Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19

Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19 The whole work is also is available at author’s sites: https://numbers-magic.com/?p=17179 Inder J. Taneja1 Abstract This work brings double-digit or double-layer algebraic magic squares of odd orders from 5 to 19 for reduced entries. This study include three types of algebraic magic squares, i.e., cyclic-type, flat-type and corner-type. The cyclic-type and the flat-type are two different ways of writing double-digit magic squares. In this work, we always use magic rectangles of width 2 except in the middle or corner, where there are magic squares of order 3 or 5. The idea of double-digit and cornered magic squares for sequential entries is already studied by the authors. For details refer the author’s work on double-digit [23, 31, 25, 26, 27, 28, 29] and cornered [30, 31, 32, 33, 34, 35, 36]. Combing three works of author [37, 38, 39], we have algebraic magic squares of even and odd orders from 4 to 20. It includes, double-digits,cornered and striped magic squares. For similar kind of work for different orders in different styles and designs, 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 Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 Contents 1 Introduction 3 2 Double-Digit Algebraic Striped Magic Squares 3 2.1 Double-Digit Algebraic Magic Square of Order 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Cornered................................................................. 3 2.2 Double-Digit Algebraic Magic Squares of Order 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Cyclic-Type ............................................................... 5 2.2.2 Flat-Type................................................................. 6 2.2.3 Cornered................................................................. 7 2.3 Double-Digit Algebraic Magic Squares of Order 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Cyclic-Type ............................................................... 8 2.3.2 Flat-Type................................................................. 10 2.3.3 Cornered................................................................. 11 2.4 Double-Digit Algebraic Magic Squares of Order 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.1 Cyclic-Type ............................................................... 12 2.4.2 Flat-Type................................................................. 14 2.4.3 Cornered................................................................. 15 2.5 Double-Digit Algebraic Magic Squares of Order 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5.1 Cyclic-Type ............................................................... 17 2.5.2 Flat-Type................................................................. 18 2.5.3 Cornered................................................................. 20 2.6 Double-Digit Algebraic Magic Squares of Order 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.1 Cyclic-Type ............................................................... 22 2.6.2 Flat-Type................................................................. 24 2.6.3 Cornered................................................................. 26 2.7 Double-Digit Algebraic Magic Squares of Order 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7.1 Cyclic-Type ............................................................... 28 2.7.2 Flat-Type................................................................. 31 2.7.3 Cornered................................................................. 33 2.8 Double-Digit Algebraic Magic Squares of Order 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.8.1 Cyclic-Type ............................................................... 35 2.8.2 Flat-Type................................................................. 37 2.8.3 Cornered................................................................. 40 3 Author’s Contribution to Magic Squares and Recreation of Numbers 43 2 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 1 Introduction This work brings double-digit or double-layer algebraic magic squares of odd orders from 5 to 19 for reduced entries. This study include three types of algebraic magic squares, i.e., cyclic-type, flat-type and corner-type.Cyclic-type and flat-type are two different ways of writing as double-digit magic squares. In this work, we always use magic rectangles of width 2 except in the middle or corner, where there is a magic square of order 3 or 5. The idea of double-digit and cornered magic squares for sequential entries is already studied by the authors. For details see the reference lists given at the end of this work. 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 By algebraic magic squares we understand that the entries of a magic square are variables and their combinations. Instead of sequential entries, we have non-sequential entries. These can be positive, negative or decimal numbers. Summarizing , this work brings double-digit or double-layer algebraic magic squares of odd orders from 5 to 19 for reduced entries. Sometimes, these types of magic squares, we call as self-made, because they are complete in themselves. Just choose the entries and magic sum, we always get a magic square. The idea of double-digit for magic squares for sequential entries is studied by the author [29] for the first time. This work is for nonsequential entries. Moreover, we have considered the magic rectangles in a cyclic,flat or cornered way. 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]. For double-digit work for sequential entries refer [23, 24, 25, 26, 27, 28, 29]. Combing all the three works appearing [37, 38, 39], finally, we have algebraic magic squares of even and odd orders from 4 to 20. It includes, double-digits,cornered and striped magic squares. 2 Double-Digit Algebraic Striped Magic Squares The section bring results and examples of reduced entries algebraic magic squares complete in itself for odd orders from 5 to 19. These are of three types: cyclic,flat and corner. From order 7 onwords there are three different ways of writing different magic squares in each case. 2.1 Double-Digit Algebraic Magic Square of Order 5 2.1.1 Cornered Result 2.1. An algebraic cornered magic square of order 5 is given by 3 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic cornered magic square of order 5 composed of two magic rectangles of orders 2×5and 2×5. In this case, the magic sums are S3×3:=Sand S5×5:=5S 3. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 3 should be multiple of 3. Below are two examples. Example 2.1. Let’s consider following two examples based on the Result 2.1: The magic sums are: •First example: S3×3:=30, S5×5:=45 and m:=18. •Second example: S3×3:=33, S5×5:=55 and and m:=22. There are few more types of algebraic magic squares of order 5. For details refer Taneja [13, 15]. 2.2 Double-Digit Algebraic Magic Squares of Order 7 Below are three different ways of writing magic square of order 7, i.e., cyclic,flat and cornered. 4 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 2.2.1 Cyclic-Type Result 2.2. Let’s consider an algebraic magic square of order 7 with reduced entries: •Details It is an algebraic magic square of order 7 composed of four equal sums magic rectangles of orders 2×5and embedded with a magic square of order 3. Since the external four strips are of equal sums, we can name it is as an algebraic cyclic-type magic square of order 7. In this case, the magic sums are S3×3:=Sand S7×7:=7S 3. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 3 should be multiple of 3. Below are two examples. Example 2.2. Let’s consider following two examples based on the Result 2.2: 5 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 The magic sums are: •First example: S3×3:=21, S7×7:=49 and m:=14. •Second example: S3×3:=24, S7×7:=56 and m:=16. 2.2.2 Flat-Type Result 2.3. Let’s consider an algebraic magic square of order 7 with reduced entries: •Details It is an algebraic magic square of order 7 composed of two equal sums magic rectangles of orders 2×7and two equal sum magic rectangles of order 2×3embedded with a magic square of order 3. For simplicity, this types of magic sequares we call as flat-type. Thus we have an algebraic flat-type magic square of order 7. In this case, the magic sums are S3×3:=Sand S7×7:=7S 3. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 3 should be multiple of 3. Below are two examples. Example 2.3. Let’s consider following two examples based on the Result 2.3: 6 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 The magic sums are: •First example: S3×3:=57, S7×7:=133 and m:=38. •Second example: S3×3:=60, S7×7:=140 and m:=40. 2.2.3 Cornered Result 2.4. Let’s consider an algebraic magic square of order 7 with reduced entries: 7 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic cornered, where the magic squares of orders 3 and 5 are at the upper-left corner. In this case, the magic sums are S3×3:=S, S5×5:=5S 3and S7×7:=7S 3. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 3 should be multiple of 3. Below are two examples. Example 2.4. Let’s consider following two examples based on the Result 2.4: The magic sums are: •First example: S3×3:=51, S5×5:=85, S7×7:=119 and m:=34. •Second example: S3×3:=63, S5×5:=105, S7×7:=147 and m:=42. 2.3 Double-Digit Algebraic Magic Squares of Order 9 Below are three different ways of writing magic square of order 9, i.e., cyclic,flat and cornered. 2.3.1 Cyclic-Type Result 2.5. Let’s consider an algebraic magic square of order 9 with reduced entries: 8 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic cyclic-type magic square of order 9 composed of four equal sums magic rectangles of orders 2×7and embedded with a magic square of order 5. In this case, the magic sums are S5×5:=Sand S9×9:=9S 5. The width of magic rectangles is given as m:=2S 5. In order to avoid decimal entries the magic sum of order 5 should be multiple of 5. Below are two examples. Example 2.5. Let’s consider following two examples based on the Result 2.5: The magic sums are: 9 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic cornered, where the magic squares of orders 11, where the magic squares of order 3, 5, 7 and 9 are at the upper-left corner. In this case, the magic sums are S3×3:=S,S5×5:=5S 3,S7×7:=7S 3,S9×9:=3Sand S11×11 :=11 S/3. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 3 should be multiple of 3. Below are two examples. Example 2.10. Let’s consider following two examples based on the Result 2.10: 16 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 The magic sums are: •First example: S3×3:=54, S5×5:=90, S7×7:=126, S9×9:=162, S11×11 :=198 and m:=36. •Second example: S3×3:=57, S5×5:=95, S7×7:=133, S9×9:=171, S11×11 :=209 and m:=38. 2.5 Double-Digit Algebraic Magic Squares of Order 13 Below are three different ways of writing magic square of order 13, i.e., cyclic,flat and cornered. 2.5.1 Cyclic-Type Result 2.11. Let’s consider an algebraic magic square of order 13 with reduced entries: •Details It is an algebraic cyclic-type magic square of order 13 composed of four equal sums magic rectangles of orders 2×11. It is again composed of four equal sums magic rectangles of order 2×7having a magic square of order 3 in the middle. In this case, the magic sums are S5×5:=S,S9×9:=9S 5 and S13×13 :=13 S 5. The width of magic rectangles is given as m:=2S 5. In order to avoid decimal entries the magic sum of order 5 should be multiple of 5. Below are two examples. 17 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 Example 2.11. Let’s consider following two examples based on the Result 2.11: The magic sums are: •First example: S5×5:=65, S9×9:=117, S13×13 :=169 and m:=26. •Second example: S5×5:=70, S9×9:=126, S13×13 :=182 and m:=28. 2.5.2 Flat-Type Result 2.12. Let’s consider an algebraic magic square of order 13 with reduced entries: 18 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic flat-type magic square of order 13 composed of two equal sums magic rectangles of orders 2×13 and two equal sums magic rectangles of order 2×9embedded again with a flat-type magic square of order 9. It is again an algebraic flat-type magic square of order 9 embedded with a magic square of order 5. In this case the magic sums are S5×5:=S,S9×9:=9S 5and S13×13 :=13 S 5, where S is the magic sum of order 5. In this case, m:=2S 5is the width of magic rectangles. To avoid decimal entries the magic sums of order 5 should be multiple of 5. See below two examples: Example 2.12. Let’s consider following two examples based on the Result 2.12: 19 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 The magic sums are: •First example: S5×5:=75, S9×9:=135, S13×13 :=195 and m:=30. •Second example: S5×5:=85, S9×9:=153, S13×13 :=221 and m:=34. 2.5.3 Cornered Result 2.13. Let’s consider an algebraic magic square of order 13 with reduced entries: 20 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic cornered, where the magic squares of orders 13, where the magic squares of order 3, 5, 7, 9 and 11 are at the upper-left corner. In this case, the magic sums are S3×3:=S,S5×5:=5S 3,S7×7:=7S 3,S9×9:=3S,S11×11 :=11 S/3 and S13×13 :=13 S/3. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 3 should be multiple of 3. Below are two examples. Example 2.13. Let’s consider following two examples based on the Result 2.13: 21 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 The magic sums are: •First example: S3×3:=72, S5×5:=120, S7×7:=168, S9×9:=216, S11×11 :=264, S13×13 :=312 and m:=48. •Second example: S3×3:=63, S5×5:=105, S7×7:=147, S9×9:=189, S11×11 :=231, S13×13 :=273 and m:=42. 2.6 Double-Digit Algebraic Magic Squares of Order 15 Below are three different ways of writing magic square of order 15, i.e., cyclic,flat and cornered. 2.6.1 Cyclic-Type Result 2.14. Let’s consider an algebraic magic square of order 15 with reduced entries: 22 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic cyclic-type magic square of order 15 composed of four equal sums magic rectangles of orders 2×13 embedded with a magic square of order 11. It is again composed of four equal sums magic rectangles of order 2×9having a magic square of order 3 in the middle. In this case, the magic sums are S3×3:=S,S7×7:=7S 3,S11×11 :=11 S 3and S15×15 :=5S. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 3 should be multiple of 3. Below are two examples. Example 2.14. Let’s consider following two examples based on the Result 2.14: 23 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 The magic sums are: •First example: S3×3:=63, S7×7:=147, S11×11 :=231, S15×15 :=315 and m:=42. •Second example: S3×3:=72, S7×7:=168, S11×11 :=264, S15×15 :=360 and m:=48. 2.6.2 Flat-Type Result 2.15. Let’s consider an algebraic magic square of order 15 with reduced entries: 24 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic flat-type magic square of order 15 composed of two equal sums magic rectangles of orders 2×15 and two equal sums magic rectangles of order 2×11 embedded again with a flat-type magic square of order 11 and so on. In this case, the magic sums are S3×3:=S, S7×7:=7S 3,S11×11 :=11 S 3and S15×15 :=5S. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 3 should be multiple of 3. Below are two examples. Example 2.15. Let’s consider following two examples based on the Result 2.15: 25 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 Example 2.18. Let’s consider following two examples based on the Result 2.18: The magic sums are: 32 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •First example: S5×5:=95, S9×9:=171, S13×13 :=247, S17×17 :=323 and m:=38. •Second example: S5×5:=125, S9×9:=225, S13×13 :=325, S17×17 :=425 and m:=50. 2.7.3 Cornered Result 2.19. Let’s consider an algebraic magic square of order 17 with reduced entries: •Details It is an algebraic cornered, where the magic squares of orders 17, where the magic squares of order 3, 5, 7, 9, 11, 13 and 15 are at the upper-left corner. In this case, the magic sums are S3×3:=S,S5×5:=5S 3,S7×7:=7S 3,S9×9:=3S,S11×11 :=11 S 3,S13×13 :=13 S 3,S15×15 :=5Sand S17×17 :=17 S 3. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 3 should be multiple of 3. Below are two examples. 33 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 Example 2.19. Let’s consider following two examples based on the Result 2.19: The magic sums are: 34 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •First example: S3×3:=99, S5×5:=165, S7×7:=231, S9×9:=297, S11×11 :=363, S13×13 :=429, S15×15 :=495, S17×17 :=561 and m:=66. •Second example: S3×3:=117, S5×5:=195, S7×7:=273, S9×9:=351, S11×11 :=429, S13×13 :=507, S15×15 :=585, S17×17 :=663 and m:=78. 2.8 Double-Digit Algebraic Magic Squares of Order 19 Below are three different ways of writing magic square of order 19, i.e., cyclic,flat and cornered. 2.8.1 Cyclic-Type Result 2.20. Let’s consider an algebraic magic square of order 19 with reduced entries: 35 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic cyclic-type magic square of order 19 composed of four equal sums magic rectangles of orders 2×17 embedded with a magic square of order 15. It is again composed of four equal sums magic rectangles of order 2×13 and so on having a magic square of order 3 in the middle. In this case, the magic sums are S3×3:=S,S7×7:=7S 2S,S11×11 :=11 S 3,S15×15 :=5Sand S19×19 :=19 S 3. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 5 should be multiple of 5. Below are two examples. Example 2.20. Let’s consider following two examples based on the Result 2.20: 36 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 The magic sums are: •First example: S3×3:=195, S7×7:=455, S11×11 :=715, S15×15 :=975, S19×19 :=1235 and m:=130. •Second example: S3×3:=198, S7×7:=462, S11×11 :=726, S15×15 :=990, S19×19 :=1254 and m:=132. 2.8.2 Flat-Type Result 2.21. Let’s consider an algebraic magic square of order 19 with reduced entries: 37 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic flat-type magic square of order 19 composed of two equal sums magic rectangles of orders 2×19 and two equal sums magic rectangles of order 2×15 embedded again with a flat-type magic square of order 15 and so on. In this case, the magic sums are S3×3:=S, S7×7:=7S 2S,S11×11 :=11 S 3,S15×15 :=5Sand S19×19 :=19 S 3. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 5 should be multiple of 5. Below are two examples. Example 2.21. Let’s consider following two examples based on the Result 2.21: 38 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 39 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 The magic sums are: •First example: S3×3:=123, S7×7:=287, S11×11 :=451, S15×15 :=615, S19×19 :=779 and m:=82. •Second example: S3×3:=231, S7×7:=539, S11×11 :=847, S15×15 :=1155, S19×19 :=1463 and m:=154. 2.8.3 Cornered Result 2.22. Let’s consider an algebraic magic square of order 19 with reduced entries: 40 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Double-Digit and Cornered Magic Squares of Odd Orders from 5 to 19, Zenodo, December 08, 2025, pp. 1-46, https://doi.org/10.5281/zenodo.17859037 •Details It is an algebraic cornered, where the magic squares of orders 19, where the magic squares of order 3, 5, 7, 9, 11, 13, 15 and 17 are at the upper-left corner. In this case, the magic sums are S3×3:=S,S5×5:=5S 3,S7×7:=7S 3,S9×9:=3S,S11×11 :=11 S 3,S13×13 :=13 S 3,S15×15 :=5S, S17×17 :=17 S 3and S19×19 :=19 S 3. The width of magic rectangles is given as m:=2S 3. In order to avoid decimal entries the magic sum of order 3 should be multiple of 3. Below are two examples. Example 2.22. Let’s consider following two examples based on the Result 2.22: 41