207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729

Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729 Site link: https://numbers-magic.com/?p=17312 138th Birth Anniversary of S. Ramanujan, December 22, 1887-April 26, 1920 National Mathematics Day, India – December 22, 2025 Inder J. Taneja1 Abstract This work brings magic squares connected with S. Ramanujan’s life and with Hardy-Ramanujan number 1729. On December 22, 2025 there is 138th anniversary of S. Ramanujan. This work brings magic squares of orders 3 to 20 with magic sum either 1729 or multiple of 1729. This number is famous in the literature as Hardy-Ramanujan Number. The special entries, such as, 2212, 1887, 2025 and 138 are considered in every magic square. There are total 207 different magic squares. The construction of these magic squares is based on reduced-entry algebraic magic squares. These type of magic squares are recently studied by the author [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. The advantage of reduced-entry algebraic magic squares is that they allow us to choose entries as well as magic sum of our choice. Sometimes, we may call them as self-made magic squares. 1Formerly, Professor of Mathematics, Federal University of Santa Catarina, Florianópolis, SC, Brazil (1978-2012). E-mail: [email protected]; Web-sites: https://numbers-magic.com; https://inderjtaneja.wordpress.com; Twitter: @IJTANEJA; Instagram: @crazynumbers. 1 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 Contents 1 Introduction 3 2 Magic Squares of Order 3 3 3 Magic Squares of Order 4 4 4 Magic Squares of Order 5 5 5 Magic Squares of Order 6 5 6 Magic Squares of Order 7 7 7 Magic Squares of Order 8 8 8 Magic Squares of Order 9 11 9 Magic Squares of Order 10 15 10 Magic Squares of Order 11 21 11 Magic Squares of Order 12 30 12 Magic Squares of Order 13 44 13 Magic Squares of Order 14 46 14 Magic Squares of Order 15 49 15 Magic Squares of Order 16 55 16 Magic Squares of Order 17 57 17 Magic Squares of Order 18 59 18 Magic Squares of Order 19 67 19 Magic Squares of Order 20 69 20 Author’s Contributions to Recreating Numbers and Magic Squares 74 2 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 1 Introduction This work brings magic squares connected with S. Ramanujan’s life and with Hardy-Ramanujan number 1729. On December 22, 2025 there is 138th anniversary of S. Ramanujan. This work brings magic squares of orders 3 to 15 and order 19 having magic sum as 1729. Some special entries, such as 2212, 1887, 2025 and 138 are considered. These represents date of birth of S. Ramanujan 22.12.1887 and 138th anniversary on 22.12.2025. Total there 207 different magic squares. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. The advantage of reduced-entry algebraic magic squares is that they allow us to choose entries as well as magic sum of our choice. Sometimes, we may call it as self-made magic squares. The table below give the details of in numbers of each order magic squares studied in this work: Order Magic Squares Semi-Magic Squares Total 31 1 2 42 0 2 52 1 3 65 1 6 78 3 11 88 2 10 915 8 23 10 16 11 27 11 28 23 51 12 23 12 35 13 3 0 3 14 5 0 5 15 6 0 6 16 4 0 4 17 3 0 3 18 8 0 8 19 3 0 3 20 5 0 5 There are total 207 magic squares of orders 3 to 20 studied in this work. All the magic squares brings the four special entries, i.e., 2212, 1887, 2025 and 138. All the magic squares are with magic sum Hardy-Ramanujan number 1729 or the multiples of 1729. 2 Magic Squares of Order 3 Let’s consider following two magic squares of order 3 having the magic sum 1729: 3 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 •Details Above there are two magic squares of order 3. The first one is magic square, while the second one is semi-magic. Both are with magic sum 1729. The difference is that the first is with fractional entries, while second one is with normal entries. The first is only with two numbers 2212 and 1887, while the second one is with all the four special entries, i.e., 2212, 1887, 2025 and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [16, 18] 3 Magic Squares of Order 4 Let’s consider following two magic squares of order 4 in two different having the magic sums 1729 and 2 ×1729: •Details Above there are two examples magic squares of order 4 with magic sum 1729. The first example is in two different ways. First one is pandiagonal, but having decimal entries. The second way is not pandiagonal but don’t require decimal entries. This example is also written in two parts. Both are with two equal sums magic rectangles of orders 2×4. Here the magic sum is 2×1729. This we have considered to avoid decimal entries. The magic rectangles sums are as follows: 4 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 Some times these magic squares are called as striped magic squares. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [17] 4 Magic Squares of Order 5 Let’s consider following three magic squares of order 5 having the magic sum 1729: •Details Above there are three examples magic squares of order 5 with magic sum 1729. The first example is a pandiagonal. The second example is a cornered magic square having magic square of order 3 at the upper-left corner. The third example is a single-digit bordered semi-magic square. All 3 examples use special entries, namely the date of birth of S. Ramanujan (22.12.1887) and his 138th anniversary on 22.12.2025. These contain the four entries 2212, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [16, 18]. 5 Magic Squares of Order 6 Let’s consider following six magic squares of order 6 having the magic sum 1729: 5 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 •Details Above there are six examples magic squares of order 6 with magic sum 1729. The first example is a normal magic square of order 6. The second example is with four equal sums semi-magic squares of order 3. The third and forth examples are with magic rectangles. See below the magic sums of these magic rectangles: The fifth one is cornered magic square of order 6. The the last example is a single-digit bordered semi-magic square of order 6. 6 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 All 6 examples use special entries, namely the date of birth of S. Ramanujan (22.12.1887) and his 138th anniversary on 22.12.2025. These contain the four entries 2212, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [17, 18] . 6 Magic Squares of Order 7 Let’s consider following 11 magic squares of order 7 having the magic sum 1729: ▶First-Type 7 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Second-Type •Details Above there are 11 examples magic squares of order 7 with magic sum 1729. The first eight examples are based on the different types of magic squares. The last three examples are of semi-magic squares of order 7. All 11 examples use special entries, namely the date of birth of S. Ramanujan (22.12.1887) and his 138th anniversary on 22.12.2025. These contain the four entries 2212, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [16, 18]. 7 Magic Squares of Order 8 Let’s consider following 10 magic squares of order 8 having the magic sum 1729: 8 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶First-Type ▶Second-Type 9 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 16 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Second-Type 17 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 18 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Second-Type •Details Above there are 24 examples magic squares of order 10 with magic sum 1729. The first 12 examples are different types of magic squares. The next 13 to 23 examples are of semi-magic squares of order 10. The last example 24 is pandiagonal with four equal sums pandiagonal magic squares of order 5. The example 11 is with decimal entries. See below the sums of four equal sums magic sums of order 5: 19 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Third-Type: Striped Magic Squares Above there are three magic square of order 10 based on magic rectangles. The first and second are double-digit magic squares, while the third one is cornered-type magic square. The the width of the strip is always same, i.e., 1729. The change is only in the lengths. The magic sum for all the three magic square is same, i.e., S10×10 :=8645 =5×1729. 20 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 All 27 examples use special entries, namely the date of birth of S. Ramanujan (22.12.1887) and his 138th anniversary on 22.12.2025. These contain the four entries 2212, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [21, ?,?] 10 Magic Squares of Order 11 Let’s consider following 48 magic squares of order 11 having the magic sum 1729: ▶First-Type 21 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 22 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 23 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 24 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Second-Type 25 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 32 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 33 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Second-Type 34 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 35 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Third-Type 36 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 •Details Above there are 26 examples magic squares of order 12 with magic sum 1729. The first 13 examples are different types of magic squares. The next 10, i.e., 14 to 23 examples are of semi-magic squares of order 12. The last three examples are of different type. All the three examples are pandiagonal. The example 24 is based on 18 equal sums magic rectangles of order 2×4. The example 25 is with 9 equal sums pandiagonal magic squares of order 4. The example 26 is based on 12 equal sums magic squares of order 3. The magic sums of examples 24, 25 and 26 are S12×12 :=10374 =6×1729, S12×12 :=10374 =6×1729 and S12×12 :=20748 =12 ×1729 respectively. See below the sums of magic rectangles and magic squares of lower orders of examples 24, 25 and 26: ▶Magic Rectangles of Equal Sums of Example 24 37 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶9 Equal Sums Pandiagonal Magic Squares of Example 25 ▶Two Magic Squares of Order 3 for the Example 26 In the similar way we can make the sums other 14 magic squares of order 3 for the Example 26. All are of equal magic sums, i.e., 1729. 38 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Forth-Type This magic square of order 12 is composed of 16 equal sums semi-magic squares of order 3. Instead considering magic squares of order 3 if we consider semi-magic squares of order 3, we avoid decimal entries in a magic square of order 12. See below the partial sums of order 3: Proceeding on the same idea for the magic square of order 12 composed of pandiagonal magic squares of order 4, we can consider magic squares of order 4. See below 39 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 This magic square is composed of equal sums magic squares of order 4. By considering this way, we avoid decimal entries. See below the partial sums of order 4. ▶Fifth-Type Below are two more types of magic squares of order 12. Both are blocks magic square of order 6 but in a different way. 40 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 The magic sum of order 6 are as follows: The another type is as follows: 41 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Third-Type •Details Above there are 5 examples magic squares of order 14. The first four examples are with magic sum 1729. The last example is with magic sum double of 1729. This is due to the fact that it is divided in four equal sums pandiagonal magic squares of order 7 with magic sum 1729. See below the individual sum of pandiagonal magic square of order 7. 48 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 The last three examples we call as striped magic squares of order 14. All 5 examples use special entries, namely the date of birth of S. Ramanujan (22.12.1887) and his 138th anniversary on 22.12.2025. These contain the four numbers 2212, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [26, 27] 14 Magic Squares of Order 15 Let’s consider following 2 magic squares of order 15 having the magic sum 1729: 49 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶First-Type •Details Above there are 2 examples magic squares of order 15 with magic sums 5187 =3×1729 and 8642 =5×1729. First examples is based on 9 equal sums pandiagonal magic squares of order 5. The second example is based on 25 equal sums magic squares of order 3. It contains fractional entries. See below the details of these magic squares of orders 5 and 3: 50 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 It contains 15 equal sums pandiagonal magic squares of order 5. There are only two magic squares order 3 are written. The other 23 are also of same magic sums. ▶Second-Type In the first example given above we have fractional values, because the magic sum of order 3 should be multiple of 3. To avoid the we can have semi-magic squares of order 3. It don’t require any condition. See below the example of magic square of order 15 composed of 25 equal semi-magic sums of order 3. 51 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 Below are examples of few semi-magic squares with there semi-magic sums. The other can also be calculated in the similar way. This magic square is constructed in such a way that the blocks of orders 6, 9 and 12 in the upper-left corner are magic squares. All the blocks of order 3 are semi-magic squares. See below 52 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Third-Type Let’s consider the following three examples of magic squares of order 15: 53 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 All the six examples of magic squares of order 15 are with magic sum either 1729 or multiple of 1729. The last three examples are of different types. Except the magic squares of order 3, the others are all magic rectangles of equal width, i.e., 2×1729. The lengths are proportional according to values of the width. In all the three example the magic sum is S15×15 :=25935 =15 ×1729. All the six examples use special entries, namely the date of birth of S. Ramanujan (22.12.1887) and his 138th anniversary on 22.12.2025. These contain the four entries 2212, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [26, 27]. The last three examples are little different. 54 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 15 Magic Squares of Order 16 Let’s consider a following magic square of order 16 having the magic sum S16×16 :=6916 :=4×1729: ▶First-Type ▶Second-Type 55 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 •Details The first example is composed of 16 equal sums magic square of order 4 with magic sum S4×4:=1729. See below the magic sum of these 16 magic square 56 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 The last three example sof magic square of order 16 based on magic rectangles. The second and third are double-digit magic squares, while the third one is cornered-type magic square. The the width of the strip is always same, i.e., 1729. The change is only in the lengths. The magic sum for all the three magic square is same, i.e., S16×16 :=13832 =8×1729. All 4 examples contain the date of birth of S. Ramanujan (22.12.1887) and his 138th anniversary on 22.12.2025. These contain the four entries 2212, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [26, 27] 16 Magic Squares of Order 17 Below are three magic squares of order 17 with the magic sum as multiple of 1729. 57 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 64 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Fifth-Type ▶Sixth-Type: Striped Magic Squares 65 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 It is little different from other four examples given above. It is single-digit bordered magic square wither magic square of order 16 is embedded in the inner part. This magic square of order 16 is with 16 equal sums magic squares of order 4. In this case, the magic sums are S4×4:=1729 and S18×18 :=4.5 ×1729 =7780.5. Here the magic sum of order 18 is with decimal value. In order de avoid this, we can consider the magic sum of order 4 as double of 1729. The last three example sof magic square of order 16 based on magic rectangles. The second and third are double-digit magic squares, while the third one is cornered-type magic square. The the width of the strip is always same, i.e., 1729. The change is only in the lengths. The magic sum for all the three magic square is same, i.e., S18×18 :=15561 =9×1729 66 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 Above there are 5 examples magic squares of order 18. All 5 examples use special entries, namely the date of birth of S. Ramanujan (22.12.1887) and his 138th anniversary on 22.12.2025. These contain the four numbers 2212, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [26, 27] 18 Magic Squares of Order 19 Let’s consider following 3 magic squares of order 19 having the magic sum 1729: 67 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 •Details Above there are 3 examples magic squares of order 19 with magic sum 1729. All 3 examples use special entries, namely the date of birth of S. Ramanujan (22.12.1887) and his 138th anniversary on 22.12.2025. These contain the four numbers 2212, 68 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [26, 27] 19 Magic Squares of Order 20 Below are few examples of magic squares of order 20 having the details of S. Ramanujan. ▶First-Type: Striped Magic Squares The above magic square is compose of equal sums magic squares of order 4. The sum of each magic square of order 4 is 1729. Thus, the magic sum of order 20 is given as S20×20 :=8645 =5×1729. See below the sum of magic squares of order 4: 69 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Second-Type: Striped Magic Squares 70 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 The above magic square is compose of equal sums pandiagonal magic squares of order 5. The sum of each magic square of order 5 is 1729. Thus, the magic sum of order 20 is given as S20×20 :=6916 =4×1729. See below the magic sums of pandiagonal magic squares of order 5: 71 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 ▶Third-Type: Striped Magic Squares 72 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 207 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Hardy-Ramanujan Number 1729, Zenodo, December 27, 2025, pp. 1-77, https://doi.org/10.5281/zenodo.18064853 It is little different from other four examples given above. It is single-digit bordered magic square wither magic square of order 20 is embedded in the inner part. This magic square of order 16 is with 16 equal sums magic squares of order 4. In this case, the magic sums are S4×4:=1729 and S18×18 :=4.5 ×1729 =7780.5. Here the magic sum of order 18 is with decimal value. In order de avoid this, we can consider the magic sum of order 4 as double of 1729. The last three example sof magic square of order 20 are based on magic rectangles. The second and third are double-digit magic squares, while the third one is cornered-type magic square. The the width of the strip is always same, i.e., 1729. The change is only in the lengths. The magic sum for all the three magic square is same, i.e., S20×20 :=17290 =10 ×1729 Above there are 5 examples magic squares of order 20. All 5 examples use special entries, namely the date of birth of S. Ramanujan (22.12.1887) and his 138th anniversary on 22.12.2025. These contain the four numbers 2212, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [26, 27] 73