References: Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Patterned Single Letter Representations of Natural Numbers, Zenodo, July 02, 2020, pp. 1-110, http://doi.org/10.5281/zenodo.3928507Inder J. Taneja, Patterned Selfie Fractions, Zenodo, October 27, 2019, pp. 1-267, http://doi.org/10.5281/zenodo.3520096Inder J. Taneja, Power-Type Semi-Selfie Numbers and Patterns, Zenodo, July 16, 2019, pp. 1-130, http://doi.org/10.5281/zenodo.3338366Inder J. Taneja, Factorial-Power Selfie Expressions, Zenodo, February … Continue reading 278th, 279th and 280th Days of Year: 05.10.21, 06.10.21 and 07.10.21 – Crazy, Pythagorean Triples Patterns and Factorial-Power Representations

## 276th and 277th Days of Year: 03.10.2021 and 04.10.2021 – Crazy, Semi-Selfie, Factorial-Power and Selfie Fractions Patterns

References: Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Patterned Single Letter Representations of Natural Numbers, Zenodo, July 02, 2020, pp. 1-110, http://doi.org/10.5281/zenodo.3928507Inder J. Taneja, Patterned Selfie Fractions, Zenodo, October 27, 2019, pp. 1-267, http://doi.org/10.5281/zenodo.3520096Inder J. Taneja, Power-Type Semi-Selfie Numbers and Patterns, Zenodo, July 16, 2019, pp. 1-130, http://doi.org/10.5281/zenodo.3338366Inder J. Taneja, Factorial-Power Selfie Expressions, Zenodo, … Continue reading 276th and 277th Days of Year: 03.10.2021 and 04.10.2021 – Crazy, Semi-Selfie, Factorial-Power and Selfie Fractions Patterns

## 274th and 275th Days of Year: 01.10.2021 and 02.10.2021 – Crazy, Semi-Selfie and Selfie Fractions Patterns

References: Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Patterned Single Letter Representations of Natural Numbers, Zenodo, July 02, 2020, pp. 1-110, http://doi.org/10.5281/zenodo.3928507Inder J. Taneja, Patterned Selfie Fractions, Zenodo, October 27, 2019, pp. 1-267, http://doi.org/10.5281/zenodo.3520096Inder J. Taneja, Power-Type Semi-Selfie Numbers and Patterns, Zenodo, July 16, 2019, pp. 1-130, http://doi.org/10.5281/zenodo.3338366

## 268th and 269th Days of Year: 25.09.2021 and 26.09.2021 – Semi-Selfie, Single Letter and Selfie Fraction Patterns

References: Inder J. Taneja, Patterned Single Letter Representations of Natural Numbers, Zenodo, July 02, 2020, pp. 1-110, http://doi.org/10.5281/zenodo.3928507Inder J. Taneja, Patterned Selfie Fractions, Zenodo, October 27, 2019, pp. 1-267, http://doi.org/10.5281/zenodo.3520096Inder J. Taneja, Power-Type Semi-Selfie Numbers and Patterns, Zenodo, July 16, 2019, pp. 1-130, http://doi.org/10.5281/zenodo.3338366

## 266th and 267th Days of Year: 23.09.2021 and 24.09.2021 – Crazy, Single Letter and Selfie Fractions Patterns

References: Inder J. Taneja, Colored Patterns With 2021 On a Board of 9×9, https://inderjtaneja.com/2020/12/26/colored-pattern-with-2021-on-a-board-of-9×9/Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Patterns in Pythagorean Triples Using Single and Double Variable Procedures, Zenodo, January 19, 2019, pp. 1-134, http://doi.org/10.5281/zenodo.2544519Inder J. Taneja, Patterned Single Letter Representations of Natural Numbers, Zenodo, July 02, 2020, pp. 1-110, http://doi.org/10.5281/zenodo.3928507Inder J. Taneja, Patterned … Continue reading 266th and 267th Days of Year: 23.09.2021 and 24.09.2021 – Crazy, Single Letter and Selfie Fractions Patterns

## 264th and 265th Days of Year: 21.09.2021 and 22.09.2021 – Crazy, Semi-Selfie and Selfie Fractions Representations

References: Inder J. Taneja, Colored Patterns With 2021 On a Board of 9×9, https://inderjtaneja.com/2020/12/26/colored-pattern-with-2021-on-a-board-of-9×9/Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Power-Type Semi-Selfie Numbers and Patterns, Zenodo, July 16, 2019, pp. 1-130, http://doi.org/10.5281/zenodo.3338366Inder J. Taneja, Patterned Selfie Fractions, Zenodo, October 27, 2019, pp. 1-267, http://doi.org/10.5281/zenodo.3520096

## 262nd and 263rd Days of Year: 19.09.2021 and 20.09.2021 – Crazy, Semi-Selfie and Selfie Fractions Representations

References: Inder J. Taneja, Colored Patterns With 2021 On a Board of 9×9, https://inderjtaneja.com/2020/12/26/colored-pattern-with-2021-on-a-board-of-9×9/Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Power-Type Semi-Selfie Numbers and Patterns, Zenodo, July 16, 2019, pp. 1-130, http://doi.org/10.5281/zenodo.3338366Inder J. Taneja, Patterned Selfie Fractions, Zenodo, October 27, 2019, pp. 1-267, http://doi.org/10.5281/zenodo.3520096

## Block-Wise Bordered Magic Squares Multiples of Magic and Bordered Magic Squares of Order 6

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 102. Based on these two big magic squares inner order magic squares multiples of 6 are studied. By inner order we understand that magic squares of orders 96, 90, 84, etc. Instead of … Continue reading Block-Wise Bordered Magic Squares Multiples of Magic and Bordered Magic Squares of Order 6

## 260th and 261st Days of Year: 17.09.2021 and 18.09.2021 – Crazy, Single Letter and Pythagorean Triples Patterns

References: Inder J. Taneja, Colored Patterns With 2021 On a Board of 9×9, https://inderjtaneja.com/2020/12/26/colored-pattern-with-2021-on-a-board-of-9×9/Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Patterns in Pythagorean Triples Using Single and Double Variable Procedures, Zenodo, January 19, 2019, pp. 1-134, http://doi.org/10.5281/zenodo.2544519Inder J. Taneja, Multiple-Type Patterns and Pythagorean Triples, Zenodo, January 19, 2019, pp.1-53, http://doi.org/10.5281/zenodo.2544527Inder J. Taneja, Palindromic-Type Pandigital Patterns in … Continue reading 260th and 261st Days of Year: 17.09.2021 and 18.09.2021 – Crazy, Single Letter and Pythagorean Triples Patterns

## 258th and 259th Days of Year: 15.09.21 and 16.09.21 – Crazy, Semi-Selfie and Single Letter Representations

References: Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Power-Type Semi-Selfie Numbers and Patterns, Zenodo, July 16, 2019, pp. 1-130, http://doi.org/10.5281/zenodo.3338366Inder J. Taneja, Patterned Single Letter Representations of Natural Numbers, Zenodo, July 02, 2020, pp. 1-110, http://doi.org/10.5281/zenodo.3928507

## 256th and 257th Days of Year: 13.09.2021 and 14.09.2021 – Crazy Representations and Pythagorean Triples Patterns

References: Inder J. Taneja, Colored Patterns With 2021 On a Board of 9×9, https://inderjtaneja.com/2020/12/26/colored-pattern-with-2021-on-a-board-of-9×9/Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Patterns in Pythagorean Triples Using Single and Double Variable Procedures, Zenodo, January 19, 2019, pp. 1-134, http://doi.org/10.5281/zenodo.2544519Inder J. Taneja, Multiple-Type Patterns and Pythagorean Triples, Zenodo, January 19, 2019, pp.1-53, http://doi.org/10.5281/zenodo.2544527Inder J. Taneja, Palindromic-Type Pandigital Patterns in … Continue reading 256th and 257th Days of Year: 13.09.2021 and 14.09.2021 – Crazy Representations and Pythagorean Triples Patterns

## 254th and 255th Days of Year – 11.09.2021 and 12.09.2021: Crazy and Semi-Selfie Representations

References: Inder J. Taneja, Colored Patterns With 2021 On a Board of 9×9, https://inderjtaneja.com/2020/12/26/colored-pattern-with-2021-on-a-board-of-9×9/Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Power-Type Semi-Selfie Numbers and Patterns, Zenodo, July 16, 2019, pp. 1-130, http://doi.org/10.5281/zenodo.3338366

## 252nd and 253rd Days of Year – 09.09.2021 and 10.09.2021: Crazy Representations and Selfie Fractions Patterns

References: Inder J. Taneja, Colored Patterns With 2021 On a Board of 9×9, https://inderjtaneja.com/2020/12/26/colored-pattern-with-2021-on-a-board-of-9×9/Inder J. Taneja, Crazy Representations of Natural Numbers – The 10958 Problem, https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/Inder J. Taneja, Selfie Fractions: Addable, Subtractable, Dottable and Potentiable, Zenodo, March 24, 2019, pp. 1-260, http://doi.org/10.5281/zenodo.2604531Inder J. Taneja, Pandigital Equivalent Selfie Fractions, Zenodo, April 02, 2019, pp. 1-392, http://doi.org/10.5281/zenodo.2622028Inder J. Taneja, Repeated Digits Selfie Fractions: Two and Three … Continue reading 252nd and 253rd Days of Year – 09.09.2021 and 10.09.2021: Crazy Representations and Selfie Fractions Patterns

## Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4

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 squares inner order magic squares multiples of 4 are studied. By inner order we understand … Continue reading Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4

## Creative Magic Squares: Singele Digit Representations

This work 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 … Continue reading Creative Magic Squares: Singele Digit Representations

## Creative Magic Squares: Single Letter Representations

This work brings magic squares of orders 3 to 10 in terms of single letter "a", where a can take any value from 1 to 9. The magic squares of orders 8 and 9 are written in two ways, i.e, one as normal magic squares and another as bimagic squares. In case of order 10, we … Continue reading Creative Magic Squares: Single Letter Representations

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

This work 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 … Continue reading Magic Squares With Perfect Square Sum of Entries: Orders 3 to 31

## Minimum Perfect Square Sum Bordered and Block-Wise Bordered Magic Squares: Orders 3 to 47

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 … Continue reading Minimum Perfect Square Sum Bordered and Block-Wise Bordered Magic Squares: Orders 3 to 47

## Area Representations Magic Squares With Fraction Numbers Entries

It is well known that every magic square can be written as perfect square sum of entries. It is always possible with odd number entries starting from 1. In case of odd order magic squares we can also write with consecutive natural number entries. Still, it is unknown whether it is possible to even order … Continue reading Area Representations Magic Squares With Fraction Numbers Entries