Self-Made Algebraic Semi-Magic Squares of Order 11

Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Self-Made Algebraic Semi-Magic Squares of Order 11 Inder J. Taneja1 The whole work as pdf files is available at author’s sites: https://numbers-magic.com/?p=16767 This work is without use of any kind of programming language Abstract This work brings semi-magic squares of order 11 for reduced entries. By reduced or less entries, we understand that instead of normal n2entries of a magic square order n, we are using less number of entries. Moreover, in these situations the entries are no more sequential numbers. These entries are non-sequential positive and negative numbers. Sometimes, we call these kind of magic squares as self-made. It means that these are complete in themselves. Just put the values of entries and choose the magic sum, we get a magic square. In some cases, there maybe decimal or fractional values of entries depending on the types of magic squares. Different kind of magic squares are used to bring these self-made magic squares. These are of type, block-wise,cornered,single-digit bordered,double-digit bordered, etc. In some cases, the idea of magic rectangles is also applied. In each case, the magic rectangles are considered with equal width and length. This work for the self-made algebraic magic squares order 11. This work is available online at above given link. For similar kind of work for different orders the readers are suggested to author’s work given in references [30]-[45]. 1Formerly, Professor of Mathematics, Federal University of Santa Catarina, Florianópolis, SC, Brazil (1978-2012). E-mail: [email protected]; Web-sites: http://inderjtaneja.wordpress.com; http://numbers-magic.com; Twitter: @IJTANEJA; Instagram: @crazynumbers. 1 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Contents 1 Introduction 2 2 Magic Squares of Order 11 3 2.1 Self-MadeSemi-MagicSquaresoforder11 .................................................. 3 3 Author’s Contribution to Magic Squares and Recreation Numbers 74 1 Introduction This work brings self-made algebraic semi-magic squares of order 11. By self-made or reduced or less entries, we understand that instead of normal n2entries of a magic square order n, we are using less numbers of entries, where the magic square is complete in itself. Putting any integer values for these less entries, we shall get always a magic square. Moreover, in these situations the entries are no more sequential numbers. These entries are non-sequential positive and/or negative numbers. In some cases, these may be decimal or fractional values depending on the way of choosing the entries. Sometime to avoid decimal or fractional entries we apply certain conditions. These conditions depends on the types of magic squares. The name self-made is not known in the literature of magic squares. It is being introduced for the first time. The work is based on different types of magic squares, such as, pandiagonal, block-wise, cornered, single-digit bordered, double-digit bordered, etc. It is not necessary, but we worked with magic rectangles having equal width and length for the same category within a magic square. If we relax this condition, i.e., by considering only equality of width, still we have good results. This work for the order 11 brings magic,semi-magic and pandiagonal magic squares. It is divided in three parts. The first part is on magic squares, the second part on semi-magic squares and the third part on pandiagonal magic squares of order 11. This part is on semi-magic squares of order 11. For the first part on magic squares refer [42]. For more details on these kind of work refer author’s previous works [30]-[45]. The table below give the quantities and references for each order. Order Magic Squares Semi-Magic Squares Pandiagonal Magic Squares Total 31 1 0 2[38] 42 0 2 4[38] 52 1 3 6[38] 65 1 6 12 [38] 75 3 8 16 [38] 88 5 13 26 [39] 910 9 18 28 [40] 10 15 14 29 58 [41] 11 25 23 to be done 48 [42, 43] 12 28 25 to be done 53 [44, 45] 2 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 The author [30, 31, 32, 33, 34, 35] also worked similar kind of work but from different point of view. This work is for the magic squares of orders 3 to 12 for the dates and days of the year 2025, where the dates are few entries and days are the sums of the magic squares. 2 Magic Squares of Order 11 Below are three different examples of magic squares of order 11 for sequential entries from 1 to 121. The first example is known by cornered magic squares. The second example is famous as single-digit bordered magic square. The third example is known as double-digit borered magic square. For more details on these kind of magic squares refer author’s work [24, 25, 26, 27]. 3 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 2.1 Self-Made Semi-Magic Squares of order 11 Below are 23 examples of self-made semi-magic squares for reduced entries. In order to bring them as magic squares we shall use some condions. These conditions are given below: R:= 11 9 ×L(1) L:= 9 7 ×T(2) T:= 7 5 ×S(3) S:= 5 3 ×M(4) where the letters M, S, T, L and Rrepresents the magic or semi-magic squares of orders 3,5,7,9and 11. 4 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.1. Let’s consider following self-made semi-magic square of order 11 with reduced entries: •Details: It is a double-digit bordered magic square of order 11 embedded with a single-digit bordered magic square of order 7 having pandiagonal magic square of order 5 in the middle. The magic rectangles of orders 2×7are 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 square at one diagonal. It becomes magic square by applying the condition given in (3), i.e., T:= 7 5×S. See below two examples. One is semi-magic square, and the second one is magic square. 5 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.1. Let’s consider an example of semi-magic square of order 11 based on the Result 2.1: The semi-magic square given in Example 2.1 includes the following magic and semi-magic squares: 6 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.2. Let’s consider an example of magic square of order 11 based on the Result 2.1: It includes the following magic squares: 7 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.2. Let’s consider following self-made semi-magic square of order 11 with reduced entries: •Details: It is a double-digit bordered magic square of order 11 embedded with a single-digit bordered magic square of order 7. It again contains a cornered magic square of order 3 at the upper-left corner. The magic rectangles of orders 2×3and 2×7 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 square at one diagonal. It becomes magic square by applying the condition given in (3), i.e., T:= 7 5×S. 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 square, and the second one is magic square. 8 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.3. Let’s consider an example of semi-magic square of order 11 based on the Result 2.2: The semi-magic square given in Example 2.3 includes the following magic and semi-magic squares: 9 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.8. Let’s consider an example of magic square of order 11 based on the Result 2.4: The above Example 2.8 is obtained by applying the condition R:= 11 9×L. It includes the following magic squares: 16 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.5. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of order 11 embedded with a double-digit bordered magic squares of order 9 having a cornered 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 2×3and 2×5are 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 given in (1), i.e., R:= 11 9×L. 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 square, and the second one is magic square. 17 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.9. Let’s consider an example of semi-magic square of order 11 based on the Result 2.5: The magic square given in Example 2.9 includes the following magic and semi-magic squares: 18 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.10. Let’s consider an example of magic square of order 11 based on the Result 2.5: The above Example 2.10 is obtained by applying the condition R:= 11 9×L. It includes the following magic squares: 19 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.6. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of order 11 embedded with a double-digit bordered magic squares of order 9 having again a single-digit bordered magic square of order 5, where magic square of order 3 is in the middle. It contains magic square of order 3 at the upper-left corner. The magic rectangles of order 2×5are 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 square at one diagonal. It becomes magic square by applying the conditions given in (1) and (4), i.e., R:= 11 9×Land S:= 5 3×M. 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 square, and the second one is magic square. 20 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.11. Let’s consider an example of semi-magic square of order 11 based on the Result 2.6: The magic square given in Example 2.11 includes the following magic and semi-magic squares: 21 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.12. Let’s consider an example of magic square of order 11 based on the Result 2.6: The above Example 2.12 is obtained by applying the conditions R:= 11 9×Land S:= 5 3×M. It includes the following magic squares: 22 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.7. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of order 11 embedded with a cornered magic squares of order 9 having a doubledigit bordered 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 2×3and 2×7are of equal width and length. The letters M, T, L and R represents the magic squares of orders 3, 7, 9 and 11 respectively. It is a semi-magic square at one diagonal. It becomes magic square by applying the condition given in (1), i.e., R:= 11 9×L. 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 square, and the second one is magic square. 23 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.13. Let’s consider an example of semi-magic square of order 11 based on the Result 2.7: The magic square given in Example 2.13 includes the following magic and semi-magic squares: 24 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.14. Let’s consider an example of magic square of order 11 based on the Result 2.7: The above Example 2.14 is obtained by applying the condition R:= 11 9×L. It includes the following magic squares: 25 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.10. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of order 11 embedded with a cornered magic squares of order 9 and 7 having a single-digit bordered 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 2×5and 2×7are 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 square at one diagonal. It becomes magic square by applying the condition given in (1), i.e., R:= 11 9×L. 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 square, and the second one is magic square. 32 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.19. Let’s consider an example of semi-magic square of order 11 based on the Result 2.10: The magic square given in Example 2.19 includes the following magic and semi-magic squares: 33 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.20. Let’s consider an example of magic square of order 11 based on the Result 2.10: The above Example 2.20 is obtained by applying the condition R:= 11 9×L. Even though it is not necesary, but an extra condition S:= 5 3×Mis applied to bring a semi-magic suuare of order 5 as magic square. See below the magic squares appearing in second example of Example 2.19. 34 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.11. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of order 11 embedded with a cornered magic squares of order 9 having a magic square of order 7 at the upper-left corner. The magic rectangles of order 2×7are 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 square at one diagonal. It becomes magic square by applying the condition given in (1), i.e., R:= 11 9×L. To avoid decimal entries the pairs (T, L) should be multiple of 2. See below two examples. One is semi-magic square, and the second one is magic square. 35 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.21. Let’s consider an example of semi-magic square of order 11 based on the Result 2.11: The magic square given in Example 2.21 includes the following magic and semi-magic squares: 36 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.22. Let’s consider an example of magic square of order 11 based on the Result 2.11: The above Example 2.22 is obtained by applying the condition R:= 11 9×L. It includes the following magic squares: 37 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.12. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of order 11 embedded with a cornered magic squares of order 9 having a singledigit bordered magic square of order 7 at the upper-left corner embedded with a pandiagonal magic square of order 5 in the middle. The magic rectangles of order 2×7are 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 square at one diagonal. It becomes magic square by applying the condition given in (1), i.e., R:= 11 9×L. To avoid decimal entries the pairs (T, L) should be multiple of 2. See below two examples. One is semi-magic square, and the second one is magic square. 38 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.23. Let’s consider an example of semi-magic square of order 11 based on the Result 2.12: The magic square given in Example 2.23 includes the following magic and semi-magic squares: 39 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.24. Let’s consider an example of magic square of order 11 based on the Result 2.12: The above Example 2.24 is obtained by applying the condition R:= 11 9×L. An extra condition R:= 7 5×Lis applied to bring a semi-magic square of order 7 to a magic square. It includes the following magic squares: 40 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.13. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of order 11 embedded with a cornered magic squares of order 9 having a singledigit bordered magic square of order 7 at the upper-left corner. It is again embedded with a cornered magic square of order 5 having a magic square of order 3 at the upper-lelft corner. The magic rectangles of orders 2×3and 2×7are 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 square at one diagonal. It becomes magic square by applying the condition given in (1), i.e., R:= 11 9×L. 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 square, and the second one is magic square. 41 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.29. Let’s consider an example of semi-magic square of order 11 based on the Result 2.15: The magic square given in Example 2.7 includes the following magic and semi-magic squares: 48 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.30. Let’s consider an example of magic square of order 11 based on the Result 2.15: The above Example 2.30 is obtained by applying the conditions R:= 11 9×Land L:= 9 7×T. It includes the following magic squares: Instead of considering semi-magic squares of order 3, we can also consider magic squares of order 3. In this case we need the condition that sums of order 3 should be multiple of order 3 to avoid decimal entries. 49 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.16. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of orders 11 and 9 embedded with a double-digit bordered magic squares of order 7 having a magic square of order 3 in the middle. It is a semi-magic square at one diagonal. It becomes magic square by applying the conditions given in (1) and (2), i.e., R:= 11 9×Land L:= 9 7×T. 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 should be multiple of 3. See below two examples. One is semi-magic square, and the second one is magic square. 50 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.31. Let’s consider an example of semi-magic square of order 11 based on the Result 2.16: The magic square given in Example 2.31 includes the following magic and semi-magic squares: 51 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.32. Let’s consider an example of magic square of order 11 based on the Result 2.16: The above Example 2.32 is obtained by applying the conditions R:= 11 9×Land L:= 9 7×T. It includes the following magic squares: 52 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.17. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of orders 11 and 9 embedded with a magic squares of order 7. It is a semi-magic square at one diagonal. It becomes magic square by applying the conditions given in (1) and (2), i.e., R:= 11 9×Land L:= 9 7×T. 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 square, and the second one is magic square. 53 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.33. Let’s consider an example of semi-magic square of order 11 based on the Result 2.17: The magic square given in Example 2.33 includes the following magic and semi-magic squares: 54 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.34. Let’s consider an example of magic square of order 11 based on the Result 2.17: The above Example 2.20 is obtained by applying the conditions R:= 11 9×Land L:= 9 7×T. It includes the following magic squares: 55 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.18. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of orders 11 and 9 embedded with a cornered magic squares of order 7. It contains apandiagonal magic square of order 5 at the upper-left corner. It is a semi-magic square at one diagonal. It becomes magic square by applying the conditions given in (1) and (2), i.e., R:= 11 9×Land L:= 9 7×T. The magic rectangles of order 2×5are of equal width and length. The letter 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 square, and the second one is magic square. 56 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.35. Let’s consider an example of semi-magic square of order 11 based on the Result 2.18: The magic square given in Example 2.35 includes the following magic and semi-magic squares: 57 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.40. Let’s consider an example of magic square of order 11 based on the Result 2.20: The above Example 2.40 is obtained by applying the conditions R:= 11 9×Land L:= 9 7×T. An extra condition S:= 5 3×Mis also applied to bring block of order 5 as a magic square. It includes the following magic squares: 64 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.21. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of orders 11, 9 and 7 embedded with a pandiagonal magic squares of order 5. It is a semi-magic square at one diagonal. It becomes magic square by applying the conditions given in (1),(1), and (2) and (3), i.e., R:= 11 9×L,L:= 9 7×Tand T:= 7 5×S. 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 square, and the second one is magic square. 65 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.41. Let’s consider an example of semi-magic square of order 11 based on the Result 2.21: The magic square given in Example 2.41 includes the following magic and semi-magic squares: 66 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.42. Let’s consider an example of magic square of order 11 based on the Result 2.21: The above Example 2.42 is obtained by applying the conditions R:= 11 9×L,L:= 9 7×Tand T:= 7 5×S. It includes the following magic squares: 67 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.22. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of orders 11, 9 and 7 embedded with a cornered magic squares of order 5. It contains a magic square of order 3 at the upperleft corner. It is a semi-magic square at one diagonal. It becomes magic square by applying the conditions given in (1),(1), and (2) and (3), i.e., R:= 11 9×L,L:= 9 7×Tand T:= 7 5×S. 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 square, and the second one is magic square. 68 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.43. Let’s consider an example of semi-magic square of order 11 based on the Result 2.22: The magic square given in Example 2.43 includes the following magic and semi-magic squares: 69 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.44. Let’s consider an example of magic square of order 11 based on the Result 2.22: The above Example 2.44 is obtained by applying the conditions R:= 11 9×L,L:= 9 7×Tand T:= 7 5×S.. It includes the following magic squares: 70 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Result 2.23. Let’s consider following magic squares of order 11 with reduced entries: •Details: It is a single-digit bordered magic square of orders 11, 9, 7 and 5 embedded with a magic square of order 3. It is a semimagic 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 given in (1)-(4), i.e., R:= 11 9×L,L:= 9 7×T,T:= 7 5×Sand S:= 5 3×M. See below two examples. One is semi-magic square, and the second one is magic square. 71 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.45. Let’s consider an example of semi-magic square of order 11 based on the Result 2.23: The magic square given in Example 2.45 includes the following magic and semi-magic squares: 72 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com; Self-Made Algebraic Magic and Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822 Example 2.46. Let’s consider an example of magic square of order 11 based on the Result 2.23: The above Example 2.46 is obtained by applying the conditions R:= 11 9×L,L:= 9 7×T,T:= 7 5×Sand S:= 5 3×M. It includes the following magic squares: 73