Algebraic Cyclic, Flat and Cornered Striped Magic Squares of Even Orders from 4 to 20

Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20 The whole work is also is available at author’s site: https://numbers-magic.com/?p=17058 Inder J. Taneja1 Abstract This work brings algebraic striped magic squares for even orders from 4 to 20 for the 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 the 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. This work is on double-digit or double-layer bordered and cornered magic squares. In case of double-digit, we have two different ways of writing. One is cyclic-type and the another is flat-type. In case of cyclic-type, each layer has four equal sums magic rectangles and in case of flat-type, we have two big and two small magic rectangles. Including the corner-type, thre are three different types of magic squares for each order are studied in this work. Whole the work in in terms of equal width magic rectangles calling as striped magic squares. For the first time, in the literature of magic squares, the idea of double-digit bordered magic squares for sequential number entries is studied by the author [31]. This work is for non-sequential entries. Moreover, in this work the magic rectangles are considered as cyclic,flat or cornered. 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 Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Contents 1 Introduction 3 2 Double-Digit Algebraic Striped Magic Squares 3 2.1 Double-Digit Algebraic Striped Magic Square of Order 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Double-Digit Algebraic Striped Magic Squares of Order 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Double-Digit Algebraic Striped Magic Squares of Order 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Cyclic-Type ............................................................... 8 2.3.2 Flat-Type................................................................. 9 2.3.3 Cornered................................................................. 10 2.4 Double-Digit Algebraic Striped Magic Squares of Order 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.1 Cyclic-Type ............................................................... 12 2.4.2 Flat-Type................................................................. 13 2.4.3 Cornered................................................................. 17 2.5 Double-Digit Algebraic Striped Magic Squares of Order 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.1 Cyclic-Type ............................................................... 18 2.5.2 Flat-Type................................................................. 20 2.5.3 Cornered................................................................. 22 2.6 Double-Digit Algebraic Striped Magic Squares of Order 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.1 Cyclic-Type ............................................................... 24 2.6.2 Flat-Type................................................................. 26 2.6.3 Cornered................................................................. 28 2.7 Double-Digit Algebraic Striped Magic Squares of Order 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.7.1 Cyclic-Type ............................................................... 30 2.7.2 Flat-Type................................................................. 33 2.7.3 Cornered................................................................. 35 2.8 Double-Digit Algebraic Striped Magic Squares of Order 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.8.1 Cyclic-Type ............................................................... 38 2.8.2 Flat-Type................................................................. 40 2.8.3 Cornered................................................................. 43 2.9 Double-Digit Algebraic Striped Magic Squares of Order 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.9.1 Cyclic-Type ............................................................... 46 2.9.2 Flat-Type................................................................. 49 2.9.3 Cornered................................................................. 52 3 Author’s Contribution to Magic Squares and Recreation of Numbers 55 2 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 1 Introduction This work brings algebraic striped magic squares for even orders from 4 to 20 for the 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 work double-digit or double-layer bordered and cornered magic squares. In case of double-digit, we have two different ways of writing. One is cyclic-type and another is flat-type. In case of cyclic-type, each layer has four equal sums magic rectangles and in case of flat-type, we have two big and two small magic rectangles in each layer. Including the corner-type, we have each order in three different types of magic squares. 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. Instead of sequential entries, we have non-sequential entries. These can be positive, negative or decimal numbers. This work brings double-digit or double-layer algebraic striped magic squares of even orders from 4 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. Whole the work in terms of equal width magic rectangles calling striped magic squares. The idea of double-digit for magic squares for sequential number entries studied by the author [31] for the first time. This work is for non-sequential 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 [25, 26, 27, 28, 29, 30, 31]. 2 Double-Digit Algebraic Striped Magic Squares The section bring results and examples of reduced entries algebraic striped magic squares complete in itself for even orders from 4 to 20. These are of three types: cyclic-type,flat-type and corner-type. In case of orders 4 and 6, we don’t have much possibilities. 2.1 Double-Digit Algebraic Striped Magic Square of Order 4 Result 2.1. An algebraic striped magic square of order 4 is given by 3 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 •Details Above there are two ways of representing algebraic striped magic square of order 4 where the strips are either in a horizonal way or in a vertical way. In both the situations, the result is always a magic square. Here we use only 4 entries instead of 16, where mis the width of the magic rectangle. In this case the magic sum is S4×4:=2m. See below some examples: Example 2.1. Below are two examples of algebraic striped magic square of order 4 based on Result 2.1 are given by 4 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 In the first case, the magic sum is S4×4:=44, m:=22. In the second case, the magic sum is S4×4:=54, m:=27. 2.2 Double-Digit Algebraic Striped Magic Squares of Order 6 Below are two different ways of writing magic square of order 6. Result 2.2. An algebraic striped magic square of order 6 is given by 5 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 •Details It is an algebraic cornered-type striped magic square of order 6, where there is a magic square of order 4 at the upper-left corner formed by two equal sums strips of order 2×4. The magic sum of order 6 is S6×6:=3m, where mis the width of the magic rectangles. Whole the magic square of order 6 is formed by 3 strips of order 2×4and one strip of order 2×6. See below few examples: Example 2.2. Below are two examples of algebraic striped magic square of order 6 based on Result 2.2 are given by Above there are two ways of writing the algebraic striped magic square of order 6. The first way is of corner-type while the second one normal type. In both the cases the magic sums are: (i) First case: –First Example:S6×6:=63, S4×4:=42 and m:=21. –Second Example:S6×6:=84, S4×4:=56 and m:=28. (ii) Second case: –First Example:S6×6:=63, S4×4:=42 and m:=21. –Second Example:S6×6:=84, S4×4:=56 and m:=28. Result 2.3. An algebraic striped magic square of order 6 is given by 6 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 •Details It is an algebraic cornered-type striped magic square of order 6 composed of three equal sums magic rectangles of order 2×6. The magic sum of order 6 is S6×6:=3m, where mis the width of the magic rectangles. See below few examples: Example 2.3. Below are two examples of algebraic striped magic square of order 6 based on Result 2.2 are given by 7 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S6×6:=162, MR2×6:=54 ×162 and m:=54. (ii) Second Example:S6×6:=147, MR2×6:=49 ×147 and m:=49. 2.3 Double-Digit Algebraic Striped Magic Squares of Order 8 Below are three different ways of writing magic square of order 8, i.e., cyclic,flat and cornered. 2.3.1 Cyclic-Type Result 2.4. An algebraic striped magic square of order 8 in cyclic way is given by •Details It is an algebraic magic square of order 8 composed of four equal sums magic rectangles of orders 2×6and embedded with a magic square of order 4. This magic square of order 4 is composed of two equal sums magic rectangles of order 2×4. Since it is composed of all the strips of equal width, we can call is an algebraic striped magic square of order 8. Moreover, the external four strips are of equal sums, we can name is as a algebraic cyclic-type striped magic square of order 8. In this case the magic sums are S4×4:=2mand S8×8:=4m, where mis the width of the strip See below two examples: Example 2.4. Below are two examples of algebraic striped magic square of order 8 based on Result 2.4 are given by 8 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=26, S8×8:=52 and m:=13. (ii) Second Example:S4×4:=30, S8×8:=60 and m:=15. 2.3.2 Flat-Type Result 2.5. An algebraic striped magic square of order 8 in flat way is given by 9 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Example 2.11. Below are two examples of algebraic striped magic square of order 10 based on Result 2.9 are given by Above there are two examples. In both the cases, the magic sums are: (i) First Example:S6×6:=246, S10×10 :=410 and m:=82. (ii) Second Example:S6×6:=357, S10×10 :=595 and m:=119. 16 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 2.4.3 Cornered Result 2.10. An algebraic striped cornered magic square of order 10 is given by •Details It is an algebraic cornered striped magic square of order 10, where the magic squares of order 4, 6 and 8 are at the upper-left corner. In this case the magic sums are S4×4:=2m,S6×6:=3m,S8×8:=4mand S10×10 :=5m, where mis the width of the strip. See below two examples: Example 2.12. Below are two examples of algebraic striped magic square of order 10 based on Result 2.10 are given by 17 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=72, S6×6:=108, S8×8:=144, S10×10 :=180 and m:=36. (ii) Second Example:S4×4:=82, S6×6:=103, S8×8:=164, S10×10 :=205 and m:=41. Remark 2.1. We have given above three different ways of writing algebraic striped magic square of order 10. The more examples are given because of different combinations of algebraic striped magic squares of order 6. The following subsection brings only three types of results and examples. 2.5 Double-Digit Algebraic Striped Magic Squares of Order 12 Below are three different ways of writing magic square of order 12, i.e., cyclic,flat and cornered. 2.5.1 Cyclic-Type Result 2.11. An algebraic striped magic square of order 12 in cyclic way is given by 18 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 •Details It is an algebraic cyclic-type striped magic square of order 12 composed of four equal sums magic rectangles of orders 2×10 embedded with a magic square of order 8 with four equal sums magic rectangles of order 2×6having magic square of order 4 in the middle. It is again divided in two equal sum magic rectangles of order 2×4. In this case the magic sums are S4×4:=2m,S8×8:=4mand S12×12 :=6mwhere mis the width of the strip. See below two examples: Example 2.13. Below are two examples of algebraic striped magic square of order 12 based on Result 2.11 are given by 19 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=204, S8×8:=408, S12×12 :=612 and m:=102. (ii) Second Example:S4×4:=238, S8×8:=476, S12×12 :=714 and m:=119. 2.5.2 Flat-Type Result 2.12. An algebraic striped magic square of order 12 in flat way is given by 20 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 •Details It is an algebraic flat-type striped magic square of order 12 composed of two equal sums magic rectangles of orders 2×12 and two magic rectangles of order 2×8embedded with a magic square of order 8. It is again composed of two equal sums magic rectangles of orders 2×8and two magic rectangles of order 2×4embedded with a magic square of order 4. In this case the magic sums are S4×4:=2m,S8×8:=4mand S12×12 :=6mwhere mis the width of the strip. See below two examples: Example 2.14. Below are two examples of algebraic striped magic square of order 12 based on Result 2.12 are given by 21 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=116, S8×8:=232, S12×12 :=348 and m:=58. (ii) Second Example:S4×4:=122, S8×8:=244, S12×12 :=366 and m:=61. 2.5.3 Cornered Result 2.13. An algebraic striped cornered magic square of order 12 is given by 22 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 •Details It is an algebraic cornered striped magic square of order 12, where the magic squares of order 4, 6, 8 and 10 are at the upper-left corner. In this case the magic sums are S4×4:=2m,S6×6:=3m,S8×8:=4m,S10×10 :=5mand S12×12 :=6m, where mis the width of the strip. See below two examples: Example 2.15. Below are two examples of algebraic striped magic square of order 12 based on Result 2.13 are given by 23 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=106, S6×6:=159, S8×8:=212, S10×10 :=265, S12×12 :=318 and m:=53. (ii) Second Example:S4×4:=124, S6×6:=186, S8×8:=244, S10×10 :=310, S12×12 :=372 and m:=62. 2.6 Double-Digit Algebraic Striped Magic Squares of Order 14 Below are three different ways of writing magic square of order 14, i.e., cyclic,flat and cornered. 2.6.1 Cyclic-Type Result 2.14. An algebraic striped magic square of order 14 in cyclic way is given by 24 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 •Details It is an algebraic cyclic striped magic square of order 14 composed of four equal sums magic rectangles of orders 2×12 embedded with a magic square of order 10. It is again composed of four equal sums magic rectangles of orders 2×8embedded with a magic square of order 6. It is again composed of three equal sums magic rectangles of order 2×6. In this case the magic sums are , S6×6:=3m,S10×10 :=5mand S14×14 :=7m, where mis the width of the strip. See below two examples: Example 2.16. Below are two examples of algebraic striped magic square of order 14 based on Result 2.14 are given by 25 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: 32 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 (i) First Example:S4×4:=218, S8×8:=436, S12×12 :=654, S16×16 :=872 and m:=109. (ii) Second Example:S4×4:=244, S8×8:=488, S12×12 :=732, S16×16 :=976 and m:=122. 2.7.2 Flat-Type Result 2.18. An algebraic striped magic square of order 16 in flat way is given by •Details It is an algebraic flat-type striped magic square of order 16 composed of two equal sums magic rectangles of orders 2×16 and two magic rectangles of order 2×12 embedded with a magic square of order 12. It is again composed of two equal sums magic rectangles of orders 2×12 and two magic rectangles of order 2×8embedded with a magic square of order 8. It is again composed of two equal sums magic rectangles of orders 2×8and two magic rectangles of order 2×4embedded with a magic square of order 4. It is again composed of two equal sums magic 33 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 rectangles of order 2×4. In this case the magic sums are S4×4:=2m,S8×8:=4m,S12×12 :=6mand S16×16 :=8m, where mis the width of the strip. See below two examples: Example 2.20. Below are two examples of algebraic striped magic square of order 16 based on Result 2.18 are given by 34 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=202, S8×8:=404, S12×12 :=606, S16×16 :=808 and m:=101. (ii) Second Example:S4×4:=236, S8×8:=472, S12×12 :=708, S16×16 :=944 and m:=118. 2.7.3 Cornered Result 2.19. An algebraic striped cornered magic square of order 16 is given by 35 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 •Details It is an algebraic cornered striped magic square of order 16, where the magic squares of order 4, 6, 8, 10, 12 and 14 are at the upper-left corner. In this case the magic sums are S4×4:=2m,S6×6:=3m,S8×8:=4m,S10×10 :=5m,S12×12 :=6m,S14×14 :=7mand S16×16 :=8m, where mis the width of the strip. See below two examples: Example 2.21. Below are two examples of algebraic striped magic square of order 16 based on Result 2.19 are given by 36 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: 37 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 (i) First Example:S4×4:=172, S6×6:=264, S8×8:=352, S10×10 :=440, S12×12 :=528, S14×14 :=616, S16×16 :=704 and m:=88. (ii) Second Example:S4×4:=242, S6×6:=363, S8×8:=464, S10×10 :=585, S12×12 :=726, S14×14 :=847, S16×16 :=968 and m:=121. 2.8 Double-Digit Algebraic Striped Magic Squares of Order 18 Below are three different ways of writing magic square of order 18, i.e., cyclic,flat and cornered. 2.8.1 Cyclic-Type Result 2.20. An algebraic striped magic square of order 18 in cyclic way is given by 38 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 The internal part is exactly the same as given in Result 2.14. •Details It is an algebraic cyclic striped magic square of order 18 composed of four equal sums magic rectangles of orders 2×16 embedded with a magic square of order 14. The internal part, i.e., magic square of order 14 is as given in Result 2.14. In this case, the magic sums are S6×6:=3m, S10×10 :=5m,S14×14 :=7mand S18×18 :=9m, where mis the width of the strip. See below two examples: Example 2.22. Below are two examples of algebraic striped magic square of order 18 based on Result 2.23 are given by 39 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S6×6:=393, S10×10 :=655, S14×14 :=917, S18×18 :=1179 and m:=131. (ii) Second Example:S6×6:=444, S10×10 :=740, S14×14 :=1036, S18×18 :=1332 and m:=148. 2.8.2 Flat-Type Result 2.21. An algebraic striped magic square of order 18 in flat way is given by 40 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 The internal part is exactly the same as given in Result 2.15. •Details It is an algebraic flat striped magic square of order 18 composed of two equal sums magic rectangles of orders 2×18 and two magic rectangles of order 2×14 embedded with a magic square of order 14. This magic square of order 14 is exactly the same as given in the Result 2.15. In this case the magic sums are S6×6:=3m,S10×10 :=5m,S14×14 :=7mand S18×18 :=9m, where mis the width of the strip. See below two examples: Example 2.23. Below are two examples of algebraic striped magic square of order 18 based on Result 2.24 are given by 41 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 48 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=350, S8×8:=700, S12×12 :=1050, S16×16 :=1400, S20×20 :=1750 and m:=175. (ii) Second Example:S4×4:=372, S8×8:=744, S12×12 :=1116, S16×16 :=1488, S20×20 :=1860 and m:=186. 2.9.2 Flat-Type Result 2.24. An algebraic striped magic square of order 20 in flat way is given by 49 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 The internal part is exactly the same as given in Result 2.18. •Details It is an algebraic flat striped magic square of order 20 composed of two equal sums magic rectangles of orders 2×20 and two magic rectangles of order 2×16 embedded with a magic square of order 16. This magic square of order 16 is exactly, the same as given in Result 2.18. In this case, the magic sums are S4×4:=2m,S8×8:=4m,S12×12 :=6m,S16×16 :=8mand S20×20 :=10 m, where mis the width of the strip. See below two examples: Example 2.26. Below are two examples of algebraic striped magic square of order 20 based on Result 2.24 are given by 50 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 51 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=344, S8×8:=688, S12×12 :=1032, S16×16 :=1204, S20×20 :=1720 and m:=172. (ii) Second Example:S4×4:=362, S8×8:=724, S12×12 :=1086, S16×16 :=1448, S20×20 :=1810 and m:=181. 2.9.3 Cornered Result 2.25. An algebraic striped cornered magic square of order 20 is given by 52 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 The internal part is exactly the same as given in Result ??. •Details It is an algebraic cornered striped magic square of order 20, where the magic squares of order 4, 6, 8, 10, 12, 14, 16 and 18 are at the upper-left corner. In this case the magic sums are S4×4:=2m,S6×6:=3m,S8×8:=4m,S10×10 :=5m,S12×12 :=6m,S14×14 :=7m,S16×16 :=8m, S18×18 :=9mand S20×20 :=10 m, where mis the width of the strip. See below two examples: Example 2.27. Below are two examples of algebraic striped magic square of order 20 based on Result 2.25 are given by 53 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 54 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=330, S6×6:=495, S8×8:=660, S10×10 :=825, S12×12 :=990, S14×14 :=1155, S16×16 :=1320, S18×18 :=1485, S20×20 := 1650 and m:=165. (ii) Second Example:S4×4:=342, S6×6:=513, S8×8:=684, S10×10 :=855, S12×12 :=1026, S14×14 :=1197, S16×16 :=1368, S18×18 := 1539, S20×20 :=1710 and m:=171. 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/ •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/ 55 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 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. •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. 56 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845 [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. [24] Inder J. Taneja, Algebraic Cyclic, Flat and Cornered Striped Magic Squares for Even Orders from 4 to 20, Zenodo, December 02, 2025, pp. 1-58, https://doi.org/10.5281/zenodo.17793845. •Double-Digit Magic Squares [25] 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. [26] 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. [27] 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. [28] 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. 57