Project

# Magic Squares

Goal: The aim is to work on construction of block-wise magic and bimagic squares

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## Project log

This paper brings representations of 1729, a famous Hardy-Ramanujan number in different situations. These representations are with single digit, single letter, Selfie-Type, Running Expressions, Equivalent Fractions, Triangular , Fibonacci, Fixed Digits Repetitions Prime Numbers Patterns , Palindromic-Type, Polygonal-Type, Prime Numbers, Embedded, Repeated, etc. Ideas toward magic squares are also extended. Some quotes and historical notes on Ramanujan's life and work are also given. On a special day: January 29, 17 (17-29) An equation means nothing to me unless it expresses a thought of God.-S. Ramanujan Another famous quote of Ramanujan on his dreams: While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing. S. Ramanujan Confirmations to Ramanujan's dreams: Ono and his colleagues (Emory University, Atlanta, GA, USA) drew on modern mathematical tools that had not been developed before Ramanujan's death to prove this theory was correct. We proved that Ramanujan was right. We've solved the problems from his last mysterious letters. For people who work in this area of math, the problem has been open for 90 years. We found the formula explaining one of the visions that he believed came from his goddess. K. Ono Read more at: 1.
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  another work. This work is a reorganized version of author's previous work  done in 2018.
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, 26.
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 magic squares. In case of odd order magic squares, still we can write them with minimum perfect square sum of entries. Based on this idea of perfect square sum of entries, we have written a magic square representing areas. This is done 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 more that one way. It is due to the fact that we can always write magic squares as normal, bordered and block-bordered ways.
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.
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.
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 numbers orders. We call it as block-bordered magic squares.
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 140 and 126. Based on these two big magic squares, the inner order magic squares multiples of 14 are studied. By inner orders we understand as the magic squares of orders 112, 98, 84, etc. Instead of working in decreasing order, we worked with increasing orders, such as, orders 14, 28, 42, etc. The construction of the block-wise bordered magic squares multiples of 14 is based on equal sum blocks of magic squares of order 14. It is done in five different ways. First way is a general magic square of order 14. The second way is bordered magic squares of order 14. The last three ways are block-bordered magic squares of order 14. By block-bordered, we understand that a magic square of order 14 formed by inner block of order 12 with three different blocks. These blocks are of orders 3, 4 and 6. 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). This work brings examples for orders 14, 28, 42 and 56. The higher order examples are given in Excel file attached with the work.
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.
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. Instead of working in decreasing order, we worked with increasing orders, such as, orders 10, 20, 30, etc. The construction of the block-wise bordered magic squares multiples of 10. It is done in three different ways. One as normal magic squares of order 10, second as block-bordered magic squares of order 10, where the inner magic square of order 8 is pandiagonal formed by 4 equal sum pandiagonal magic squares of order 4. The third way is with bordered magic square of order 10. In this case, the inner blocks are magic squares of orders 8, 6 and 4. The magic squares of order 4 are pandiagonal. 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.
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. Instead of working in decreasing order, we worked with increasing orders, such as, orders 12, 24, 36, etc. The construction of the block-wise bordered magic squares multiples of 12 is based on equal sum blocks of magic squares of order 12. It is done in six different ways. First three ways are such that each magic square of order 12 is composed by blocks of orders 3, 4 and 6. The forth way is bordered magic squares. 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 block-wise bordered magic squares. Moreover, the magic squares from orders 36 onwards are block-wise bordered magic squares. The advantage in studying block-wise bordered magic squares is that when we remove external borders, still we left with magic squares with sequential entries. The bordered magic squares also have the same property. The difference is that instead of numbers here we have blocks of equal sum magic squares multiples of 12. 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.
The aim is to work on construction of block-wise magic and bimagic squares