we can get order 12 as a magic square. The magic rectangles of orders 2×10 are of equal width and length. The letters M, S, T, L and R represents the magic sums for the magic squares of orders 3, 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums of the pair (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

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

Result 31: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is also a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper left corner containing cornered<\/strong> magic square of order 6 with a magic square of order 4. The blocks of orders 8 and 10 are reduced entries semi-magic<\/strong> squares, but the way they are constructed lead us to a magic square of order 12. The magic rectangles of orders 2×4 and 2×10 are of equal width and length in each case. The letters M, S, T, L and R represents the magic sums for the magic squares of orders 4, 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the pair of magic sums (S, T) and (L, R) should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

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

Result 32: Algebraic Cornered with Single-Digit Bordered Magic Squares Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

It is also a cornered<\/strong> magic square of order 12 with single-digit bordered<\/strong> magic square of order 10 at the upper left corner containing magic square of order 4. The blocks of orders 6, 8 and 10 are reduced entries semi-magic<\/strong> squares, but the way they are constructed lead us to a magic square of order 12. The two magic rectangles of orders 2×10 are of equal width and length in each case. The letters M, S, T, L and R represents the magic sums for the magic squares of orders 4, 6, 8, 10 and 12 respectively. In order to avoid decimal entries, the magic sums L and R should be of same type, i.e., either even<\/strong> or odd<\/strong>. Below are two examples with even<\/strong> and odd <\/strong>magic sums.<\/p>\n\n\n

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

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

References<\/h3>\n\n\n\n

Part 1: Representing Days and Date<\/h3>\n\n\n\n
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  1. Inder J. Taneja<\/strong>, Magic Squares of Orders 3 to 7 in Representing Dates and Days of the Year 2025, Zenodo<\/strong>, May 04, 2025, pp. 1-474, https:\/\/doi.org\/10.5281\/zenodo.15338142<\/a>.\n