Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20

Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20 The whole work is also is available at author’s sites: https://numbers-magic.com/?p=17009 Inder J. Taneja1 Abstract This work brings double-digit cyclic-type algebraic magic squares of orders 7 to 20 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 the entries depending on the types of magic squares. The idea of double-digit [30] is applied to bring these of magic squares. Moreover, we have considered the magic rectangles in a cyclic way, i.e, all the four side in each case are of equal sums in widths and lengths. For similar kind of work for different orders in different styles and ways, the readers are suggested to see author’s work [13, 14, 15, 16, 17, 18, 19, 20, 21]. 1Formerly, Professor of Mathematics, Universidade Federal de Santa Catarina, Florian´ opolis, SC, Brazil (1978-2012). E-mail: [email protected]; Web-sites: http://inderjtaneja.wordpress.com; http://numbers-magic.com; Twitter: @IJTANEJA 1 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 Contents 1 Introduction 3 2 Double-Digit Algebraic magic squares 3 2.1 Double-Digit Bordered Algebraic Magic Square of Order 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Double-Digit Bordered Algebraic Magic Square of Order 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Double-Digit Bordered Algebraic Magic Square of Order 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Double-Digit Bordered Algebraic Magic Square of Order 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Double-Digit Bordered Algebraic Magic Square of Order 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6 Double-Digit Bordered Algebraic Magic Square of Order 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.7 Double-Digit Bordered Algebraic Magic Square of Order 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.8 Double-Digit Bordered Algebraic Magic Square of Order 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.9 Double-Digit Bordered Algebraic Magic Square of Order 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.10Double-Digit Bordered Algebraic Magic Square of Order 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.11Double-Digit Bordered Algebraic Magic Square of Order 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.12Double-Digit Bordered Algebraic Magic Square of Order 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.13Double-Digit Bordered Algebraic Magic Square of Order 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.14Double-Digit Bordered Algebraic Magic Square of Order 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 Author’s Contribution to Magic Squares and Recreation of Numbers 34 2 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 1 Introduction This work brings double-digit or double-layer algebraic magic squares of orders 7 to 20 for reduced entries. Sometimes, these types of magic squares, we call as self-made, beacause they are complete in themselves. Just choose the entries and magic sum, we always get a magic square. We know that magic sum of a magic square of order n having 1 to n2number of entries is given by Sn×n:= n×(1 + n2) 2 In this work the entries are written as variables and their combinations. The work is based on the four equal sums magic rectangles in each border with width as 2. The size of the length the magic rectangle depends on the orders of the magic squares. For simplicity, these types of magic squares we call as cyclic-type. Similar kind of study for the different styles and orders refer author’s work [13, 14, 15, 16, 17, 18, 19, 20, 21]. For double-digit work for sequential entries refer [24, 25, 26, 27, 28, 29, 30]. 2 Double-Digit Algebraic magic squares The section bring results and examples of reduced entries algebraic magic squares complete in itself for the orders 7 to 20. These are based on equal sums magic rectangles of equal width. For our study we shall make use of reduced entries algebraic magic squares of orders 3, 4, 5 and 6. These are given as follows. Result 2.1. A magic square of order 3 with reduced entries algebraic is given by •Details Knowing only two entries A1 and A2 and the magic sum M, we can construct a magic square of order 3. To avoid decimal entries, we must always consider the magic sum as a multiple of 3. This magic square can seen in the web-site of F. Gaspalou [3]. Result 2.2. Let’s consider the following magic square of order 4 3 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 •Details It is a magic square of order 4 for reduced entries. The letter S represents the magic sum of order 4. This magic square is also due to F. Gaspalou [3]. Result 2.3. Let’s consider following reduced entries magic square of order 5: •Details It is an algebraic pandiagonal magic square of order 5 with reduced entries. The letter M represents the magic sum of order 5. This magic square can seen in F. Gaspalou [3] web-site. See below two examples. Result 2.4. Let’s consider a following magic square of order 6 with reduced entries: •Details It is an algebraic magic square of order 6 for reduced entries. It constructed based on four equal sums magic squares order 3. The letter S represents the magic sum of order 3. 4 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 We shall frequently use these for magic square in our work based on magic rectangles. These four magic squares shall also be in the middle of the magic square. Magic squares of orders 3, 4 and 5 can be seen in F. Gaspalou’s site [3]. The work on order 6 is given by the author [14, 15]. 2.1 Double-Digit Bordered Algebraic Magic Square of Order 7 Result 2.5. Let’s consider a following magic square of order 7 with reduced entries: •Details It is a composed of four equal sums magic rectangles of order 2×5embedded with a magic square of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we have decimal entries, then the magic sum of order 7 is also multiple of 3. The magic sum of order 7 is given as S7×7:= 7S 3, where Sis the magic sum of order 3. Below are two examples based on the Result 2.5. Example 2.1. Let’s consider following two examples based on the Result 2.5: 5 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 The magic sums are: •First example: S3×3:= 21 and S7×7:= 49. •Second example: S3×3:= 24 and S7×7:= 56. 2.2 Double-Digit Bordered Algebraic Magic Square of Order 8 Result 2.6. Let’s consider a following magic square of order 8 with reduced entries: •Details It is a composed of four equal sums magic rectangles of order 2×6embedded with a magic square of order 4. Since the magic square of order 4 requires magic sum as multiple of 2, otherwise we have decimal entries, then the magic sum of order 8 is also 6 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 multiple of 2. The magic sum of order 8 is given as S8×8:= 2 S, where Sis the magic sum of order 4. Below are two examples based on the Result 2.6. Example 2.2. Let’s consider following two examples based on the Result 2.6: The magic sums are: •First example: S4×4:= 26 and S8×8:= 52. •Second example: S4×4:= 34 and S8×8:= 68. 2.3 Double-Digit Bordered Algebraic Magic Square of Order 9 Result 2.7. Let’s consider a following magic square of order 9 with reduced entries: 7 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 •Details It is a composed of four equal sums magic rectangles of order 2×7embedded with a pandiagonal magic square of order 5. Here the magic sum of order 5 don’t have any condition, but the magic sum of order 9 depends on number 5, i.e., S9×9:= 9S 5, where Sis the magic sum of order 5. This requires the magic sum of order 9 should be multiple of 5, otherwise we may have decimal entries. Below are two examples based on the Result 2.7. Example 2.3. Let’s consider following two examples based on the Result 2.7: 8 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 The magic sums are: •First example: S5×5:= 45 and S9×9:= 81. •Second example: S5×5:= 50 and S9×9:= 90. 2.4 Double-Digit Bordered Algebraic Magic Square of Order 10 Result 2.8. Let’s consider a following magic square of order 10 with reduced entries: •Details It is a composed of four equal sums magic rectangles of order 2×8embedded with a magic square of order 6. This magic square of order 6 is again composed of four equal sums magic squares of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we have decimal entries, then the magic sum of order 10 is also a multiple of 3. The magic sum of order 10 is given as S10×10 := 10 S 3, where Sis the magic sum of order 3. In this case the magic sum of order 6 is given as S6×6:= 2 S. Below are two examples based on the Result 2.8. Example 2.4. Let’s consider following two examples based on the Result 2.8: 9 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 The magic sums are: •First example: S5×5:= 65,S9×9:= 117 and S13×13 := 169 •Second example: S5×5:= 70,S9×9:= 126 and S13×13 := 182. 2.8 Double-Digit Bordered Algebraic Magic Square of Order 14 Result 2.12. Let’s consider a following magic square of order 14 with reduced entries: 16 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 •Details It is a composed of four equal sums magic rectangles of orders 2×8and 2×12 (equal in each case) embedded with a magic square of order 6. This magic square of order 6 is again composed of four equal sums magic squares of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we may have decimal entries. The magic sums of orders 10 and 14 are also multiple of 3. The magic sum of orders 10 and 14 are given as S10×10 := 10 S 3and S14×14 := 14 S 3, where Sis the magic sum of order 3. In this case the magic sum of order 6 is given as S6×6:= 2 S. Below are two examples based on the Result 2.12 Example 2.8. Let’s consider following two examples based on the Result 2.12: 17 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 The magic sums are: •First example: S3×3:= 66,S6×6:= 132,S10×10 := 220 and S14×14 := 108. •Second example: S3×3:= 75,S6×6:= 150,S10×10 := 250 and S14×14 := 350. 2.9 Double-Digit Bordered Algebraic Magic Square of Order 15 Result 2.13. Let’s consider a following magic square of order 15 with reduced entries: 18 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 •Details It is a composed of four equal sums magic rectangles of orders 2×8and 2×12 (equal in each case) embedded with a magic square of order 6. This magic square of order 6 is again composed of four equal sums magic squares of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we may have decimal entries. The magic sums of orders 10 and 14 are also multiple of 3. The magic sum of orders 10 and 14 are given as S10×10 := 10 S 3and S14×14 := 14 S 3, where Sis the magic sum of order 3. In this case the magic sum of order 6 is given as S6×6:= 2 S. Below are two examples based on the Result 2.13 Example 2.9. Let’s consider following two examples based on the Result 2.13: 19 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 The magic sums are: •First example: S3×3:= 63,S7×7:= 147,S11×11 := 231 and S15×15 := 315. •Second example: S3×3:= 72,S7×7:= 168,S11×11 := 264 and S15×15 := 360. 2.10 Double-Digit Bordered Algebraic Magic Square of Order 16 Result 2.14. Let’s consider a following magic square of order 16 with reduced entries: 20 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 •Details It is a composed of four equal sums magic rectangles of orders 2×6,2×10 and 2×14 (equality in each case) embedded with a magic square of order 4. Since the magic square of order 4 requires magic sum as multiple of 2, otherwise we have decimal entries, then the magic sum of orders, 8, 12 and 16 are also multiple of 2. The magic sum of orders 16, 12 and 8 ares given as S16×16 := 4 S,S16×16 := 3 Sand S16×16 := 2 S, where Sis the magic sum of order 4. Below are two examples based on the Result 2.14 Example 2.10. Let’s consider following two examples based on the Result 2.14: 21 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 The magic sums are: •First example: S4×4:= 102,S8×8:= 204,S12×12 := 312 and S16×16 := 408. •Second example: S4×4:= 110,S8×8:= 220,S12×12 := 330 and S16×16 := 440. 2.11 Double-Digit Bordered Algebraic Magic Square of Order 17 Result 2.15. Let’s consider a following magic square of order 17 with reduced entries: 22 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 •Details It is a composed of four equal sums magic rectangles of orders 2×7,2×11 and 2×15 (equality in each case) embedded with a pandiagonal magic square of order 5. The magic square of order 5 don’t requires any condition but the magic sums of orders 17, 13 and 9 depends on five. Thus, we must have these magic sums as multiples of 5 to avoid decimal entries. The magic sum of orders 17, 13 and 9 ares given as S17×17 := 17 S 5,S13×13 := 13 S 5, and S9×9:= 9S 5, where Sis the magic sum of order 5. Below are two examples based on the Result 2.15 Example 2.11. Let’s consider following two examples based on the Result 2.15: 23 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 and 24 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 The magic sums are: •First example: S5×5:= 85,S9×9:= 153,S13×13 := 221 and S17×17 := 289. •Second example: S5×5:= 105,S9×9:= 189,S13×13 := 273 and S17×17 := 357. 2.12 Double-Digit Bordered Algebraic Magic Square of Order 18 Result 2.16. Let’s consider a following magic square of order 18 with reduced entries: 25 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 The internal part is the exactly the same as given in the Result 2.14. The complete structure can be seen in a web-site link: https://numbersmagic.com/?p=17009. •Details It is a composed of four equal sums magic rectangles of orders 2×6,2×10,2×14 and 2×18 (equality in each case) embedded with a magic square of order 4. Since the magic square of order 4 requires magic sum as multiple of 2, otherwise we have decimal entries, then the magic sum of orders, 8, 12 and 16 are also multiple of 2. The magic sum of orders 20, 16, 12 and 8 ares given as S20×20 := 5 S,S16×16 := 4 S,S12×12 := 3 Sand S8×8:= 2 S, where Sis the magic sum of order 4. Below are two examples based on the Result 2.18 32 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 Example 2.14. Let’s consider following two examples based on the Result 2.18: and 33 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 The magic sums are: •First example: S4×4:= 102,S8×8:= 204,S12×12 := 312,S16×16 := 408 and S20×20 := 510. •Second example: S4×4:= 152,S8×8:= 304,S12×12 := 456,S16×16 := 608 and S20×20 := 760. 3 Author’s Contribution to Magic Squares and Recreation of Numbers For author’s contribution to magic squares and recreation of numbers please see the links below: •Inder J. Taneja, Magic Squares, (i) https://numbers-magic.com/?p=668 (ii) https://inderjtaneja.wordpress.com/2019/06/27/publications-magic-squares/ 34 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 •Inder J. Taneja, Recreation of Numbers, (i) https://numbers-magic.com/?p=671 (ii) https://inderjtaneja.wordpress.com/2019/06/27/publications-recreation-of-numbers/ References [1] A. de Winkel, The magic Encyclopedia, http://home.wanadoo.nl/aaledewinkel/Encyclopedia/index.html [2] C. Boyer, Multimagic Squares and Cubes, http://www.multimagie.com [3] F. Gaspalou, “Magic Squares” http://www.gaspalou.fr/magic-squares/ [4] W. Trump,http://www.trump.de/magic-squares [5] H. White, Bordered Magic Squares - http://budshaw.ca/Download.html [6] W.S. Andrews, Magic squares and Cubes, Dover Publications, New York •Reduced Entries Algebraic Magic Squares: Dates and Days of the Year [7] Inder J. Taneja, Magic Squares of Orders 3 to 7 Representing Dates and Days of the Year 2025, Zenodo, May 04, 2025, pp. 1-474, https://doi.org/10.5281/zenodo.15338142. [8] Inder J. Taneja, Magic Squares of Order 8 Representing Days and Dates of the Year 2025, Zenodo, May 04, 2025, pp. 1-134, https://doi.org/10.5281/zenodo.15338246. [9] Inder J. Taneja, Magic Squares of Order 9 Representing Days and Dates of the Year 2025, Zenodo, May 09, 2025, pp. 1-132, https://doi.org/10.5281/zenodo.15375349. [10] Inder J. Taneja, Magic Squares of Order 10 Representing Days and Dates of the Year 2025, Zenodo, May 21, 2025, pp. 1-59, https://doi.org/10.5281/zenodo.15481738. [11] Inder J. Taneja, Magic Squares of order 11 Representing Days and Dates of the Year 2025, Zenodo, June 02, 2025, pp. 1-111, https://doi.org/10.5281/zenodo.15576562. [12] Inder J. Taneja, Magic Squares of order 12 Representing Days and Dates of the Year 2025, Zenodo, June 10, 2025, pp. 1-43, https://doi.org/10.5281/zenodo.15631884. 35 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 •Reduced Entries Algebraic Magic Squares: Different Styles [13] Inder J. Taneja, Reduced Entries Magic and Semi-Magic Squares of Orders 3, 5, 7 and 9, Zenodo, July 01, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.15783321. [14] Inder J. Taneja, Reduced Entries Magic and Semi-Magic Squares of Orders 4, 6, 8 and 10, Zenodo, July 05, 2025, pp. 1-85, https://doi.org/10.5281/zenodo.15814675. [15] Inder J. Taneja,Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Orders 3 to 7, Zenodo, September 29, 2025, pp. 1-59, https://doi.org/10.5281/zenodo.17219769. [16] Inder J. Taneja, Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 8, Zenodo, September 23, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17186001. [17] Inder J. Taneja, Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 9, Zenodo, August 27, 2025, pp. 1-92, https://doi.org/10.5281/zenodo.16955571. [18] Inder J. Taneja, Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 10, Zenodo, September 21, 2025, pp. 1-132, https://doi.org/10.5281/zenodo.17171790. [19] Inder J. Taneja, Self-Made Algebraic Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17330815. [20] Inder J. Taneja, Self-Made Algebraic Semi-Magic Squares of Order 11, Zenodo, October 12, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17330822. [21] Inder J. Taneja, Reduced Entries Algebraic Magic and PanMagic Squares of Order 12, Zenodo, July 23, 2025, pp. 1-74, https://doi.org/10.5281/zenodo.16370556. [22] Inder J. Taneja, Reduced Entries Algebraic Semi-Magic Squares of Order 12, Zenodo, July 23, 2025, pp. 1-60, https://doi.org/10.5281/zenodo.15692014. [23] Inder J. Taneja, Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032. •Double-Digit Magic Squares [24] Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 10, 14, 18 and 22, Zenodo, April, 30, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.7880931. [25] Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 26 and 30, Zenodo, April, 30, 2023, pp. 1-45, https://doi.org/10.5281/zenodo.7880937. 36 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Double-Digit Cyclic-Type Bordered Reduced Entries Algebraic Magic Squares of Orders 7 to 20, Zenodo, November 21, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.17675032 [26] Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 36 and 40, Zenodo, May, 04, 2023, pp. 1-41, https://doi.org/10.5281/zenodo.7896709. [27] Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 34 and 38, Zenodo, May 10, 2023, pp. 1-45, https://doi.org/10.5281/zenodo.7922571. [28] Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 28 and 32, Zenodo, April, 26, 2023, pp. 1-36, https://doi.org/10.5281/zenodo.7866981. [29] Inder J. Taneja, Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24, Zenodo, April, 26, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.7866956. [30] Inder J. Taneja, New Concepts in Magic Squares: Double Digits Bordered Magic Squares of Orders 7 to 108, Zenodo, August 09, 2023, pp. 1-30, https://doi.org/10.5281/zenodo.8230214. ————————————————- 37