Different Types and Aspects of Magic Squares of Order 16

Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Different Types and Aspects of Magic Squares of Order 16 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 16. 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 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Contents 1 Introduction 3 2 Different Types of Magic Squares 4 2.1 Block-WiseMagicSquares................................................ 5 2.2 Block-WiseBimagicSquare ............................................... 7 2.3 Single-DigitBorderedMagicSquares.......................................... 8 2.4 Embedded Single-Digit Bordered Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 CorneredMagicSquare ................................................. 13 2.6 Double-DigitBorderedMagicSquares......................................... 14 2.7 StripedMagicSquaresofOrder16 ........................................... 17 2.8 DifferentStylesofMagicSquares............................................ 21 2.9 Four-DigitsBorderedSquares.............................................. 25 3 Different Aspects of Magic Squares 29 3.1 LatinSquareDistributions................................................ 29 3.2 Palindromic with Equal Sum Blocks of Order 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 PalindromicBimagicSquare............................................... 33 3.4 PerfectSquareEntriesSum ............................................... 34 3.4.1 UniformityProperty............................................... 34 3.4.2 PythagoreanTriple................................................ 36 3.4.3 Minimum Perfect Square Entries Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Upside-Down, Mirror-Looking and Water Reflection 39 4.1 Upside-DownandMirrorLooking........................................... 40 4.2 Upside-Down and Mirror Looking with Magic Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.3 WaterReflectionMagicSquares............................................. 54 4.4 Self-MadeAlgebraicMagicSquare........................................... 62 4.5 Double-Digit Algebraic Striped Magic Squares of Order 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.5.1 Cyclic-Type .................................................... 64 4.5.2 Flat-Type...................................................... 67 4.5.3 Cornered...................................................... 70 5 Author’s Contribution to Magic Squares and Recreation of Numbers 74 2 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 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 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 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 According to Equation (1), the magic sum of order 16 is given by S16×16 :=16 ×(1+162) 2=2056. We can write, 16 :=42=4×4. This gives the possibility of blocks of order 4 and 8. Let’s see division of 2056 by 4 and 2: (i)2056 4=514 =⇒equal blocks of order 4; (ii)2056 2=1024 =⇒equal blocks of order 8. Below are few examples of different types of magic squares of order 16. 4 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 2.1 Block-Wise Magic Squares Example 2.1. It is a block-wise pandiagonal magic square of order 16 with equal sums pandiagonal magic squares of order 4. 5 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 2.2. It is a block-wise magic square of order 16, where the blocks are single-digit bordered magic squares of order 8. 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 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 2.2 Block-Wise Bimagic Square Example 2.3. It is a bimagic square of order 16. The blocks of order 4 are equal sums magic square of order 4. For more details refer author’s work [11] 7 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 2.3 Single-Digit Bordered Magic Squares Example 2.4. It is a single-digit bordered magic square of order 16. Removing the higher borders still we are left with magic squares of lower orders, such as of orders 14, 12, 10,... etc. The internal block is a magic square of order 4. For more details refer author’s work [22, 24] 8 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 2.4 Embedded Single-Digit Bordered Magic Squares Example 2.5. It is a single-digit bordered magic square of order 16 embedded with a magic square of order 12 formed by equal sums semi-magic squares of order 3. 9 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 2.12. It is a alsodouble-digit bordered magic square of order 16 written in little different way. In each blocks the magic rectangles are of equal width and length. We call these types as cyclic magic rectangles. Removing the higher borders still we are left with magic squares of lower orders, such as of orders 12, 8 and 4. The internal block is a pandiagonal magic square of order 4. For more details refer author’s work [33, 34] 16 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 2.7 Striped Magic Squares of Order 16 Example 2.13. It is a also double-digit bordered magic square of order 16 formed by magic rectangle strips of equal width. The only difference is in the length of each magic rectangle. These types if magic squares we call as striped magic squares. 17 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 2.14. It is a also double-digit bordered magic square of order 16 formed by magic rectangle strips of equal width. The only difference is in the length of each magic rectangle. These types if magic squares we call as striped magic squares. 18 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 2.15. It is a also double-digit corner type magic square of order 16 formed by magic rectangle strips of equal width. The only difference is in the length of each magic rectangle. These types if magic squares we call as striped magic squares. 19 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 2.16. It is a also striped magic square of order 16 as it is formed by equal width and length of magic rectangles, i.e., 2×4. For more details refer author’s work [39, 41, 45]. 20 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 2.8 Different Styles of Magic Squares Example 2.17. It is a magic square of order 16 formed by magic, bordered magic 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 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 2.18. It is a magic square of order 16 formed by magic squares and magic rectangles. 22 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 2.19. It is a magic square of order 16 formed by magic rectangles. 23 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 2.20. It is a magic square of order 16 formed by magic rectangles. For mor details refer author’s work [6, 7, 8, 10] 24 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 2.9 Four-Digits Bordered Squares Example 2.21. It is a four-digit bordered magic square of order 16. Four digits means border formed by magic squares of order 4. In this case case easily replace the inner part of the magic square foming magic square of order 8. 25 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 3.2 Palindromic with Equal Sum Blocks of Order 4 Example 3.2. Apandiagonal magic square of order 16 with equal sum pandiagonal magic squares of order 4 is given by In this case, the magic sum is S16×16 =479960. All the 4×4blocks are pandiagonal magic squares of order 4 with equal magic sums, S4×4=119990. 32 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 3.3 Palindromic Bimagic Square Example 3.3. Abimagic square of order 16 is given by In this case, the magic and bimagic sums are S16×16 :=479960 and Sb16×16 :=16484528520 respectively. Details of this bimagic square can be seen in author’s work [?]. The blocks of order 4 are also magic squares with equal magic sums S4×4:=119990 For more details refer author’s work [12, 13] 33 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 3.4 Perfect Square Entries Sum Below are five different magic squares of order 16 resulting in uniformity,Pythagorean triples and minimum perfect square sum properties. Out of these five, three of them are with fraction numbers entries. 3.4.1 Uniformity Property Example 3.4. Ablock-wise pandiagonal magic square of order 16 for consecutive odd numbers entries {1, 3, 5, ..., 509, 511}is given by 34 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 3.5. Ablock-wise pandiagonal magic square of order 16 for consecutive fraction numbers entries {257/2, 259/2, . . . , 763/2, 767/2} is given by Both the Examples 3.4 and 3.5 are with equal magic sums. The blocks of order 4 are pandiagonal magic squares with equal magic sums. See the details below: S16×16 =4096 =163;T256 :=16 ×4096 =65536 =2562=164 S4×4:=1024; T16 :=4×1024 =4096 =642. The Examples 3.4 and 3.5 also satisfy the uniformity property, i.e., 16, 162, 163, 164. 35 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 3.4.2 Pythagorean Triple Example 3.6. Ablock-wise pandiagonal square of order 16 for consecutive odd numbers entries {69, 71, . . . , 577, 579}is given by 36 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 3.7. Ablock-wise pandiagonal square of order 16 for consecutive fraction numbers entries {393/2, 395/2, . . . , 901/2, 903/2} is given by Both the Examples 3.6 and 3.7 are with equal magic sums. The blocks of order 4 are pandiagonal magic squares with equal magic sums. See the details below: S16×16 =5184; T256 :=16 ×5184 =82944 =2882; S4×4=1296; T16 :=4×1296 =5184 =722. Both the Examples 3.6 and 3.7 are generated by Pythagorean triple (34, 288, 290), i.e., 342+2882=2902with least possible entries resulting in perfect square entries sum. 37 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 3.4.3 Minimum Perfect Square Entries Sum Example 3.8. Ablock-wise pandiagonal magic square of order 16 for consecutive fraction numbers entries {33/2, 35/2, . . . , 541/2, 543/2} is given by The entries sum is minimum perfect square. The blocks of order 4 are pandiagonal magic squares with equal magic sums. See below the details: S16×16 =2304; T256 :=16 ×2304 =36864 =1922 S4×4=576; T16 :=4×576 =2304 =482. For more details refer author’s work [14, 15, 16, 17, 18]. 38 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 4 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: •180oRotation In this case, the readable numbers are 0, 1, 2, 5, 6, 8 and 9 where the number 6 becomes 9 and 9 as 6. Thus the survival numbers after 180orotation are 0, 1, 2, 5, 6, 8 and 9. Sometimes we call them as upside-down numbers. •Mirror Looking It is same as horizontal flip as describe above. In this case, we have 39 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 In this case, the readable numbers are 0, 1, 2, 5 and 8, where 2 becomes 5 and 5 as 2. Thus the survival numbers after horizontal flip are 0, 1, 2, 5 and 8. Sometimes we call them as mirror-looking numbers. •Water Reflection It is same as vertical flip as describe above. In this case, we have In this case, the readable numbers are 0, 1, 2, 3, 5 and 8, where 2 becomes 5 and 5 as 2. Thus the survival numbers after vertical flip are 0, 1, 2, 3, 5 and 8. For the first time we call these numbers as water reflection or water reflexive numbers. We observe that the numbers 0, 1, 2, 5 and 8 are in all the three situations, i.e., these are upside-down,mirror-looking and water reflexive. We call them as universal numbers provided they are written in digital form. There left only one number 3. It is only water reflexive. While the numbers 6 and 9 are only upside-down. There is a lot of work by author on upside-down and mirror-looking numbers. This work is concentrated only towards magic squares having water reflexive numbers, i.e., 0, 1, 2, 3, 5 and 8. The numbers 0, 1, 2, 5 and 8 are already studied previously. This work brings magic squares of order 14 to 16 specially in number 3 along with 0, 1, 2, 5 and 8. 4.1 Upside-Down and Mirror Looking This section brings magic and bimagic squares of order 16 written in such way that, if we make 180orotation and/or see in the mirror, still we are able to read the entries again resulting in magic squares. It is possible only with the digits 0, 1, 2, 5, 6, 8 and 9 written in digital forms. Example 4.1. The magic squares of order 16 for the digits (1, 2, 5, 8)is given by 40 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 The above magic square for the digits (1, 2, 5, 8)is upside-down and mirror-looking, i.e., it is a universal magic squares of order 16 with magic sum S16×16(1, 2, 5, 8):=71104. The blocks of order 4 are magic squares with different magic sums. 41 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Below are three examples of bimagic squares of order 16 for the digits (1, 8),(2, 5)and (6, 9). Example 4.8. The bimagic squares of order 16 for the digits (1, 8)is given by The magic and bimagic sums of above magic square are S16×16(1, 8):=799999992 and Sb16×16(1, 8):=59797978997979800. The blocks of order 4 are magic squares with equal magic sums. It is upside-down and mirror-looking, i.e., universal magic square. 48 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 4.9. The bimagic squares of order 16 for the digits (2, 5)is given by The magic and bimagic sums of above magic square are S16×16(2, 5):=622222216 and Sb16×16(2, 5):=27833894016610552. The blocks of order 4 are magic squares with equal magic sums. It is upside-down and mirror-looking, i.e., universal magic square. 49 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 4.10. The bimagic squares of order 16 for the digits (6, 9)is given by The magic and bimagic sums of above magic square are S16×16(6, 9):=1333333320 and Sb16×16(6, 9):=114747472525252536. The blocks of order 4 are magic squares with equal magic sums. It is only upside-down. 50 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 4.2 Upside-Down and Mirror Looking with Magic Rectangles Example 4.11. The magic squares of order 16 formed by magic rectangles of order 2×4for the digits (1, 8)is given by The magic sums of above magic square is S16×16(1, 8):=799999992. The blocks of order 4 are pandiagonal magic squares with equal magic sums. It is upside-down and mirror-looking, i.e., universal magic square . It is also pandiagonal. 51 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 4.12. The magic squares of order 16 formed by magic rectangles of order 2×4for the digits (2, 5)is given by The magic sums of above magic square is S16×16(2, 5):=622222216. The blocks of order 4 are pandiagonal magic squares with equal magic sums. It is upside-down and mirror-looking, i.e., universal magic square . It is also pandiagonal. 52 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 4.13. The magic squares of order 16 formed by magic rectangles of order 2×4for the digits (6, 9)is given by The magic sums of above magic square is S16×16(6, 9):=1333333320. The blocks of order 4 are pandiagonal magic squares with equal magic sums. It is only only upside-down, It is also pandiagonal. 53 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 4.3 Water Reflection Magic Squares Below are few examples of magic square with water reflection. These examples are for 4-digits: (2,3,5,8), (1,2,3,5), (0,2,3,5) and (0,1,3,8) with water reflection property. Example 4.14. Let’s consider a magic square of order 16 with 4-digits (2,3,5,8) given by The above magic squares is bimagic. The magic sums are S16×16(2, 3, 5, 8) = 79992 and Sb16×16(2, 3, 5, 8) = 484768488. The blocks of order 4 are magic squares with equal magic sums, i.e., S4×4(2, 3, 5, 8) = 19998. 54 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 •Water Reflection Applying the vertical flip over the above magic square, we get Thus, the above magic square of order 16 with digits is (2,3,5,8) is water reflexive and bimagic. The magic sums are S16×16(2, 3, 5, 8) = 79992 and Sb16×16(2, 3, 5, 8) = 484768488. The blocks of order 4 are magic squares with equal magic sums, i.e., S4×4(2, 3, 5, 8) = 19998. Let’s see below few more examples. 55 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 4.15. Let’s replace 8 by 1 in Example 4.14, we get It is bimagic square. The magic sums are S16×16(1, 2, 3, 5) = 48884 and Sb16×16(1, 2, 3, 5) = 184706376. The blocks of order 4 are magic squares with equal magic sums, i.e., S4×4(1, 2, 3, 5) = 12221. 56 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 •Water Reflection Applying the vertical flip over the above magic square, we get It is bimagic square. The magic sums are S16×16(1, 2, 3, 5) = 48884 and Sb16×16(1, 2, 3, 5) = 184706376. The blocks of order 4 are magic squares with equal magic sums, i.e., S4×4(1, 2, 3, 5) = 12221. Thus, the above magic square is water reflexive with equal magic sums. The blocks of order 4 are magic squares with equal magic sums. 57 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Example 4.19. Let’s consider a following magic square based on the Result 1 Each block of order 4 is a magic square of order 4 with magic sum S4×4:=47 resulting in magic sum of order 16 as S16×16 :=188. For more studied toward this direction see the author’s work [57, 58, 59, 60, 61, 62, 63, 64]. 4.5 Double-Digit Algebraic Striped Magic Squares of Order 16 Below are three different ways of writing magic square of order 16, i.e., cyclic,flat and cornered. 4.5.1 Cyclic-Type Result 2. An algebraic striped magic square of order 16 in cyclic 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 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 •Details It is an algebraic cyclic striped magic square of order 16 composed of four equal sums magic rectangles of orders 2×14 embedded with a magic square of order 12. It is again composed of four equal sums magic rectangles of orders 2×10 embedded with a magic square of order 8. It is again composed of four equal sums magic rectangles of orders 2×6embedded with a magic square of order 4. This magic square of order 4 is with equal sums two magic rectangles of order 2×4. In this case the magic sums are S4×4:=2m,S8×8:=4m,S12×12 :=6mand S16×16 :=8m, where mis the width of the strip. See below two examples: Example 4.20. Below are two examples of algebraic striped magic square of order 16 based on Result 2 are given by 65 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 66 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=218, S8×8:=436, S12×12 :=654, S16×16 :=872 and m:=109. (ii) Second Example:S4×4:=244, S8×8:=488, S12×12 :=732, S16×16 :=976 and m:=122. 4.5.2 Flat-Type Result 3. An algebraic striped magic square of order 16 in flat way is given by 67 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 •Details It is an algebraic flat-type striped magic square of order 16 composed of two equal sums magic rectangles of orders 2×16 and two magic rectangles of order 2×12 embedded with a magic square of order 12. It is again composed of two equal sums magic rectangles of orders 2×12 and two magic rectangles of order 2×8embedded with a magic square of order 8. It is again composed of two equal sums magic rectangles of orders 2×8and two magic rectangles of order 2×4embedded with a magic square of order 4. It is again composed of two equal sums magic rectangles of order 2×4. In this case the magic sums are S4×4:=2m,S8×8:=4m,S12×12 :=6mand S16×16 :=8m, where mis the width of the strip. See below two examples: Example 4.21. Below are two examples of algebraic striped magic square of order 16 based on Result 3 are given by 68 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 69 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=202, S8×8:=404, S12×12 :=606, S16×16 :=808 and m:=101. (ii) Second Example:S4×4:=236, S8×8:=472, S12×12 :=708, S16×16 :=944 and m:=118. 4.5.3 Cornered Result 4. An algebraic striped cornered magic square of order 16 is given by 70 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 •Details It is an algebraic cornered striped magic square of order 16, where the magic squares of order 4, 6, 8, 10, 12 and 14 are at the upper-left corner. In this case the magic sums are S4×4:=2m,S6×6:=3m,S8×8:=4m,S10×10 :=5m,S12×12 :=6m, S14×14 :=7mand S16×16 :=8m, where mis the width of the strip. See below two examples: Example 4.22. Below are two examples of algebraic striped magic square of order 16 based on Result 4 are given by 71 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 72 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 Above there are two examples. In both the cases, the magic sums are: (i) First Example:S4×4:=172, S6×6:=264, S8×8:=352, S10×10 :=440, S12×12 :=528, S14×14 :=616, S16×16 :=704 and m:=88. (ii) Second Example:S4×4:=242, S6×6:=363, S8×8:=464, S10×10 :=585, S12×12 :=726, S14×14 :=847, S16×16 :=968 and m:=121. For more details refer Taneja [65, 66]. 73 Inder J. Taneja https://inderjtaneja.wordpress.com; https://numbers-magic.com Different Types and Aspects of Magic Squares of Order 16, Zenodo, December 03, 2025, pp. 1-80, https://doi.org/10.5281/zenodo.17806619 •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 ————————————————- 80