153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729

Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 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 15 and order 19 having magic sum as 1729. The special entries, such as, 2212, 1887, 2025 and 138 are considered in each magic square. There are total 153 different magic squares. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. 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 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 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 9 8 Magic Squares of Order 9 11 9 Magic Squares of Order 10 15 10 Magic Squares of Order 11 18 11 Magic Squares of Order 12 26 12 Magic Squares of Order 13 33 13 Magic Squares of Order 14 35 14 Magic Squares of Order 15 36 15 Magic Squares of Order 19 38 2 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 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 153 different magic squares. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. The table below give the total number of 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 10 912 8 20 10 9 4 13 11 25 23 48 12 13 13 26 13 3 0 3 14 4 0 4 15 2 0 2 19 3 0 3 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 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 •Details Above there are two magic squares of order 3. The first one is magic square, while the second one is semimagic. 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 [15, 17] 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 4 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 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: 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 [16] 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 upperleft 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 [15, 17]. 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 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 •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: 6 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 The fifth one is cornered magic square of order 6. The the last example is a single-digit bordered semimagic square of order 6. 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 [16, 17] . 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 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 ▶Second-Type 8 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 •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 [15, 17]. 7 Magic Squares of Order 8 Let’s consider following 10 magic squares of order 8 having the magic sum 1729: ▶First-Type 9 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 ▶Second-Type 16 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 ▶Third-Type •Details Above there are 13 examples magic squares of order 10 with magic sum 1729. The first 9 examples are different types of magic squares.. The last 4 examples are of semi-magic squares of order 10. The example 9 is written in two different ways. One the decimal entries are only in the outer part of order 6. The second one is with decimal entries in the inner part, i.e, in the magic square of order 6. The examples 7 and are the 17 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 cornered magic squares, where the magic square of order 8 is in the upper-left corner. In the example 7 it is made of four equal sums magic squares of order 4. In the example 8 there eight equal sums strips of order 2×4. All 13 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 [20] 10 Magic Squares of Order 11 Let’s consider following 48 magic squares of order 11 having the magic sum 1729: ▶First-Type 18 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 19 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 20 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 21 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 ▶Second-Type 22 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 23 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 24 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 25 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 •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 se based on 12 equal sums magic squares of order 3. The example 25 is with decimal entries, while the example 26 is with fractional entries. Instead of magic square of order 3, if we consider semi-magic squares of order 3 (as in example of order 9), we can have magic square of order 12 without fractional entries. 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 32 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 ▶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. All 26 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 [23, 24] 12 Magic Squares of Order 13 Let’s consider following 3 magic squares of order 13 having the magic sum 1729: 33 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 34 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 •Details Above there are 3 examples magic squares of order 13 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, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [25, 26] 13 Magic Squares of Order 14 Let’s consider following 4 magic squares of order 14 having the magic sum 1729: 35 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 •Details Above there are 4 examples magic squares of order 14 with magic sum 1729. All 4 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 [25, 26] 14 Magic Squares of Order 15 Let’s consider following 2 magic squares of order 15 having the magic sum 1729: 36 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 •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: 37 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 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. Both 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 [25, 26] 15 Magic Squares of Order 19 Let’s consider following 3 magic squares of order 19 having the magic sum 1729: 38 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 39 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 •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, 1887, 2025, and 138. The construction of these magic squares is based on reduced-entry algebraic magic squares recently studied by the author [25, 26] •Author’s Contributions to Recreating Numbers and Magic Squares For author’s contribution to magic squares and recreation numbers please see the links below: •Inder J. Taneja, Magic Squares, 1. https://inderjtaneja.wordpress.com/2019/06/27/publications-magic-squares/ 2. https://numbers-magic.com/?p=668 •Inder J. Taneja, Recreation of Numbers, 1. https://inderjtaneja.wordpress.com/2019/06/27/publications-recreation-of-numbers/ 2. https://numbers-magic.com/?p=671 40 Inder J. Taneja https://numbers-magic.com; www.inderjtaneja.wordpress.com 153 Magic Squares in Honor of the 138th Anniversary of S. Ramanujan with Magic Sum 1729, Zenodo, December 21, 2025, pp. 1-44, https://doi.org/10.5281/zenodo.18011320 References [1] A. de Winkel, The magic Encyclopedia, http://home.wanadoo.nl/aaledewinkel/Encyclopedia/index.html [2] C. Boyer, Multimagic Squares and Cubes, http://www.multimagie.com [3] F. Gaspalou, “Magic Squares” http://www.gaspalou.fr/magic-squares/ [4] W. Trump,http://www.trump.de/magic-squares [5] H. White, Bordered Magic Squares - http://budshaw.ca/Download.html [6] W.S. Andrews, Magic squares and Cubes, Dover Publications, New York •General References [7] Inder J. Taneja, Hardy-Ramanujan Number – 1729, Zenodo, December 22, 2021, pp. 1-106, https://doi.org/10.5281/zenodo.5799640. [8] Inder J. Taneja, Numbers and Magic Squares Representations of Hardy-Ramanujan Number-1729, Zenodo, December 20, 2024, pp. 1-127, https://doi.org/10.5281/zenodo.14538297. •Reduced Entries Algebraic Magic Squares: Dates and Days of the Year [9] Inder J. Taneja, Magic Squares of Orders 3 to 7 Representing Dates and Days of the Year 2025, Zenodo, May 04, 2025, pp. 1-474, https://doi.org/10.5281/zenodo.15338142. [10] Inder J. Taneja, Magic Squares of Order 8 Representing Days and Dates of the Year 2025, Zenodo, May 04, 2025, pp. 1-134, https://doi.org/10.5281/zenodo.15338246. [11] Inder J. Taneja, Magic Squares of Order 9 Representing Days and Dates of the Year 2025, Zenodo, May 09, 2025, pp. 1-132, https://doi.org/10.5281/zenodo.15375349. [12] Inder J. Taneja, Magic Squares of Order 10 Representing Days and Dates of the Year 2025, Zenodo, May 21, 2025, pp. 1-59, https://doi.org/10.5281/zenodo.15481738. [13] Inder J. Taneja, Magic Squares of order 11 Representing Days and Dates of the Year 2025, Zenodo, June 02, 2025, pp. 1-111, https://doi.org/10.5281/zenodo.15576562. 41