Preprints in Zenodo.org – Inder J. Taneja – Recreating Numbers and Magic Squares

Work in 2025

277. New Concepts in Magic Squares: Cornered Magic Squares of Orders 5 to 108

Inder J. Taneja

For the first time in history of magic squares, this work brings different kinds of magic squares. These are double digits cornered magic squares. The work is for the orders 5 to 108. From order 82 onwards the magic squares are still semi-magic. The author also worked on cornered magic squares in different ways up to order 24. Complete details are available at author’s site. The examples given in this work are only up to order 30. Further orders are available as excel files. These are in two parts. One from order 5 to 81, and the second part from order 82 to 108. The second part is still as semi-magic squares. Uploaded on January 29, 2025

276. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Order 25

Inder J. Taneja

There are many ways of representing magic squares with palindromic type entries. Also, we can write magic squares in the composite forms based on pair of Latin squares. Based on palindromic and composite magic squares we have written upside-down and/or mirror looking magic squares. By upside-down, we understand that making 180 degrees rotation still we get a magic square. By mirror looking, we understand that putting magic square in front of mirror, still we get another magic square. When the magic square is of both type, upside-down and mirror looking, we call it as universal magic square. It is a revised and enlarged version of author’s previous works on digital-type fonts magic squares. In case of upside-down situation, the number 6 becomes 9 and 9 as 6. In case of mirror looking, the numbers 2 becomes 5 and 5 as 2 (writing as digital fonts). Total work of magic squares of orders 3 to 16 and order 20. It is divided in parts. First is for orders 3 to 6, and second part for orders 7 to 10, and third part is for the 11 to 13. The forth part is for the orders 14 to 16. The fifth part is for the orders 17 to 20. The sixth part is for the orders 21 to 25. The seventh part is for the order 24. The work is eight part of the complete project and is for the order 25. It is bimagic with pandiagonal magic squares of blocks of order 5. For more details, follow the online links of author’s web-site (3to6, 7to10, 11to15, 16, 20, 21, 24 and 25). January 20, 2025 (v2)

275. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Order 24

Inder J. Taneja

There are many ways of representing magic squares with palindromic type entries. Also, we can write magic squares in the composite forms based on pair of Latin squares. Based on palindromic and composite magic squares we have written upside-down and/or mirror looking magic squares. By upside-down, we understand that making 180 degrees rotation still we get a magic square. By mirror looking, we understand that putting magic square in front of mirror, still we get another magic square. When the magic square is of both type, upside-down and mirror looking, we call it as universal magic square. It is a revised and enlarged version of author’s previous works on digital-type fonts magic squares. In case of upside-down situation, the number 6 becomes 9 and 9 as 6. In case of mirror looking, the numbers 2 becomes 5 and 5 as 2 (writing as digital fonts). Total work of magic squares of orders 3 to 16 and order 20. It is divided in parts. First is for orders 3 to 6, and second part for orders 7 to 10, and third part is for the 11 to 15. The next parts are for orders 16, 20 and 21. This work is for order 24. We worked with blocks of magic squares of orders 3, 4, 6 and 8. For more details follow online link of authors web-site (3to6, 7to10, 11to15, 16, 20, 21 and 24). January 20, 2025 (v2)

274. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 21 to 23

Inder J. Taneja

There are many ways of representing magic squares with palindromic type entries or composite forms based on pair of Latin squares. Based on palindromic and composite magic squares we have written upside-down and/or mirror looking magic squares. By upside-down, we understand that making 180 degrees rotation still we have a magic square. Applying the upside-down property, the numbers 0, 1, 2, 5, 6, 8 and 9 remains the same, where 6 becomes 9 and 9 as 6. In this case, the numbers are written in digital/special fonts. The mirror looking property is same as horizontal flip. In this case, the numbers 0, 1, 2, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. There is one more property, known by vertical flip. For simplicity, let’s call it as water reflection. In this case, the numbers 0, 1, 2, 3, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. Thus the numbers 0, 1, 2 5 and 8 are available in all the three properties. The numbers those satisfy all the three properties, we call them as universal. The same is with magic squares, i.e., the magic squares containing the numbers 0, 1, 2, 5 and 8 are known by universal magic squares. Finally, in case of upside-down, the number 6 becomes 9 and 9 as 6, and in case of water reflection, the number 3 remains the same. In this paper we worked with magic squares of orders 21 to 23, satisfying one or all the above three properties. For more details following online link of authors web-site (3to6, 7to10, 11to13, 14to16 and 17to20, 21to23). January 17, 2025 (v2)

273. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 17 to 20

Inder J. Taneja

There are many ways of representing magic squares with palindromic type entries or composite forms based on pair of Latin squares. Based on palindromic and composite magic squares we have written upside-down and/or mirror looking magic squares. By upside-down, we understand that making 180 degrees rotation still we have a magic square. Applying the upside-down property, the numbers 0, 1, 2, 5, 6, 8 and 9 remains the same, where 6 becomes 9 and 9 as 6. In this case, the numbers are written in digital/special fonts. The mirror looking property is same as horizontal flip. In this case, the numbers 0, 1, 2, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. There is one more property, known by vertical flip. For simplicity, let’s call it as water reflection. In this case, the numbers 0, 1, 2, 3, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. Thus the numbers 0, 1, 2 5 and 8 are available in all the three properties. The numbers those satisfy all the three properties, we call them as universal. The same is with magic squares, i.e., the magic squares containing the numbers 0, 1, 2, 5 and 8 are known by universal magic squares. Finally, in case of upside-down, the number 6 becomes 9 and 9 as 6, and in case of water reflection, the number 3 remains the same. In this paper we worked with magic squares of orders 17 to 20, satisfying one or all the above three properties. For more details see the reference list. For more details following online link of authors web-site (3to6, 7to10, 11to13, 14to16 and 14to20). January 15, 2025 (v2)

272. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 14 to 16

Inder J. Taneja

There are many ways of representing magic squares with palindromic type entries or composite forms based on pair of Latin squares. Based on palindromic and composite magic squares we have written upside-down and/or mirror looking magic squares. By upside-down, we understand that making 180 degrees rotation still we have a magic square. Applying the upside-down property, the numbers 0, 1, 2, 5, 6, 8 and 9 remains the same, where 6 becomes 9 and 9 as 6. In this case, the numbers are written in digital/special fonts. The mirror looking property is same as horizontal flip. In this case, the numbers 0, 1, 2, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. There is one more property, known by vertical flip. For simplicity, let’s call it as water reflection. In this case, the numbers 0, 1, 2, 3, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. Thus the numbers 0, 1, 2 5 and 8 are available in all the three properties. The numbers those satisfy all the three properties, we call them as universal. The same is with magic squares, i.e., the magic squares containing the numbers 0, 1, 2, 5 and 8 are known by universal magic squares. Finally, in case of upside-down, the number 6 becomes 9 and 9 as 6, and in case of water reflection, the number 3 remains the same. In this paper we worked with magic squares of orders 14 to 16, satisfying one or all the above three properties. For more details see the reference list. For more details following online link of authors web-site (3to6, 7to10 and 11to13). January 15, 2025 (v4)

271. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 11 to 13

Inder J. Taneja

There are many ways of representing magic squares with palindromic type entries or composite forms based on pair of Latin squares. Based on palindromic and composite magic squares we have written upside-down and/or mirror looking magic squares. By upside-down, we understand that making 180 degrees rotation still we have a magic square. Applying the upside-down property, the numbers 0, 1, 2, 5, 6, 8 and 9 remains the same, where 6 becomes 9 and 9 as 6. In this case, the numbers are written in digital/special fonts. The mirror looking property is same as horizontal flip. In this case, the numbers 0, 1, 2, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. There is one more property, known by vertical flip. For simplicity, let’s call it as water reflection. In this case, the numbers 0, 1, 2, 3, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. Thus the numbers 0, 1, 2 5 and 8 are available in all the three properties. The numbers those satisfy all the three properties, we call them as universal. The same is with magic squares, i.e., the magic squares containing the numbers 0, 1, 2, 5 and 8 are known by universal magic squares. Finally, in case of upside-down, the number 6 becomes 9 and 9 as 6, and in case of water reflection, the number 3 remains the same. In this paper we worked with magic squares of orders 11 to 13, satisfying one or all the above three properties. For more details see the reference list. For more details following online link of authors web-site (3to6, 7to10 and 11to13). January 7, 2025 (v3)

270. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 7 to 10

Inder J. Taneja

There are many ways of representing magic squares with palindromic type entries or composite forms based on pair of Latin squares. Based on palindromic and composite magic squares we have written upside-down and/or mirror looking magic squares. By upside-down, we understand that making 180 degrees rotation still we have a magic square. Applying the upside-down property, the numbers 0, 1, 2, 5, 6, 8 and 9 remains the same, where 6 becomes 9 and 9 as 6. In this case, the numbers are written in digital/special fonts. The mirror looking property is same as horizontal flip. In this case, the numbers 0, 1, 2, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. There is one more property, known by vertical flip. For simplicity, let’s call it as water reflection. In this case, the numbers 0, 1, 2, 3, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. Thus the numbers 0, 1, 2 5 and 8 are available in all the three properties. The numbers those satisfy all the three properties, we call them as universal. The same is with magic squares, i.e., the magic squares containing the numbers 0, 1, 2, 5 and 8 are known by universal magic squares. Finally, in case of upside-down, the number 6 becomes 9 and 9 as 6, and in case of water reflection, the number 3 remains the same. In this paper we worked with magic squares of orders 7 to 10, satisfying one or all the above three properties. For more details see the reference list. January 7, 2025 (v3)

Work in 2024

269. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 3 to 6

Inder J. Taneja

There are many ways of representing magic squares with palindromic type entries or composite forms based on pair of Latin squares. Based on palindromic and composite magic squares we have written upside-down and/or mirror looking magic squares. By upside-down, we understand that making 180 degrees rotation still we have a magic square. Applying the upside-down property, the numbers 0, 1, 2, 5, 6, 8 and 9 remains the same, where 6 becomes 9 and 9 as 6. In this case, the numbers are written in digital/special fonts. The mirror looking property is same as horizontal flip. In this case, the numbers 0, 1, 2, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. There is one more property, known by vertical flip. For simplicity, let’s call it as water reflection. In this case, the numbers 0, 1, 2, 3, 5 and 8 remains the same, where 2 becomes 5 and 5 as 2. Thus the numbers 0, 1, 2 5 and 8 are available in all the three properties. The numbers those satisfy all the three properties, we call them as universal. The same is with magic squares, i.e., the magic squares containing the numbers 0, 1, 2, 5 and 8 are known by universal magic squares. Finally, in case of upside-down, the number 6 becomes 9 and 9 as 6, and in case of water reflection, the number 3 remains the same. In this paper we worked with magic squares of orders 3 to 6, satisfying one or all the above three properties. For more details see the reference list. December 20, 2024 (v3)

268. Numbers and Magic Squares Representations of Hardy-Ramanujan Number-1729

Inder J. Taneja

This paper brings representations of 1729, a famous Hardy-Ramanujan number in different ways. These representations are of crazy-type, single digit, single letter, Selfie-type, running expressions, selfie fractions, equivalent fractions, Triangular, Fibonacci, fixed digits repetitions prime numbers patterns, Pythagorean triples, palindromic-type, polygonal-type, prime numbers, embedded, etc. Some quotes and historical notes on Ramanujan’s life are also specified. This work brings magic squares connected with S. Ramanujan’s life and with Hardy-Ramanujan number 1729. There are three different types of magic squares. The first type is based on some historical years of S. Ramanujan’s life. The second type is with magic sum as 1729. In this case the magic squares constructed are of orders 7, 13, 14 and 19. The third type is magic squares based on Pythagorean tripes where the number 1729 is one the entry. In this case the magic squares constructed are of orders 6, 7, 11 and 19. December 20, 2024 (v2)

267. Reflexive Year 25: Mathematics of 25 and 2025 in Numbers and Magic Squares

Inder J. Taneja

This work brings numerical and magics squares representations for 25 and 2025 in different ways. The number representations are of crazy-type, pyramid-type3, single digit, single letter, running numbers, Triangular, Fibonacci, palindromic-type, prime numbers, embedded prime patterns, repeated digits prime patterns, selfie, semi-selfie, narcissistic, etc. Some interesting patterns for 25 and 2025 are also included in the work. The magic square representations are of there are three types. One is universal magic squares with 25 and 2025. By universal, we understand that the magic square is upside-down and mirror looking. In this case, we have constructed magic squares of orders 3, 4, 5, 6, 7, 8, 9 and 10. The second types of magic squares are with \textbf{magic sums} 25. In this case, there are only three types of magic squares, i.e., of orders 5, 10 and 25. The third type is magic squares with magic sum 2025. In this case, the magic squares constructed are of orders 3, 5, 6, 9, 10, 15, 18 and 25. In second and third type we have used the sequential entries with positive and/or negative numbers. December 20, 2024

266. Crazy Representations of Half a Million Natural Numbers: Part 5 – 460.001 to 500.000

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done up to number 300.000. For details refer author’s web-sites (site1, site2). This paper brings natural numbers from 460.001 to 500.000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorials. Few numbers are by use of extra operation square-root. Total work of crazy representations of 200.000 numbers from 300.001 to 500.000 is divided in five parts. This is Part 5. For more details see author’s web-site links (link1, link2). December 5, 2024 (v1)

265. Crazy Representations of Half a Million Natural Numbers: Part 4 – 420.001 to 460.000

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done up to number 300.000. For details refer author’s web-sites (site1, site2). This paper brings natural numbers from 420.001 to 460.000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorials. Few numbers are by use of extra operation square-root. Total work of crazy representations of 200.000 numbers from 300.001 to 500.000 is divided in five parts. This is Part 4. For more details see author’s web-site links (link1, link2). December 5, 2024 (v1)

264. Crazy Representations of Half a Million Natural Numbers: Part 3 – 380.001 to 420.000

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done up to number 300.000. For details refer author’s web-sites (site1, site2). This paper brings natural numbers from 380.001 to 420.000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorials. Few numbers are by use of extra operation square-root. Total work of crazy representations of 200.000 numbers from 300.001 to 500.000 is divided in five parts. This is Part 3. For more details see author’s web-site links (link1, link2). December 5, 2024 (v1)

263. Crazy Representations of Half a Million Natural Numbers: Part 2 – 340.001 to 380.000

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done up to number 300.000. For details refer author’s web-sites (site1, site2). This paper brings natural numbers from 340.001 to 380.000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorials. Few numbers are by use of extra operation square-root. Total work of crazy representations of 200.000 numbers from 300.001 to 500.000 is divided in five parts. This is Part 2. For more details see author’s web-site links (link1, link2). December 5, 2024 (v1)

262. Crazy Representations of Half a Million Natural Numbers: Part 1 – 300.001 to 340.000

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done up to number 300.000. For details refer author’s web-sites (site1, site2). This paper brings natural numbers from 300.001 to 340.000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorials. Few numbers are by use of extra operation square-root. Total work of crazy representations of 200.000 numbers from 300.001 to 500.000 is divided in five parts. This is Part 1. For more details see author’s web-site links (link1, link2). October 17, 2024 (v2)

261. Universal and Upside-down Magic Cubes

Inder J. Taneja

This work brings magic cubes in universal or upside-down way. It is based on 6 magic squares for six sides of the cube. By upside-down, we understand that making 180 degrees rotation still we have a magic square. By mirror looking, we understand that putting magic square in front of mirror, still we get a magic square. When the magic square is of both type, i.e., upside-down and mirror looking, we call it as universal magic square. In case of upside-down situation, then 6 becomes 9 and 9 as 6. In case of mirror looking, the numbers 2 becomes 5 and 5 as 2 (using digital fonts). The work is only for the magic squares of orders 3, 4, 5, 7 and 8. Whole work can be accessed at author’s web-site. October 17, 2024 (v2)

260. Magic Cubes Based on Magic Squares

Inder J. Taneja

This work brings magic cubes based on magic squares constructed with sequential numbers. Each magic cube is with six magic squares representing each face with different magic square. The total work is for the orders 3 to 10. In each order, some different examples are considered. These examples are based on different types of magic squares, such as, single-digit bordered, double-digits bordered, striped, cornered, etc. Whole work can be accessed at author’s web-site. Uploaded on October 17, 2024 (v1)

259. Striped Magic Squares of Order 12 – Revised

Inder J. Taneja

This work brings magic squares of Order 12 based on equal width magic rectangles. These are magic rectangles of of type 2×4, 2×6, 2×8, etc. Alternatively, we can write it as $2xn, where n is the length of magic rectangle. Magic squares constructed based on equal width magic rectangles, we call as striped magic squares. For the striped magic squares of orders 4, 6, 8, 10, 12 and 14 refer the author’s work. The previous work was only with 72 striped magic squares of order 12, while this version is with 220 striped magic squares. The whole work with 220 striped magic squares is available as pdf file at author’s web-site. Uploaded on September 7, 2024

258. Striped Magic Squares of Order 16 – Revised

Inder J. Taneja

This work brings magic squares of order 16 based on equal width magic rectangles. These are magic rectangles of of type 2×4, 2×6, 2×8, etc. Alternatively, we can write it as 2xn, where n is the length of magic rectangle. Magic squares constructed based on equal width magic rectangles, we call them as striped magic squares. For the striped magic squares of orders 4, 6, 8, 10 and 12 refer author’s work. The whole work as pdf file is available at author’s web-site as pdf file. This revised version brings 2104 striped magic squares of order 16 divided in 9 parts. Uploaded on September 7, 2024 (v2)

257. Crazy Representations of Natural Numbers From 260001 to 280000 – Revised

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done from 0 to 200000. For details see the references. This work brings natural numbers from 260001 to 280000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorial. Few numbers are by use of square-root. For more details see author’s web-site links (link1, link2). Uploaded on August 5, 2024

256. Crazy Representations of Natural Numbers From 240001 to 260000 – Revised

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done from 0 to 200000. For details see the references. This work brings natural numbers from 240001 to 260000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorial. Few numbers are by use of square-root. For more details see author’s web-site links (link1, link2). Uploaded on August 5, 2024

255. Crazy Representations of Natural Numbers From 220001 to 240000 – Revised

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done from 0 to 200000. For details see the references. This work brings natural numbers from 220001 to 240000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorial. Few numbers are by use of square-root. For more details see author’s web-site links (link1, link2). Uploaded on August 5, 2024 (v2)

254. Crazy Representations of Natural Numbers From 200001 to 220000 – Revised

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done from 0 to 200000. For details see the references. This work brings natural numbers from 200001 to 220000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorial. Few numbers are by use of square-root. For more details see author’s web-site links (link1, link2). Uploaded on August 5, 2024 (v2)

253. Crazy Representations of Natural Numbers From 180001 to 200000 – Revised

Inder J. Taneja

This paper brings natural numbers from 180001 to 200000 written in \textbf{ascending} and \textbf{descending} orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorial. For previous results see author’s web-site links (link1, link2). This revised version brings some better results. Uploaded on August 2, 2024 (v2)

252. Crazy Representations of Natural Numbers From 160001 to 180000 – Revised

Inder J. Taneja

This paper brings natural numbers from 160001 to 180000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorial. For previous results see author’s web-site links (link1, link2). This revised version brings some better results. Uploaded on August 2, 2024 (v2)

251. Crazy Representations of Natural Numbers From 140001 to 160000 – Revised

Inder J. Taneja

This paper brings natural numbers from 140001 to 160000 written in \textbf{ascending} and \textbf{descending} orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorial. For previous results see author’s web-site links (link1, link2). This revised version brings some better results. Uploaded on August 2, 2024

250. Crazy Representations of Natural Numbers From 120001 to 140000 – Revised

Inder J. Taneja

This paper brings natural numbers from 120001 to 140000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorial. For previous results see author’s web-site links (link1, link2). This revised version brings some better results. Uploaded on August 2, 2024 (v2)

249. Crazy Representations of Natural Numbers From 100001 to 120000 – Revised

Inder J. Taneja

This paper brings natural numbers from 100001 to 120000 written in \textbf{ascending} and \textbf{descending} orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorial. For previous results see author’s web-site links (link1, link2). This revised version brings some better results. Uploaded on August 2, 2024

248. Striped Magic Squares of Order 18

Inder J. Taneja

This work brings magic squares of order 18 based on equal width magic rectangles. These are magic rectangles of of type 2×4, 2×6, 2×8, etc. Alternatively, we can write it as 2xn, where n is the length of magic rectangle. Magic squares constructed based on equal width magic rectangles, we call them as striped magic squares. For the striped magic squares of orders 4, 6, 8, 10, 12, 14 and 16 refer author’s work (work1, work2). The whole work as pdf file is available at author’s web-site. Uploaded on June 13, 2024 (v2)

247. Multiple Choice Patterns in Selfie Numbers – II

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. There are many ways of representing selfie numbers. It can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, etc. Also we can use binomial coefficients, quadratic (square), cubic functions, etc. In the past author worked with these functions separately. For more details see the author’s works given in references. Also refer recent work, where the the author worked with selfie numbers having together these functions. This work brings patterns in selfie numbers derived from the recent work of author recent work. The work is for numbers 3001 to 10000. For the previous work up to 3000 click here. Uploaded on April 23, 2024 (v1)

246. Multiple Representations of Selfie Numbers – II

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. There are many ways of representing selfie numbers. These can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, Square numbers, Cubic numbers, etc. Also we can use binomial coefficients. In the past author worked with these functions separately. For more details see the author’s work This work is combined study of previous applied functions with different combinations. These are done using single, double, triple functions, etc. In this situation, instead of single representation, we have multiples representations. This work covers the numbers from 1000 to 3000. For the previous work up to 999 click here. Uploaded on April 15, 2024 (v4)

245. Multiple Choice Patterns in Selfie Numbers – I

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. There are many ways of representing selfie numbers. It can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, etc. Also we can use binomial coefficients, quadratic (square), cubic functions, etc. In the past author worked with these functions separately. For more details see the author’s works given in references. Also refer recent work, where the the author worked with selfie numbers having together these functions. This work brings patterns in selfie numbers derived from the recent work of author recent work. The work is extended up to 3000 numbers instead of three digits. Work on higher digits shall be done later on. Uploaded on April 15, 2024

244. Crazy Representations of Natural Numbers from 0 to 10000 Using Factorial and Square Root

Inder J. Taneja

This work brings natural numbers from 0 to 10.000 written in two different forms. Increasing and decreasing orders of 1 to 9 and 9 to 1. This is done using factorial and square-root along with basic operations. Previously, the author worked with same numbers using only basic operations. The previous work is up to number 300.000. Extending more than 10000, in some cases, we need factorial and/or square-root. For more details refer author’s work in reference list. Similar kind of work using Triangular numbers, Fibonacci sequence, square function and cubic function is done in another works. Uploaded on April 7, 2024

243. Crazy Representations of Natural Numbers from 0 to 10000 Using Cubic Function

Inder J. Taneja

This work brings natural numbers from 0 to 10.000 written in two different forms. Increasing and decreasing orders of 1 to 9 and 9 to 1. This is done using cubic function along with basic operations. In some cases, square-root and factorial are also used. Previously, the author also worked with numbers up to 300.000 using only basic operations along with square-root, factorial, etc. For more details refer author’s work in reference list. Similar kind of work is given with Triangular numbers, Fibonacci sequence values and square function. This kind of work using cubic function is never seen before. Not all the representations are by use of cubic function. Uploaded on March 30, 2024

242. Crazy Representations of Natural Numbers from 0 to 10000 Using Square Function

Inder J. Taneja

This work brings natural numbers from 0 to 10.000 written in two different forms. Increasing and decreasing orders of 1 to 9 and 9 to 1. This is done using square function along with basic operations. Previously, the author also worked with numbers up to 300.000 using only basic operations along with square-root, factorial, etc. For more details refer author’s work in reference list. Similar kind of work is given with Triangular numbers and Fibonacci sequence values. This kind of work using square function is never seen before. Uploaded on March 28, 2024

241. Fixed Power Representation of Numbers

Inder J. Taneja

This short work brings numbers written in terms of fixed powers. For minimum length permutable among bases and powers see author’s work. Multiple representations pyramid style are given in author’s another works. For power representations with powers see author work. Uploaded on February 22, 2024

240. Unified Study of Narcissistic Numbers without and with Division

Inder J. Taneja

This paper brings unified study of narcissistic numbers without and with division. This work is in done in two parts. One for without division and second for with division. It is combination of author’s previous two works (work1, work2). The previous are seperated in different situations, while this one com together. Uploaded on February 15, 2024

239. Multiple Representations of Selfie Numbers – I

Inder J. Taneja

238. Pyramid-Type Representations of Natural Numbers from 1001 to 10000

Inder J. Taneja

This work brings natural numbers from 1001 to 10000 written with permutations of bases and powers having same digits. These representations are of pyramid-type. This means representation of each numbers is 3 to 7 ways. It extends the author’s previous work. The previous work up to number 1000. Uploaded on January 16, 2024

237. Crazy Representations of Natural Numbers From 280001 to 300000 – Revised

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done from 0 to 200000. For details see the references. This work brings natural numbers from 280001 to 300000 written in ascending and descending orders of 1 to 9 and 9 to 1. The numbers are obtained by using basic operations along with factorial. Few numbers are by use of square-root. For more details see author’s web-site links (link1, link2). Uploaded on August 5, 202419094

236. Crazy Representations of Natural Numbers 0 to 10000 Using Triangular Numbers

Inder J. Taneja

This work brings natural numbers from 0 to 10.000 written in two different forms. Increasing and decreasing orders of 1 to 9 and 9 to 1. This is done using Triangular numbers along with basic operations. Previously, the author worked with same numbers using only basic operations. For the numbers some extra operations are used, such as, square-root, factorial, etc. The previous work is up to number 300.000. For more details refer author’s site. This kind of work using Triangular numbers is never seen before. The results are up to number 10.000. Uploaded on January 16, 2024

235. Crazy Representations of Natural Numbers from 0 to 10000 Using Fibonacci Sequence Values

Inder J. Taneja

This work brings natural numbers from 0 to 10000 written in two different forms. Increasing and decreasing orders of 1 to 9 and 9 to 1. This is done using Fibonacci sequence values along with basic operations. Peviously, the author worked with similar kind of work using only basic operations. For the higher values, some extra operations, such as, square-root, factorial, etc. also done by the author. The previous total work is up to number 300.000. For more details refer author’s site. This kind of work is nover seen before using Fibonacci sequence values. The results are up to number 10000. Uploaded on January 13, 2024

Work in 2023

234. Mathematical Representations of the Last Day of the Year 23 Written American Style: 12.31.23 (123123)

Inder J. Taneja

If write last day of this year, i.e., December 31, 23 in american style, it stands as 12.31.23, i.e., 123123. In this work we shall write representions of the number 123123 in different ways. These repreentations are of crazy-type, single digit, single letter, base and power permutable, selfie representations, upside down and mirror looking, etc. Magic squares of orders 3 to 7 are also written having the number 123123 as well as having magic sum as 123123. Uploaded on December 19, 2023

233. Different Types of Magic Squares of Order 14 Using Bordered Magic Rectangles

Inder J. Taneja

It is revised version of author’s previous work. It brings new ideas of construction of magic squares. This work is for the order 14. The new ideas used to bring these magic square are: bordered magic rectangles, bordered double digits magic rectangles, cornered magic rectangles, striped magic rectangles, etc. When the length and width are equal these becomes as magic squares. Another idea used is of algebraic formula (a+b)^2. Here we consider small blocks of magic squares and magic rectangles, such as a^2, b^2, axb and bxa. These are available as pdf file at author’s site. For previous works of orders 6, 8, 10 and 12 (click here) . Uploaded on November 18, 2023

232. Striped Magic Squares of Even Orders 6, 8, 10, 12 and 14

Inder J. Taneja

This work brings striped magic squares of even orders 6, 8, 10, 12 and 14> These are constructed based on equal width magic rectangles. Equal width magic rectangles are of type 2×4, 2×6, 2×8, etc. Magic squares constructed based on equal width magic rectangles, we call as striped magic squares. Uploaded on November 10, 2023

231. Different Types of Magic Squares of Orders 6, 8, 10 and 12

Inder J. Taneja

It is revised version of author’s previous work. It brings new ideas of construction of magic squares. This work is for magic squares of even orders 6, 8, 10 and 12. The ideas ideas used to bring these magic square are: bordered magic rectangles, bordered double digits magic rectangles, cornered magic rectangles, striped magic rectangles, etc. When the length and width are equal these magic rectangles becomes as magic squares. Another idea used is of algebraic formula (a+b)^2. Here we consider small blocks of magic squares and magic rectangles, such as a^2, b^2, axb and bxa. We are able to bring 6 magic squares of order 6, 30 magic squares of order 8, 175 magic squares of order 10 and 634 magic squares of order 12. These are available in author’s site, whose links are given above.Uploaded on November 7, 2023

230. Cornered Magic Squares in Construction of Magic Squares of Orders 16, 20, 24 and 28

Inder J. Taneja

Recently, the author worked with magic squares of orders 14, 18, 22, 26, 30 and 34 using different types of magic rectangles. In this work magic squares of orders 16, 20, 24 and 28 are constructed with four equal sums cornered magic squares. Since, this work is for orders multiples of 4, it makes easy to work with cornered magic squares. Uploaded on October 30, 2023

229. Different Types of Magic Rectangles in Construction of Magic Squares of Order 34

Inder J. Taneja

This work brings magic squares of order 34. These are constructed based on different types of magic rectangles. These types are bordered, double digits and cornered magic rectangles. For more details on these types of magic rectangles refer author’s work. The construction of magic squares is based on four equal sums magic rectangles of orders 14×20, 12×22, 10×24, 8×26, 6×28 and 4×30. In case of cornered magic rectangles, three different types are magic squares are studies. Uploaded on September 10, 2023

228. Different Types of Magic Rectangles in Construction of Magic Squares of Order 30

Inder J. Taneja

This work brings magic squares of order 30. These are constructed based on different types of magic rectangles. These types are bordered, double digits and cornered magic rectangles. For more details on these types of magic rectangles refer author’s work. The construction of magic squares is based on four equal sums magic rectangles of orders 12×18, 10×20, 8×22, 6×24 and 4×26. In case of cornered magic rectangles, three different types are magic squares are studies. Uploaded on September 10, 2023

227. Different Types of Magic Rectangles in Construction of Magic Squares of Order 26

Inder J. Taneja

This work brings magic squares of order 26. These are constructed based on different types of magic rectangles. These types are bordered, double digits and cornered magic rectangles. For more details on these types of magic rectangles refer author’s work. The construction of magic squares is based on four equal sums magic rectangles of orders 10×16, 8×18, 6×20 and 4×22. In case of cornered magic rectangles, three different types are magic squares are studies. Uploaded on September 10, 2023

226. Different Types of Magic Rectangles in Construction of Magic Squares of Order 22

Inder J. Taneja

This work brings magic squares of order 22. These are constructed based on different types of magic rectangles. These types are bordered, double digits and cornered magic rectangles. For more details on these types of magic rectangles refer author’s work. The construction of magic squares is based on four equal sums magic rectangles of orders 8×14, 6×16 and 4×18. In case of cornered magic rectangles, three different types are magic squares are studies. Uploaded on September 10, 2023

225. Different Types of Magic Rectangles in Construction of Magic Squares of Orders 14 and 18

Inder J. Taneja

This work brings magic squares of orders 14 and 18. These are constructed based on different types of magic rectangles. These types are bordered, double digits and cornered magic rectangles. For more details on these types of magic rectangles refer author’s work. The construction of magic squares is based on four equal sums magic rectangles. In case of cornered magic rectangles, three different types of magic squares are studies. Uploaded on September 10, 2023

224. Different Types of Magic Rectangles

Inder J. Taneja

This work brings the idea of different types of magic rectangles. These includes bordered magic rectangles, double digits magic rectangles and cornered magic rectangles. These kinds are frequently used in construction of different types of magic squares. Main idea is in construction magic squares based on four equal order and equal sums of magic rectangles. For details, see author’s work. Uploaded on September 4, 2023

223. A Simplified Procedure to Construct Pandiagonal Magic Squares Multiples of 4

Inder J. Taneja

This work is revised version of author’s previous work. It brings pandiagonal magic squares of type 4k, i.e., multiple of 4. This means that it is possible to write pandiagonal magic squares of orders 4, 8, 12, etc. with equal sums magic squares of order 4. The procedure is based on half-sequential numbers entries. This works brings pandiagonal magic squares up to order 32. Total work is up to order 108. An excel file of complete work is attached for download. Uploaded on August 10, 2023

222. New Concepts in Magic Squares: Cornered Magic Squares of Orders 5 to 81

Inder J. Taneja

For the first time in history of magic squares, this work brings different kinds of magic squres. These are two double digits cornered magic squares. The work is for the orders 5 to 81. Few months back, the author worked on these kind of magic squares up to order 24. Complete details are available at author’s site. Here the examples are only up to order 30. Further orders are available in an excel file attached with this work. It contains the cornered magic squares of orders 5 to 81. Uploaded on August 9, 2023

221. New Concepts in Magic Squares: Double Digits Bordered Magic Squares of Orders 7 to 108

Inder J. Taneja

For the first time in history of magic squares, this work brings different kinds of magic squres. These are two digits or double digits bordered magic squares. The work is for the orders 7 to 108. Few months back, the author worked on these kind of magic squares but only for even orders from order 8 to 40. Work on odd numbers orders from orders 7 to 31 is also done. Complete details for even and odd orders are available at author’s site. Here the examples are only up to order 30. Further orders are available in excel files attached with this work. It contains the double digits magic squares of orders 7 to 108. Uploaded on August 9, 2023

220. Magic Squares of Order 31

Inder J. Taneja

Recently, author worked with cornered magic squares of order 6. Later this work is extended up to order 24 (paper1, paper2, paper3). This work is for odd order magics squares. It for the orders 31. In this work we have covered two types of magic squares. One centered magic squares. These are traditionally previous known magic squares. The second one is double digits bordered magic squares. Some examples of embedded type magic squares are also given. For previou work on odd orders from 3 to 29 refer author’s website. Further studies are for the orders 33, 35, etc. Uploaded on August 6, 2023

219. Magic Squares of Order 29

Inder J. Taneja

Recently, author worked with cornered magic squares of order 6. Later this work is extended up to order 24 (paper1, paper2, paper3). This work is for odd order magics squares. It for the orders 29. In this work we have covered two types of magic squares. One centered magic squares. These are traditionally previous known magic squares. The second one is double digits bordered magic squares. Some examples of embedded type magic squares are also given. For previou work on odd orders from 3 to 27 refer author’s website. Further studies are for the orders 31, 33, etc.Uploaded on August 6, 2023

218. Magic Squares of Order 27

Inder J. Taneja

Recently, author worked with cornered magic squares of order 6. Later this work is extended up to order 24 (paper1, paper2, paper3). This work is for odd order magics squares. It for the orders 27. In this work we have covered three types of magic squares. One centered magic squares. These are traditionally previous known magic squares. The second one is double digits bordered magic squares. The third one is cornered magic squares. Some embedded examples are also given where one type is embedded in another type. For previou work on odd orders from 3 to 25 refer author’s website. Further studies are for the orders 29, 31, etc.Uploaded on August 6, 2023

217. Patterns in Splitted Selfie Fractions

Inder J. Taneja

By selfie fractions, we understand that a fraction, where numerator and denominators are represented by same digits, with basic operation. For more details refer author’s work . Patterned selfie fractions are understand as selfie fractions extendable in symmetric way. There are two types of patterned selfie fractions. One is multiplicative type and another is splitted type. This paper brings patterns in splitted selfie fraction where we use the repetition of digits.Uploaded on July 30, 2023

216. Bordered and Pandiagonal Magic Squares Multiples of 20

Inder J. Taneja

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with bordered magic squares of orders 140 and 110. Based on these two big magic squares, the inner order magic squares multiples of 20 are studied. It is done with 57 different types of magic squares of order 20. Two pandiagonal magic square of order 20 are also considered. These are composed of pandiagonal magic squares of orders 4 and 5. Similar kind of work for the multiples of orders 4, 6, 8, 10, 12, 14, 16 and 18 is already done by the author (multiples-4, multiples-6, multiples-8, multiples-10, multiples-12, multiples-14, multiple-16, multiple-18 ). The further multiples, such as multiples, for order 22, shall be done in another works. This work brings examples bordered and pandiagonal magic squares up to order 60. Higher order examples are given in Excel files attached with the work.Uploaded on July 28, 2023

215. Bordered Magic Squares Multiples of 18

Inder J. Taneja

214. Bordered and Pandiagonal Magic Squares Multiples of 16

Inder J. Taneja

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with bordered magic squares of orders 144 and 128. Based on these two big magic squares, the inner order magic squares multiples of 16 are studied. It is done with 24 different types of magic squares of order 16. Two blocks of order 16 composed with small blocks of 4 are pandiagonal. This lead us to write all orders multiples of 16 as pandiagonal magic squares. The only difference is that the pandiagonal magic squares multiples of 16 are no more bordered magic squares. Similar kind of work for the multiples of orders 4, 6, 8, 10 and 12 is already done by the author (multiples-4, multiples-6, multiples-8, multiples-10, multiples-12, multiples-14 ). The further multiples, such as multiples, for order 18, shall be done in another works. This work brings examples bordered and pandiagonal magic squares up to order 48. Higher order examples are given in Excel files attached with the work. Uploaded on July 27, 2023

213. Bordered Magic Squares Multiples of 14

Inder J. Taneja

212. Bordered and Pandiagonal Magic Squares Multiples of 12

Inder J. Taneja

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 120 and 108. Based on these two big magic squares, the inner order magic squares multiples of 12 are studied. By inner orders we understand as the magic squares of orders 96, 84, 72, etc. The construction of the bordered magic squares multiples of 12 is based on equal sum blocks of magic squares of order 12. It is done in 16 different ways. Two blocks of order 12 composed with small blocks of order 3 and 4 are pandiagonal. This lead us to write all orders multiples of 12 as pandiagonal magic squares. The only difference is that the pandiagonal magic squares multiples of 12 are no more bordered magic squares. The bordered magic squares also have the same property. For multiples of orders 4, 6, 8 and 10, see author’s recent works (multiples-4, multiples-6, multiples-8 and multiples-10). The further multiples, such as multiples, for order 14, shall be done in another works. This work brings examples only up to order 48. Higher order examples are given in Excel files attached with the work. Uploaded on July 27, 2023

211. Block-Wise Bordered Magic Squares Multiples of 10

Inder J. Taneja

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 120 and 110. Based on these two big magic inner order magic squares multiples of 10 are studied. By inner orders we understand that magic squares of orders 100, 90, 80, etc. The construction of the bordered magic squares multiples of 10. It is done in 14 different ways. The advantage in studying block-wise bordered magic squares is that when we remove external border, still we left with magic squares with sequential entries. It is the same property of bordered magic squares. The difference is that instead of numbers here we have blocks of equal sum magic squares of order 10. For multiples of orders 4, 6 and 8 see author’s recent works. The further multiples, such as multiples of 12, 14, etc. shall be done in another works. This work brings examples only up to order 40. Higher orders examples can be seen in Excel file attached with the work. The total work is up to orders 120. Uploaded on July 26, 2023.

210. Bordered and Pandiagonal Magic Squares Multiples of 8

Inder J. Taneja

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 120 and 112. Based on these two big magic inner order magic squares multiples of 8 are studied. By inner orders we understand that magic squares of orders 96, 88, 80, etc. The construction of the bordered magic squares multiples of 8 is based on equal sum blocks of bordered magic squares of order 8. The advantage in studying block-wise bordered magic squares is that when we remove external border, still we left with magic squares with sequential entries. It is the same property of bordered magic squares. In this work we considered 6 different types of magic squares of order 8. For multiples of orders 4, 6 and 10 see author’s recent works. The further multiples, such as multiples of 12, 14, etc. shall be done in another works. This work brings examples only up to order 40. Higher orders examples can be seen in Excel file attached with the work. The total work is up to orders 120. Uploaded on July 26, 2023.

209. Bordered Magic Squares Multiples of Order 6

Inder J. Taneja

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 120 and 114. Based on these two big magic inner order magic squares multiples of 6 are studied. By inner orders we understand that magic squares of orders 114, 108, 102, 96, 90, 84, etc. The construction of the bordered magic squares multiples of 6 is based on equal sum blocks of magic squares of order 6. It is done in three different types of magic squares of order 6. The advantage in studying bordered magic squares is that when we remove external border, still we left with magic squares of lower orders with sequential entries. This work is for multiples of order 6. For multiples of order 4 see author’s recent work. The further multiples, such as multiples, 8, 10, 12, etc. shall be done in another works. This work brings examples only up to order 36. Higher orders examples can be seen in Excel file attached with the work. The total work is up to order 120. Uploaded on July 25, 2023

208. Bordered Magic Squares Multiples of 17

Inder J. Taneja

During past years author worked with bordered magic squares of even number blocks. These are based on equal sums magic squares of orders 4, 6, 8, 10, etc. This type of work is an extension of classical bordered magic squares. In case of multiples of 4, the extension is made for pandiagonal magic squares. The work is extended for the multiples of orders 6, 8, etc. Recently, author worked on odd number multiples of orders 3, 5, 7, 9, 11, 13 and 15. These are based on different sums of magic squares of order 3, 5, 7, 9, 11, 13 and 15. This work is for multiples of 17. This we have done with 28 different types of magic squares of order 17. This work brings few examples for the magic squares of order order 51 multiples of 17. Higher order examples can be seen in Excel files attached with the work. The total work is up to magic squares of order 136. Uploaded on July 25, 2023

207. Bordered Magic Squares Multiples of 19

Inder J. Taneja

During past years author worked with bordered magic squares of even number blocks. These are based on equal sums magic squares of orders 4, 6, 8, 10, etc. This type of work is an extension of classical bordered magic squares. In case of multiples of 4, the extension is made for pandiagonal magic squares. The work is extended for the multiples of orders 6, 8, etc. Recently, author worked on odd number multiples from 3 to 17 with different types of magic squares. This work is for multiples of 19. This we have done with 42 different types of magic squares of order 19. This work brings few examples for the magic squares of orders 19 and 57. Higher order examples can be seen in Excel files attached with the work. The total work is up to magic squares of order 133. Uploaded on July 25, 2023

206. Bordered Magic Squares Multiples of 15

Inder J. Taneja

During past years author worked with bordered magic squares of even number blocks. These are based on equal sums magic squares of orders 4, 6, 8, 10, etc. This type of work is an extension of classical bordered magic squares. In case of multiples of 4, the extension is made for pandiagonal magic squares. The work is extended for the multiples of orders 6, 8, etc. Recently, author worked on odd number multiples of 3, 5, 7, 9, 11 and 13. These are based on different sums of magic squares of order 3, 5, 7, 9, 11 and 13. This work is for multiples of 15. This we have done with 36 different types of magic squares of order 15. This work brings only up to magic squares of order 60 multiples of 15. Higher order examples can be seen in Excel files attached with the work. The total work is up to magic squares of order 150. Uploaded on July 24, 2023

205. Bordered Magic Squares Multiples of 13

Inder J. Taneja

During past years author worked with bordered magic squares of even number blocks. These are based on equal sums magic squares of orders 4, 6, 8, 10, etc. This type of work is an extension of classical bordered magic squares. In case of multiples of 4, the extension is made for pandiagonal magic squares. The work is extended for the multiples of orders 6, 8, etc. Recently, author worked on odd number multiples of 3, 5, 7, 9 and 11. These are based on different sums of magic squares of order 3, 5, 7, 9 and 11. This work is for multiples of 13. This we have done with 25 different types of magic squares of order 13. This work brings only up to magic squares of order 52 multiples of 13. Higher order examples can be seen in Excel files attached with the work. The total work is up to magic squares of order 130. Uploaded on July 24, 2023

204. Bordered Magic Squares Multiples of 11

Inder J. Taneja

During past years author worked with bordered magic squares of even number blocks. These are based on equal sums magic squares of orders 4, 6, 8, 10, etc. This type of work is an extension of classical bordered magic squares. In case of multiples of 4, the extension is made for pandiagonal magic squares. The work is extended for the multiples of orders 6, 8, etc. Recently, author worked on odd number multiples of 3, 5, 7 and 9. These are based on different sums of magic squares of order 3, 5, 7 and 9. This work is for multiples of 11. It is revised and extended version of author’s previous work on bordered magic squares multiples 11. This we have done with 20 different types of magic squares of order 11. The brings only up to magic squares of order 44 multiples of 11. Higher order examples can be seen in Excel files attached with the work. The total work is up to order 154. Uploaded on July 24, 2023

203. Bordered Magic Squares Multiples of 9

Inder J. Taneja

During past years author worked with bordered magic squares of even number blocks. These are based on equal sums magic squares of orders 4, 6, 8, 10, etc. This type of work is an extension of classical bordered magic squares. In case of multiples of 4, the extension is made for pandiagonal magic squares. The work is extended for the multiples of orders 6, 8, etc. Recently, author worked on multiples of 3, 5 and 7. These are based on different sums of magic squares of order 3, 5 and 7. This work is for multiples of 9. It is revised and extended verison of previous work. This we have done with eleven different types of magic squares of order 9. Examples are written only up to order 35. Higher order examples can be seen in Excel file attached with the work. The total work is up to order 144.Uploaded on July 24, 2023

202. Bordered and Pandiagonal Magic Squares Multiples of 7

Inder J. Taneja

During past years author worked with block-wise bordered magic squares of even orders. It includes blocks of orders 4, 6, 8, 10, etc. Most of the cases are with equal sums magic squares. This type of work is an extension of classical bordered magic squares. In case of multiples of 4, the extension is made for pandiagonal magic squares (click here). For multiples of order 6 refer Taneja (click here). For the first time, we are presenting here bordered magic squares of odd number blocks. Recently, author worked on mutiples of 3 and 5, based on different sums magic squares of orders 3 and 5 (order3, order5). This work is for borders of magic squares of order 7. It is done with two types of magic squares of order 7. One type is pandiagonal magic squares, and another as bordered magic squares. This work is up to order 49. Higher orders examples can be seen in Excel file attached with the work. The total work is up to order 140. Pandiagonal magic squares based on equal sums pandiagonal magic squares of order 7 are also included in Excel file.Uploaded on July 23, 2023

201. Bordered and Pentagonal Magic Squares Multiples of 5

Inder J. Taneja

During past years author worked with block-wise bordered magic squares of even orders. It includes blocks of orders 4, 6, 8, 10, etc. Most of the cases are with equal sums magic squares. This type of work is an extension of classical bordered magic squares. In case of multiples of 4, the extension is made for pentagonal magic squares (click here). For multiples of order 6 refer Taneja (click here). For the first time, we are presenting here bordered magic squares of odd number blocks. Recently, author worked on mutiples of 3, based on different sums magic squares of order 3 (click here). This work is for borders of magic squares of order 5. It is the revison over the previous work. It is done with three types of magic squares of order 5. One type is pandiagonal magic squares, the second type is bordered magic square. The third type is cornered magic square. This work is up to order 35. Higher orders examples can be seen in Excel file attached with the work. The total work is up to order 150. Pandiagonal magic squares based on equal sums pandiagonal magic squares of order 5 are also included in Excel file. Uploaded on July 23, 2023

200. Magic Squares of Orders 25

Inder J. Taneja

Recently, author worked with cornered magic squares of order 6. Later this work is extended up to order 24 (paper1, paper2, paper3). This work is for odd order magics squares. It for the orders 25. In this work we have covered three types of magic squares. One centered magic squares. These are traditionally previous known magic squares. The second one is double digits bordered magic squares. The third one is cornered magic squares. Some embedded examples are given where one type is embedded in another type. For previou work on odd orders from 3 to 23 refer author’s work (paper1, paper2, paper3). Further studies are for the orders 27, 29, etc.Uploaded on June 15, 2023

199. Magic Squares of Orders 21 and 23

Inder J. Taneja

Recently, author worked with cornered magic squares of order 6. Later this work is extended up to order 24 (paper1, paper2, paper3). This work is for odd order magics squares. It for the orders 21 to 23. In this work we have covered three types of magic squares. One centered magic squares. These are traditionally previous known magic squares. The second one is double digits bordered magic squares. The third one is cornered magic squares. Some embedded examples are given where one type is embedded in another type. For previou work on odd orders from 3 to 19 refer author’s work (paper1, paper2). Further studies are for orders 25, 27, etc. Uploaded on June 15, 2023

198. Magic Squares of Orders 17 and 19

Inder J. Taneja

Recently, author worked with cornered magic squares of order 6. Later this work is extended up to order 24 (paper1, paper2, paper3). This work is for odd order magics squares. It for the orders 17 to 19. In this work we have covered three types of magic squares. One centered magic squares. These are traditionally previous known magic squares. The second one is double digits bordered magic squares. The third one is cornered magic squares. Some embedded examples are given where one type is embedded in another type. For previou work on orders 3 to 15 refer author’s work (click here). Further studies are for orders 21, 23, etc. Uploaded on June 15, 2023

197. Odd Order Magic Squares: Orders 3 to 15

Inder J. Taneja

Recently, author worked with cornered magic squares of order 6. Later this work is extended up to order 24 (paper1, paper2, paper3). This work is for odd order magics squares. It for the orders 3 to 15. In this work we have covered three types of magic squares. One centered magic squares. These are traditionally previous known magic squares. The second one is double digits bordered magic squares. The third one is cornered magic squares. Some embedded examples are given where one type is embedded in another type. Further studies are for the orders 17, 19, etc. Uploaded on June 15, 2023

196. Multiple Order Bordered Magic Squares

Inder J. Taneja

During past years author worked with block-wise bordered magic squares of even and odd number blocks. In case of even number orders, these are with equal sums magic squares of orders 4, 6, 8, 10, 12 and 14. It is an extension of classical bordered magic squares. Recently, the author also worked with odd orders bordered magic squares. This work is for the blocks of orders 3, 5, 7, 9 and 11. This work is little different. It brings multiple types borders for magic squares. Initially, we start with magic square of order 12 formed by blocks of magic squares of order 3. Then external border is made with blocks of pandiagonal magic squares of order 4 resulting in magic squares of order 20. Proceeding in the same way we get magic squares of orders 30, 42, 56, 72, 90, 110 and 132 based on borders of magic squares of orders 5, 6, 7, 8, 9, 10 and 11 respectively. The same procedure can be applied for further orders. The whole work can be seen in excel files attached with the work. Uploaded on June 9, 2023

195. Cornered Magic Squares of Orders 14 to 24

Inder J. Taneja

Recently, author (click here) worked with cornered magic squares of orders 5 to 13. This type of study is new in the literature of magic squares, and it is brought for the first time in this work. These types of magic squares we call as cornered magic squares. In each case , there is one magic square known by nested corner magic square. This work brings cornered magic squares of orders 14 to 24. The order 14 is with 82 examples, and are in the same direction. Similar to previous works we can rotate it in other three directions. The magic squares of orders 15 to 24 are with single example in each case. These examples are known as nested corner magic squares.Uploaded on June 3, 2023

194. Cornered Magic Squares of Orders 5 to 13

Inder J. Taneja

Recently, author (click here) worked with cornered magic squares of order 6. This work extend it to magic squares of orders 5 to 13. This type of study is new in the literature of magic squares, and it is brought for the first time in this work. These types of magic squares we call as cornered magic squares. In each case one of them is known by nested corner magic square. This work brings just cornered magic squares in different ways. The results are written in four directions. Further study for the orders 14 to 24 shall be given in another work.Uploaded on June 3, 2023

193. Cornered Magic Squares of Order 6

Inder J. Taneja

This work brings for the first time in the history of magic squares a new kind of magic square of order 6. It is called cornered magic square of order 6. It contains at the corner a pandiagonal magic square of order 4. Before to this the author (click here) also introduced a new kind of magic square of order 6 based on magic rectangle of order 4×6. Using pandiagonal magic square of order 4, we have written four different kinds of cornered magic squares of order 6. Based on these four cornered magic squares of order 6, we have written some examples of magic squares of orders 8, 10, 12, 14 and 16. Uploaded on May 23, 2023

192. Two Digits Bordered Magic Squares of Orders 34 and 38

Inder J. Taneja

For the first time in history, this work brings a new kind of magic squares. These are of type, 4k+2, k>1. These are two digits bordered magic squares of orders 10, 14, etc. This part brings magic squares of orders 26 and 30. The first part is for the orders 10, 14, 18 and 22, 26 and 30. Whole the work with pdf files can be accessed at the author’s site (orders4to22, orders26and30). The single digit bordered magic squares are already known in the literature. For three digits, we work with magic squares of order 3. For 4 digits we work with magic squares of order 4, for five digits we worked with magic squares of order 5, and so on. Uploaded on May 10, 2023

191. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 3

Inder J. Taneja

During past years author worked with \textbf{block-wise bordered} magic squares of even orders. It includes blocks of orders 4, 6, 8, 10, etc. Most of the cases are with equal sums magic squares. This type of work is an extension of classical bordered magic squares. In case of multiples of 4, the extension is made for pandiagonal magic squares (click here) . For multiples of order 6 refer (click here). For the first time, we are presenting here bordered magic squares of odd number blocks. Specially in this work, we give bordered with different sum magic squares of order 3. Pandiagonal magic squares multiples of 3 are also given. These we get for all orders starting from order 3, except orders 18, 30, 42, etc. The work is up to order 36. Higher orders examples can be seen in Excel file is attached with the work. It also include pandiagonal magic squares of equal sums up to order 117. The total work is up to order 120. Uploaded on May 5, 2023

190. Two digits Bordered Magic Squares of Orders 36 and 40

Inder J. Taneja

For the first time in history, this work brings a new kind of magic squares. These are multiples of 4 starting from order 8. These are two digits bordered magic squares. This part brings magic squares of orders 36 and 40. For the previous orders from order 8 to 32 click the two links (link1, link2). Whole the work with pdf files can be accessed at the author’s site (order 36, order 40). The single digit bordered magic squares are already known in the literature. For three digits, we work with magic squares of order 3. For four digits we work with magic squares of order 4.Uploaded on May 4, 2023

189. Two Digits Bordered Magic Squares of Orders 26 and 30

Inder J. Taneja

For the first time in history, this work brings a new kind of magic squares. These are of type, 4k+2, k>1. These are two digits bordered magic squares of orders 10, 14, etc. This part brings magic squares of orders 26 and 30. The first part is for the orders 10, 14, 18 and 22. Whole the work with pdf files can be accessed at the author’s site. The single digit bordered magic squares are already known in the literature. For three digits, we work with magic squares of order 3. For four digits we work with magic squares of order 4.Uploaded on April 30, 2023

188. Two Digits Bordered Magic Squares of Orders 10, 14, 18 and 22

Inder J. Taneja

For the first time in history, this work brings a new kind of magic squares. These are of type, 2k+2, k>3. These are two digits bordered magic squares of orders 10, 14, etc. This part brings magic squares of orders 10, 14, 18 and 22. The second part is for the orders 26 and 30. Whole the work with pdf files can be accessed at the author’s site. The single digit bordered magic squares are already known in the literature. For three digits, we work with magic squares of order 3. For four digits we work with magic squares of order 4. Uploaded on April 30, 2023

187. Two Digits Bordered Magic Squares of Orders 28 and 32

Inder J. Taneja

For the first time in history, this work brings a new kind of magic squares. These are multiples of 4 starting from order 8. These are two digits bordered magic squares based on magic rectangles of order 2×4. This part brings magic squares of orders 28 and 32. For the previous orders from order 8 to 24 click here. Whole the work with pdf files can be accessed at the author’s site. The single digit bordered magic squares are already known in the literature. For three digits, we work with magic squares of order 3. For four digits we work with magic squares of order 4. Uploaded on April 26, 2023

186. Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24

Inder J. Taneja

For the first time in history, this work brings a new kind of magic squares. These are multiples of 4 starting from order 8. These are two digits bordered magic squares based on magic rectangles of order 2×4. This part brings magic squares of orders 8, 12, 16, 20 and 24. The second part is for the orders 28 and 32. Whole the work with pdf files can be accessed at the author’s site. The single digit bordered magic squares are already known in the literature. For three digits, we work with magic squares of order 3. For four digits we work with magic squares of order 4. Uploaded on April 26, 2023

185. 8000+ Magic Squares of Order 22 in Different Styles, Models and Designs

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. This work is for the magic square of order 22. Total there are 8071 magic squares of order 22. Specially, these are based on small blocks of magic squares, bordered magic squares and bordered magic rectangles. This work revised the author’s previous work on magic squares of order 22. It is in different styles, designs and models. This work is with few few examples. The whole work as pdf files is at author’s site. The work is done in different ways including the application of the formula (a+b)^2. Uploaded on April 8, 2023

184. Closed Double-Crossed Bordered Magic Rectangles and Magic Squares of Order 42

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. A systematic procedure to construct these magic squares is given. The complete work is up to order 42. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. In this work we have considered crossed bordered magic rectangles for the magic squares of order 40. It is done with double-cross bordered magic rectangles. The work is with few examples. The pdf files of full work can be downloaded at author’s site. Uploaded on March 3, 2023

183. Double-cross Bordered Magic Rectangles and Magic Squares of Order 42

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. In this work we have considered crossed bordered magic rectangles for the magic squares of order 40. It is done with double-cross bordered magic rectangles. The work is with few examples. The pdf files of full work can be downloaded at author’s site. Uploaded on March 3, 2023

182. Single-Cross Bordered Magic Rectangles and Magic Squares of Order 42

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. In this work we have considered crossed bordered magic rectangles for the magic squares of order 42. It is done with single-cross bordered magic rectangles. The double-cross bordered magic rectangles are done in another work. The work is with few examples. The pdf file of full work can be downloaded at author’s site. Uploaded on March 3, 2023

181. Magic Squares of Order 42 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares of order 42. Pdf file os complete work of magic squares of order 42 can be downloaded at author’s site. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on March 3, 2023

180. Double Crossed Bordered Magic Rectangles and Magic Squares of Order 40

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. In this work we have considered crossed bordered magic rectangles for the magic squares of order 40. It is done with double crossed bordered magic rectangles. The single crossed bordered magic rectangles are done in the previous work. The work is with few examples. The pdf files of full work can be downloaded at author’s site. Uploaded on January 30, 2023

179. Single Crossed Bordered Magic Rectangles and Magic Squares of Order 40

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. In this work we have considered crossed bordered magic rectangles for the magic squares of order 40. It is done with single crossed bordered magic rectangles. The double crossed bordered magic rectangles are done in another work. The work is with few examples. The pdf file of full work can be downloaded at author’s site. Uploaded on January 24, 2023

178. Figured Magic Squares of Order 40 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of order 40. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on January 3, 2023

177. Figured Magic Squares of Order 38 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of order 38. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on January 3, 2023

Work in 2022

176. Figured Magic Squares of Order 36 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of order 36. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on December 27, 2022

175. Figured Magic Squares of Order 34 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of order 34. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on December 27, 2022

174. 23 and 2023 in Numbers and Patterns

Inder J. Taneja

This work brings representations of 23 and 2023 in different ways. These representations are of crazy-type, running numbers, single digit, single letter, Triangular, Fibonacci, palindromic-type, prime numbers, embedded, repeated digits, magic squares, etc. Among two numbers 23 and 2023, the number 23 is a prime number. The digits 2, 0, 2 and 3 of 2023 are written in 45 equal sums magic squares of order 4 using consecutive numbers from 1 to 720. Uploaded on December 22, 2022

173. Figured Magic Squares of Order 32 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of order 32. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on December 22, 2022

172. Figured Magic Squares of Order 30 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of order 30. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on December 2, 2022

171. Figured Magic Squares of Order 28 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of order 28. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on December 2, 2022

170. Figured Magic Squares of Order 26 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of order 26. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on November 29, 2022

169. Figured Magic Squares of Order 24 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of order 24. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on November 29, 2022

168. Figured Magic Squares of Order 22 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of order 22. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on November 29, 2022

167. Figured Magic Squares of Orders 18 and 20 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of orders 18 and 20. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. Uploaded on November 29, 2022

166. Figured Magic Squares of Orders 6, 10, 12, 14 and 16 Using Bordered Magic Rectangles: A Systematic Procedure

Inder J. Taneja

Recently, author constructed even order magic squares from orders 6 to 20 with different styles and models, for examples the order 20 is with 1616 magic squares, order 18 with 810 magic squares, etc. For details see the link. The aim is to proceed further orders of magic squares. In this work there are few examples of magic squares given only in figures of orders 6, 8, 10, 12, 14 and 16. A systematic procedure to construct these magic squares is given. It is based on the magic squares and bordered magic rectangles (BMR) of orders 4, 6, 8 etc. forming external borders. The inner blocks are filled with previous known magic squares. For the orders multiples of 4, we can always write magic squares with equal sums blocks of magic squares of order 4. This procedure is very helpful for the orders of type 2p, where p is a prime number, for examples, 14, 22, 26, 34, 38, etc. For the orders like 18, 30, etc., we can make good external blocks with order 4, and for orders like 16, 20, 28, 32, etc. we can make good external borders of order 6, and so on. The explanations of constructions are given for the orders 14 and 16.Uploaded on November 29, 2022

165. Different Styles of Magic Squares of Order 20 Using Bordered Magic Rectangles

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For the first time in this work, two new concepts in construction of magic square are considered. It is based on bordered magic rectangles. and the second process is based on the algebraic formula (a+b)^2. Both the processes uses small blocks of bordered magic rectangles. This work is revised a version of author’s previous work, and brings magic squares of order 20. This work is with much more magic squares then previous work. The previous work is on orders 6, 8, 10, 12, 14, 16 and 18. For summary see the links (link1, link2).Uploaded on November 15, 2022

164. Different Styles of Magic Squares of Order 18 Using Bordered Magic Rectangles

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For the first time in this work, two new concepts in construction of magic square are considered. It is based on bordered magic rectangles. and the second process is based on the algebraic formula (a+b)^2. Both the processes uses small blocks of bordered magic rectangles. This work is revised a version of author’s previous work, and brings magic squares of order 18. This work is with much more magic squares then previous work. The previous work is on orders 6, 8, 10, 12, 14 and 18 The further work is on order 20. For summary see the links (link1, link2). Uploaded on November 15, 2022

163. Different Styles of Magic Squares of Order 16 Using Bordered Magic Rectangles

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For the first time in this work, two new concepts in construction of magic square are considered. It is based on bordered magic rectangles. and the second process is based on the algebraic formula (a+b)^2. Both the processes uses small blocks of bordered magic rectangles. This work is revised a version of author’s previous work, and brings magic squares of order 16. This work is with much more magic squares then previous work. The previous work is on orders 6, 8, 10, 12 and 14. The further works are on orders 18 and 20. For summary see the links (link1, link2). Uploaded on November 15, 2022

162. Few Examples of Magic Squares of Even Orders 20 to 30 Using Bordered Magic Rectangles

Inder J. Taneja

The author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For the first time in recent works (paper1, paper2, paper3, paper4), the author considered two new concepts in construction of magic squares. The first one is by using small blocks of magic or bordered magic rectangles. It is not possible always to write bordered magic rectangles for the non-sequential entries. In this case, the simple magic rectangles are considered. The second aspect is based on the algebraic formula (a+b)^2. The work is still limited to even order magic squares. It is done for the magic squares of orders 8, 10, 12, 14, 16 and 18. This brings few examples of similar type for the orders 20, 22, 24, 26, 28 and 30. The extended study shall be given elsewhere. Uploaded on October 19, 2022

161. Few Examples of Magic Squares of Even Orders 6 to 18 Using Bordered Magic Rectangles

Inder J. Taneja

The author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For the first time in recent works (paper1, paper2, paper3, paper4), the author considered two new concepts in construction of magic squares. The first one is by using small blocks of magic or bordered magic rectangles. It is not possible always to write bordered magic rectangles for the non-sequential entries. In this case, the simple magic rectangles are considered. The second aspect is based on the algebraic formula (a+b)^2. The work is still limited to even order magic squares. It is done for the magic squares of orders 8, 10, 12, 14, 16 and 18. This work summaries few examples from the even orders 6 to 18 done previously. Uploaded on October 19, 2022

160. Magic Rectangles in Construction of Magic and Block Bordered Magic Squares

Inder J. Taneja

In this work, blocks of magic rectangles with equal and unequal sums are used to construct magic squares. The equal sum blocks magic rectangles lead us to even order magic squares, such as, of orders 8, 12, 16, 20, 24, 28, 32, 36, 40, 42, 44 and 48. These magic squares are constructed using magic rectangles of orders 2×4, 2×14, 4×6, 6×10, 6×14, etc. There is only one example of even order magic square with different sums blocks of magic rectangles, i.e., of order 18. The odd order magic squares are with unequal sum blocks magic rectangles. These are of orders 15, 21, 27, 33, 39 and 45. These are constructed using magic rectangles of orders 3×5, 3×7, 3×9, 3×11, 3×13, 3×15, 5×7, etc. The magic squares of orders 12, 20, 24, 28, 42, 45 and 48 are written in two different ways. These magic squares are used to bring block bordered magic squares. These block bordered are of orders 10, 14, 17, 19, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46 and 47. Most of the magic squares from order 8 to 48 are studied in this work, except the orders 9, 11, 13 and 25. These are already studied by author in another work. Uploaded on June 7, 2022

159. Different Types of Multiple Style Magic Squares of Order 30

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For details refer author’s work. It brings multiples blocks of orders 4, 6, 8, 10, 12 and 14. This work brings magic squares of orders 28 in different types. It combines all the four types, such as block-wise, bordered, block-bordered and block-wise bordered. There are total 612 types of magic squares of order 32. It is revised and extended version of author’s previous work with corrections. For previous works from orders 10 to 26, 28, 30, 36 and 40 refer author’s work (work1, work2, work3, work4). The previous work was only with 110 magic squares of order 30. Total there are 612 different types of magic squares of order 30 given in this work. This paper is with few examples. The completed examples with pdf files are available at author’s web-sites (site1, site2). Uploaded on May 3, 2022

158. Different Types of Multiple Style Magic Squares of Order 28

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For details refer author’s work. It brings multiples blocks of orders 4, 6, 8, 10, 12 and 14. This work brings magic squares of orders 28 in different types. It combines all the four types, such as block-wise, bordered, block-bordered and block-wise bordered. There are total 238 types of magic squares of order 28. It is revised and extended version of author’s previous work with corrections. For previous works from orders 10 to 26, 28, 30, 36 and 40 refer author’s work (work1, work2, work3, work4). The previous work was only with 102 magic squares of order 28. Total there are 238 different types of magic squares of order 28 given in this work. This paper is with few examples. The completed examples with pdf files are available at author’s web-sites (site1, site2). Uploaded on May 1, 2022

157. Different Types of Multiple Style Magic Squares of Order 32

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For details refer author’s work. It brings multiples blocks of orders 4, 6, 8, 10, 12 and 14. This work brings magic squares of orders 32. It combines all the four types, such as block-wise, bordered, block-bordered and block-wise bordered. For previous works from orders 10 to 26, 28, 30, 36 and 40 refer author’s work (work1, work2, work3, work4). It is revised and extended version of author’s previous work with corrections. The previous work was only with 324 magic squares of order 32. Total there are 1309 different types of magic squares of order 32 given in this work. This paper is with few examples. The completed examples with pdf files are available at author’s web-sites (site1, site2). Uploaded on May 1, 2022

156. Multiple Style Different Types of Magic Squares of Order 36

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For details refer author’s work. It brings multiples blocks of orders 4, 6, 8, 10, 12 and 14. This work brings magic squares of orders 36. It combines all the four types, such as block-wise, bordered, block-bordered and block-wise bordered. For previous works from orders 10 to 26, 28, 30 and 32 refer author’s recent works (work1, work2, work3, work4). It is revised and extended version of author’s previous work with corrections. The previous work was only with 565 magic squares of order 36. This paper is with few examples. The completed pdf files are available at author’s web-sites (site1, site2). Uploaded on April 27, 2022

155. Different Types of Multiple Style Magic Squares of Order 40

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For details refer author’s work. It brings multiples blocks of orders 4, 6, 8, 10, 12 and 14. This work brings differet types of magic squares of order 40. It is a combination of block-wise, bordered, block-bordered and block-wise bordered magic squares. For previous works from orders 10 to 26, 28, 30, 32 and for order 36 refer author’s recent works (work1, work2, work3, work4, work5) Total there are 3557 different types of magic squares of order 40 given in this work. It is revised version of author’s previous work with corrections. This paper brings only few examples. For complete details of 3557 examples, see the 9 pdf files attached with this work or at author’s sites (site1, site2). The work is in the file Diff-Style-40a.pdf. Uploaded on April 23, 2022

154. Different Types of Magic Squares of Order 32: Part 4

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For details refer author’s work. It brings multiples blocks of orders 4, 6, 8, 10, 12 and 14. This work brings magic squares of orders 32. It combines all the four types, such as block-wise, bordered, block-bordered and block-wise bordered. There are total 324 types of magic squares of order 32 divided in four parts (Part1, Part2, Part3, Part4). For previous works from orders 10 to 26, 28 and for order 30 refer author’s recent works (work1, work2, work3). As the orders of magic squares increases, the number of possibilities to write different types of magic squares also increases. For more idea of this work, the excel file is also attached. This excel file contains all the four parts. Uploaded on March 27, 2022

153. Different Types of Magic Squares of Order 32: Part 3

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For details refer author’s work. It brings multiples blocks of orders 4, 6, 8, 10, 12 and 14. This work brings magic squares of orders 32. It combines all the four types, such as block-wise, bordered, block-bordered and block-wise bordered. There are total 324 types of magic squares of order 32 divided in four parts (Part1, Part2, Part3, Part4). For previous works from orders 10 to 26, 28 and for order 30 refer author’s recent works (work1, work2, work3). As the orders of magic squares increases, the number of possibilities to write different types of magic squares also increases. For more idea of this work, the excel file is also attached. This excel file contains all the four parts. Uploaded on March 27, 2022

152. Different Types of Magic Squares of Order 32: Part 2

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For details refer author’s work. It brings multiples blocks of orders 4, 6, 8, 10, 12 and 14. This work brings magic squares of orders 32. It combines all the four types, such as block-wise, bordered, block-bordered and block-wise bordered. There are total 324 types of magic squares of order 32 divided in four parts (Part1, Part2, Part3, Part4). For previous works from orders 10 to 26, 28 and for order 30 refer author’s recent works (work1, work2, work3). As the orders of magic squares increases, the number of possibilities to write different types of magic squares also increases. For more idea of this work, the excel file is also attached. This excel file contains all the four parts. Uploaded on March 27, 20227

151. Different Types of Magic Squares of Order 32: Part 1

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. For details refer author’s work. It brings multiples blocks of orders 4, 6, 8, 10, 12 and 14. This work brings magic squares of orders 32. It combines all the four types, such as block-wise, bordered, block-bordered and block-wise bordered. There are total 324 types of magic squares of order 32 divided in four parts (Part1, Part2, Part3, Part4). For previous works from orders 10 to 26, 28 and for order 30 refer author’s recent works (work1, work2, work3). As the orders of magic squares increases, the number of possibilities to write different types of magic squares also increases. For more idea of this work, the excel file is also attached. This excel file contains all the four parts. Uploaded on March 27, 2022

150. Different Types of Magic Squares: Even Number Orders From 10 to 26

Inder J. Taneja

Author worked on different kinds of magic squares. These include block-wise, bordered, block-bordered and block-wise bordered magic squares. The block-wise bordered magic squares are very much similar to bordered magic squares, where the numbers are replaced by blocks of magic squares. This work brings magic squares of orders 10 to 26 in different types combining all the four types, such as block-wise, bordered, block-bordered and block-wise bordered. In case of orders 14, 22 and 26, there are no new as these are based on results of magic squares of orders 12, 20 and 24 respectively. As the orders of magic squares increases, the number of possibilities to write different types of magic squares also increases, for example, on case of orders 36, 40, etc. there are much more than 500 possibilities in each case. Uploaded on March 26, 2022

149. Single Digit Representations of Natural Numbers From 40001 to 50000

Inder J. Taneja

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper bring numbers 40001 to 50000 in terms of digits 1 to 9 separately. For the representations of the numbers from 1 to 40000 refer author’s previous work ((work1, work2, work3, work4, work5, work6). Uploaded on March 23, 2022

148. Single Digit Representations of Natural Numbers From 30001 to 40000

Inder J. Taneja

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper bring numbers 30001 to 40000 in terms of digits 1 to 9 separately. For the representations of the numbers from 1 to 30000 refer author’s previous works (work1, work2, work3, work4, work5). Uploaded on March 23, 2022

147. Single Digit Representations of Natural Numbers From 20001 to 30000

Inder J. Taneja

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper bring numbers 20001 to 30000 in terms of digits 1 to 9. For the representations of the the numbers from 1 to 20000 refer author’s previous work (first, second, third and forth). Uploaded on March 21, 2022

146. 100-Magic Squares of Order 42 With Numbers 00-99

Inder J. Taneja

During past years author worked with block-wise, bordered and block-bordered magic squares. In this work, we shall 100-write magic squares of order 42 with numbers from 00 to 99. These numbers are equal sums with bordered magic squares of order 6. Uploaded on February 10, 2022

145. Representation of Numbers from 1 to 20000 in Terms of Palindromic Digits 1357-9-7531

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done from 1 to 200000. For details see the references. This work is little different. It brings crazy representations of numbers from 1 to 20000 in terms of palindromic digits of 1357-9-7531. For each number 9 digits, 1, 3, 5, 7, 9, 7, 5, 3, 1 are used. This we have done by the help of extra operations or functions, such as factorial, square-root. Similar kind of work for the numbers from 1 to 1729 in terms of digits 1729271 see author’s work (click here) and for the numbers 1-10000 in terms of 2022-2022 (click here). Uploaded on January 6, 2022

144. Representation of Numbers from 1 to 10000 in Terms of Palindromic Digits 2022-2202

Inder J. Taneja

During past years author worked with representations of numbers in terms of 1 to 9 and 9 to 1. These representations called as crazy representations, and are done from 1 to 200000. For details see the references. This work is little different. It brings crazy representations of numbers from 1 to 10000 in terms of palindrodic digits of 2022-2202. For each number 8 digits are used. This we have done by the help of extra operations or functions, such as factorial, square-root and triangular numbers. Similar kind of work for the numbers from 1 to 1729 in terms of digits 1729271 see author’s work (here). For different representations of 2022 see the author’s recent work. Uploaded on January 2, 2022

Work in 2021

143. Block-Wise and Block-Bordered Magic Squares With Magic Sum 2022

Inder J. Taneja

During past years, the author worked with magic squares in different situations. These are of digital-type, block-wise, block-bordered, bordered, creative-type, 2-digits, magic-crosses, magic letters and magic numbers, etc. The detailed study can be seen in the reference list. The magic square summing 2020, 2021 etc. are also studied with different properties. In this work, block-wise and block-bordered magic squares are constructed in such a way that the final magic sum in each case is always 2022. The work is for orders 3 to 22. The entries are either fractional, decimal and whole numbers with positive and/or negative values. Uploaded on December 28, 2021

142. Mathematical Beauty of 2022

Inder J. Taneja

This work brings representations of 2022 in different ways. These representations are of crazy-type, running numbers, single digit, single letter, Triangular, Fibonacci, palindromic-type, prime numbers, embedded, repeated digits, colored patterns, magic squares, etc. Interesting, this year there will be a day with 11 times repetition of single digit 2. It will happens on: 22h, 22m, 22s, February 22, 22, i.e., 22.22.22.22.2.22. Uploaded on December 26, 2021

141. Crazy Running Equality Expressions With Factorial and Square-Root

Inder J. Taneja

In previous works, the author wrote equality expressions in terms of 1 to 9 and 9 to 1 or 9 to 0 separated by single or double equality signs. These expressions are based on the idea of crazy representations of natural numbers. These equality expressions are based on basic operations along with factorial, square-root, Fibonacci sequence and triangular number values. These types of equalities, we called as crazy running equality expressions. This work is revised and enlarged version of author’s previous works, where we considered basic operations along with factorial and square-root. It is not possible to write all the natural numbers in terms of crazy running equality expressions. Only the possible numbers are written. The work is up to 6 digits. Uploaded on December 6, 2021

140. Crazy Representations of Natural Numbers From 20001 to 40000

Inder J. Taneja

This paper brings natural numbers from 20001 to 40000 written in \textbf{ascending} and \textbf{descending} orders of 1 to 9. The numbers are obtained by using basic operations, and \textbf{factorial}. For previous results see author’s works [work1, work2, work3]. For more details and comments see author’s site link. Uploaded on November 3, 2021

139. Crazy Representations of Natural Numbers From 80001 to 100000

Inder J. Taneja

This paper brings natural numbers from 80001 to 100000 written in ascending and descending orders of 1 to 9. The numbers are obtained by using basic operations, and factorial. For previous results see author’s works [work1, work2, work3]. For more details and comments see author’s site linkUploaded on November 3, 2021

138. Crazy Representations of Natural Numbers From 60001 to 80000

Inder J. Taneja

This paper brings natural numbers from 60001 to 80000 written in ascending and descending orders of 1 to 9. The numbers are obtained by using basic operations, and factorial. For previous results see author’s works [work1, work2, work3]. For more details and comments see author’s site linkUploaded on November 3, 2021

137. Crazy Representations of Natural Numbers From 40001 to 60000

Inder J. Taneja

This paper brings natural numbers from 40001 to 60000 written in ascending and descending orders of 1 to 9. The numbers are obtained by using basic operations, and \textbf{factorial}. For previous results see author’s works [work1, work2, work3]. For more details and comments see author’s site link. Uploaded on November 3, 2021

136. Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4

Inder J. Taneja

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 108 and 104. Based on these two big magic inner order magic squares multiples of 4 are studied. By inner order we understand that magic squares of orders 100, 96, 92, etc. Instead of working in decreasing order, we worked with increasing orders, such as, orders 4, 8, 12, etc. The construction of the block-wise bordered magic squares multiples of 4 is based on equal sum blocks of pandiagonal magic squares of order 4. The block-wise bordered magic squares studied are not pandiagonal. Redistributing the same blocks in each case, we get pandiagonal magic squares of order 4, 8, 12, etc. This work is only for the multiples of order 4. The further multiples, such as multiples, 6, 8, 10, etc. shall be done in another works. Examples are given only up to order 48. Higher orders examples can be seen in Excel file attached with the work. The total work is up to order 108. Uploaded on August 31, 2021

135. Creative Magic Squares: Area Representations With Fraction Numbers Entries

Inder J. Taneja

It is well known that every magic square can be written as perfect square sum of entries. It is always possible with consecutive odd numbers entries starting from 1. In case of odd order magic squares we can also write with consecutive natural numbers entries. In case of even order magic squares it is possible with consecutive fraction numbers entries. Still we can have minimum perfect square sum of entries in two different ways, i.e., one with consecutive natural numbers for odd order magic squares and secondly with consecutive fraction numbers entries for even order magic squares. Based on this idea of perfect square sum of entries, magic square are written as area-representations of each number resulting always in perfect square sum of entries. The work is for the magic squares of orders 3 to 11. In the case of magic squares of orders 10 and 11 the images are not very clear, as there are a lot of numbers. To have a clear idea, the magic squares are also written in numbers. In all the cases, the area representations are given in more that one way. It is due to the fact that we can always write magic squares as normal, bordered and block-bordered ways. This work is revised version of author’s previous work. This work brings more results with \textbf{fraction numbers} entries. In future the work shall be extended for higher order magic squares. Uploaded on August 16, 2021

134. Magic Squares With Perfect Square Sum of Entries: Orders 3 to 47

Inder J. Taneja

This paper shows how to create magic squares with perfect square sum of entries. There are three types of entries, consecutive odd numbers, natural numbers and fraction numbers. The consecutive odd numbers entries are for all order magic squares. The consecutive natural numbers entries are for odd order magic squares or the consecutive fraction numbers entries are for even order magic squares resulting in equal sum magic squares. This process is applied in three types of situations. One is uniformity, i.e., k, k^2, k^3, k^4, where k is the order of a magic square. This property means, we have magic square of order k, with k^2 entries. The magic sum is k^3 and the total sum of entries is k^4. The second process is generating magic squares from Pythagorean triples with least possible sum of entries. The third process is having directly magic squares with minimum perfect square sum of entries. For each order, there are five types of magic squares with perfect square sum of entries having total three different types of magic sums. The sum is minimum possible only in the third case. The work is for the magic square orders 3 to 47. It include the revised version of of author’s previous works for orders 3 to 31. The idea of this work is applied to area-representations of magic squares. Uploaded on August 16, 2021

133. Minimum Perfect Square Sum Bordered and Block-Wise Bordered Magic Squares: Orders 32 to 47

Inder J. Taneja

There are many ways of writing magic or bordered magic squares, where the sum of entries is always a perfect square. In one of the possibility, the magic sums are such that they satisfy uniformity property. Another way is to write magic squares generated by Pythagorean triples. Based on these idea, we can always write magic squares or bordered magic squares where the sum of entries is always a perfect square. Still, these sums are not minimum. There is a third way to write magic squares or bordered squares such that we get minimum perfect square sum of entries. This what we have done in this work with bordered and block-wise bordered magic squares. In case of even order magic squares, we can always write block-wise bordered magic squares with equal sum sub-blocks. In case of odd order magic squares, we can write them as block-wise bordered magic squares, but with different sub-blocks sums. This work is for the magic squares of orders 32 to 47. The work on magic squares of orders 3 to 31 is given in another work. Recently, author applied this idea of perfect square sum of entries to write area representations of magic squares. Uploaded on July 20, 2021

132. Minimum Perfect Square Sum Bordered and Block-Wise Bordered Magic Squares: Orders 3 to 31

Inder J. Taneja

There are many ways of writing magic or bordered magic squares, where the sum of entries is always a perfect square. In one of the possibility, the magic sums are such that they satisfy uniformity property. Another way is to write magic squares generated by Pythagorean triples. Based on these idea, we can always write magic squares or bordered magic squares where the sum of entries is always a perfect square. Still, these sums are not minimum. There is a third way to write magic squares or bordered squares such that we get minimum perfect square sum of entries. This what we have done in this work with bordered and block-wise bordered magic squares. In case of even order magic squares, we can always write block-wise bordered magic squares with equal sum sub-blocks. In case of odd order magic squares, we can write them as block-wise bordered magic squares, but with different sub-blocks sums. This work is for the orders 3 to 31. The extension to bordered magic squares of orders 32 to 47 is given in another work. Recently, author applied this idea of perfect square sum of entries to write area representations of magic squares. Uploaded on July 20, 2021

131. Magic Squares With Perfect Square Sum of Entries: Orders 3 to 31

Inder J. Taneja

This paper shows how to create magic squares with a perfect square number for the total sum entries. This has been done in five ways. Initially, the two ways are using entries as consecutive odd numbers and consecutive natural numbers for odd order magic squares and consecutive fraction numbers for even order magic squares. This process sastify an interesting property known by uniformity property. The second way is also give two kind of magic squares with entries sum as perfect square. It is based on magic squares generated by Pythagorean triples. This procedure give many magic squares based on interval and order of magic squares. In this work, only the first value considered giving least possible perfect square sum of entries. In this way also there are two possibilities, one using entries as consecutive odd numbers and second using entries as consecutive natural numbers for odd order magic squares and consecutive fraction numbers for even order magic squares. In all the four possibilities given above not even a single one give us minimum perfect square sum of entries. By using the entries as nonnegative numbers, a fifth procedure is considered to get minimum perfect square sum of entries. These entries are either consecutive natural numbers for odd order magic squares and consecutive fraction numbers for even order magic squares. For each order, the work is divided in three parts resulting in five magic squares with perfect square sum of entries. Recently, author applied this idea of perfect square sum of entries to write area representations magic squares. This work is for the magic squares of orders 3 to 31. Further orders from 32 to 47 are given in another work. Uploaded on July 19, 2021

130. Sequential Pythagorean Triples and Perfect Square Sum Magic Squares

Inder J. Taneja

The Pythagoras theorem is very famous in the literature of mathematics. It led us to many Pythagorean triples. In this work we have considered all possible Pythagorean triples between 3 to 1000 and generated magic squares, where the sum of entries is a perfect square. The Pythagorean triples are considered in sequential form from number 3 onwards. Fortunately, we have Pythagorean triples for all numbers except number 4. Em total we have 5803 triples. These triples generate 2172 magic squares. These magic squares are written with possible entries. In the second part of the work these magic squares are analysed according to their orders, and found interesting relations with Pythagorean triples. Also, it is observed that there is always a triple with the first element as an even number generating a magic square. These magic squares are written in last section of the work. Uploaded on June 21, 2021

129. Generating Pythagorean Triples and Magic Squares: Orders 3 to 31

Inder J. Taneja

This paper shows how to create magic squares with a perfect square number for the total sum of their entries. This has been done in two ways: Firstly, by using the sum of consecutive odd numbers, and secondly, by using consecutive natural numbers. In the first case, for all orders of magic squares, one can always obtain a perfect square entries sum. In the second case, magic squares with perfect square magic sums do exist, but only for odd order magic squares. For the even order magic squares, such as, 4, 6, 8, etc. it is not possible to write consecutive natural number magic squares with perfect square entries sums. A simplified idea is introduced to check when it is possible to obtain minimum perfect square entries sums. Also, a uniform method is presented so that, if k is the order of a magic square, then the magic sum of the square is k^3, and the sum of all entries of the magic square is k^4. Based on these aspects, connections with Pythagorean triples are also made. The work is for the magic squares of orders 3 to 31. Further orders shall be dealt later on. In another work, the magic squares are generated based on Pythagorean triples. Uploaded on May 28, 2021

128. Block-Wise and Block-Bordered Magic Squares Generated by Pythagorean Triples: Orders 3 to 47

Inder J. Taneja

Pythagorean triples are very famous in the literature of mathematics. Special kind of magic squares can be constructed based Pythagorean triples and vice-versa. Recently, author studied different kind of magics squares including block-wise, block-bordered, block-wise-bordered, etc. In this paper, some special kind of Pythagorean triples are generated, and then based on these Pythagorean triples magic squares of orders 3 to 47 are constructed with consecutive odd numbers entries. The odd order magic squares also lead to consecutive natural number entries with same sum resulting in Pythagorean triples. We know that we can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. In these type of cases, block-bordered magic squares are constructed. These are of orders 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46 and 47. In other cases, block-wise magic squares are constructed. These are of orders 8, 9, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44 and 45. In another work, Pythagorean triples are generated based on consecutive odd number entries magic squares starting from 1. Uploaded on May 28, 2021

127. Creative Magic Squares: Increasing and Decreasing Orders Crazy Representations

Inder J. Taneja

This paper brings magic squares of orders 3 to 10 in terms of crazy representations. These representations are of three types. One in increasing order of digits starting from 1. The second in decreasing order of digits ending in 1. The third is also in decreasing orders ending in 0. These representations are neither ending or starting from 9. Minimum possible representation way is applied. For example, we can write 123 as 123 or 4!*3!-21 or 3+((2+1)!-0!)!. These representations are with two extra operations, such as factorial and square-root. The magic squares from the order 3 to 9 are written once, and the magic square of order 10 is written twice. One as a normal way, the second as block-bordered, where inner part as block-wise pandiagonal magic square of order 8 with equal sum blocks of order 4. Previous works of similar kind is done with single digit, single letter and permutable base-powers. Uploaded on May 26, 2021

126. Creative Magic Squares: Permutable Base-Power Digits Representations

Inder J. Taneja

This paper brings magic squares of orders 3 to 10 in terms of permutable base-power digits. It means that in a same numbers, there are same digits in powers and bases. In case of order 10 there are two possibilities are given. One as general normal magic squares and another as block-bordered magic square, with inner block as pandiagonal magic square of order 8. Previous works of similar kind is done with single digit and single letter. Uploaded on April 3, 2021

125. Creative Magic Squares: Single Letter Representations

Inder J. Taneja

This paper brings magic squares of orders 3 to 10 in terms of single letter. In case of orders 8 and 9 there are two possibilities, i.e, one as normal magic square and another as bimagic square. In this situation, the magic squares are written in both the possibilities. In case of order 10, again there are also two possibilities. One without any blocks, and second as blocks of order 8, divided again in four blocks of order 4, known as block-bordered magic squares. Writing in terms of single digit “a”, where a can be 1, 2,…, 9. For the orders 3 to 9, few exercises are also written. Uploaded on March 25, 2021

124. Creative Magic Squares: Single Digit Representations

Inder J. Taneja

This paper brings traditional magic squares of orders 3 to 10 in terms of single digit. In this case, the magic squares are written separately for each digit, i.e., for the digits 1 to 9. This has been done for all the orders 3 to 10. In case of orders 8 and 9 there are two possibilities, i.e, one as normal magic squares and another as bimagic squares. In case of magic square of order 10, two different ways are written. One as a general magic square without any block. Another as block-bordered magic squares with inner magic square of order 8. Again the inner magic square can be written in two ways, i.e., one just pandiagonal and another pandiagonal and bimagic. In case of single digit the representations of numbers are not uniform. Writing in terms of single letter “a”, we can get uniformity in representations of numbers. It is done in another work. Uploaded on March 25, 2021

123. Pandigital-Type and Pythagorean Triples Patterns

Inder J. Taneja

In this paper we worked with generating Pythagorean triples by use of two variables Pythagoras theorem. Based on these triples pythagorean patterns are also calculated by use of same formula of two variables Pythagoras theorem. Again by use of same formula, pandigital-type pythagorean triples are obtained. In other words, three ways study on the same pythagorean triples is done. One generating them. Second, writing patterns, and third to bring pandigital-type patterns. The pandigital-type patterns are written in two different ways, one is like, 12345678987654321, and second is of type 102030405060708090807060504030201. This work is revised form of author’s previous two works. Uploaded on March 17, 2021

122. Patterns in Pythagorean Triples

Inder J. Taneja

The Pythagoras theorem is very famous in the literature of mathematics. It lead us to many Pythagorean triples. Some of these triples allows us to write as patterns. The aim of this work is to write patterns in Pythagorean triples in a sequential way, i.e., starting from 3 to 1000. Patterns in Pythagorean triples are also studied by author in previous works, but in different ways. For details refer to reference list. The work is from 3 to 1000 with lot of gaps, i.e., still we don’t have patterns for them. This will be completed in near future. Also some patterns are not so good to check their further values. These shall be removed in future study. Uploaded on March 13, 2021

121. Block-Wise Universal Bimagic and Semi-Bimagic Squares With Digits 1 and 8

Inder J. Taneja

In many papers, author worked with two digits such as, {1, 8}, {2, 5} and {6, 9}. Bimagic and semi-bimagic squares of order 8, 16, 24, 25 and 32 are studied previous works. This work brings bimagic and semi-bimagic squares of orders 48, 64 and 128 using only the digits 1 and 8. The order 64 is as multiples of blocks of orders 8 and 16, while the orders 48 and 128 are as multiples of orders 16. The previous results such as of orders 8, 16, 24 and 32 are also written again. The magic squares studied are upside-down and reflexive, i.e., universal magic squares. Uploaded on March 11, 2021

120. Odd Order Multiples Universal Magic Squares With 1 and 8

Inder J. Taneja

In previous works, the author worked with magic squares multiples of 4, 6 and 12 for the magic squares of orders 33 to 128. These works are for the two digits of type {1, 8}; {2, 5} and {6, 9}. For the same digits, the work also done for the magic squares of orders 3 to 32. All these works refers to even order multiples special from order 33 onwards. In this work we studied odd order multiples 3 to 19 using only the digits 1 and 8. Each magic square is with 12-digits cell entries. In all the situations the magic squares are universal, i.e., upside-down and mirror looking or reflexive. Uploaded on March 10, 2021

119. Bordered and Block-Wise Bordered Magic Squares: Odd Order Multiples

Inder J. Taneja

In the previous work we did block-wise bordered and block-wise block-bordered magic squares mutiples of even orders. The even order multiples considered are of orders 6, 8, 10, 12, 14, 16, 18 and 22. In this case, we have blocks or block-wised bordered magic squares of equal sums. There is only two cases where we don’t have equal sums is of orders 3 and 5. In case of odd order multiples, still we don’t have equal sum blocks. In this situation, this work is with different block sums of magic squares resulting in a final magic squares. For examples, multiple of order 5, lead us to block-wise bordered magic squares of order 15, 20, 25, etc. with different sub- block sums. The same is with the multiples of orders 5, 7, 9, 11, 13 and 15. Finally, we wrote block-wise bordered and block-wise block-bordered magic squares of orders 15, 20, 21, 25, 27, 28, 30, 33, 35, 36, 39, 40, 42, 44 and 45. Uploaded on February 10, 2021

118. Bordered and Block-Wise Bordered Magic Squares: Even Order Multiples

Inder J. Taneja

We know that we can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. In the previous works, the author combine the both, i.e., bordered and block-wise magic squares, calling block-bordred magic squares. We can always write either block-wise or block-bordered magic squares. This work brings block-wise bordered and block-bordered magic square. This means, the bordered and block-bordered magic squares are written as sub-blocks of equal or different sums. The magic squares studied are of orders 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42 and 44. These are written as blocks of orders 6, 8, 10, 12, 14, 16, 18 and 22. Uploaded on February 10, 2021

117. Fixed Digits Repetitions Prime Patterns for 5-Digits Prime Numbers

Inder J. Taneja

The author worked with patterns in prime numbers in different situations, i.e., in terms of lengths, such as 10, 9, 8, 7 and 6. The prime patterns are understood as fixed digits repetitions along with prime number resulting again in a new prime number. These types of patterns are of fixed length. In this work, the prime patterns are written for the 5-digits prime numbers, i.e., from 10007 to 9991. In this case there are total 8363 prime numbers. Due to high quantity of prime numbers, the results are written only in length 6 with maximum up to 7 digits repetitions. The work for the prime numbers for the 3 and 4 digits are done by author in previous works. Moreover, the fixed repetition numbers are always multiples of 3. Uploaded on January 17, 2021

116. Block-Wise and Block-Bordered Magic and Bimagic Squares of Orders 10 to 47

Inder J. Taneja

We can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to combine the both, i.e., bordered and block-wise magic squares. Whenever possible, we shall write block-wise magic squares, otherwise block-bordered magic squares. This work is combination of authors previous work in both the direction. The work brings magic squares of orders 10 to 47, either block-wise or block-bordered. This work is a combinations of author’s many previous papers. Future plans are to go till magic squares of order 108. Uploaded on January 14, 2021

Work in 2020

115. 21 Mathematical Highlights for 2021

Inder J. Taneja

This short work brings 21 main representations of 2021 in different ways. These representations are of crazy-type, running numbers, single digit, single letter, Triangular, Fibonacci, palindromic-type, prime numbers, embedded, repeated digits, colored patterns, magic squares, etc. Uploaded on December 26, 2020

114. Block-Wise and Block-Bordered Magic and Bimagic Squares With Magic Sums 21, 21^2 and 2021

Inder J. Taneja

We know that we can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to bring both bordered and block-wise magic squares. These block-wise and block-bordered magic squares are of orders 8 to 26. All these magic squares are with magic sums as 21, 21^2 and 2021. In some cases, such as magic squares of orders 8,9, 16 and 25 are also bimagic squares. In case of order 24 with the blocks of orders 3 and 8 the magics squares also turns semi-bimagic squares. This work is based on authors three paper (Paper-I, Paper-II, Paper-III). Uploaded on December 20, 2020

113. Factorial-Type Numerical Calender 2021

Inder J. Taneja

This short work brings factorial-type numerical representations of the calender. It is based on a functional equation of three variables representing day, month and year. The results are based on 5! and 6!. First half of the year (January to June) is with 5!, and the second half of the year (July to December) is with 6!. It is given for the years ending in 21 and 2021. The the dates of other years can be calculated on similar lines. Uploaded on December 16, 2020

112. Fractional and Decimal Type Bordered Magic Squares With Magic Sum 2021

Inder J. Taneja

The idea of bordered magic squares is well known in the literature. In this work, bordered magic squares are constructed in such a way that the final magic sum of each bordered magic square is 2021. The work is for the orders 3 to 26. The work include fractional and decimal numbers entries having positive and/or negative signs. In some cases, the sum-magic sums lead us to Pythagorean triples. It happens with the even order magic squares starting from order 10, such as, orders 10, 12, …, 24 and 26. Uploaded on December 16, 2020

111. 3 and 5-Digits Multiple Choice Embedded Palprimes

Inder J. Taneja

The embedded palindromic prime (palprime) numbers are generally represented in the form of pyramid or tree. These types of embedded palprimes are very famous in the literature, where previous palprime is in the middle of next, and so on. In these situations there is no limit where it ends, because always we find next palprime containing previous one. In this work, we brought embedding procedure for the palprimes of 3 and 5 digits. There are total 15 palprimes of 3-digits, and 93 palprimes of 5-digits. The results obtained are of multiple choices. In case of 3-digits, the palprimes are written in 6 lines, and in case of 5-digits, the palprimes are written in 5 lines. There are 17 palprimes of 5-digits those contains some of the 3-digits palprimes. The results for the 7-digits palprimes shall be given in another work. Uploaded on December 5, 2020

110. 4-Digits Prime Numbers in Fixed Digits Repetition Prime Patterns

Inder J. Taneja

In past years, the author worked with patterns in prime numbers in different situations. It means, in terms of lengths, such as 10, 9, 8, 7 and 6. The prime patterns are understood as fixed digits repetitions along with prime number resulting again in a new prime number. These types of patterns are of fixed length. In this work, the prime patterns are written for the 4-digits prime numbers, i.e., . i.e., from 1009 to 9973. In this case there are total 1061 prime numbers. Most of the results are obtained for the prime patterns of length 7 up to 8 digits repetition of fixed digits. There are very few examples of length 8 and length 9. The work for the prime numbers up to 3-digits is recently done by author. The continuation of the same work for 5-digits prime numbers are under study. Moreover, the fixed repetition numbers are always multiples of 3. Uploaded on November 29, 2020

109. Prime Numbers in Fixed Digits Repetitions Prime Patterns

Inder J. Taneja

In preivous years, the author worked with patterns in prime numbers in different situations. It means, in terms of lengths, such as, 10, 9, 8, 7 and 6. The prime patterns are understood as fixed digits repetitions along with prime number resulting again in a prime number. These types of patterns are of fixed length. In this work, the prime patterns are written for prime numbers up to 3 digits, i.e., from 2 to 977, except 3. For prime number 3 the results are little different, and are given in another work. The continuation of the same work for four digits primes are under study. Uploaded on November 10, 2020

108. Upside-Down Magic Squares of Orders 128, 126 and 120 With Digits 6 and 9

Inder J. Taneja

his work brings magic squares of type 4k, 6k and 12k using the digits 6 and 9. Each magic square contains 14-digits in each cell. Just with two digits, one can have exactly 16384 different possibilities of 14-digits combinations. In case of 4k, we have written magic squares of orders, 4, 8,…., 60, 64,…, 124, 128. In case 6k, we have written magic squares of orders 6, 12, …, 54, 60, 120, 126. In case of 12k, we have written magic squares of orders 12, 24, 36, 48 , 60, 72, 84, 96, 108 and 120. In each case, all the blocks of magic squares of orders 4, 6 and 12 are with equal magic sums. The same work for the digits 1 and 8 is done in Taneja (click here), and for the digits 2 and 5 see Taneja (click here). The work on multiples of 3, 5, 7 etc. shall be done elsewhere. Later, the idea is to extend the same work up to magic square of order 256 having 16-digits in each cell just with two digits: {1,8}, {2,5} and {6,9\}. In the first two cases, the work is upside-down and mirror looking, while in the third case the work is only upside-down. When the magic squares are upside-down and mirror looking, we call them as universal magic squares. In this work, all the magic squares are only upside-down. The whole work is without use of any programming language. It is just based on the number’s combinations. Uploaded on October 31, 2020

107. Universal Magic Squares of Orders 128, 126 and 120 With Digits 2 and 5

Inder J. Taneja

This work brings magic squares of type 4k, 6k and 12k using the digits 2 and 5 written in digital form. Each magic square contains 14-digits in each cell. Just with two digits, one can have exactly 16384 different possibilities of 14-digits combinations from two digits. In case of 4k, we have written magic squares of orders, 4, 8,…., 60, 64,…, 124, 128. In case 6k, we have written magic squares of orders 6, 12, …, 54, 60, 120, 126. In case of 12k, we have written magic squares of orders 12, 24, 36, 48 , 60, 72, 84, 96, 108 and 120. In each case, all the blocks of magic squares of orders 4, 6 and 12 are with equal magic sums. The same work for the digits 1 and 8 is done in Taneja (click here), and for the digits 6 and 9 is under prepartion. The work on multiples of 3, 5, 7 etc. shall be done elsewhere. Later, the idea is to extend the same work up to magic square of order 256 having 16-digits in each cell just with two digits: {1,8}, {2,5} and {6,9\}. In the first two cases, the work is upside-down and mirror looking, while in the third case the work is only upside-down. When the magic squares are upside-down and mirror looking, we call them as universal magic squares. The whole work is without use of any programming language. It is just based on the number’s combinations. Uploaded on October 31, 2020

106. Universal Magic Squares of Orders 128, 126 and 120 With Digits 1 and 8

Inder J. Taneja

This work brings magic squares ot type 4k, 6k and 12k using the digits 1 and 8. Each magic square contains 14-digits in each cell. Just with two digits, one can have exactly 16384 different possibilities of 14-digits combinations from two digits. In case of 4k, we have written magic squares of orders, 4, 8,…., 60, 64,…, 124, 128. In case 6k, we have written magic squares of orders 6, 12, …, 54, 60, 120, 126. In case of 12k, we have written magic squares of orders 6, 12, 24, 36, 48 , 60, 72, 84, 96, 108 and 120. In each case, all the blocks of magic squares of orders 4, 6 and 12 are with equal magic sums. The same work for the digits 2 and 5, and 6 and 9 is done in another works. The work on multiples of 3, 5, 7 etc. shall be done elsewhere. Later, the idea is to extend the same work up to magic square of order 256 having 16-digits in each cell just with two digits: {1,8}, {2,5} and {6,9}. In the first two cases, the work is always upside-down and mirror looking, while in the third case the work is only upside-down. When the magic squares are upside-down and mirror looking, we call them as universal magic squares. The whole work is without use of any programming language. It is just based on the number’s combinations. Uploaded on October 26, 2020

105. Geometrical, Numerical, and Symmetrical Representations for the Days of 2020

Inder J. Taneja

In this work, the days of year 2020 are represented in two different ways. One way is numerical, and another way is geometrical. In numerical representations, four different forms are considered. The first one is of crazy type, and the second one is of power type. In these two cases the year ends in 20. The third one is representations in terms of single letter “a”. This is done only for the months of April and August. The forth numerical representation is of factorial type. In this case, four different ways are considered. One ending in 20, and other three ending in 2020. For these representations, we used the idea of 5!, 6! and 8!. The geometrical representation is new, and is not done so far. In this case the representations are for the year ending in 2020. This we have done for each day separately. The first 364 days are organized within a square of 9×9. The last two days of year, i.e., 30.12.2020 and 31.12.2020 are organized in a square of 11×11. All the geometrical representations are organized in symmetrical way. In this situation five types of symmetries are defined, such as, color design symmetry, design symmetry, half-design symmetry, half-color symmetry and half-color half-design symmetry. All these symmetries are based on well known reflection symmetry. By no means, we can say that these representations are unique. There are much more possibilities. In future, we shall improve them by putting week’s days together. Uploaded on October 4, 2020

104. Block-Bordered Magic Squares of Prime and Double Prime Orders – III

Inder J. Taneja

We know that we can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to combine the both, i.e., bordered and block-wise magic squares, for the magic squares of prime and double prime orders. We call it as block-bordered magic squares. The magic squares considered in this work are of orders orders 41, 43, 46, 47 and 51. In order to bring these block-bordered magic squares, we make use of author’s previous works (work1, work2) on block-wise constructions of magics squares, such as, of orders, 39, 40, 42, 44, 45, 49 and 51. This is the third part of the work. The first and second parts (part1, part2) works with orders, 10, 11, 13, 14, 17, 19, 22, 26, 29, 31, 34, 37 and 38. The forth part of the work shall be on magic squares of orders 58, 59 and 61. Uploaded on September 2, 2020

103. Block-Bordered Magic Squares of Prime and Double Prime Numbers – II

Inder J. Taneja

We know that we can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to combine bordered and block-wise magic squares, for the magic squares of prime and double prime orders. We call it as block-bordered magic squares. The magic squares considered in this work are of orders 34, 37 and 38. In order to bring these block-bordered magic squares, we make use of author’s previous works on block-wise constructions of magics squares, such as, of orders, 28, 30, 32, 35 and 36. The first part of this work brings block-bordered magic squares of orders 11, 13, 14, 17, 19, 22, 26, 29 and 31. The third part is for the magic squares of orders 41, 43, 46 and 47. Uploaded on August 18, 2020

102. Block-Bordered Magic Squares of Prime and Double Prime Numbers – I

Inder J. Taneja

We can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to combine bordered and \block-wise magic squares, for the magic squares of \textbf{prime} and \textbf{double prime numbers} orders. We call it as block-bordered magic squares. The magic squares considered in this work are of orders 10, 11, 13, 14, 17, 19, 22, 23, 26, 29 and 31. In order to bring these block-bordered magic squares, we make use of author’s previous works on block-wise constructions of magics squares, such as, of orders, 8, 9, 12, 16, 18, 20, 21, 24, 25 and 27. The second part of this work is for the block-bordered magic squares of orders 34, 37 and 38 \cite{ta41}. The third part is for the magic squares of orders 41, 43, 46 and 47. Uploaded on August 18, 2020

101. Same Digits Embedded Palprimes of Lengths 3, 5 and 7

Inder J. Taneja

The idea of embedding palindromic prime (palprime) numbers in the form of pyramid or tree is very famous in the literature, where the previous palprime is in the middle of next, and so on. In these situations there is no limit where it ends, because always we find a next palprime containing previous one. In this work, we brought palprimes of lengths 3, 5 and 7 in such a way that pyramids are with same digits as of palprime. In this case, we have total 15 for length 3, 93 for length 5 and 668 for length 7 palprimes. Uploaded on August 8, 2020

100. Patterned Single Digits Representations of Natural Numbers

Inder J. Taneja

This work brings patterned representations of natural numbers from 1 to 1000 in terms of single digits from 1 to 9. To bring these results, only basic operations, such as addition, subtraction, multiplication and division are used. This work is an extension of author’s previous works in patterned form. Patterns representations are some what similar to symmetrical extensions. Uploaded on July 4, 2020

99. Patterned Single Letter Representations of Natural Numbers

Inder J. Taneja

This work brings patterned representations of natural numbers from 1 to 1000 in terms of single letter a. For any value of letter a from 1 to 9, the results are always same. To bring these results, only basic operations, such as addition, subtraction, multiplication and division are used. This work is extension of author’s previous work done in 2019, but is in patterned form. Patterns representations are some what similar to symmetrical extensions. Uploaded on July 2, 2020

98. Upside-Down Magic Squares of Type 4k, 6k and 12k Using the Digits 6 and 9

Inder J. Taneja

This work brings magic squares of multiples of 4k, 6k and 12k using the digits 6 and 9. Each magic square contains 12-digits in each cell. Just with two digits, one can have exactly 1024 different possibilities of 12-digits combinations from two digits. In case of 4k, we have written magic squares of orders: 4, 8,…., 60 and 64. In case 6k, we have written magic square of orders: 6, 12, …, 54 and 60. In case of 12k, we have written magic square of orders: 6, 12, 24, 36, 48 and 60. All the blocks of magic squares of orders 4, 6 and 12 are with equal sums in each case. Similar works for the digits 1 and 8, and the digits 2 and 5 are done in another papers. The work on multiples of 3, 5, 7, etc. shall be done elsewhere. Later, the work shall be extended up to magic squares of order 128 having 14-digits in each cell just with two digits: {1,8}, {2,5} and {6,9}. In the first two cases the work is always upside-down and mirror looking, while in the third case for the digits 6 and 9, the work is only upside-down. When the magic squares are upside-down and mirror looking, we call them as universal magic squares. In this paper, all the results are upside-down. The whole work is without use of any programming language. It is just based on the number’s combinations. Uploaded on June 28, 2020

97. Universal Magic Squares of Type 4k, 6k and 12k Using the Digits 1 and 8

Inder J. Taneja

This work brings magic squares of multiples of 4k, 6k and 12k using the digits 1 and 8. Each magic square contains 12-digits in each cell. Just with two digits, one can have exactly 1024 different possibilities of 12-digits combinations from two digits. In case of 4k, we have written magic squares of orders: 4, 8,…., 60 and 64. In case 6k, we have written magic square of orders: 6, 12, …, 54 and 60. In case of 12k, we have written magic square of orders: 6, 12, 24, 36, 48 and 60. All the blocks of magic squares of orders 4, 6 and 12 are with equal sums in each case. Similar works for the digits 2 and 5, and the digits 6 and 9 are done in another papers. The work on multiples of 3, 5, 7, etc. shall be done elsewhere. Later, the work shall be extended up to magic squares of order 128 having 14-digits in each cell just with two digits: {1,8}, {2,5} and {6,9}. In the first two cases the work is always upside-down and mirror looking, while in the third case for the digits 6 and 9 the work is only upside-down. When the magic squares are upside-down and mirror looking, we call them as universal magic squares. The whole work is without use of any programming language. It is just based on the number’s combinations. Uploaded on June 28, 2020

96. Universal Magic Squares of Type 4k, 6k and 12k Using the Digits 2 and 5

Inder J. Taneja

This work brings magic squares of multiples of 4k, 6k and 12k using the digits 2 and 5. Each magic square contains 12-digits in each cell. Just with two digits, one can have exactly 1024 different possibilities of 12-digits combinations from two digits. In case of 4k, we have written magic squares of orders: 4, 8,…., 60 and 64. In case 6k, we have written magic square of orders: 6, 12, …, 54 and 60. In case of 12k, we have written magic square of orders: 6, 12, 24, 36, 48 and 60. All the blocks of magic squares of orders 4, 6 and 12 are with equal sums in each case. Similar works for the digits 1 and 8, and the digits 6 and 9 are done in another papers. The work on multiples of 3, 5, 7, etc. shall be done elsewhere. Later, the work shall be extended up to magic squares of order 128 having 14-digits in each cell just with two digits: {1,8}, {2,5} and {6,9}. In the first two cases the work is always upside-down and mirror looking, while in the third case for the digits 6 and 9, the work is only upside-down. When the magic squares are upside-down and mirror looking, we call them as universal magic squares. In order to have universal magic squares, the digits 2 and 5 are written in digital form. The whole work is without use of any programming language. It is just based on the number’s combinations. Uploaded on June 28, 2020

95. Upside-Down Magic and Bimagic Squares of Orders 17 to 32 With Digits 6 and 9

Inder J. Taneja

By universal we understand that the magic squares are upside-down and mirror looking independent of magic square sums. This work brings upside-down magic and bimagic squares of order 17 to 32 using only two digits 6 and 9. In case of upside-down, the number 6 becomes 9, and 9 as 6. For the digits {1, 8} and 2, 5}, the magic squares are universal, and given in by author (work1, work2). The same work can easily be extended for the digits {0,1} and {0,8} Similar kind of work for the magic and bimagic squares of orders 3-16 can be seen in author’s another work. In some cases, the block-wise constructions are also given. These are for the magic squares of orders, 18, 20, 21, 24, 25, 27, 28, 30 and 32. Moreover, in case of order 25, we have bimagic square, while for the case of order 24, we have semi-bimagic square. The whole work is without use of any programming language. It is just based on the number’s combinations. Uploaded on May 30, 2020

94. Universal Magic and Bimagic Squares of Orders 17 to 32 With Digits 2 and 5

Inder J. Taneja

This work brings universal magic squares of order 17 to 32. By universal we understand that the magic squares are upside-down and mirror looking independent of magic square sums. The work for the digits as 2 and 5. For the digits 1,8 and 6,9 is given in another papers. The same work can be extended for the digits 0,1 and 0,8. Similar kind of work for orders 3-16 is already done before. The block-wise constructions for the magic squares of orders 18, 20, 21, 24, 25, 27, 28, 30 and 32 are also included. Moreover, in case of orders 25 and 32, we have bimagic squares, while for the case of order 24, it is semi-bimagic square. The whole work is without use of any programming language. It is just based on the number’s combinations. Uploaded on May 30, 2020

93. Universal Magic and Bimagic Squares of Orders 17 to 32 With Digits 1 and 8

Inder J. Taneja

This work brings universal magic squares of order 17 to 32. By universal we understand that the magic squares are upside-down and mirror looking independent of magic square sums. The work for the digits 1 and 8. For the digits 2,5 and 6,9 is given in another papers. The same work can be extended for the digits 0,1 and 0,8. Similar kind of work for orders 3-16 is already done before. The block-wise constructions for the magic squares of orders 18, 20, 21, 24, 25, 27, 28, 30 and 32 are also included. Moreover, in case of orders 25 and 32, we have bimagic squares, while for the case of order 24, it is semi-bimagic square. The whole work is without use of any programming language. It is just based on the number’s combinations. Uploaded on May 30, 2020

92. Fixed and Flexible Powers Narcissistic Numbers with Division

Inder J. Taneja

This paper brings extensions of narcissistic numbers with division. These extensions are done in different situations, such as, with positive and negative coefficients, fixed and flexible powers. Comparison with previous known numbers are also given. Uploaded on May 11, 2020

91. 2-Digits Universal and Upside-Down Palindromic Magic and Bimagic Squares: Orders 3 to 16

Inder J. Taneja

This work brings universal and upside-down magic squares of order 3 to 16. The work is for two digits as {1,8}, {2,5} and 6,9. It can easily to extended for the digits {0,1} and {0,8}. In case of orders 8, 9 and 16 the bimagic squares are also constructed. The block-wise constructions for the orders 8, 9, 12, 15 and 16 are also done. In case of order 15, two ways are given. One as blocks of pandiagonal magic squares of order 5 and another as semi-magic squares of order 3. In both the cases, the magic square of order 15 is semi-magic. The extension of this work to magic squares of orders 17 to 32 is given in another paper. The whole work is without use of any programming language. It is just based on the combination of numbers. Uploaded on April 7, 2020

90. Factorial-Type Numerical Calendar

Inder J. Taneja

This short work brings factorial-type numerical representations of the Calendar. It is based on a functional equation of three variables representing “day”, “month” and “year”. It is given for the years ending in 20 and 2020. The other years can be calculated on similar bases. Uploaded on March 24, 2020

89. Pyramid-Type Representations of Natural Numbers

Inder J. Taneja

This work brings natural numbers from 0 to 1000 written in two different forms. One in terms of power and bases, and another in decreasing order of numbers ending in 0. These representations are of pyramid-type. This work is combined version of author’s two previous works done in 2016. Uploaded on February 5, 2020

88. Universal Palindromic Day and Two Digits Magic Squares

Inder J. Taneja

This work brings magic squares of orders 3 to 16 just for two digits 0 and 2 having the numbers 02022020, 20022020 and 22022020 representing respectively the dates 02.02.2020, 20.02.2020 and 22.02.2020. Also the the palindromic number 02022020, writing in reverse side becomes as 20200202, Writing in American style the last two dates are written as 02.20.2020 and 02.22.2020. Thus, there are total six numbers, 02022020, 20022020, 22022020, 20200202, 02.20.2020 and 02.22.2020. From 8th order onward, the magic square constructed brings these six numbers. For orders less than 8, only some of these six numbers appears in the magic squares. Most of the magic squares are upside down and mirror looking. Uploaded on February 2, 2020

87. Bordered Magic Squares With Order Square Magic Sums

Inder J. Taneja

The idea of nested magic squares is well known in the literature, generally known by bordered magic squares. In this work, nested magic squares are studied in such a way that the magic sums are equal to the order of the magic square. The study include integer values. In some cases decimal entries with positive and negative entries are also used. The magic sums of sub-magic squares lead us to a general formula. Uploaded on January 20, 2020

86. Fractional and Decimal Type Bordered Magic Squares With Magic Sum 2020

Inder J. Taneja

The idea of bordered magic squares is well known in the literature. In this work, the bordered magic squares are constructed in such a way that the final magic sum of each bordered magic square is 2020. The work is for the orders 3 to 25. In each case, a symmetric result is found for the magic sums of sub-magic squares. The work include fractional, decimal and whole numbers with positive and negative signs. Uploaded on January 20, 2020

85. Same Digits Equality Expressions: Power and Plus

Inder J. Taneja

This paper brings numbers in such a way that both sides of the expressions are with the same digits. One side is numbers with powers, while other side just with numbers having same digits, such as, a^b+c^d+… = ab+cd+…, etc. The the expressions studied are with positive coefficients. The work is the increasing order of numbers maximum up to 3 digits numbers, i.e., up to 999. Between 10 to 99 there are many numbers are not available. Between 100 to 999, only 104 is not available. This we have written in terms of positive negative coefficients. The rest work is in positive coefficients. Uploaded on January 4, 2020

84. 2020 In Numbers: Mathematical Style – Revised

Inder J. Taneja

This short work brings representations of 2020 in different situations. These representations are of crazy-type, running numbers, single digit, single letter, Triangular, Fibonacci, palindromic-type, prime numbers, embedded, repeated digits, magic squares, etc..Uploaded on January 1, 2020

Work in 2019

83. Different Digits Equivalent Fractions: Three Digits Numerator

Inder J. Taneja

This work brings equivalent fractions without repetition of digits. The work is for three digits numerators. Uploaded on November 19, 2019

82. Different Digits Equivalent Fractions: Two Digits Numerator

Inder J. Taneja

This work brings equivalent fractions without repetition of digits. The work is for two digits numerators. Sor single digits numerator see the link. For the higher digits numerators the work in given in another papers. Uploaded on November 15, 2019

81. Different Digits Equivalent Fraction: Single Digit Numerator

Inder J. Taneja

This work brings equivalent fractions without repetition of digits. This work is for single digit numerators. For the two and higher digits numerators the work in given in another papers. Uploaded on November 15, 2019

80. Patterned Selfie Fractions

Inder J. Taneja

By selfie fractions, we understand that a fraction, where numerator and denominators are represented by same digits, with basic operation. This paper brings patterned selfie fractions. Patterned selfie fractions are understand as selfie fractions extendable in symmetric way. This work is with repeated digits.Uploaded on October 27, 2019

79. Fibonacci Sequence Type Selfie Numbers with Factorial: Reverse Order of Digits

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci sequence, Triangular numbers, etc. These operations are applied for single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factorial and square-root. This paper extends authors previous work for Fibonacci sequence type selfie numbers with factorial in reverse order of digits. The work is up to 5-digits numbers. Uploaded on October 13, 2019

78. Fibonacci Sequence Type Selfie Numbers with Factorial: Digit’s Order

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci sequence, Triangular numbers, etc. These operations are applied for single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factorial and square-root. This paper extends authors previous work for Fibonacci sequence type selfie numbers with factorial. The work is up to 5-digits numbers. Uploaded on October 13, 2019

77. Fibonacci Sequence Type Selfie Numbers With Square-Root

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci sequence, Triangular numbers, etc. These operations are applied for single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factorial and square-root. This paper extends authors previous work for Fibonacci sequence type selfie numbers with square-root. The work is up to 5-digits numbers. Uploaded on October 10, 2019

76. Different Digits Selfie Fractions: Five Digits Numerator – Pandigital

Inder J. Taneja

The addable fractions are proper fractions where addition can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for other operations, such as, addition, subtraction, multiplication, potentiation, etc. This work brings selfie fractions with single and/or multiple representations in different digits with all operations. The numerator values are with five digits numbers giving pandigital fractions. The results are in increasing order of numerator values. Also, numerator is always less than denominator. Uploaded on October 6, 2019

75. Different Digits Selfie Fractions: Four Digits Numerator

Inder J. Taneja

The addable fractions are proper fractions where addition can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for other operations, such as, addition, subtraction, multiplication, potentiation, etc. This work brings selfie fractions with single and/or multiple representations in different digits with all operations. The numerator values are with four digits numbers. The denominator values considered are maximum up to 6-digits. The results are in increasing order of numerator values. Also, numerator is always less than denominator. Uploaded on October 6, 2019

74. Different Digits Selfie Fractions: Two and Three Digits Numerators

Inder J. Taneja

The addable fractions are proper fractions where addition can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for other operations, such as, addition, subtraction, multiplication, potentiation, etc. This work brings selfie fractions with single and/or multiple representations in different digits with all operations. The numerator values are with two and three digits numbers. The denominator values considered are maximum up to 6-digits. The results are in increasing order of numerator values. Also, numerator is always less than denominator. Uploaded on October 6, 2019

73. Repeated Digits Selfie Fractions: Two and Three Digits Numerators

Inder J. Taneja

The addable fractions are proper fractions where addition can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for other operations, such as, addition, subtraction, multiplication, potentiation, etc. This work brings selfie fractions with single and/or multiple representations having repetition of digits using basic operations without subtraction. The numerator values are with two and three digits numbers. The denominator values considered are maximum up to 5-digits. The results are in increasing order of numerator values. Uploaded on September 12, 2019

72. Square-Root Type Selfie Numbers

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basic operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, quadratic numbers, cubic numbers, etc. This paper brings selfie numbers with square-root up to seven digits. Uploaded on July 26, 2019

71. Prime Numbers in Prime Numbers Up To 5 Digits

Inder J. Taneja

This work brings prime numbers formed digits appearing in prime number. We have considered all the prime numbers from single digits to 5 digits. For higher digits refer to further works. This type of work is known in the literature but still is not aware of references in this direction. These shall be filled as long as we know them. Uploaded on July 16, 2019

70. Power-Type Semi-Selfie Numbers and Patterns

Inder J. Taneja

Author studied many ways of writing selfie numbers. There are numbers very much near to selfie-numbers, but are not selfie numbers. These types of numbers, referred as semi-selfie numbers, where numbers are written in terms of expressions with positive and negative signs having same digits on both sides of the expressions, except the power values. This paper brings semi-selfie numbers with power-type representations. The idea is extended to some patterns also. The general study of semi-selfie numbers with patterns can be seen in author’s work. Uploaded on July 16, 2019

69. Selfie Numbers With Binomial Coefficients, Triangular Numbers and Factorial

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. These are by use of single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factorial and square-root. This paper brings binomial coefficient triangular type selfie numbers, i.e., binomial coefficients and triangular numbers together. This is done by use of only basic operations and factorial. The work is in digit’s order and in reverse order of digits. The work is up to 5-digits numbers. Uploaded on July 9, 2019

68. General Sum Symmetric and Positive Entries Nested Magic Squares

Inder J. Taneja

The idea of nested magic squares are well known in the literature, generally known by bordered magic squares. In this paper, we worked with general sum for the nested magic squares. Considering the idea of magic sum 0, the nested magic squares are studied in different situations. The work is for the orders from 20 to 3. Symmetric positive nested magic squares are also obtained. Uploaded on July 4, 2019

67. Symmetric Properties of Nested Magic Squares

Inder J. Taneja

The idea of nested magic squares is well known in the literature, generally known by bordered magic squares. In this work, nested magic squares are studied for the consecutive natural numbers for the orders 5 to 25. Properties like, sub-magic squares sums, total entries sums, borders entries sums, etc. are studies. Final results lead us to symmetric properties. The nested magic squares for the consecutive odd numbers entries refer author’s another work.Uploaded on June 29, 2019

66. Nested Magic Squares With Perfect Square Sums, Pythagorean Triples, and Borders Differences

Inder J. Taneja

The idea of nested magic squares is well known in the literature, generally known by bordered magic squares. Instead, considering entries as consecutive numbers, we considered consecutive odd numbers entries. This gives us perfect square sum magic squares. We worked with magic squares of orders 3 to 25. Finally, the result can be written in a symmetric way, i.e., T(m x k):= m^2 x k^2, where m is the order of nested magic square, k is the order of sub-magic squares in each case, and T is the sum of total entries. Also the entries sums are connected with Pythagorean triples and borders. In each case, the difference among consecutive border sums is always a fixed value. Uploaded on June 14, 2019

65. Block-Wise Constructions of Magic and Bimagic Squares of Orders 8 to 108

Inder J. Taneja

This paper brings idea how we can organize block-wise constructions of magic and bimagic squares. This we have done for the magic squares of orders 8 to 108. By use of the formula for the calculations of magic squares, we can check the existence of blocks of equal or unequal sums. In case of blocks of order 3, either we have equal sums of semi-magic blocks or unequal blocks of magic squares of order 3. While, in case of blocks of order 4, 5 and 7, the magic squares can be written as pandiagonal. For the case of blocks of order 6, the blocks are of equal magic sums, but the magic squares are not pandiagonal. Uploaded on May 15, 2019

64. Selfie Numbers With Binomial Coefficients, Triangular Numbers and Square-Root

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. These are by use of single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factorial and square-root. This paper brings binomial coefficient triangular type selfie numbers, i.e., binomial coefficients and triangular numbers together. This is done by use of only basic operations and square-root. The work is in digit’s order and in reverse order of digits. The work is up to 5-digits numbers. Uploaded on May 10, 20191

63. Perfect Square Sum Magic Squares

Inder J. Taneja

This paper shows how to create magic squares with a perfect square number for the total sum of their entries. This has been done in two ways: Firstly, by using the sum of consecutive odd numbers, and secondly, by using consecutive natural numbers. In the first case, for all orders of magic squares, one can always obtain a perfect square sum. In the second case, magic squares with perfect square magic sums do exist, but only for odd order magic squares. For the even order magic squares, such as 4, 6, 8, etc. it is not possible to write consecutive natural number magic squares with perfect square sums of their entries. A simplified idea is introduced to check when it is possible to obtain minimum perfect square sums. Also, a uniform method is presented so that, if k is the order of a magic square, then the magic sum of the square is k^3, and the sum of all entries of the magic square is k^4. Examples are given for the magic squares of orders 3 to 25. Uploaded on April 29, 2019

62. Fibonacci Sequence Type Selfie Numbers: Basic Operations

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci sequence, Triangular numbers, etc. These operations are applied for single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factorial and square-root. This paper extends authors previous work for Fibonacci sequence type selfie numbers in basic operations. For the operations, such as, factorial and square-root, the work shall be given elsewhere. The work is in digit’s order and in reverse order of digits, and is up to 5-digits numbers. This extends considerably, author’s previous work. Uploaded on April 28, 2019

61. Binomial Coefficients Triangular Type Selfie Numbers: Basic Operations

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. These are by use of single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factorial and square-root. This paper brings binomial coefficient triangular type selfie numbers, i.e., binomial coefficients and triangular numbers together. This is done by use of only basic operations. The work is in digit’s order and in reverse order of digits. The work is up to 5-digits numbers. Uploaded on April 25, 2019

60. Triangular-Type Selfie Numbers: Reverse Order of Digits

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, S-gonal values, centered polygonal numbers, quadratic numbers, cubic numbers, etc. This paper brings selfie numbers with triangular numbers in reverse order of digits. The previous work of author is up to 4-digits numbers. This work extends to 5-digits numbers. The work on digit’s order in 5-digits numbers can be seen in [32]. Uploaded on April 14, 2019

59. Triangular-Type Selfie Numbers: Digit’s Order

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, S-gonal values, centered polygonal numbers, quadratic numbers, cubic numbers, etc. This paper brings selfie numbers with triangular numbers in digit’s order. The author’s previous work is up to 4-digits numbers. This work extends to 5-digits numbers. The work on reverse order is given in a separate paper. Uploaded on April 11, 2019

58. Pandigital Equivalent Selfie Fractions

Inder J. Taneja

The addable fractions are proper fraction where addition can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for other operations, such as, addition, multiplication, potentiation, etc. For more details refer author’s work [10]. This work bring mixed selfie fractions with all operations using 10 digits, known by pandigital selfie fractions. This work brings maximum up to 91 equivalent fractions. The fractions are without repetition of digits. Uploaded on April 2, 2019

57. Selfie Numbers With Binomial Coefficients and Fibonacci Numbers

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. These are by use of single variable. In two variables, work is with binomial coefficients type selfie numbers. This paper brings combine results by use of binomial coefficient type selfie numbers and Fibonacci sequence values. This is done by use of basic operations, factorial, and square-root. The work is in digit’s order and in reverse order of digits. Uploaded on March 30, 2019

56. Selfie Numbers: Basic Operations

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, quadratic numbers, cubic numbers, etc. This paper brings selfie numbers with basic opeartions up to 7 digit’s numbers. The previous work up to 6 digits numbers. Uploaded on March 26, 2019

55. Different Digits Equivalent Fractions – II

Inder J. Taneja

This work brings equivalent fractions for 10-digits without repetition of digits. Sometimes known as pandigital. For the 3 to 9 digits see first part of this work. For repetition of digits, the number of equivalent fractions increases too much. These shall be dealt elsewhere. Uploaded on March 24, 2019

54. Different Digits Equivalent Fractions – I

Inder J. Taneja

This work brings equivalent fractions from 3 to 9 digits without repetition of digits. For repetition of digits the the number of equivalent fractions increases too much. The 10-digits equivalent fractions are in the second part.Uploaded on March 24, 2019

53. Selfie Fractions: Addable, Subtractable, Dottable and Potentiable

Inder J. Taneja

The numerator and denominator of a fraction represented by same digits with certain operations, we call as seflie fractions. These operations can be done by use of operations as addition, subtraction, multiplication, potentiation, etc. For example, in case of addition, let’s call it as addable selfie fractions, in case of multiplications, let’s call it as dottable selfie fractions, etc. The same is true for other operations, such as, substraction, potentiation, etc. These operations can be used individually or together. This work brings single representations, and multiple representations of selfie fractions having these operations. The results are from 4 to 10 digits without repetitions. There are much more examples with pandigital selfie equivalent fractions. It is given in another work.Uploaded on March 24, 2019

52. Selfie Numbers With Binomial Coefficients

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. This paper works with binomial coefficient type selfie numbers. This work is reorganized version of author’s previous work done in 2018 reference [17]. Uploaded on March 17, 2019

51. Cubic-Type Selfie Numbers

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. In two variables, the selfie numbers are obtained by use of binomial coefficients, S-gonal numbers, centered polygonal numbers, etc. This paper brings selfie numbers in terms of cubic numbers. Some cubic-type patterned selfie numbers are also studied. The work is up to 5-digits numbers in digit’s order and reverse order of digits. Uploaded on March 12, 2019

50. Concatenation-Type Selfie Numbers With Factorial and Square-Root

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, etc. In this work, we have written selfie numbers by use of concatenation, along with factorial and square-root. The concatenation idea is used in a very simple way. The work is limited to 5-digits numbers. Work on higher digits shall be dealt elsewhere. Uploaded on March 8, 2019

49. Factorial-Type Selfie Numbers in Reverse Order of Digits

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, etc. In this work, we have re-written selfie numbers in reverse order of digits using, factorial. The work is up to 6 digits numbers. The digit’s order selfie numbers with factorial can be seen in author’s another work. Uploaded on March 6, 2019

48. Factorial-Type Selfie Numbers in Digit’s Order

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, etc. In this work, we have re-written selfie numbers in digit’s order using factorial. The work is up to 6 digits numbers. The reverse order selfie numbers with factorial can be seen in authors’s another work. Uploaded on March 6, 2019

47. Amicable Numbers With Patterns in Products and Powers

Inder J. Taneja

There are many ways of writing amicable numbers. One with divisions and sums. The other with pair of powers of each other. There is another way to represent is in product. In this paper, we brings amicable numbers in pairs in terms of products and powers. The idea of self-amicable is also introduced. Few blocks of symmetrical amicable numbers multiples of 10 are also given. Some interesting patterns among amicable numbers are also given. Uploaded on March 5, 2019

46. Magic Squares Type Palprimes of Orders 5×5, 7×7 and 9×9

Inder J. Taneja

In this paper, we worked with magic square type palindromic primes numbers of order a x a, in such a way that rows, columns and principal diagonals are palprimes along with extended row of rows are also palprimes. These types of distributions are named as magic square type palprimes or palprime distributions of ordera a x a. This work brings results for palprime distributions of orders 5×5, 7×7 and 9×9. This work is combination of authors three papers [12], [13] and [14] studied in 2017. Uploaded on February 27, 2019

45. Quadratic-Type Selfie Numbers

Inder J. Taneja

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. In two variables, the selfie numbers are obtained by use of binomial coefficients, S-gonal numbers, centered polygonal numbers, etc. Quadratic numbers are particular cases of S-gonal numbers in one variable. This paper brings selfie numbers in terms of quadratic numbers. The work is up to 5 digits numbers. The work is done in digit’s order and reverse order of digits. Uploaded on February 25, 2019

44. Natural Numbers From 1 to 20000 in Terms of Fibonacci Sequence and Triangular Numbers

Inder J. Taneja

This paper brings natural numbers from 1 to 20000 written in terms of Fibonacci sequence and Triangular numbers. The numbers are obtained just with operation of addition. This work is revised and extended version of authors previous works done in 2018. Uploaded on February 21, 2019

43. Selfie Expressions With Factorial, Fibonacci and Triangular Values

Inder J. Taneja

Selfie expressions are written in such a way that both sides of the expressions are with same digits. This work brings expressions where one side with factorial, and other side with Fibonacci and/or with triangular numbers having same digit’s order. This we have done in different ways. One expressions with Factorial, Fibonacci and Triangular values. Second, expressions with Factorial and Fibonacci values. Third, expressions with Factorial and Triangular numbers. Forth, expressions with Fibonacci sequence and Triangular numbers. The operations used are addition, subtraction and multiplication. Uploaded on February 20, 2019

42. Simultaneous Representations of Selfie Numbers in Terms of Fibonacci and Triangular Numbers

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. There are many ways of representing selfie numbers. It can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, etc. In this work, we have written selfie numbers in such a way that these are simultaneously equal by use of Fibonacci sequence as well as Triangular numbers. This is done by use of basic operations along with factorial and/or square-root.Uploaded on February 20, 2019

41. Factorial-Power Selfie Expressions

Inder J. Taneja

This paper brings numbers in such a way that both sides of the expressions are with same digits and in same order. One side is digits with factorial and another side are with same digits with respective powers. These types of expressions, we call as selfie expressions. Three types of expressions are studied. One when digits involved are distinct, second when there is a repetition of digits but only with positive sign. The third type is with repetition of digits with positive and negative signs. In all the cases the digits follow the same order but not the operations. Operations used are only addition, subtraction and multiplication. Uploaded on February 20, 2019

40. Same Digits Equalities Expressions

Inder J. Taneja

This paper brings numbers in such a way that both sides of the expressions are with the same digits. One side is numbers with powers, while other side just with numbers having same digits, such as, a^b+c^d+… =ab+cd+…, etc. The the expressions studied are with positive and negative coefficients. Work is done for 2 to 10 terms expressions. From 5 terms expressions onwards, the results are only for positive coefficients. Uploaded on February 19, 2019

39. Flexible Powers Narcissistic-Type Numbers

Inder J. Taneja

This paper works with extensions of narcissistic numbers as flexible powers narcissistic numbers with positive, and positive-negative coefficients. This is reorganized version of author’s previous work [12] done in 2017.Uploaded on February 19, 2019

38. Fibonacci Sequence and Selfie Numbers

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as Selfie Numbers. There are many ways of representing Selfie Numbers, such as, numbers written in digit’s order or its reverse. It can also be represented in increasing and/or decreasing order of digits. This is generally obtained by use of basis operations along with factorial, square-root, Triangular numbers, Fibonacci sequence, etc. In this work selfie numbers are written using Fibonacci sequence values in digit’s order and reverse order of digits. In some situations, the results are up to 4, 5 or 6 digits numbers. Results by use of factorial are also calculated.Uploaded on February 19, 2019

37. Triangular-Type Selfie Numbers

Inder J. Taneja

Numbers represented by their own digits by certain operations are considered as selfie numbers. There are many ways of writing selfie numbers, such as, numbers written in digit’s order or its reverse just with basic operations. We can extend them by use of other operations, such as, factorial, square-root, Triangular numbers, Fibonacci sequence values, etc. In this work, the selfie numbers are written by use of triangular numbers in digit’s order and reverse. Uploaded on February 17, 2019

36. Permutable Powers Selfie Numbers

Inder J. Taneja

This paper works with representations of numbers in such a way that we have same digits on both sides of the expressions. One side is just number and other side formed by bases and exponents with same digits as of numbers, but with different permutations. The expressions are by use of operations of addition and/or subtraction. These numbers are called permutable powers selfie numbers. In this paper, we worked with number up to 9-digits. Up to 6 digits the work is with repetition of digits, while 7, 8 and 9 digits numbers are without repetitions. This work is a combination of author’s three previous papers given in references [15], [16] and [17} done in done in 2016. Uploaded on February 15, 2019

35. Multiple Choice Power Expressions

Inder J. Taneja

Pythagoras theorem is famous in the literature. It works with squares of numbers, i.e., a square of number equals other two numbers with square on each. This paper bring numbers with equality expressions with powers of same order, for example, 7^2:=2^2+3^2+6^2, 6^3:=3^3+4^3+5^3, etc. The work is for the powers from 2 to 6.Uploaded on February 15, 2019

34. Patterns in Selfie and Semi-Selfie Numbers

Inder J. Taneja

Author studied many ways of writing selfie numbers, sometimes known by wild narcissistic numbers. There are numbers very much near to selfie-number, but are not selfie numbers. These types of numbers, referred as semi-selfie numbers, where numbers are written in terms of expressions with positive and negative signs having same digits on both sides of the expressions, except the power values. This paper brings interesting patterns with selfie and semi-selfie numbers. This work is combination of author’s previous two works done in 2015 and 2017. See the references [7] and [17] in the reference list. Uploaded on February 12, 2019

33. Semi-Selfie Numbers

Inder J. Taneja

Author studied many ways of writing selfie numbers (reference [13]). By selfie numbers, we understand those numbers that can be expressed in terms of same digits as of number, either in digit’s order or reverse order of digits. These numbers are obtained by use of basic operations along with factorial, square-root, etc. There are numbers very much near to selfie-number, but are not selfie numbers. These types of numbers, referred as semi-selfie numbers, where the numbers are written in terms of expressions with positive and negative signs having same digits on both sides of the expressions, except the power values. This paper brings semi-selfie numbers in different situations. In some situations, they are expressed with positive sign, and positive and negative signs. The work is limited to 11-digits numbers. This is revised and enlarged version of author’s previous two works appearing in reference lists as [17] and [19]. Uploaded on February 12, 2019464106

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32. Fixed Digits Repetitions Prime Patterns of Length 6

Inder J. Taneja

In 2017, theauthor worked with patterns in prime numbers in such a way that they can be extended by inserting digits in repeated ways, and still remains prime numbers. These types of prime numbers are called fixed digits repetitions prime patterns . In this work, multiple choice prime patterns are considered. Multiple choice means that the patterns formed by same prime number can be represented by more than one way. The work is for prime patterns of length 6. When we talk of length 6, it means that the pattern at length 7 is no more a prime number. A complete list of prime patterns with possible number of digits repetitions is written in reduced form. This work is a revised and enlarged version of author’s previous three works done in 2017. Similar kind of work on lengths 10, 9, 8 and 7 can seen in [14] and [15].Uploaded on February 9, 2019

31. Fixed Digits Repetitions Prime Patterns of Length 7

Inder J. Taneja

The author in 2017, worked with patterns in prime numbers in such a way that they can be extended by inserting digits in repeated ways, and still remains prime numbers. These types of prime numbers are called fixed digits repetitions prime patterns . In this work, multiple choice prime patterns are considered. Multiple choice means that the same pattern can be represented by more than one way. The work is for prime patterns of length 7. When we talk of length 7, it means that the pattern at length 8 is no more a prime number. A complete list of primes patterns with possible number of digits repetitions is written in reduced form. This work is revised and enlarged version of author’s previous two works done in 2017. The revised and enlarge version of work on lengths 10, 9 and 8 can be seen in [14]. Uploaded on February 8, 2019

30. Fixed Digits Repetitions Prime Patterns of Lengths 10, 9 and 8

Inder J. Taneja

The author in 2017, worked with patterns in prime numbers in such a way that they can be extended by inserting digits in repeated ways, and still remains prime numbers. These types of prime numbers are called fixed digits repetitions prime patterns . In this work, multiple choice prime patterns are considered. Multiple choice means that the same pattern can be represented by more than one way. The work is for prime patterns of lengths 10, 9 and 8. When we talk of length 10, it means that the pattern at length 11 is no more a prime number. It happens same for the patterns of lengths 9 and 8. A complete list of primes with possible number of digits repetitions is written in reduced form. It is done at the end of each section on lengths 10, 9 and 8. This work is revised and enlarged version of author’s previous works done in 2017. Uploaded on February 8, 2019

29. Single Letter Patterned Representations and Fibonacci Sequence Values

Inder J. Taneja

This work brings representations of palindromic and number patterns in terms of single letter “a”. Some examples of prime patterns are also considered. Different classifications of palindromic patterns are considered, such as palindromic decomposition, double symmetric patterns, number patterns decompositions, etc. Number patterns with powers are also studied. Some extensions to Pythagorean triples are also given. Study towards Fibonacci sequence and its extensions are also made. This work is revised and enlarged version of author’s previous work done in 2015.Uploaded on February 6, 2019

28. Single Letter Representations of Natural Numbers from 1 to 11111

Inder J. Taneja

The natural numbers form 1 to 11111 are written in terms of single letter “a” in two different ways. One running-type expressions, and second fraction-type expressions. In this paper, we worked with running-type expressions. It means the numbers 1 to 11111 are written in terms of single letter “a” . The letter “a” can have any value from 1 to 9, and the final results is always same. To bring these results, only basic operations, such as, addition, subtraction, multiplication and division are used. In the end some extra results using potentiation are also given. The fraction-type single letter representations can be see in author’s [17] another work. This is a reorganized version of author’s previous work [16]. Uploaded on February 5, 2019

27. Fraction-Type Single Letter Representations of Natural Numbers From 1 to 11111

Inder J. Taneja

The natural numbers form 1 to 11111 are written in terms of single letter “a” in two different ways. One is running-type expressions, and second is fraction-type expressions. In this work, we considered the fraction-type way. It means the numbers 1 to 11111 are written as fraction-type using only the single letter “a” . The single letter “a” can have any valure from 1 to 9, and the final result is always same. To bring these results, only basic operations, such as, addition, subtraction, multiplication and division are used. The idea of potentiation is not considered here. In another work, few numbers are written using potentiation. The running-type single letter representations can be see in author’s [16] another work. This work is a reorganized version of author’s previous work [17] done in 2018. Uploaded on February 4, 2019

26. Block-Wise Magic and Bimagic Squares of Orders 39 to 45

Inder J. Taneja

This paper brings block-wise construction of magic squares of order 39 to 45. In order to construct these magic squares we used the previous known magic squares of orders 3 to 14, except order 12. In each case these are written again. Specially, in case of magic square of order 40, one of the possibility is bimagic square. The block-wise construction of magic squares of orders 8 to 36 can be seen in author’s previous work [29]. Uploaded on February 2, 2019

25. Palindromic, Patterned Magic Sums, Composite, and Colored Patterns in Magic Squares

Inder J. Taneja

There are many ways of representing magic squares with palindromic type entries. This paper works with magic squares of order 3 to 10. In each case, the magic squares written for the entries from the digits 3 to 8 with all palindromic numbers. These entries are arranged in such a way that their magic sum turns number patterns. In some cases, using digital type letters, these palindromic magic squares are made in such a way that they becomes upside down and mirror looking. The study is extended to composite magic squares and double colored patterns. Some particular case resulting in upside down and mirror looking magic squares are also considered. In case of magic square of order 9, we have considered two different situations. One with bimagic and another with pandiagonal magic squares. In case of magic square of order 10, to make it palindromic-type, we considered some numbers as 010, 020, etc. In some other examples, we also used the idea of symmetry as 010, 020, etc. This is reorganized version of author’s previous works 15, 16, 30] done in 2015 and 2018.Uploaded on February 2, 2019

24. Block-Wise Magic and Bimagic Squares of Orders 12 to 36

Inder J. Taneja

This paper summarize some of the results done before by author on block-wise constructions of magic squares. In this paper, we shall rewrite some these results without details. The details can be seen in the reference list. This is done for the magic squares of orders 12 to 36, i.e., for the orders 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35 and 36. In some cases, the magic squares are bimagic or semi-bimagic. We tried to bring all the possible combinations in each case. In all the cases, at least one of the block-wise representation is pandiagonal except the orders 18 and 30. Magic squares for the prime numbers and double of prime numbers, such as, 11, 13, 22, 26, etc. are not considered. Uploaded on February 1, 2019

23. Different Digits Magic Squares and Number Patterns

Inder J. Taneja

In this work, we shall brings number patterns based on magic square sums for the magic and bimagic squares constructed with different digits. In each case, as the digits increases in cells, the values of magic and bimagic squares sums also increases giving interesting number patterns. This is done for the magic squares of orders 5, 7, 8, 9 and 10.Uploaded on February 1, 2019

22. Representations of Letters and Numbers With Equal Sums Magic Squares of Orders 4 and 6

Inder J. Taneja

This work brings 26 letters from A to Z and 10 numbers from 0 to 9 in terms of blocks of magic squares of orders 4 and 6. Letters and numbers ares constructed with blocks of equal sums magic squares of orders 4 and 6. In each case, consecutive natural numbers are used starting from 1, and there is no repetition of numbersUploaded on February 1, 2019

21. Block-Wise Unequal Sums Magic Squares

Inder J. Taneja

There are magic square where it impossible to divide in sub-blocks with equal sums magic squares. For example, in case of order 12, it is impossible to divide in equal sums magic squares of order 3. In case of order 20, it is impossible to divide equal sums magic squares of order 5. In case of order 28, it is impossible to divide equal sums magic squares of order 7, etc. This paper brings constructions of magic squares with sub-blocks of unequal sums. The work is for the magic squares of orders, 12,18, 20, 30 and 36. It is revised version of author’s previous work [28] done in 2017. Uploaded on February 1, 2019

20. Block-Wise Equal Sums Magic Squares of Orders 3k and 6k

Inder J. Taneja

This paper brings block-wise magic squares multiple of 3 and 6. The multiples of 6 means even order multiples of 3, such as, orders 12, 18, 24, 30, 36, etc. From magic squares of order 9 to 36, all the cases are of pandiagonal magic squares except the orders 18 and 30. The constructions are in such a way that in each case the sub-blocks are of same magic sums or sum of all entries are same. The work on pandiagonal squares of orders 4k is done by author [22]. In these cases, all the sub-blocks are of perfect pandiagonal magic squares of order 4 with same magic sums. In case of order 30, three different ways are given. The pandiagonal magic square of order 36 is done two different ways, one with 81 sub-blocks of pandiagonal magic squares of order 4, and second with 16 blocks of order 9 pandiagonal magic squares with different magic sums. This work is a combined version of author’s previous two papers [23, 24] done in 2017. Uploaded on February 1, 2019

19. Magic Crosses: Repeated and Non Repeated Entries

Inder J. Taneja

The idea of magic rectangles is well known in the literature. Using this idea we brought for the first time in history a new concept on magic crosses. The work is divided in two groups. One on orders (odd, odd) and another on orders (even, even). Within the orders (odd, odd), the work is on magic crosses of type (3, 2n+3), (5, 2n+5),… n=1, 2,… Within orders orders (even, even), the work is on magic crosses of orders (4n, 4m), (4n, 2n+2), 2(even, odd), etc. In all the case, we used the same number of entries as of magic rectangles to bring magic squares. In case of lower rows and columns of magic crosses the entries are repeated. For non repeated entries we worked with orders (4,12), (5,15), (6,18), (8,24) and (10,30). In this case, the, the magic squares are of equal magic sums. The inspiration of this is due to classical magic square of Naranyana in 14th century (1356AD). Uploaded on February 1, 2019

18. Magic Rectangles in Construction of Block-Wise Pandiagonal Magic Squares

Inder J. Taneja

The idea of magic rectangles is used to bring pandiagonal magic squares of orders 15, 21, 27 and 33, where 3×3 blocks are with equal sums entries and are semi-magic squares of order 3 (in rows and columns). The magic squares of order 9, 12, 18, 24, 30 and 36 are calculated, with the property that 3×3 blocks are magic squares of order 3 with different magic sums. All the magic squares constructed are pandiagonal except the orders 18 and 30. Exceptionally, the pandiagonal magic square of order 35 is of type 5×7 or 7×5. It is constructed in two different approaches. One with 25 blocks of equal sums magic squares of order 7 and second with 49 blocks of equal sums magic squares of order 5. This work is same as done by author in 2017. Uploaded on January 31, 2019

17. Permutable Power Minimum Length Representations of Natural Numbers from 0 to 20000

Inder J. Taneja

This paper works with representations of natural numbers from 0 to 20000 written in terms of expressions with addition, subtraction and exponents. Digits from 0 to 9 and 1 to 9 are used in such a way that for each number represents with same digits in bases and exponents with different permutations. Some numbers can be written in more than one way, but we have chosen with less possible expression, calling as minimum length. This work extends the author’s previous work done in 1 to 9 digits to 0 to 9 digits. Some comparisons among the digits 0 to 9 and 1 to 9 are also given. Uploaded on January 30, 2019

16. Single Digit Representations of Natural Numbers From 15001 to 20000

Inder J. Taneja

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper bring numbers 15001 to 20000 in terms of each digit. The total worEk up to 20000 numbers divided in four parts. For other parts refer reference list.Uploaded on January 26, 2019

15. Single Digit Representations of Natural Numbers From 10001 to 15000

Inder J. Taneja

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper bring numbers 10001 to 15000 in terms of each digit. The total work up to 20000 numbers divided in four parts. For other parts can be seen in reference list. Uploaded on January 26, 2019

14. Generating Pythagorean Triples, Patterns, and Magic Squares

Inder J. Taneja

This paper brings simplified and symmetric procedure to generate Pythagorean triples. These triples are obtained in different procedures. First procedure is given in three blocks. The second procedure is the extension of first procedure, but in little different way. These triples are applied to generate perfect square sums magic squares of consecutive odd numbers, and patterned Pythagorean triples. The patterned Pythagorean triples are obtained in two different way. One is a general way, and the second procedure give us Palindromic-Type Pandigital Pythagorean Triples in two different forms. As examples, the magic squares of orders 3 to 20 are given. The sum of entries of magic squares always give a perfect square resulting in a Pythagorean triple. Uploaded on January 20, 2019

13. Palindromic-Type Pandigital Patterns in Pythagorean Triples

Inder J. Taneja

This paper brings examples of Palindromic-Type Pandigital Patterns in Pythagorean Triples. These are constructed in a padronized way. This means that in all the patterns we have pandigital palindromic-type expressions. The only change appears is in the middle terms and the last numbers of the first and third values. The results are obtained in such a way that we have patterns as: 9, 99, 999, 9999, 99999, etc. The construction is based on a procedure well known in the literature. There is very much uniformity among the results. In this work we have two different types of palindromic-type expressions, such as blocks of 121, 12321, 1234321, …, and blocks of 10201, 102030201, 1020304030201, …. This work is a combinations of author’s previous two works. Uploaded on January 20, 2019

12. Multiple-Type Patterns and Pythagorean Triples

Inder J. Taneja

The Pythagoras theorem is very famous in the literature of mathematics. This paper brings patterns obtained by multiplication by natural numbers to known Pythagorean triples resulting again in patterns. This gives many patterns of similar kind. The constructions and details of the procedures can be seen in author’s work given in references.Uploaded on January 19, 2019

11. Patterns in Pythagorean Triples Using Single and Double Variable Procedures

Inder J. Taneja

The Pythagoras theorem is very famous in the literature of mathematics. The aim of this work is to extend in a symmetrical way the some Pythagorean triples resulting in patterns. These symmetric extensions are in such a way that we reach to good patterns. In some cases, the final sums also give a good pattern. In some cases examples are with interesting pandigital palindromic-type patterns. The patterns are obtained based on five procedures for single variable functions, and four for double variable functions. This work is a combination of authors previous two works. Uploaded on January 19, 2019

10. Crazy Representations of Natural Numbers From 11112 to 20000

Inder J. Taneja

This paper brings natural numbers from 11112 to 20000 written in ascending and descending orders of 1 to 9. Most of the numbers are obtained by using basic operations, except few. It is revised version of author’s previous work done in 2018, where for the missing numbers factorial and square-root are used . In this work, only \textbf{factorial} is used for missing numbers.The representations for the numbers 20001 to 30000 are given in the next work. The representations from 0 to 11111 author’s work done in 2013-2014. In the previous work, all the numbers are written with basic operations except one, i.e., 10958 (in the increasing case). For more details on number 10958 refer author’s web-site link. Uploaded on January 18, 2019

9. Palindromic-Type Squared Expressions with Palindromic and Non-Palindromic Sums – I

Inder J. Taneja

In the previous papers, the author worked with palindromic-type expressions. These expressions are by use of operations of addition and multiplications. When the operations are removed, we get normal even order palindromes. The previous works are limited to particular cases including patterns. The final sums are either palindromes or non-palindromes. The aim of this paper is to write palindromic-type squared expressions when final sums are either palindromes or non-palindromes. This work is up to 21 digits final sums. The higher digits shall be dealt elsewhere.Uploaded on January 15, 2019

8. Palindromic-Type Non-Palindromes – I

Inder J. Taneja

In the previous papers, the author worked with palindromic-type expressions. These expressions are by use of operations addition and multiplications. When the operations are removed, we get even order palindromes. In the previous works, it is limited to particular cases including patterns. The final sum is either palindrome or non-palindrome. The aim of this paper, we have written palindromic-type expressions, where final sums are non-palindromes. The results are with multiple choices representations, i.e., in some cases palindromes can be written in terms of different palindromic-type expressions. Due to high quantity of numbers, the work is limited to 6 digits non-palindromes. In the another part of the work (https://doi.org/10.5281/zenodo.2541174), the palindromic-type expressions are given where the final sums are palindromes. Further extension shall be given later on.Uploaded on January 15, 2019

7. Palindromic-Type Palindromes – I

Inder J. Taneja

In the previous papers, the author worked with palindromic-type expressions. These expressions are by use of operations addition and multiplications. When the operations are removed, we get normal even order palindromes. In the previous works, it is limited to particular cases including patterns. The final sum is either palindromic or non palindromic. The aim of this paper is to write palindromic-type expressions when final sums are palindromes. The results are with multiple choices representations, i.e., in some cases palindromes can be written in terms of different palindromic-type expressions. Due to high quantity of numbers, the work is limited to 7 digits palindromes. In the another work, palindromic-type expressions are given, where the final sums are non-palindromes. The extensions to higher order digits given later on.Uploaded on January 15, 2019

6. All Digits Flexible Power Representations of Natural Numbers From 30001 to 50000

Inder J. Taneja

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper extends the authors previous work on representation of natural numbers in terms of flexible powers using all digits with different permutations. The previous work is up to 30000. This paper bring numbers 30001 to 50000 in terms of flexible powers using all digits with different permutations. Uploaded on January 14, 2019

5. All Digits Flexible Power Representations of Natural Numbers From 11112 to 30000

Inder J. Taneja

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper extends the authors previous work on representation of natural numbers in terms of flexible powers using all digits with different permutations. The previous work is up to 11111. This paper bring numbers 11112 to 30000 in terms of flexible powers using all digits with different permutations.Uploaded on January 14, 2019

4. Single Digit Representations of Natural Numbers From 5001 to 10000

Inder J. Taneja

There are different ways of representing natural numbers, such as, writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper extends the authors previous work on representation of natural numbers in terms of single digit. This paper bring numbers 5001 to 10000 in terms of each digit. The total work up to 20000 numbers divided in four parts. Uploaded on January 14, 2019

3. Single Digit Representations of Natural Numbers From 1 to 5000

Inder J. Taneja

There are different ways of representing natural numbers, such as, writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper extends the authors previous work on representation of natural numbers in terms of single digit. The previous work is upto 1000. This paper bring numbers 1 to 5000 in terms of each digit. The total work up to 20000 numbers divided in four parts. Uploaded on January 14, 2019

Work in 2018

2. 2019 In Numbers

Inder J. Taneja

This short paper brings representations of 2019 in different situations. These representations are of crazy-type, running numbers, single digit, single letter, Triangular, Fibonacci, palindromic-type, prime numbers, embedded, repeated, magic squares, etc. Uploaded on December 31, 2018

1. Running Expressions with Triangular Numbers – I

Inder J. Taneja

In previous works, running equalities are written in terms of 1 to 9 and 9 to 1 or 9 to 0 separated by single or double equality signs. Each digit is used with basic operations, along with factorial, square-root and Fibonacci sequence. These types of equalities, we called as running expressions. This work brings double and triple equality type running expressions with triangular numbers along with basic operations. The work is up to 3 digits in increasing and decreasing orders. For 4 digits onwards the results shall be given later on. Uploaded on December 21, 2018

Post Views: 38

Preprints in Zenodo.org – Inder J. Taneja

ByInder J. Taneja

Work in 2025

277. New Concepts in Magic Squares: Cornered Magic Squares of Orders 5 to 108

276. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Order 25

275. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Order 24

274. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 21 to 23

273. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 17 to 20

272. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 14 to 16

271. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 11 to 13

270. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 7 to 10

Work in 2024

269. Upside-Down, Mirror Looking and Water Reflection Magic Squares: Orders 3 to 6

268. Numbers and Magic Squares Representations of Hardy-Ramanujan Number-1729

267. Reflexive Year 25: Mathematics of 25 and 2025 in Numbers and Magic Squares

266. Crazy Representations of Half a Million Natural Numbers: Part 5 – 460.001 to 500.000

265. Crazy Representations of Half a Million Natural Numbers: Part 4 – 420.001 to 460.000

264. Crazy Representations of Half a Million Natural Numbers: Part 3 – 380.001 to 420.000

263. Crazy Representations of Half a Million Natural Numbers: Part 2 – 340.001 to 380.000

262. Crazy Representations of Half a Million Natural Numbers: Part 1 – 300.001 to 340.000

261. Universal and Upside-down Magic Cubes

260. Magic Cubes Based on Magic Squares

259. Striped Magic Squares of Order 12 – Revised

258. Striped Magic Squares of Order 16 – Revised

257. Crazy Representations of Natural Numbers From 260001 to 280000 – Revised

256. Crazy Representations of Natural Numbers From 240001 to 260000 – Revised

255. Crazy Representations of Natural Numbers From 220001 to 240000 – Revised

254. Crazy Representations of Natural Numbers From 200001 to 220000 – Revised

253. Crazy Representations of Natural Numbers From 180001 to 200000 – Revised

252. Crazy Representations of Natural Numbers From 160001 to 180000 – Revised

251. Crazy Representations of Natural Numbers From 140001 to 160000 – Revised

250. Crazy Representations of Natural Numbers From 120001 to 140000 – Revised

249. Crazy Representations of Natural Numbers From 100001 to 120000 – Revised

248. Striped Magic Squares of Order 18

247. Multiple Choice Patterns in Selfie Numbers – II

246. Multiple Representations of Selfie Numbers – II

245. Multiple Choice Patterns in Selfie Numbers – I

244. Crazy Representations of Natural Numbers from 0 to 10000 Using Factorial and Square Root

243. Crazy Representations of Natural Numbers from 0 to 10000 Using Cubic Function

242. Crazy Representations of Natural Numbers from 0 to 10000 Using Square Function

241. Fixed Power Representation of Numbers

240. Unified Study of Narcissistic Numbers without and with Division

239. Multiple Representations of Selfie Numbers – I

238. Pyramid-Type Representations of Natural Numbers from 1001 to 10000

237. Crazy Representations of Natural Numbers From 280001 to 300000 – Revised

236. Crazy Representations of Natural Numbers 0 to 10000 Using Triangular Numbers

235. Crazy Representations of Natural Numbers from 0 to 10000 Using Fibonacci Sequence Values

Work in 2023

234. Mathematical Representations of the Last Day of the Year 23 Written American Style: 12.31.23 (123123)

233. Different Types of Magic Squares of Order 14 Using Bordered Magic Rectangles

232. Striped Magic Squares of Even Orders 6, 8, 10, 12 and 14

231. Different Types of Magic Squares of Orders 6, 8, 10 and 12

230. Cornered Magic Squares in Construction of Magic Squares of Orders 16, 20, 24 and 28

229. Different Types of Magic Rectangles in Construction of Magic Squares of Order 34

228. Different Types of Magic Rectangles in Construction of Magic Squares of Order 30

227. Different Types of Magic Rectangles in Construction of Magic Squares of Order 26

226. Different Types of Magic Rectangles in Construction of Magic Squares of Order 22

225. Different Types of Magic Rectangles in Construction of Magic Squares of Orders 14 and 18

224. Different Types of Magic Rectangles

223. A Simplified Procedure to Construct Pandiagonal Magic Squares Multiples of 4

222. New Concepts in Magic Squares: Cornered Magic Squares of Orders 5 to 81

221. New Concepts in Magic Squares: Double Digits Bordered Magic Squares of Orders 7 to 108

220. Magic Squares of Order 31

219. Magic Squares of Order 29

218. Magic Squares of Order 27

217. Patterns in Splitted Selfie Fractions

216. Bordered and Pandiagonal Magic Squares Multiples of 20

215. Bordered Magic Squares Multiples of 18

214. Bordered and Pandiagonal Magic Squares Multiples of 16

213. Bordered Magic Squares Multiples of 14

212. Bordered and Pandiagonal Magic Squares Multiples of 12

211. Block-Wise Bordered Magic Squares Multiples of 10

210. Bordered and Pandiagonal Magic Squares Multiples of 8

209. Bordered Magic Squares Multiples of Order 6

208. Bordered Magic Squares Multiples of 17

207. Bordered Magic Squares Multiples of 19

206. Bordered Magic Squares Multiples of 15

205. Bordered Magic Squares Multiples of 13

204. Bordered Magic Squares Multiples of 11

203. Bordered Magic Squares Multiples of 9