Different Types and Aspects of Magic Squares of Order 15

Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Different Types and Aspects of Magic Squares of Order 15 The whole work as pdf files is available at author’s sites: https://numbers-magic.com/?p=16911 Inder J. Taneja1 Abstract This work summarizes author’s previous works on magic squres of order 15. 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 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Contents 1 Introduction 3 2 Different Types of Magic Squares of Order 15 4 2.1 Block-WiseMagicSquares................................................ 4 2.2 Single-DigitBorderedMagicSquares.......................................... 7 2.3 CorneredMagicSquares................................................. 9 2.4 Double-Digits Bordered Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Double-Digits Embedded Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Triple-DigitBorderedMagicSquares.......................................... 15 3 Different Aspects of Magic Squares of Order 15 17 3.1 LatinSquareDistributions................................................ 17 3.2 PerfectSquareEntriesSum ............................................... 20 3.2.1 UniformityProperty............................................... 20 3.2.2 PythagoreanTriple................................................ 22 3.2.3 Minimum Perfect Square Sum of Entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Upside-Down, Mirror Looking and Water Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3.1 Upside-Down and Mirror Looking Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.2 Water Reflection and Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Self-Made Algebraic Magic Squares of Order 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4.1 Equal Sum Magic Squares of Order 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4.2 Equal Sum Magic Squares of Order 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.3 Double-DigitBordered.............................................. 56 4 Author’s Contribution to Magic Squares and Recreation of Numbers 59 2 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 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. 3 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 17. Algebraic Maic 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 15 According to Equation (1), the magic sum of order 15 is given by S15×15 :=15 ×(1+152) 2=1695. We can write, 15 :=3×5. This gives the possibilities of blocks of orders 3 and 5. Let’s see the divisions of magic square sum 1695 by 5 and 3: (i)1595 5=339 =⇒equal or unequal sum blocks of order 3; (ii)29679 3=565 =⇒equal sum blocks of order 5. 2.1 Block-Wise Magic Squares Example 2.1. 4 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 5 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 2.2. 6 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 2.2 Single-Digit Bordered Magic Squares Example 2.3. 7 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 2.4. 8 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 2.3 Cornered Magic Squares Example 2.5. 9 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 2.12. 16 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 3 Different Aspects of Magic Squares of Order 15 Below are few examples of magic square of order 14 representing different aspects. Initially we shall present the idea of Latin square distributions. 3.1 Latin Square Distributions Example 3.1. A pair of Latin square decomposition of magic square of order 15 given in Example 3.35 is as follows: A 120 1 6 7 12 13 3 4 9 10 15 2 5 8 11 14 120 12 13 1 6 7 10 15 3 4 9 11 14 2 5 8 120 6 7 12 13 1 4 9 10 15 3 5 8 11 14 2 120 13 1 6 7 12 15 3 4 9 10 14 2 5 8 11 120 7 12 13 1 6 9 10 15 3 4 8 11 14 2 5 120 3 4 9 10 15 2 5 8 11 14 1 6 7 12 13 120 10 15 3 4 9 11 14 2 5 8 12 13 1 6 7 120 4 9 10 15 3 5 8 11 14 2 6 7 12 13 1 120 15 3 4 9 10 14 2 5 8 11 13 1 6 7 12 120 9 10 15 3 4 8 11 14 2 5 7 12 13 1 6 120 2 5 8 11 14 1 6 7 12 13 3 4 9 10 15 120 11 14 2 5 8 12 13 1 6 7 10 15 3 4 9 120 5 8 11 14 2 6 7 12 13 1 4 9 10 15 3 120 14 2 5 8 11 13 1 6 7 12 15 3 4 9 10 120 8 11 14 2 5 7 12 13 1 6 9 10 15 3 4 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 17 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 B 120 2 11 5 14 8 3 10 4 15 9 1 12 6 13 7 120 5 14 8 2 11 4 15 9 3 10 6 13 7 1 12 120 8 2 11 5 14 9 3 10 4 15 7 1 12 6 13 120 11 5 14 8 2 10 4 15 9 3 12 6 13 7 1 120 14 8 2 11 5 15 9 3 10 4 13 7 1 12 6 120 1 12 6 13 7 2 11 5 14 8 3 10 4 15 9 120 6 13 7 1 12 5 14 8 2 11 4 15 9 3 10 120 7 1 12 6 13 8 2 11 5 14 9 3 10 4 15 120 12 6 13 7 1 11 5 14 8 2 10 4 15 9 3 120 13 7 1 12 6 14 8 2 11 5 15 9 3 10 4 120 3 10 4 15 9 1 12 6 13 7 2 11 5 14 8 120 4 15 9 3 10 6 13 7 1 12 5 14 8 2 11 120 9 3 10 4 15 7 1 12 6 13 8 2 11 5 14 120 10 4 15 9 3 12 6 13 7 1 11 5 14 8 2 120 15 9 3 10 4 13 7 1 12 6 14 8 2 11 5 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 By application of the formula 15 ×(A−1) + B, we get a magic square given in Example 2.1. Grid 1. Let’s write the above Latin squares in terms of a Grid representing the numbers in alphabets: a f g l m c d i j p b e h k n l m a f g j p c d i k n b e h f g l m a d i j p c e h k n b m a f g l p c d i j n b e h k g l m a f i j p c d h k n b e c d i j p b e h k n a f g l m j p c d i k n b e h l m a f g d i j p c e h k n b f g l m a p c d i j n b e h k m a f g l i j p c d h k n b e g l m a f b e h k n a f g l m c d i j p k n b e h l m a f g j p c d i e h k n b f g l m a d i j p c n b e h k m a f g l p c d i j h k n b e g l m a f i j p c d A b k e n h c j d p i a l f m g e n h b k d p i c j f m g a l h b k e n i c j d p g a l f m k e n h b j d p i c l f m g a n h b k e p i c j d m g a l f a l f m g b k e n h c j d p i f m g a l e n h b k d p i c j g a l f m h b k e n i c j d p l f m g a k e n h b j d p i c m g a l f n h b k e p i c j d c j d p i a l f m g b k e n h d p i c j f m g a l e n h b k i c j d p g a l f m h b k e n j d p i c l f m g a k e n h b p i c j d m g a l f n h b k e A 18 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 It is the general procedure to write magic squares of order 15 by applying the condition 15 ×(A−1) + Band choosing properly the value of letters. 19 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 3.2 Perfect Square Entries Sum Below are five different magic squares of order 15 resulting in uniformity,Pythagorean triples and minimum perfect square entries sum properties. 3.2.1 Uniformity Property Example 3.2. For the consecutive odd numbers entries {1, 3, 5, . . . , 447, 449}, a pandiagonal magic square of order 15 is given by 20 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.3. For the consecutive natural numbers entries {113, 114, 115, . . . , 336, 337}, a pandiagonal magic square of order 15 is given by In both the Examples 3.2 and 3.3, the magic sums are equal. The blocks of order 5 are pandiagonal magic squares with equal magic sums. See below the details: S15×15 :=3375; T225 :=15 ×3375 =50625 =2252=154; S5×5=1125; T25 :=5×1125 =5625 =752. The Examples 3.2 and 3.3 also satisfy the uniformity property, i.e., 15, 152, 153, 154. 21 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 3.2.2 Pythagorean Triple Example 3.4. For the consecutive odd numbers entries {65, 67, . . . , 511, 513}, a pandiagonal magic square of order 15 is given by 22 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.5. For the consecutive natural numbers entries {177, 178, . . . , 400, 401}, a pandiagonal magic square of order 15 is given by In both the Examples 3.4 and 3.5, the magic sums are equal. The blocks of order 3 are semi-magic squares with equal semi-magic sums. See below the details: S15×15 :=4335; T225 :=15 ×4335 =65025 =2552 Sm3×3:=867; T225 :=3×867 =2601 =512. Both the Examples 3.4 and 3.5 are generated by Pythagorean triple (32, 255, 257), i.e., 322+2552=2572with least possible sum of entries resulting in a perfect square sum. 23 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 3.2.3 Minimum Perfect Square Sum of Entries Example 3.6. For the consecutive natural numbers entries {9, 10, 11, . . . , 232, 233}, a pandiagonal magic square of order 15 is given by The entries sum is minimum perfect square. The blocks of order 5 are pandiagonal magic squares with equal magic sums. See below the details: S15×15 =1815; T729 :=T225 :=15 ×1815 =27225 =1652 S5×5:=605; T25 :=5×605 =3025 =552. 24 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 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: 25 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 •8-Digits Cell: The Numbers 1 and 8 - Blocks of Order 3 Example 3.12. Let’s consider a magic square of order 15 with 2-digits (1,8) with blocks of order 3: The magic square of order 15 given in Example 3.12 with 2-digits (1,8) is universal and semi-magic with semi-magic sum: S15×15(1, 8):=711111104. The blocks of order 3 are semi-magicsquares with different semi-magic sums. 32 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 •8-Digits Cell: The Numbers 2 and 5 - Blocks of Order 5 Example 3.13. Let’s consider a magic square of order 15 with 2-digits (2,5) given by The magic square of order 15 given in Example 3.13 with 2-digits (2,5) is universal and semi-magic with semi-magic sums: S15×15(2, 5):=566666661 and S15×15(2, 5):=599999994. The sum S15×15(2, 5):=599999994 refers to mirror looking version. Blocks of order 5 are pandiagonal magic squares with different magic sums. 33 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 •8-Digits Cell: The Numbers 2 and 5 - Blocks of Order 3 Example 3.14. Let’s consider a magic square of order 15 with 2-digits (2,5) with blocks of order 3: The magic square of order 15 given in Example 3.14 with 2-digits (2,5) is universal and semi-magic with semi-magic sums: S15×15(2, 5):=566666661 and S15×15(2, 5):=599999994. The sum S15×15(2, 5):=599999994 refers to mirror looking version. Blocks of order 3 are semi-magic squares with different semi-magic sums. 34 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 •8-Digits Cell: The Numbers 6 and 9 - Blocks of Order 5 Example 3.15. Let’s consider a magic square of order 15 with 2-digits (6,9) is given by The semi-magic square of order 15 given in Example 3.15 with 2-digits (6,9) is universal and semi-magic with semi-magic sums: S15×15(6, 9):=1233336654. Blocks of order 5 are pandiagonal magic squares with different magic sums. 35 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 •8-Digits Cell: The Numbers 6 and 9 - Blocks of Order 3 Example 3.16. Let’s consider a magic square of order 15 with 2-digits (6,9) with blocks of order 3: The semi-magic square of order 15 given in Example 3.16 with 2-digits (6,9) is upside-down and semi-magic with semi-magic sums: S15×15(6, 9):=1233336654. Blocks of order 3 are semi-magic squares with different semi-magic sums. 36 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 3.3.2 Water Reflection and Magic Squares bullet 4-Digits Magic Squares In this subsection we shall bring four magic squares of order 15 of 4-digits (2,3,5,8), (1,2,3,5), (0,2,3,5) and (0,1,3,8). Example 3.17. Let’s consider a magic square of order 15 with 4-digits (2,3,5,8) given by The above semi-magic square of order 15 with digits is (2,3,5,8) is water reflexive with different semi-magic sums. The blocks of order 5 are pandiagonal magic squares with different magic sums. Let’s see below few more examples. 37 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.18. Let’s replace 8 by 1 in Example 3.17, we get The above semi-magic square is water reflexive with equal semi-magic sums. The blocks of order 5 are pandiagonal magic squares with different magic sums. 38 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.19. Let’s replace 1 by 0 in Example 3.18, we get The above semi-magic square is water reflexive with semi-magic sum S15×15(0, 2, 3, 5):=41107. 39 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.20. Let’s replace 2 by 1 and 5 by 8 in Example 3.19, we get The above semi-magic square is water reflexive with semi-magic sum S15×15(0, 1, 3, 8):=49995. Finally, we have four semi-magic squares of order 15 with 4-digits (2,3,5,8), (1,2,3,5), (0,2,3,5) and (0,1,3,8) results in water reflection semi-magic squares with same semi-magic sums except the first with digits (2,3,5,8). The blocks of order 5 are pandiagonal magic squares with different magic sums. 40 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 bullet 3-Digits Magic Squares In this subsection we shall bring four magic squares of order 15 with 3-digits (2,3,5), (1,3,8), (0,3,8) and (0,1,3). Example 3.21. Let’s consider a magic square of order 15 with 4-digits (2,3,5) given by Thus, the above semi-magic square of order 15 with digits is (2,3,5) is water reflexive with different semi-magic sum. The blocks of order 5 are pandiagonal magic squares with different magic sums. Let’s see below few more examples. 41 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 bullet 2-Digits Magic Squares: Blocks of Order 3 In this subsection, we shall bring four magic squares of order 15 with 2-digits (3,8), (1,3) and (0,3). Here instead of blocks of order 5, we have blocks of order 3 as semi-magic squares of order 3. Example 3.28. Let’s consider a magic square of order 15 with 2-digits (3,8) given by The semi-magic sum of above semi-magic square is S15×15(3, 8):=888888880. 48 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.29. Let’s replace 8 by 1 in Example 3.28, we get The semi-magic sum of above semi-magic square is S15×15(1, 3):=344444441. 49 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.30. Let’s replace 1 by 0 in Example 3.29, we get The semi-magic sum of above semi-magic square is S15×15(0, 3):=266666664. Finally, we have three semi-magic squares of order 15 with 2-digits (3,8), (1,3) and (0,3) results in water reflection semimagic squares with same semi-magic sums. The blocks of order 3 are semi-magic squares with different semi-magic sums. 3.4 Self-Made Algebraic Magic Squares of Order 15 3.4.1 Equal Sum Magic Squares of Order 5 Below is are two results of a magic square order 15 made from the variables in such a way that blocks of orders 5 and 3 are of equal sums magic squares of order 5 and 3. In case of order 5 it is pandiagonal. 50 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Result 1. Below is an algebraic magic square of order 15 •Details It is constructed in such a way that each block of order 5 is a pandiagonal magic square of order 5. The letter S represents the sum of each block of order 5. The magic sum of order 15 is 3×S. We call it a self-made as choosing any numbers for A1 to A72 and the magic sum, we always get a magic square of order 15. Sometimes we may call it as algebraic magic square for reduced entries. It requires only 72 entries instead of 225. See below two examples: 51 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.31. Let’s consider a following magic square based on the Result 1 Each block of order 5 is apandiagonal magic square of order 5 with magic sum S5×5:=76 resulting in magic sum of order 15 as S15×15 :=228. 52 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.32. Let’s consider a following magic square based on the Result 1 Each block of order 5 is apandiagonal magic square of order 5 with magic sum S5×5:=81 resulting in magic sum of order 15 as S15×15 :=243. 3.4.2 Equal Sum Magic Squares of Order 3 Result 2. Below is an algebraic magic square of order 15 53 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 •Details It is constructed in such a way that each block of order 3 is a magic square of order 3. The letter S represents the magic sum of each block of order 3. The magic sum of order 15 is 5×S. We call it a self-made as choosing any numbers for A1 to A50 and the magic sum, we always get a magic square of order 15. Sometimes we may call it as algebraic magic square for reduced entries. It requires only 50 entries instead of 225. See below two examples: 54 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.33. Let’s consider a following magic square based on the Result 1 The blocks of order 3 are a magic squares of order 3 with equal magic sum S3×3:=42 resulting in magic sum of order 15 as S15×15 :=210. 55 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 Example 3.34. Let’s consider a following magic square based on the Result 1 The blocks of order 3 are a magic squares of order 3 with equal magic sum S3×3:=51 resulting in magic sum of order 15 as S15×15 :=255. For more studied toward this direction see the author’s work [57, 58, 59, 60, 61, 62, 63, 64]. 3.4.3 Double-Digit Bordered Result 3. Let’s consider a following double-digit bordered algebraic magic square of order 15 with reduced entries: 56 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 •Details It is a composed of four equal sums magic rectangles of orders 2×8and 2×12 (equal in each case) embedded with a magic square of order 6. This magic square of order 6 is again composed of four equal sums magic squares of order 3. Since the magic square of order 3 requires magic sum as multiple of 3, otherwise we may have decimal entries. The magic sums of orders 10 and 14 are also multiple of 3. The magic sum of orders 10 and 14 are given as S10×10 :=10 S 3and S14×14 :=14 S 3, where Sis the magic sum of order 3. In this case the magic sum of order 6 is given as S6×6:=2S. Below are two examples based on the Result 3 Example 3.35. Let’s consider following two examples based on the Result 3: 57 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 [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. 64 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 15, Zenodo, December 03, 2025, pp. 1-65, https://doi.org/10.5281/zenodo.17806610 •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 ————————————————- 65