{"id":16447,"date":"2025-07-24T20:35:25","date_gmt":"2025-07-24T23:35:25","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=16447"},"modified":"2025-12-02T22:53:35","modified_gmt":"2025-12-03T01:53:35","slug":"reduced-entries-algebraic-semi-magic-squares-of-order-12","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=16447","title":{"rendered":"Reduced Entries Algebraic Semi-Magic 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 rectangles 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 second part on semi-magic squares of order 12. For the first part on magic<\/strong> and panmagic<\/strong> squares refer the link<\/a>. <\/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

This work on semi-magic<\/strong> squares. Every semi-magic<\/strong> squrare is brought to a magic<\/strong> square based on certain conditions. These conditions given by
1. L = 5*R\/6
2. T = 4*L\/5
3. S = 3*T\/4
4. M = 2*S\/3
The letter M, S, T, L and R represents the magic or semi-magic sums of orders 4, 6, 8, 10 and 12. All the reduced entries semi-magic appearing in this work are semi-magic<\/strong> only in one diagonal<\/strong>, i.e., upper-diagonal<\/strong>. <\/p>\n\n\n\n

Based on above conditions for semi-magic<\/strong> squares to bring as magic squares, the work is divided in four sub-groups<\/strong>:
1. Single-Condition Magic Squares.
2. Double-Conditions Magic Squares.
3. Three-Conditions Magic Squares .
4. Four-Conditions Magic Squares.<\/p>\n\n\n\n

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

Single Condition Algebraic Semi-Magic Squares of Order 12<\/h3>\n\n\n\n

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

Below are few examples of reduced entries algebraic semi-magic<\/strong> squares of order 12. In order to bring them as magic squares we applied one<\/strong> of the above four<\/strong> conditions.<\/p>\n\n\n\n

Result 1: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square of order 12 embedded with a pandiagonal<\/strong> magic square of order 10 composed by four equal sums pandiagonal<\/strong> magic squares of order 5. The letters S, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 5, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 1<\/strong> as given above, i.e., L=5*R\/6, where L and R are magic sums of orders 10 and 12 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 2: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square of order 12 embedded with a double-digit bordered<\/strong> magic square of order 10 having a magic square of order 6 in the middle. The letters S, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 6, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 1<\/strong> as given above, i.e., L=5*R\/6, where L and R are magic sums of orders 10 and 12 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 3: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square of order 12 embedded with a double-digit bordered<\/strong> magic square of order 10 having a pandiagonal<\/strong> magic square of order 6 composed of four equal sums semi-magic<\/strong> squares of order 3. The letters M, S, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 3, 6, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 1<\/strong> as given above, i.e., L=5*R\/6, where L and R are magic sums of orders 10 and 12 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 4: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> semi-magic<\/strong> square of order 12 embedded with double-digit<\/strong> magic square of order 10. The inner part is a cornered<\/strong> magic square of order 6 with magic square of order 4. The letters M, S, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 6, 10 and 12. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 1<\/strong> as given above, i.e., L=5*R\/6, where L and R are magic sums of orders 10 and 12 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 5: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> semi-magic<\/strong> square of order 12 embedded with cornered<\/strong> magic squares of orders 6, 8 and 10. The letters M, S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 6, 8, 10 and 12. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 1<\/strong> as given above, i.e., L=5*R\/6, where L and R are magic sums of orders 10 and 12 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 6: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> semi-magic<\/strong> square of order 12 embedded with cornered<\/strong> magic square of orders 8 and 10, where the magic square of order 6 is in the upper-left corner. The letters S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 1<\/strong> as given above, i.e., L=5*R\/6, where L and R are magic sums of orders 10 and 12 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 7: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> semi-magic<\/strong> square of order 12 embedded with cornered<\/strong> magic square of orders 8 and 10. It contains a pandiagonal<\/strong> magic square of order 6 at upper-left corner composed of four equal sums semi-magic<\/strong> squares of order 3. The letters M, S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 3, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 1<\/strong> as given above, i.e., L=5*R\/6, where L and R are magic sums of orders 10 and 12 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 8: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> semi-magic<\/strong> square of order 12 embedded with a cornered<\/strong> magic square of order 10, where in the upper-left corner we have a double-digit bordered<\/strong> magic square of order 8 having magic square of order 4 at the center. The letters M, S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 6, 8, 10 and 12. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 1<\/strong> as given above, i.e., L=5*R\/6, where L and R are magic sums of orders 10 and 12 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 9: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> semi-magic<\/strong> square of order 12 embedded with a cornered<\/strong> magic square of order 10, where in the upper-left corner we have a striped pandiagonal<\/strong> magic square of order 8. The letters T, L and R represents respectively the magic<\/strong> or semi-magic sums<\/strong> of orders 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 1<\/strong> as given above, i.e., L=5*R\/6, where L and R are magic sums of orders 10 and 12 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 10: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> semi-magic<\/strong> square of order 12 embedded with a cornered<\/strong> magic square of order 10, where in the upper-left corner we have a pandiagonal<\/strong> magic square of order 8 composed by four equal sums pandiagonal<\/strong> magic squares of order 4. The letters S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 1<\/strong> as given above, i.e., L=5*R\/6, where L and R are magic sums of orders 10 and 12 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 11: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a double-digit bordered semi-magic<\/strong> square embedded with single-digit bordered semi-magic<\/strong> square of order 8. The inner block is a magic square of order 6. It is semi-magic<\/strong> only due to order 8 being a semi-magic<\/strong>. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 3<\/strong> as given above, i.e., S=3*T\/4, where S and T are magic sums of orders 6 and 8 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 12: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a double-digit bordered semi-magic<\/strong> square embedded with single-digit bordered<\/strong> semi-magic<\/strong> square of order 8. The inner block is a pandiagonal<\/strong> magic square of order 6 composed with four equal sums semi-magic<\/strong> squares of order 3. It is semi-magic<\/strong> only due to order 8 being a semi-magic<\/strong>. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 3<\/strong> as given above, i.e., S=3*T\/4, where S and T are magic sums of orders 6 and 8 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 13: Single Condition Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a double-digit bordered semi-magic<\/strong> square embedded with single-digit bordered<\/strong> semi-magic<\/strong> square of order 8. The inner block is a cornered<\/strong> magic square of order 6 with magic square of order 4 at the upper-left corner. It is semi-magic<\/strong> only due to order 8 being a semi-magic<\/strong>. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the condition 3<\/strong> as given above, i.e., S=3*T\/4, where S and T are magic sums of orders 6 and 8 respectively.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Double Conditions Algebraic Semi-Magic Squares of Order 12<\/h3>\n\n\n\n

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

Below are few examples of reduced entries algebraic semi-magic<\/strong> squares of order 12. In order to bring them as magic squares we applied two<\/strong> of the above four<\/strong> conditions.<\/p>\n\n\n\n

Result 14: Double Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic squares of orders 10 and 12 embedded with a pandiagonal<\/strong> magic square of order 8 composed by four equal sums magic squares of order 4. The letters S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the conditions <\/strong>1 and 2 as given above, i.e., L=5*R\/6 and T=4*L\/5.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 15: Double Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square of orders 10 and 12 embedded with a striped pandiagonal<\/strong> magic square of order 8. It is composed by 8 equal sums magic rectangles of order 2×4. The letters T, L and R represents the magic<\/strong> or semi-magic<\/strong> sums of orders 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the conditions <\/strong>1 and 2 as given above, i.e., L=5*R\/6 and T=4*L\/5.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 16: Double Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square of orders 10 and 12 embedded with a cornered<\/strong> magic square of order 8 having a magic square of order 6 at upper-left corner. The letters S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the conditions <\/strong>1 and 2 as given above, i.e., L=5*R\/6 and T=4*L\/5.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 17: Double Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square of orders 10 and 12 embedded with a cornered<\/strong> magic square of order 8 with a pandiagonal<\/strong> magic square of order 6 at upper-left corner. This magic square of order 6 is composed of four equal sums semi-magic <\/strong>squares of order 3. The letters M, S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 3, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the conditions <\/strong>1 and 2 as given above, i.e., L=5*R\/6 and T=4*L\/5.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 18: Double Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square embedded with a cornered magic squares of orders 6 and 8 having magic square of order 4 at upper-left corner. The letters M, S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the conditions <\/strong>1 and 2 as given above, i.e., L=5*R\/6 and T=4*L\/5.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 19: Double Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square of order 10 and 12 embedded with cornered<\/strong> magic squares of order 8 with single-digit bordered<\/strong> magic square of order 6 at the upper-left corner having magic square of order 4. The letters M, S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and second is magic<\/strong> satisfying the conditions <\/strong>1 and 2 as given above, i.e., L=5*R\/6 and T=4*L\/5.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 20: Double Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square of order 12 embedded with double-digit bordered<\/strong> magic square of order 10. The inner part is again a single-digit bordered<\/strong> magic square of order 6 with magic square of order 4. The letters M, S, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 6, 10 and 12. Below are two examples. First is semi-magic<\/strong> and the second is magic<\/strong> satisfying the conditions <\/strong>1 and 4 as given above, i.e., L=5*R\/6 and M=2*S\/3.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 21: Double Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a double-digit bordered semi-magic<\/strong> square embedded with a single-digit bordered semi-magic<\/strong> squares of orders 6 and 8. The inner block is a magic square of order 4. The letters M, S, T and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 6, 8 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and the second is magic<\/strong> satisfying the conditions <\/strong>3 and 4 as given above, i.e., S = 3*T\/4 and M=2*S\/3.<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Triple Conditions Algebraic Semi-Magic Squares of Order 12<\/h3>\n\n\n\n

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

Below are few examples of reduced entries algebraic semi-magic<\/strong> squares of order 12. In order to bring them as magic squares we applied three <\/strong>of the above four<\/strong> conditions.<\/p>\n\n\n\n

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

Result 22: Triple Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square of orders 8, 10 and 12 embedded with a magic square of order 6. The letters S, T, L and R represents the magic<\/strong> or semi-magic<\/strong> sums of orders 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and the second is magic<\/strong> satisfying the conditions <\/strong>1, 2 and 3 as given above, i.e., L = 5*R\/6, T = 4*L\/5 and S = 3*T\/4. <\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 23: Triple Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic <\/strong>square of orders 8, 10 and 12 embedded with a pandiagonal <\/strong>magic square of order 6. This magic square of order 6 is again composed of four equal sums semi-magic<\/strong> squares of order 3. The letters M, S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 3, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and the second is magic<\/strong> satisfying the conditions <\/strong>1, 2 and 3 as given above, i.e., L = 5*R\/6, T = 4*L\/5 and S = 3*T\/4. <\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 24: Triple Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered semi-magic<\/strong> square so order 8, 10 and 12 embedded with a cornered magic square of order 6 having magic square of order 4 at upper-left corner. The letters M, S, T, L and R represents respectively the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and the second is magic<\/strong> square satisfying the conditions <\/strong>1, 2 and 3 as given above, i.e., L = 5*R\/6, T = 4*L\/5 and S = 3*T\/4. <\/p>\n\n\n\n

<\/p>\n\n\n

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

Triple Conditions Algebraic Semi-Magic Squares of Order 12<\/h3>\n\n\n\n

Below is a single example of a reduced entries algebraic semi-magic<\/strong> squares of order 12. In order to bring it as magic square we applied all the four <\/strong>conditions given above.<\/p>\n\n\n\n

Result 25: Four Conditions Algebraic Semi-Magic Squares of Order 12<\/h4>\n\n\n
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It is a single-digit bordered semi-magic squares of orders 6, 8, 10 and 12 having magic square of order 4 at the inner part. The letters M, S, T, L and R represents the magic<\/strong> or semi-magic<\/strong> sums of orders 4, 6, 8, 10 and 12 respectively. Below are two examples. First is semi-magic<\/strong> and the second is magic<\/strong> satisfying the conditions <\/strong>1, 2, 3 and 4 as given above, i.e., L = 5*R\/6, T = 4*L\/5, S = 3*T\/4 and M=2*S\/3. <\/p>\n\n\n\n

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References<\/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