{"id":16767,"date":"2025-10-12T00:30:11","date_gmt":"2025-10-12T03:30:11","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=16767"},"modified":"2025-10-12T03:31:11","modified_gmt":"2025-10-12T06:31:11","slug":"self-made-algebraic-semi-magic-squares-of-order-11","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=16767","title":{"rendered":"Self-Made Algebraic Semi-Magic Squares of Order 11"},"content":{"rendered":"\n

This work brings self-made algebraic<\/strong> magic squares of order 11 for reduced entries<\/strong>. By reduced <\/strong>or less entries<\/strong>, we understand that instead of normal n2<\/sup><\/em><\/strong> entries of a magic square order n<\/em><\/strong>, we are using less number of entries. Moreover, in these situations the entries are no more sequential numbers<\/strong>. These entries are non-sequential positive<\/strong> and negative numbers<\/strong>. Sometimes, we call these kind of magic squares as self-made<\/strong>. It means that these are complete in themselves. Just put the values of entries<\/strong> and choose the magic sum<\/strong>, we get a magic square. In some cases, there maybe decimal<\/strong> or fractional <\/strong>values of the entries depending on the types of magic squares. Different kind of magic squares are used to bring these self-made magic squares<\/strong>. These are of type, block-wise<\/strong>, cornered<\/strong>, single-digit bordered<\/strong>, double-digit bordered<\/strong>, etc. In some cases, the idea of magic rectangles<\/strong> is also applied. In each case, the magic rectangles<\/strong> are considered with equal width and length. For similar kind of work for different orders the readers are suggested to see author\u2019s work given in references.<\/p>\n\n\n\n

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

Once again, self-made<\/strong> means that they are complete in themselves: once you choose the entries and the magic sum, a magic square will always result. These squares can contain integer<\/strong>, decimal<\/strong>, or fractional<\/strong> values.<\/p>\n\n\n\n

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

For more details see the link given below:

Inder J. Taneja<\/strong>, Self-Made Algebraic Magic Squares of Order 11, Zenodo<\/strong>, October 12, 2025, pp. 1-58, https:\/\/doi.org\/10.5281\/zenodo.17330815<\/a>.
Inder J. Taneja<\/strong>, Self-Made Algebraic Semi-Magic Squares of Order 11, Zenodo<\/strong>, October 12, 2025, pp. 1-77, https:\/\/doi.org\/10.5281\/zenodo.17330822<\/a>.

See below the details of the work with Examples<\/p>\n\n\n\n

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

Self-Made Algebraic Semi-Magic Squares of Order 11<\/h3>\n\n\n\n

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

Below are 23 self-made<\/strong> algebraic magic squares of order 11 for reduced entries<\/strong>.<\/p>\n\n\n\n

Result 1: Double-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a double-digit bordered<\/strong> magic square of order 11 embedded with a single-digit bordered<\/strong> magic square of order 7 having pandiagonal magic square of order 5 in the middle. The magic rectangles of orders 2\u00d77 are of equal width and length. The letters S, T and R represents the magic squares of orders 5, 7 and 11 respectively. The difference between R and T should be multiple of 4 to avoid decimal entries. It is a semi-magic <\/strong>square at one diagonal. It becomes magic square by applying the condition T := (7\/5)* S<\/strong>. Also the difference between R and T should be multiple of 4<\/em><\/em>. Below are two examples. First<\/strong> one is semi-magic square<\/strong>, and the second<\/strong> one is magic square<\/strong>. <\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition T:=(7\/5)* S<\/strong>, where S and T are the magic squares of orders 5 and 7. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 2: Double-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a double-digit bordered <\/strong>magic square of order 11 embedded with a single-digit bordered <\/strong>magic square of order 7. It again contains a cornered <\/strong>magic square of order 3 at the upper-left corner. The magic rectangles of orders 2\u00d73 and 2\u00d77 are of equal width and length in each case. The letters M, S, T and R represents the magic squares of orders 3, 5, 7 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the condition T:=(7\/5)*S<\/strong>. To avoid decimal entries the magic square of order 3 should be multiple of 3. Also the difference between R and T should be multiple of 4. See below two examples. One is semi-magic<\/strong> square<\/strong>, and the second one is magic square<\/strong>.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition T:=(7\/5)* S<\/strong>, where S and T are the magic squares of orders 5 and 7. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 3: Double-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a double-digit bordered<\/strong> magic square of order 11 embedded with a single-digit bordered <\/strong>magic squares of order 7 and 5 having magic square of order 3 in the middle. The magic rectangles of order 2\u00d77 are of equal width and length. The letters M, S, T and R represents the magic squares of orders 3, 5, 7 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic <\/strong>square by applying the conditions T:=(7\/5)*S<\/strong> and S:=(5\/3)*M<\/strong>. Moreover the magic square of order 3 should be multiple of 3 to avoid decimal entries. Also the difference between R and T should be multiple of 4. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square. <\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the conditions T:=(7\/5)*S<\/strong> and S:=(5\/3)*M<\/strong><\/em><\/strong>, where M, S and T are the magic squares of orders 3, 5 and 7. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 4: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of order 11 embedded with a double-digit bordered<\/strong> magic squares of order 9 having a pandiagonal magic square of order 5 in the middle. The magic rectangles of order 2\u00d75 are of equal width and length. The letters S, L and R represents the magic squares of orders 5, 9 and 11 respectively. It is a semi-magic <\/strong>square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)*L<\/strong>. To avoid decimal entries the difference between R and L should be multiple of 2. See below two examples. One is semi-magic<\/strong> square, and the second one is magic <\/strong>square. <\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition R:=(11\/9)*L<\/strong><\/em><\/strong>, where L and R are the magic squares of orders 9 and 11. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 5: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit<\/strong> bordered magic square of order 11 embedded with a double-digit bordered<\/strong> magic squares of order 9 having a cornered<\/strong> magic square of order 5 in the middle. It contains magic square of order 3 at the upper-left corner. The magic rectangles of order 2\u00d73 and 2\u00d75 are of equal width and length in each case. The letters M, S, L and R represents the magic squares of orders 3, 5, 9 and 11 respectively. It is a semi-magic square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)* L.<\/strong> To avoid decimal entries the difference between S and L should be multiple of 4. Moreover, the magic square of order 3 should be multiple of 3. See below two examples. One is semi-magic <\/strong>square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition R:=(11\/9)* L<\/strong><\/em><\/strong>, where L and R are the magic squares of orders 9and 11. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 6: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of order 11 embedded with a double-digit bordered<\/strong> magic squares of order 9 having again a single-digit bordered <\/strong>magic square of order 5, where magic square of order 3 is in the middle. The magic rectangles of order 2\u00d75 are of equal width and length. The letters M, S, L and R represents the magic squares of orders 3, 5, 9 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the conditions R:=(11\/9)* L <\/strong>and S:=(5\/3)*M<\/strong>. To avoid decimal entries the difference between S and L should be multiple of 4. Moreover, the magic square of order 3 should be multiple of 3. See below two examples. One is semi-magic <\/strong>square, and the second one is magic<\/strong> square. <\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the conditions R:=(11\/9)* L <\/strong>and S:=(5\/3)*M<\/strong><\/strong>, where S, M, L and R are the magic squares of orders 3, 5, 9 and 11. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 7: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of order 11 embedded with a cornered <\/strong>magic squares of order 9 having a double-digit bordered<\/strong> magic square of order 7 at the upper-left corner. It contains magic square of order 3 in the middle. The magic rectangles of orders 2\u00d73 and 2\u00d77 are of equal width and length in each case. The letters M, T, L and R represents the magic squares of orders 3, 7, 9 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)*L<\/strong>. To avoid decimal entries the difference between T and S should be multiple of 4. Moreover, the magic square of order 3 should be multiple of 3. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition R:=(11\/9)*L<\/strong>,<\/strong> where L and R are the magic squares of orders 9 and 11 respectively. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 8: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of order 11 embedded with a cornered<\/strong> magic squares of order 9 and 7 having a pandiagonal magic square of order 5 at the upper-left corner. The magic rectangles of orders 2\u00d75 and 2\u00d77 are of equal width and length in each case. The letters S, T, L and R represents the magic squares of orders 5, 7, 9 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)*L<\/strong>. To avoid decimal entries the difference between T and L, and L and R should be multiple of 2. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition R:=(11\/9)*L<\/strong>, where S and T are the magic squares of orders 5 and 7. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 9: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of order 11 embedded with a cornered<\/strong> magic squares of order 9, 7 and 5 having a magic square of order 3 at the upper-left corner. The magic rectangles of orders 2\u00d73, 2\u00d75 and 2\u00d77 are of equal width and length in each case. The letters M, S, T, L and R represents the magic squares of orders 3, 5, 7, 9 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)*L<\/strong>. To avoid decimal entries the pairs (M, S), (S, T) and (T, L) should be multiple of 2. Moreover, the magic sum of order 3 should be multiple of 3. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition R:=(11\/9)*L<\/strong>, where S and T are the magic squares of orders 5 and 7. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 10: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of order 11 embedded with a cornered<\/strong> magic squares of order 9 and 7 having a single-digit bordered<\/strong> magic square of order 5 at the upper-left corner. It contains a magic square of order 3 in the middle. The magic rectangles of orders 2\u00d75 and 2\u00d77 are of equal width and length in each case. The letters M, S, T, L and R represents the magic squares of orders 3, 5, 7, 9 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)*L<\/strong>. To avoid decimal entries the pairs (M, S), (S, T) and (T, L) should be multiple of 2. Moreover, the magic sum of order 3 should be multiple of 3. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition R:=(11\/9)*L<\/strong><\/em><\/strong>, where L and R are the magic squares of orders 9 and 11. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 11: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of order 11 embedded with a cornered<\/strong> magic squares of order 9 having a magic square of order 7 at the upper-left corner. The magic rectangles of order 2\u00d77 are of equal width and length. The letters T, L and R represents the magic squares of orders 7, 9 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)*L<\/strong>. To avoid decimal entries the pairs (T, L) should be multiple of 2. See below two examples. One is semi-magic<\/strong> square, and the second one is a magic <\/strong>square. <\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition R:=(11\/9)*L<\/strong>, where L and R are the magic squares of orders 9 and 11. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

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

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

It is a single-digit bordered<\/strong> magic square of order 11 embedded with a cornered <\/strong>magic squares of order 9 having a single-digit bordered <\/strong>magic square of order 7 at the upper-left corner embedded with a pandiagonal <\/strong>magic square of order 5 in the middle. The magic rectangles of order 2\u00d77 are of equal width and length. The letters S, T, L and R represents the magic squares of orders 5, 7, 9 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)*L<\/strong>. To avoid decimal entries the pairs (T, L) should be multiple of 2. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition R:=(11\/9)*L<\/strong>, where R and L are the magic squares of orders 11 and 9. An extra conditon T:=(7\/5)*S<\/strong> is also used to bring block of order 7 as a magic square. This magic square of order 11 contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 13: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered <\/strong>magic square of order 11 embedded with a cornered<\/strong> magic squares of order 9 having a single-digit bordered<\/strong> magic square of order 7 at the upper-left corner. It is again embedded with a cornered<\/strong> magic square of order 5 having a magic square of order 3 at the upper-lelft corner. The magic rectangles of orders 2\u00d73 and 2\u00d77 are of equal width and length in each case. The letters M, S, T, L and R represents the magic squares of orders 3, 5, 7, 9 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)*L<\/strong>. To avoid decimal entries the pairs (M, S) and (T, L) should be multiple of 2. The magic square of order 3 should also be multiple of 3. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic or semi-magic<\/strong> squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition S:=(11\/9)*L<\/strong>, where S and T are the magic squares of orders 5 and 7. An extra conditon T:=(7\/5)*S<\/strong> is also used to bring block of order 7 as a magic square. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 14: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of order 11 embedded with a cornered<\/strong> magic squares of order 9 having a single digit bordered<\/strong> magic squares of orders 7 and 5 at the upper-left corner embedded with a magic square of order 3 in the middle. The magic rectangles of order 2\u00d77 are of equal width and length. The letters M, S, T, L and R represents the magic squares of orders 3, 5, 7, 9 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)* L<\/strong>. To avoid decimal entries the pairs (T, L) should be multiple of 2. Moreover, the magic square of order 3 should be multiple of 3. See below two examples. One is semi-magic <\/strong>square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition R:=(11\/9)*L<\/strong>, where R and L are the magic squares of orders 11 and 9. Extra conditons T:=(7\/5)*S<\/strong> and An extra conditon S:=(5\/3)*M<\/strong> are considered to bring block of order 7 as a magic square. The magic square of order 11 contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 15: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit<\/strong> bordered magic square of order 11 embedded with a block-wise<\/strong> magic squares of order 9 having nine equal sums semi-magic<\/strong> magic squares of order 3. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the condition R:=(11\/9)*L<\/strong>. The letter M, L and R represents the magic squares of orders 3, 9 and 11 respectively, where L:=3*M<\/strong>. The equal sums blocks of order 3 as semi-magic squares are considered to avoid considering magic square of order 3 as multiple of 3. See below two examples. One is semi-magic<\/strong> square, and the second one is magic <\/strong>square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the condition R:=(11\/9)*L<\/strong>, where R and L are the magic squares of orders 11 and 9. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 16: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered <\/strong>magic square of orders 11 and 9 embedded with a double-digit bordered<\/strong> magic squares of order 7 having a magic square of order 3 in the middle. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the conditions R:=11\/9)*L<\/strong> and L:=(9\/7)*T<\/strong>. The letter M, T, L and R represents the magic squares of orders 3, 7, 9 and 11 respectively. To avoid decimal entries we must have the pairs (T, L) and (L, R) as multiple of 2. Also magic square of order 3 as a multiple of 3. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square. <\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the conditions R:=11\/9)*L<\/strong> and L:=(9\/7)*T<\/strong><\/strong>, where R, L and T are the magic squares of orders 11, 9 and 7 respectively. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 17: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of orders 11 and 9 embedded with a magic squares of order 7. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the conditions R:=(11\/9)*L<\/strong> and L:=(9\/7)*T<\/strong>. The letter T, L and R represents the magic squares of orders 7, 9 and 11 respectively. To avoid decimal entries we must have the pairs (T, L) and (L, R) as multiple of 2. See below two examples. One is semi-magic <\/strong>square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the conditions R:=(11\/9)*L<\/strong> and L:=(9\/7)*T<\/strong>. The letter T, L and R represents the magic squares of orders 7, 9 and 11 respectively. . It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 18: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of orders 11 and 9 embedded with a cornered magic squares of order 7. It contains a pandiagonal <\/strong>magic square of order 5 at the upper-left corner. The magic rectangles of order 2\u00d75 are of equal width and length. The letter S, T, L and R represents the magic squares of orders 5, 7, 9 and 11 respectively. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the conditions R:=11\/9)*L<\/strong> and L:=(9\/7)*T<\/strong>. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square. <\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the conditions R:=11\/9)*L<\/strong> and L:=(9\/7)*T<\/strong><\/strong>, where T, L and R are the magic squares of orders 7, 9 and 11. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 19: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of orders 11 and 9 embedded with a cornered magic squares of order 7 and 5. It contains a magic square of order 3 at the upper-left corner. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the conditions R:=11\/9)*L<\/strong> and L:=(9\/7)*T<\/strong>. The magic rectangles of orders 2\u00d73 and 2\u00d75 are of equal width and length in each case. The letter M, S, T, L and R represents the magic squares of orders 3, 5, 7, 9 and 11 respectively. To avoid decimal entries we must have the pair (S, T) as multiple of 2. Also the magic square of order 3 should be multiple of order 3. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square. <\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the conditions R:=11\/9)*L<\/strong> and L:=(9\/7)*T<\/strong><\/strong>, whereT, L and R are the magic squares of orders 7, 9 and 11. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 20: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered <\/strong>magic square of orders 11 and 9 embedded with a cornered <\/strong>magic squares of order 7. It contains a single-digit bordered<\/strong> magic square of order 5 embedded with a magic square of order 3 in the middle. It is a semi-magic <\/strong>square at one diagonal. It becomes magic square by applying the conditions R:=(11\/9)*L<\/strong> and L:=9\/7)*T<\/strong>. The magic rectangles of order 2\u00d75 are of equal width and length. The letter M, S, T, L and R represents the magic squares of orders 3, 5, 7, 9 and 11 respectively. To avoid decimal entries we must have the pair (S, T) as multiple of 2. See below two examples. One is semi-magic<\/strong> square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the conditions R:=11\/9)*L<\/strong> and L:=(9\/7)*T<\/strong><\/strong>, whereT, L and R are the magic squares of orders 7, 9 and 11. Also we applied an extra condition S:=(5\/3)*M<\/strong><\/strong> bring a magic square of order 5, where M and S are magic squares of orders 3 and 5 respectively. This magic square of order 11 contains the following sub-magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 21: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of orders 11, 9 and 7 embedded with a pandiagonal<\/strong> magic squares of order 5. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the conditions R:=(11\/9)*L<\/strong>, L:=(9\/7)*T<\/strong> and T:=7\/5)*S<\/strong>. The letters S, T, L and R represents the magic squares of orders 5, 7, 9 and 11 respectively. See below two examples. One is semi-magic<\/strong> square, and the second one is magic square. <\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the conditions R:=(11\/9)*L<\/strong>, L:=(9\/7)*T<\/strong> and T:=7\/5)*S<\/strong>, where S, T, L and R represents the magic squares of orders 5, 7, 9 and 11 respectively. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 22: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of orders 11, 9 and 7 embedded with a cornered <\/strong>magic squares of order 5. It contains a magic square of order 3 at the upper- left corner. It is a semi-magic<\/strong> square at one diagonal. It becomes magic square by applying the conditions R:=(11\/9)*L<\/strong>, L:=(9\/7)*T<\/strong> and T:=7\/5)*S<\/strong>. The letters S, T, L and R represents the magic squares of orders 5, 7, 9 and 11 respectively. To avoid decimal entries we must have the (S,M) as multiple of 2. Moreover, the magic square of order 3 should also be multiple of 3. See below two examples. One is semi-magic<\/strong> square, and the second one is magic <\/strong>square. <\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the conditions R:=(11\/9)*L<\/strong>, L:=(9\/7)*T<\/strong> and T:=7\/5)*S<\/strong>, where S, T, L and R represents the magic squares of orders 5, 7, 9 and 11 respectively. It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

Result 23: Single-Digit Bordered Algebraic Semi-Magic Square of Order 11<\/h4>\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

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

It is a single-digit bordered<\/strong> magic square of orders 11, 9, 7 and 5 embedded with a magic square of order 3. It is a semi-magic<\/strong> square at one diagonal. The letters M, S, T, L and R represents the magic squares of orders 3, 5, 7, 9 and 11 respectively. It becomes magic square by applying the conditions R:=(11\/9)*L<\/strong>, L:=(9\/7)*T<\/strong>, T:=7\/5)*S<\/strong> and S:=(5\/3)*M<\/strong>. The letters M, S, T, L and R represents the magic squares of orders 3, 5, 7, 9 and 11 respectively. . To avoid decimal entries the magic square of order 3 should be multiple of 3. See below two examples. One is semi-magic <\/strong>square, and the second one is magic<\/strong> square.<\/em><\/p>\n\n\n\n

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

First Example: Semi-Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It contains the following magic squares:<\/p>\n\n\n

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

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

Second Example: Magic Square<\/h4>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

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

It is obtained by applying the conditions R:=(11\/9)*L<\/strong>, L:=(9\/7)*T<\/strong>, T:=7\/5)*S<\/strong> and S:=(5\/3)*M<\/strong>, where M, S, T, L and R represents the magic squares of orders 3, 5, 7, 9 and 11 respectively. . It contains the following magic squares:<\/p>\n\n\n\n

<\/p>\n\n\n

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

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

References<\/h3>\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