Different Types and Aspects of Magic Squares of Order 18

Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Different Types and Aspects of Magic Squares of Order 18 The whole work is also is available at author’s sites: https://numbers-magic.com/?p=17001 Inder J. Taneja1 Abstract This work summarizes author’s previous works on magic squares of order 18. It brings different types of magic square such as block-wise, single-digit bordered, double-digit bordered, cornered, designs, styles, etc. In terms os aspects, we have considered the idea of Pythagorean triples, perfect square sum entries, Latin square distributions, upside-down, mirror-looking, water reflection, algebraic magic squares, etc. 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 Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Contents 1 Introduction 3 2 Different Types of Magic Squares 4 2.1 Block-WiseMagicSquares................................................ 5 2.2 Single-LayerBorderedSquares ............................................. 7 2.3 CorneredMagicSquare ................................................. 12 2.4 Double-LayerBorderedMagicSquares......................................... 13 2.5 StripedBorderedMagicSquares ............................................ 16 2.6 DifferentDesignedMagicSquares ........................................... 20 2.7 MagicRectanglesandMagicSquares.......................................... 21 2.8 Triple-LayerMagicSquares ............................................... 24 2.9 Palindromic-Number Entries Magic Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Different Aspects of Magic Squares 29 3.1 PerfectSquareEntriesSum ............................................... 29 3.1.1 UniformityProperty............................................... 29 3.1.2 PythagoreanTriple................................................ 31 3.1.3 Minimum Perfect Square Sum of Entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 LatinSquareDistributions................................................ 34 3.3 Upside-Down, Mirror Looking and Water Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Upside-DownandMirror-Looking........................................... 43 3.5 WaterReflectionMagicSquares............................................. 50 3.6 Self-MadeAlgebraicMagicSquare........................................... 56 3.7 Double-Digit Algebraic Striped Magic Squares of Order 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.7.1 Cyclic-Type .................................................... 61 3.7.2 Flat-Type...................................................... 64 3.7.3 Cornered...................................................... 67 4 Author’s Contribution to Magic Squares and Recreation of Numbers 71 2 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 1 Introduction In the previous works [19, 20, 21, 22, 23, 24, 25], the author worked with block-wise constructions of magic squares. The work is from the orders 8 to 45. In each case, all the possibilities are considered. These possibilities are based on divisions of magic squares. The magic sums of order nof consecutive numbers from 1 to n2is given by Sn×n:=n×(1+n2) 2,n≥3. (1) This formula is applied to all order magic squares. Based on this formula we shall brink blocks for the block-wise magic squares. That is, whenever is possible, we shall try to bring blocks of equal sum magic squares. In some cases, they are magic,semi-magic,pandiagonal, etc. When the question come to bimagic squares, in some cases, we have semi-bimagic squares. On the other hand the idea of bordered magic squares is well explained in the work by H.White [4, 5]. Few results in this direction can be seen in author’s work [26, 27, 28, 29, 30, 31]. In some case, the magic rectangles are also used to write it in different styles. Most of the author’s work on magic squares is summarized below in details according to topic: 1. Digital Fonts: Upside-down and Mirror Looking. 2. Two Digits Universal Magic Squares. 3. Different Digits Magic Squares. 4. Pythagorean Triples Magic Squares. 5. Block-Wise Magic Squares. 6. Selfie and Palindromic Type Magic Squares. 7. Block Bordered Magic Squares. 8. Block-Wise Bordered Magic Squares. 9. Magic Crosses, Letters and Numbers. 10. Cornered Magic Squares. 11. Single-Digit Bordered Magic Squares. 12. Double-Digit Bordered Magic Squares. 13. Multiple-Digit Bordered Magic Squares. 14. Striped Magic Squares. 15. Upside-down and Mirror Looking. 16. Water reflection Magic Squares. 17. Algebraic Magic Squares. 3 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 The aim of this work is to write magic squares of order 18 in different ways and styles using the aspects of blocks, bordered,block-bordered and magic rectangles,corner-type,single-digit,double-digit, etc. More details on these works on magic squares can be seen in author’s web-sites: (i) https://numbers-magic.com/?p=668 (ii) https://inderjtaneja.wordpress.com/2019/06/27/publications-magic-squares/ 2 Different Types of Magic Squares According to Equation (1), the magic sum of order 18 is given by S18×18 :=18 ×(1+182) 2=2925. We can write, 18 :=2×3×3. This gives the possibility of blocks of orders 3 and 6. Let’s see division of 2925 by 6 and 3: (i)2925 6=975 2=⇒non-equal blocks of order 3; (ii)2925 3=975 =⇒equal blocks of order 6. Below are few examples of different types of magic squares of order 18. 4 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 2.1 Block-Wise Magic Squares Example 2.1. Let’s consider a magic square of order 18 given by It is composed of magic squares of order 3 with different magic sums 5 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 2.2. Let’s consider a magic square of order 18 given by It is composed of equal sums magic squares of order 6. For more details refer author’s work [20, 22]. 6 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 2.2 Single-Layer Bordered Squares Example 2.3. Let’s consider a magic square of order 18 given by It is single-layer bordered magic square. Removing the external bordered, still we left with less order magic squares, such as orders 16, 14, 12, etc. 7 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 2.4. Let’s consider a magic square of order 18 given by It is composed of equal sums single-layer bordered magic squares of order 6. 8 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 2.5. Let’s consider a magic square of order 18 given by It is a single-layer bordered magic square embedded with equal sums pandiagonal magic squares of order 4. The magic squares of order 4 are also pandiagonal. 9 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 2.5 Striped Bordered Magic Squares Example 2.12. Let’s consider a magic square of order 18 given by It is a double-layer striped magic square, where the width is always 2. The difference is only in the length. It composed magic rectangles. 16 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 2.13. Let’s consider a magic square of order 18 given by It is also a double-layer striped magic square but written in cyclic way composed magic rectangles. 17 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 2.14. Let’s consider a magic square of order 18 given by It is also a double-layer striped magic square but written in corner-type magic square. 18 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 2.15. Let’s consider a magic square of order 18 given by It is a striped magic square made of equal width and length magic rectangles. For more details refer author’s work [40, 41, 42, 43, 44, 45, 46]. 19 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 2.6 Different Designed Magic Squares Example 2.16. Let’s consider a magic square of order 18 given by It is mixed-type symmetric designed magic square. For more details refer author’s work [6, 7, 8, 10, 53]. 20 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 2.7 Magic Rectangles and Magic Squares Example 2.17. Let’s consider a magic square of order 18 given by It is mixed-type magic square composed of magic squares and bordered magic rectangles. 21 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 2.18. Let’s consider a magic square of order 18 given by It is mixed-type magic square composed of bordered magic rectangles. 22 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 2.19. Let’s consider a magic square of order 18 given by It is mixed-type magic square composed of magic squares and bordered magic rectangles. For more details refer author’s work [6, 7, 8, 10, 53] 23 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 2.8 Triple-Layer Magic Squares Example 2.20. Let’s consider a magic square of order 18 given by It is a triple-layer bordered magic square. It is composed of magic squares of order 3 in such a way that if we remove external border still we are left with magic squares of lower orders. 24 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 2.21. Let’s consider a magic square of order 18 given by It is a triple-layer bordered magic square. embedded with four equal sums single-layer magic squares of order 6. 25 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 The block-wise magic squares of order 18 given in Examples 3.3 and 3.4 are respectively with equal magic sums. The blocks of order 6 are magic squares with equal magic sums. See below the details: S18×18 =7200; T324 :=18 ×7200 =129600 =3602; S6×6=2400; T36 :=6×2400 =14400 =1202. Both the Examples 3.3 and 3.4 are generated by Pythagorean triple (38,360,362), i.e., 382+3602=3622with least possible sum of entries resulting in a perfect square. 32 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 3.1.3 Minimum Perfect Square Sum of Entries Example 3.5. Ablock-wise magic square of order 18 for consecutive fraction numbers entries {15/2, 17/2, . . . , 659/2, 661/2} is given by The block-wise magic square of order 18 given in Example 3.5 is with minimum perfect square sum of entries. The entries sum is minimum perfect square. The blocks of order 6 are magic squares with equal magic sums. See below the details S18×18 =3042; T324 :== 54756 =2342; S6×6=1014; T36 :=6084 =782. 33 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 For more details refer author’s work [14, 15, 16, 17, 18]. 3.2 Latin Square Distributions Let’s consider the Latin square distributions of magic square of order 18 given in Examples 2.1 and 2.2. Example 3.6. Let’s consider a pair of Latin square decomposition as given below: and 34 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 By application of the formula 18 ×(A−1) + B, we get the magic square given in Example 3.25. Example 3.7. Let’s consider a pair of Latin square decomposition as given below: 35 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 and 36 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 By application of the formula 18 ×(A−1) + B, we get the magic square given in Example 3.26. We observe from above two examples that the Latin square distribution is not orthogonally diagonalized. Below is a third example, where the distribution is ”SOLDS”. See below Example 3.8. Let’s consider a pair of Latin square decomposition as given below: 37 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 and 38 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 By application of the formula 18 ×(A−1) + B, we get the following magic square of order 18: 39 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 The magic square given in Example 3.27 shall be used in a topic on upside-down, mirror looking and water reflection magic squares. For more details refer author’s work [49]. The next subsection is based on the Latin square distributions. 40 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 3.3 Upside-Down, Mirror Looking and Water Reflection Let’s consider the following image: Source: https://www.mathsisfun.com/definitions/vertical-flip.html From the above image we understand that horizontal flip is same as mirror looking image and the vertical flip is same as water reflection image. The same terms are given in paint brush of microsoft. Let’s see how it works on numbers. Let’s consider following 9 digits written in digital form: 41 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 In each case all the blocks of order 6 are magic squares with equal magic sums. The magic sums are given by S18×18(2, 5):=69999999993; S6×6(2, 5):=23333333331 It is only upside-down magic square. Example 3.14. The magic squares of order 18 for the digits (1, 8),(2, 5)and (6, 9)are given by 48 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 In each case all the blocks of order 6 are magic squares with equal magic sums. The magic sums are given by S18×18(6, 9):=149999999985; S6×6(6, 9):=49999999995. It is only upside-down magic square. For more studied with different options, see the author’s work [47, 48, 49]. 49 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 3.5 Water Reflection Magic Squares Below are few examples of magic square with water reflection property •4-Digits Magic Squares Example 3.15. Let’s consider a magic square of order 18 with 4-digits (1,2,3,5) given by The magic sum of above magic square is given as S18×18(1, 2, 3, 5) = 62216 . 50 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 3.16. Let’s consider a magic square of order 18 with 5-digits (2,3,5,8) given by The magic sum of above magic square is given as S18×18(2, 3, 5, 8) = 69993. The magic squares given in Examples 3.15 and 3.16 are with water reflection property 51 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 •3-Digits Magic Squares Example 3.17. Let’s consider a magic square of order 18 with 3-digits (2,3,5) given by The magic sum of above magic square is given as S18×18(2, 3, 5) = 6795789 . 52 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 3.18. Let’s consider a magic square of order 18 with 3-digits (1,3,8) given by The magic sum of above magic square is given as S18×18(1, 3, 8) = 8368360. The magic squares given in Examples 3.17 and 3.18 are with water reflection property 53 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 •2-Digits Magic Squares Example 3.19. Let’s consider a magic square of order 18 with 2-digits (1,3) given by The magic sum of above magic square is given as S18×18(1, 3) = 39998399980 . 54 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Example 3.20. Let’s consider a magic square of order 18 with 2-digits (3,8) given by The magic sum of above magic square is given as S18×18(3, 8) = 110004000029 . The magic squares given in Examples 3.19 and 3.20 are with water reflection property. For more studied with different options, see the author’s work [47, 48, 49]. 55 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 3.6 Self-Made Algebraic Magic Square Below is a result of a magic square order 18 made from the variables A1 to A128 and a magic sum. Result 1. Below is an algebraic magic square of order 18 56 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 •Details It is single-digits bordered magic square of order 18 embedded with a magic square of order 16 formed by equal sums magic sums of order 4. We call it a self-made as choosing any numbers for A1 to A128 and the magic sum S, we always get a magic square of order 18. Sometimes we may call it as algebraic magic square for reduced entries. It requires only 96 entries instead of 324. See below two examples: Example 3.21. Let’s consider a following magic square based on the Result 1 It is single-digits bordered magic square of order 18 embedded with a magic square of order 16 formed by equal sums magic sums of order 4. 57 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 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. 3.7.2 Flat-Type Result 4. An algebraic striped magic square of order 18 in flat way is given by 64 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 The internal part of order 14 can be seen in [68] •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 ??. 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 3.26. Below are two examples of algebraic striped magic square of order 18 based on Result 4 are given by 65 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 66 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S6×6:=417, S10×10 :=695, S14×14 :=973, S18×18 :=1251 and m:=139. (ii) Second Example:S6×6:=456, S10×10 :=760, S14×14 :=1064, S18×18 :=1368 and m:=152. 3.7.3 Cornered Result 5. An algebraic striped cornered magic square of order 18 is given by 67 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 The internal part of order 14 can be seen in [68] •Details It is an algebraic cornered striped magic square of order 18, where the magic squares of order 4, 6, 8, 10, 12, 14 and 16 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 :=8mand S18×18 :=9m, where mis the width of the strip. See below two examples: Example 3.27. Below are two examples of algebraic striped magic square of order 18 based on Result 5 are given by 68 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 69 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=262, S6×6:=393, S8×8:=524, S10×10 :=655, S12×12 :=786, S14×14 :=917, S16×16 :=1048, S18×18 := 1179 and m:=131. (ii) Second Example:S4×4:=288, S6×6:=432, S8×8:=576, S10×10 :=720, S12×12 :=864, S14×14 :=1008, S16×16 := 1152, S18×18 :=1296 and m:=144. For more details refer Taneja [67, 68]. 70 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 4 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/ References [1] G. Bogdan, Numbers Glaxy: Magic Squares, https://www.number-galaxy.eu/ [2] Dwane H. Campbell and Keith A. Campbell, Welcome to Magic Cube Generator, http://magictesseract.com. [3] H. Heinz, Magic Squares, Magic Stars and Other Patterns, http://www.magicsquares.net. [4] H. White, Magic Squares, https://budshaw.ca/Download.html [5] H. White, Block Magic Squares, https://budshaw.ca/BlockSquares.html •Different Types of Magic Rectangles and Magic Squares [6] Inder J. Taneja, Magic Rectangles in Construction of Block-Wise Pandiagonal Magic Squares, Zenodo, January 31, 2019, pp. 1-49, http://doi.org/10.5281/zenodo.2554520. [7] Inder J. Taneja, Figured Magic Squares of Orders 6, 10, 12, 14 and 16 Using Bordered Magic Rectangles: A Systematic Procedure, Zenodo, November 29, 2022, pp. 1-31, https://doi.org/10.5281/zenodo.7377674. [8] Inder J. Taneja, Different Styles of Magic Squares of Order 16 Using Bordered Magic Rectangles, Zenodo, [9] Inder J. Taneja, 2022, pp. 1-63, https://doi.org/10.5281/zenodo.7320116. [10] Inder J. Taneja, Different Types of Magic Squares: Even Number Orders From 10 to 26, Zenodo, March 26, 2022, pp. 1-167, https://doi.org/10.5281/zenodo.6386742. 71 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 •Bimagic Squares [11] Inder J. Taneja, Block-Wise Construction of Bimagic Squares: Multiples of Orders 8 and 16. •Selfie and Palindromic-Type Magic Squares [12] Inder J. Taneja, Selfie Palindromic Magic Squares, RGMIA Research Report Collection, 18(2015), Art. 98, pp. 1-15. https://rgmia.org/papers/v18/v18a98.pdf. [13] Inder J. Taneja, Palindromic, Patterned Magic Sums, Composite, and Colored Patterns in Magic Squares. Zenodo, February 2, 2019, pp. 1-99, https://doi.org/10.5281/zenodo.2555741. •Perfect Square Sums and Pythagorean Triples Magic Squares [14] Inder J. Taneja, Block-Wise and Block-Bordered Magic Squares Generated by Pythagorean Triples: Orders 3 to 47, May 28, 2021, pp. 1-119, Zenodo,https://doi.org/10.5281/zenodo.4837454. [15] Inder J. Taneja, Generating Pythagorean Triples and Magic Squares: Orders 3 to 31, Zenodo, May 28, 2021, pp. 1-153, https://doi.org/10.5281/zenodo.4837491. [16] Inder J. Taneja, Sequential Pythagorean Triples and Perfect Square Sum Magic Squares, Zenodo, June 21, 2021, pp. 1-595, https://doi.org/10.5281/zenodo.5009204. [17] Inder J. Taneja, Magic Squares with Perfect Square Sum of Entries: Orders 3 to 31, Zenodo, July 19, pp. 1-181, 2021, https://doi.org/10.5281/zenodo.5115214. [18] Inder J. Taneja, Minimum Perfect Square Sum Bordered and Block-Wise Bordered Magic Squares: Orders 3 to 31, Zenodo, July 20, 2021, pp. 1-82, https://doi.org/10.5281/zenodo.5116408. •Block-Wise, Bordered and Block-Bordered Magic Squares [19] Inder J. Taneja, Block-Wise Constructions of Magic and Bimagic Squares of Orders 8 to 108, May 15, 2019, pp. 1-43, Zenodo,https://doi.org/10.5281/zenodo.2843326. [20] Inder J. Taneja, Block-Wise Equal Sums Pandiagonal Magic Squares of Order 4k, Zenodo, January 31, 2019, pp. 1-17, https://doi.org/10.5281/zenodo.2554288. 72 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 [21] Inder J. Taneja, Magic Rectangles in Construction of Block-Wise Pandiagonal Magic Squares, Zenodo, January 31, 2019, pp. 1-49, https://doi.org/10.5281/zenodo.2554520. [22] Inder J. Taneja, Block-Wise Magic and Bimagic Squares of Orders 12 to 36, Zenodo, February 1, 2019, pp. 1-53, https://doi.org/10.5281/zenodo.2555343. [23] Inder J. Taneja, Block-Wise Magic and Bimagic Squares of Orders 39 to 45, Zenodo, February 2, 2019, pp. 1-73, http://doi.org/10.5281/zenodo.2555889. [24] Inder J. Taneja, Block-Wise and Block-Bordered Magic and Bimagic Squares of Orders 10 to 47. Zenodo, January 14, 2021, pp. 1-185, https://doi.org/10.5281/zenodo.4437783. [25] Inder J. Taneja, Bordered Magic Squares With Order Square Magic Sums, Zenodo, January 20, 2020, pp. 1-26, http://doi.org/10.5281/zenodo.3613690. [26] Inder J. Taneja, Magic Squares with Perfect Square Sum of Entries: Orders 3 to 47. Zenodo. August 16, 2021, pp. 1-317, https://doi.org/10.5281/zenodo.5205214. [27] Inder J. Taneja, Symmetric Properties of Nested Magic Squares, Zenodo, June 29, 2019, pp. 1-55, https://doi.org/10.5281/zenodo.3262170. [28] Inder J. Taneja, Bordered and Block-Wise Bordered Magic Squares: Even Order Multiples, Zenodo, February 10, 2021, pp. 1-96, https://doi.org/10.5281/zenodo.4527746. [29] Inder J. Taneja, Nested Magic Squares with Perfect Square Sums, Pythagorean Triples, and Borders Differences, Zenodo, June 14, 2019, pp. 1-59, https://doi.org/10.5281/zenodo.3246586. [30] Inder J. Taneja, Magic and Semi-Magic Squares with Blocks of Magic Rectangles, May 28, 2022, pp. 1-27, Zenodo, https://doi.org/10.5281/zenodo.6590637. [31] Inder J. Taneja, Magic Rectangles in Construction of Magic and Block Bordered Magic Squares, June 03, 2022, pp. 1-70, Zenodo,https://doi.org/10.5281/zenodo.6621071. •Multiple Orders Bordered Magic Squares [32] Inder J. Taneja, Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4, Zenodo, August 31, 2021, pp. 1-148, https://doi.org/10.5281/zenodo.5347897. [33] Inder J. Taneja, Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 3, Zenodo, May 05, pp. 1-29, 2023, https://doi.org/10.5281/zenodo.7898383. 73