Different Types and Aspects of Magic Squares of Order 14

Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Different Types and Aspects of Magic Squares of Orders 14 The whole work as pdf files is available at author’s sites: https://numbers-magic.com/?p=16991 Inder J. Taneja1 Abstract This work summarizes author’s previous works on magic squres of order 14. 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, upside-down, mirror-looking, water reflection, perfect square sum entries, Latin square distributions, 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 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Contents 1 Introduction 3 2 Different Types of Magic Squares of Order 14 4 2.1 GeneralMagicSquares.................................................. 5 2.2 Single-DigitBorderedMagicSquares.......................................... 6 2.3 CorneredMagicSquares................................................. 11 2.4 Double-DigitBorderedMagicSquares......................................... 12 2.5 StripedMagicSquares .................................................. 15 2.6 DifferentStylesofMagicSquares............................................ 18 2.7 EmbeddedMagicSquares ................................................ 21 3 Different Aspects of Magic Squares of Order 14 23 3.1 LatinSquareDistributions................................................ 23 3.2 PerfectSquareEntriesSum ............................................... 25 3.2.1 UniformityProperty............................................... 25 3.2.2 PythagoreanTriple................................................ 27 3.2.3 Minimum Perfect Square Sum of Entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Upside-Down, Mirror Looking and Water Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.1 Upside-DownandMirrorLooking....................................... 32 3.3.2 WaterReflection ................................................. 39 3.4 Self-MadeAlgebraicMagicSquare........................................... 50 3.5 Double-Digit Algebraic Striped Magic Squares of Order 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5.1 Cyclic-Type .................................................... 55 3.5.2 Flat-Type...................................................... 57 3.5.3 Cornered...................................................... 59 4 Author’s Contribution to Magic Squares and Recreation of Numbers 62 2 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 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 in on different types, and is summarized below in details according to topics: 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. 3 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 16. Water reflection Magic Squares. 17. Algebraic Magic Squares. The aim of this work is to write magic squares of order 16 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 of Order 14 According to Equation (1), the magic sum of order 14 is given by S14×14 :=14 ×(1+142) 2=1379. Since order 14 is double of prime number, it doesn’t give us block-wise construction of magic square of order 14. Below are few examples of different ways of writing magic squares of order 14. 4 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 2.1 General Magic Squares Example 2.1. It is constructed based on a pair of self-orthogonalized latin squares. For more details refer author’s work [22, 23, 24] 5 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 2.2 Single-Digit Bordered Magic Squares Example 2.2. It is well-known single-digit bordered magic square of order 14, where removing the upper borderes, still are left with magic squres of orders 13,12,11, etc. 6 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 2.3. It is a single-digit bordered magic square of order 14 embedded with a pandiagonal magic square of order 12. It is composed of equal sums semi-magic squares of order 3. 7 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 2.4. It is a single-digit bordered magic square of order 14 embedded with a magic square of order 12. It is composed 9 equal sums pandiagonal magic squares of order 4. 8 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 2.5. It is a single-digit bordered magic square of order 14 embedded with a magic square of order 12. It is composed 4 equal sums magic squares of order 6. 9 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 2.12. 16 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 2.13. The magic squares given in Examples 2.11, 2.12 and 2.13 we call as striped magic squares. We call them as striped as these are based on the equal width of order 2. The only different is in the length of the each magic rectangles. For more details refer author’s work [39, 41, 45]. 17 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 2.6 Different Styles of Magic Squares Example 2.14. It is composed of bordered magic rectangles having single-digit borered magic square of order 6 in the middle. 18 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 2.15. It is composed of bordered magic rectangles and magic squares. 19 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 2.16. It is composed of three bordered magic rectangles. 20 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 2.7 Embedded Magic Squares Example 2.17. It is double-digit cyclic bordered magic square of order 14 embedded with a single-digit bordered magic square of order 10. 21 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 2.18. It is double-digit bordered magic square of order 14 embedded with a double-digit cyclic bordered magic square of order 10. It again contains a single-digit bordered magic square of order 6. For mor details refer author’s work [6, 7, 8, 10, 49] 22 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 3 Different Aspects of Magic Squares of Order 14 3.1 Latin Square Distributions Let’s consider the Latin square distributions of magic square of order 14 given in Example 3.2. Example 3.1. Let’s consider a magic square of 14 with Latin square decomposition as given below: A 105 1 4 11 5 8 2 9 3 7 14 12 13 6 10 105 14 2 5 11 6 3 10 4 8 12 13 7 1 9 105 12 14 3 6 11 4 1 5 9 13 8 2 10 7 105 13 12 14 4 7 5 2 6 10 9 3 1 8 11 105 10 13 12 14 5 6 3 7 1 4 2 9 11 8 105 7 8 9 10 1 11 13 14 12 65432105 9 10 1 2 3 14 12 11 13 87654105 6 7 8 9 10 12 14 13 11 54321105 8 9 10 1 2 13 11 12 14 76543105 3 11 4 7 9 1 8 2 6 10 14 12 13 5 105 11 3 6 8 4 10 7 1 5 2 9 14 12 13 105 2 5 7 3 13 9 6 10 4 11 1 8 14 12 105 4 6 2 13 12 8 5 9 3 1 11 10 7 14 105 5 1 13 12 14 7 4 8 2 3 10 11 9 6 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 23 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 B 105 1 13 10 14 11 6 7 9 8 3 4 2 12 5 105 4 2 13 1 14 7 8 10 9 5 3 12 6 11 105 6 5 3 13 2 8 9 1 10 4 12 7 11 14 105 5 7 6 4 13 9 10 2 1 12 8 11 14 3 105 12 6 8 7 5 10 1 3 2 9 11 14 4 13 105 9 10 1 2 3 11 14 12 13 87654105 2345613 12 14 11 1 10 9 8 7 105 3456714 11 13 12 2 1 10 9 8 105 7 8 9 10 1 12 13 11 14 65432105 13 9 14 11 4 5 6 8 7 10 2 3 1 12 105 8 14 11 3 12 4 5 7 6 13 9 1 2 10 105 14 11 2 12 9 3 4 6 5 7 13 8 10 1 105 11 1 12 8 10 2 3 5 4 14 6 13 7 9 105 10 12 7 9 8 1 2 4 3 11 14 5 13 6 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 By application of the formula 14 ×(A−1) + B, we get the magic square given in Example 3.2. Grid 1. Let’s write the above Latin squares in terms of a Grid representing the numbers in alphabets: a g n h j f e c b l k m i d k b h m n j g i a d e f c l i l c j m e f a d b g h k n n h k d l i a f c m b g j e l f d a i g c h j e m b n k h a g c b l m k n f d j e i b c j e a k n l m g i d f h d e f g c n k m l i j a h b c d e f g m l n k h a i b j e k l b f d h g i j n c a m f n m i e b j d g a h k l c j m a l k h i b f n c e d g m j i n d c b e h k f l g a g i b k h a d j e c l n m f A a k i n l h b d c e f j m g g b l h f a c e d k n m j i n h c k d g j f e l m a i b h m j d a c e g f b i l n k j n m l i b a c g f e k d h f j e i g l k n m d b h c a e g f a c m n k l h j i b d c i a f h k l m n g d b e j b a d c j n m l k i g f h e l d b m e f g i h j a n k c k e g b m d i j a n h c f l m f h g b j d a i c k e l n i c k j n e f h b a l d g m d l n e k i h b j m c g a f B For more details refer author’s work [46] 24 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 3.2 Perfect Square Entries Sum Below are five different magic squares of order 14 resulting in uniformity,Pythagorean triples and minimum perfect square sum properties. Out of these five, three of them are with fraction numbers entries. 3.2.1 Uniformity Property Example 3.2. For the consecutive odd numbers entries {1, 3, . . . , 389, 391}, a magic square of order 14 is given by 25 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 3.3.1 Upside-Down and Mirror Looking •4-Digits Cell Entries Below are two examples of upside-down and/or mirror looking magic squares of order 14 with four digits (0,2,5,8) and (1,6,8,9). Example 3.7. This Example 3.29 is upside-down and mirror looking with magic sum, S14×14(0, 2, 5, 8):=58883. 32 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.8. This Example 3.30 is only upside-down with magic sum, S14×14(1, 6, 8, 9):=96657. 33 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 •6-Digits Cell Entries Below are two examples of upside-down and/or mirror looking magic squares of order 14 with three digits (1,6,9) and (2,5,8) Example 3.9. This Example 3.9 is upside-down and mirror looking with magic sum S14×14(2, 5, 8):=8048040. 34 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.10. This Example 3.10 is only upside-down with magic sum, S14×14(1, 6, 9):=9453444. 35 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 •8-Digits Cell Below are three examples of upside-down and/or mirror looking magic squares of order 14 with two digits (1,8), (2,5) and (6,9) Example 3.11. This Example 3.11 is upside-down and mirror looking with magic sum S14×14(1, 8):=699999993. The internal block is a magic square of order 4. 36 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.12. This Example 3.12 is upside-down and mirror looking with magic sum S14×14(2, 5):=544444439. The internal block is a magic square of order 4. 37 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.13. This Example 3.13 is only upside-down with magic sum S14×14(6, 9):=1166666655. The internal block is a magic square of order 4. For more details refer author’s work [46, 47]. 38 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 3.3.2 Water Reflection •4-Digits Magic Squares In this subsection we shall bring four magic squares of order 14 of 4-digits (2,3,5,8), (1,2,3,5), (0,2,3,5) and (0,1,3,8). Example 3.14. Let’s consider a magic square of order 14 with 4-digits (2,3,5,8) given by The above magic square of order 14 with digits is (2,3,5,8) is water reflexive. Let’s see below few more examples. 39 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.15. Let’s replace in Example 3.14, we get The magic sum of above magic square is S14×14(1, 2, 3, 5):=44440. 40 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.16. Let’s replace in Example 3.14, we get The above sum of above magic square is S14×14(0, 2, 3, 5):=41107. 41 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.23. Let’s replace in Example 3.22, we get The magic sum of above magic square is S14×14(1, 3):=311111108. 48 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.24. Let’s replace in Example 3.22, we get The above sum of above magic square is S14×14(0, 3):=233333331. Finally, we have four magic squares of order 14 with 2-digits (3,8), (1,3) and (0,3) results in water reflection magic squares with same magic sums. For more details refer author’s work [46]. 49 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 3.4 Self-Made Algebraic Magic Square Below are two result of a magic squares order 14 made in such a way that a magic square of order 12 is embedded in it. Result 1. Below is an algebraic magic square of order 14 •Details It is a single-digit bordered magic square of order 14 embedded with a pandiagonal magic square of order 12. It is composed of 9 equal sums pandiagonal magic squares of order 4. The letter S represents the magic sum of magic square of order 4. In this case the magic sum of order 14 is 7 2×S. We call it a self-made as it is complete in itself. Just choose values for A1 to A52 and the magic sum S, we always get a magic square of order 14. Sometimes we may call it as algebraic magic square for reduced entries. It requires only 52 entries instead of 196. To avoid decimal entries we need to choose even number for the magic sum of order 4. See below two examples: 50 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.25. Let’s consider a magic square The magic sum of above magic square is S14x14 :=168. The internal block is pandiagonal magic square of order 12 formed by equal sums pandiagonal magic squares of order 4 with magic sum S4x4:=48 51 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.26. Let’s consider a magic square The magic sum of above magic square is S14x14 :=252. The internal block is pandiagonal magic square of order 12 formed by equal sums pandiagonal magic squares of order 4 with magic sum S4x4:=72 52 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Result 2. Below is an algebraic magic square of order 14 •Details It is a single-digit bordered magic square of order 14 embedded with a magic square of order 12. It is composed of 9 equal sums magic squares of order 4. The letter S represents the magic sum of magic square of order 4. In this case the magic sum of order 14 is 7 2×S. We call it a self-made as it is complete in itself. Just choose values for A1 to A78 and the magic sum S, we always get a magic square of order 14. Sometimes we may call it as algebraic magic square for reduced entries. It requires only 78 entries instead of 196. To avoid decimal entries of magic square of order 14 we need to choose even number for the magic sum of order 4. See below two examples: Example 3.27. Let’s consider a magic square 53 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 The magic sum of above magic square is S14x14 :=147. The internal block is a magic square of order 12 formed by equal sums pandiagonal magic squares of order 4 with magic sum S4x4:=42 54 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Example 3.28. Let’s consider a magic square The magic sum of above magic square is S14x14 :=182. The internal block is a magic square of order 12 formed by equal sums pandiagonal magic squares of order 4 with magic sum S4x4:=52 Remark 3.1. The different between two Results 1 and 2 is that in the first result the magic square of order 12 is pandiagonal with magic square of order 4 also pandiagonal, while in case of second result the magic squares of orders 12 and 4 are just magic square not pandiagonal. The second example allows us to bring magic square of order 14 with odd order sum without decimal entries. For more studied toward this direction see the author’s work [57, 58, 59, 60, 61, 62, 63, 64]. 3.5 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. 3.5.1 Cyclic-Type Result 3. An algebraic striped magic square of order 14 in cyclic way is given by 55 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 •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 3.29. Below are two examples of algebraic striped magic square of order 14 based on Result 3 are given by 56 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S6×6:=213, S10×10 :=355, S14×14 :=497 and m:=71. (ii) Second Example:S6×6:=246, S10×10 :=410, S14×14 :=574 and m:=82. 3.5.2 Flat-Type Result 4. An algebraic striped magic square of order 14 in flat way is given by 57 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 [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. 64 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 •Double Digits and Cornered Magic Squares [33] Inder J. Taneja, Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24, Zenodo, April, 26, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.7866956. [34] Inder J. Taneja, New Concepts in Magic Squares: Double Digits Bordered Magic Squares of Orders 7 to 108, Zenodo, August 09, 2023, pp. 1-30, https://doi.org/10.5281/zenodo.8230214. [35] Inder J. Taneja, Cornered Magic Squares of Orders 14 to 24, Zenodo, June 03, 2023, pp. 1-39, https://doi.org/10.5281/zenodo.8000471. [36] Inder J. Taneja, New Concepts in Magic Squares: Cornered Magic Squares of Orders 5 to 81, Zenodo, August 09, 2023, pp. 1-27, https://doi.org/10.5281/zenodo.8231157. [37] Inder J. Taneja, Cornered Magic Squares in Construction of Magic Squares of Orders 16, 20, 24 and 28, August 23, 2023, https://numbers-magic.com/?p=10172. [38] Inder J. Taneja, New Concepts in Magic Squares: Cornered Magic Squares of Orders 5 to 108, Zenodo, January 29, 2025, pp. 1-33, https://doi.org/10.5281/zenodo.14759238. •Striped Magic Squares [39] Inder J. Taneja, Striped Magic Squares of Even Orders 4, 6, 8 and 10, Zenodo, November 10, 2023, pp. 1-34, https://doi.org/10.5281/zenodo.15228903. [40] Inder J. Taneja, Striped Magic Squares of 12 – Revised, Zenodo, September 07, 2024, pp. 1-30, https://zenodo.org/records/13725031. [41] Inder J. Taneja, 5600+ Striped Magic Squares of Order 16, Zenodo, February 05, 2025, pp. 1-52, https://doi.org/10.5281/zenodo.14807639. [42] Inder J. Taneja, Striped Magic Squares of 18, Zenodo, June 13, 2024, pp. 1-34, https://doi.org/10.5281/zenodo.11629567. [43] Inder J. Taneja, 8000+ Striped Magic Squares of 20, Zenodo, March 15, 2025, pp. 1-37, https://doi.org/10.5281/zenodo.15032524. [44] Inder J. Taneja, Striped and Semi-Striped Double Digits Bordered Magic Squares: Orders 7 to 50, Zenodo, March 13, 2025, pp. 1-30, https://doi.org/10.5281/zenodo.15021581. 65 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 [45] Inder J. Taneja, Striped and Semi-Striped Cornered Magic Squares of Orders 6 to 50 – Recreating Numbers and Magic Squares, March 17, 2025, https://numbers-magic.com/?p=15048 •Upside-down, Mirror Looking and Water Reflection Magic and Bimagic Squares [46] Inder J. Taneja, Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 14 to 16, Zenodo, January 15, 2024, pp. 1-140, https://doi.org/10.5281/zenodo.14649519. [47] Inder J. Taneja, 2-Digits Universal and Upside-Down Palindromic Magic and Bimagic Squares: Orders 3 to 16, Zenodo, April 07, 2020, pp. 1-103, https://doi.org/10.5281/zenodo.3743362. •Different Styles of Magic Squares [48] Inder J. Taneja, Different Styles of Magic Squares of Orders 6, 8, 10 and 12 Using Bordered Magic Rectangles, Zenodo, November 14, 2022, pp. 1-26, https://doi.org/10.5281/zenodo.7319985. [49] Inder J. Taneja, Different Styles of Magic Squares of Order 14 Using Bordered Magic Rectangles, Zenodo, November 14, 2022, pp. 1-40, https://doi.org/10.5281/zenodo.7319787. [50] Inder J. Taneja, Different Styles of Magic Squares of Order 16 Using Bordered Magic Rectangles, Zenodo, November 14, 2022, pp. 1-63, https://doi.org/10.5281/zenodo.7320116. [51] Inder J. Taneja, Different Styles of Magic Squares of Order 18 Using Bordered Magic Rectangles, Zenodo, November 14, 2022, pp. 1-85, https://doi.org/10.5281/zenodo.7320131. [52] Inder J. Taneja, Different Styles of Magic Squares of Order 20 Using Bordered Magic Rectangles, Zenodo, November 15, 2022, pp. 1-88, https://doi.org/10.5281/zenodo.7320877. [53] Inder J. Taneja, Different Styles of Magic Squares of Orders 8, 10 and 12 Using Bordered Magic Rectangles and the Formula (a+b)2,Zenodo, September 18, 2022, pp. 1-30, https://doi.org/10.5281/zenodo.7090737. Also see the following links: [54] Inder J. Taneja, Different Styles of Magic Squares of Order 14 Using Bordered Magic Rectangles and the Formula (a+b)2,Zenodo, September 18, 2022, pp. 1-45, https://doi.org/10.5281/zenodo.7090764. [55] Inder J. Taneja, Different Styles of Magic Squares of Order 16 Using Bordered Magic Rectangles and the Formula (a+b)2,Zenodo, September 18, 2022, pp. 1-85, https://doi.org/10.5281/zenodo.7090770. 66 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 [56] Inder J. Taneja, Different Styles of Magic Squares of Order 18 Using Bordered Magic Rectangles and the Formula (a+b)2,Zenodo, September 23, 2022, pp. 1-136, https://doi.org/10.5281/zenodo.7108694. Also see the following links: Different Styles of Magic Squares of Order 18 Using Bordered Magic Rectangles and the Formula (a+b)2. •Self-Made Algebraic Magic Squares [57] 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. [58] 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. [59] 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. [60] Inder J. Taneja. Self-Made Algebraic Magic, Semi-Magic and Pandiagonal Magic Squares of Order 10, Zenodo, September 18, 2025, pp. 1-112, https://doi.org/10.5281/zenodo.17149185. [61] 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. [62] 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. [63] 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. [64] 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. [65] 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. [66] 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. 67 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 14, Zenodo, November 10, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602 •Different Types and Aspects of Magic Squares [67] Inder J. Taneja, Different Types and Aspects of Magic Squares of Order 14, Zenodo, December 03, 2025, pp. 1-68, https://doi.org/10.5281/zenodo.17806602. [68] Inder J. Taneja, Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610. [69] Inder J. Taneja, Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619. [70] Inder J. Taneja, Different Types and Aspects of Magic Squares of Order 18, Zenodo, December 03, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.17806626 ————————————————- 68