Ramanujan Magic Square: Definition, Construction & Solved Examples

Ramanujan Magic Square is a matrix of numbers in which every row, column and diagonal adds up to the same number. A square with many distinct integers placed in such a way that the sum or total of the numbers is the same in every row, column, and main diagonal, as well as typically on some or all of the other diagonals, is known as a magic square.

Srinivasa Ramanujan had a special affinity toward numbers. A Mathematician without parallel, he made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. His works have been collected and analyzed throughout the world.

What is Ramanujan Magic Square?

In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2. The term “magic square” is also sometimes used to refer to any of the various types of word square.

Magic squares may be classified into three types:

(i) Additive magic square (standard magic square): Its elements are arranged in such a way that they add to give a magic sum (S) along rows, columns, and diagonals.

(ii) Multiplicative magic square: Its elements are arranged to multiply, giving a magic product (P) along rows, columns, and diagonals.

(iii) Additive-multiplicative magic square: Its elements are arranged to add and multiply to give S and P, respectively, along rows, columns, and diagonals.

Mathematician Ramanujan created a Magic Square: one of its kind fascinating mathematical object that has a deep and mysterious history that has been so far unmanageable for researchers and mathematicians to decipher. Knowing and learning about this magic square can be helpful and rewarding for students.

Ramanujan magic square is a special kind of magic square that was invented by the Indian mathematician Srinivasa Ramanujan. It is a 3×3 grid in which each of the nine cells contains a number from 1 to 9, and each row, column, and diagonal have the same sum.

The above picture is a 3-by-3 magic square from Ramanujan’s notebook. The elements in the middle row, middle column and each diagonal are in arithmetic progression, and all columns, rows and diagonals add up to 15.

Ramanujan constructed different magic squares of the same size but with different magic constants. For instance, he constructed an even square of 4 × 4 with magic constants equal to 34 and 35. For an odd order, Ramanujan constructed a 5 × 5 square with magic constants equal to 65 and 66. Ramanujan built 7 × 7 squares with magic sums 170 and 175 in two different problems. He

also constructed magic rectangles.

It is possible to create a Ramanujan magic square by starting with any 3×3 magic square and adding or deleting the same number from each cell. This implies that there are an endless number of Ramanujan magic squares that could exist! It is said that he discovered this method while working on his famous notebook.

There are many things that make this method of creating magic squares unique.

• Unlike most other methods, it takes a unique approach. Ramanujan’s strategy makes use of geometry rather than algebra or mathematics. This improves its visual attractiveness and makes it simpler to comprehend. Additionally, compared to other techniques, it has the advantage of producing larger magic squares.
• The way the Ramanujan magic square is put together is another distinctive feature. The majority of other ways to make magic squares employ a predetermined configuration. This indicates that the numbers in the square are set up in a particular order. On the other hand, Ramanujan’s approach makes use of a flexible structure. As a result, the square can be built with greater originality and variety.
• Last but not least, Ramanujan magic square is special because it can be utilized to make various magic squares. Most other techniques can only produce one kind of magic square. Ramanujan’s technique is significantly more robust and adaptable thanks to this flexibility.

How Ramanujan Magic Square was Constructed

Here’s how one possibility of Ramanujan’s square can be constructed.

The important dates in the life of Srinivasa Ramanujan were compiled from various sources. These dates were taken two digits at a time, representing either the date of the month or the month or the first/second half of the four-digit year. As an example, Ramanujan’s date-of-birth 22-12-1887 is taken as four separate entries as 22 12 18 and 87. In short, Ramanujan’s entire life history is reproduced here, from his birth to till date in Ramanujan style. In India, the day December 22’ has been declared as National Mathematics Day.

Since the sum of the birth date is 139, we need to make sure that all the other rows and columns are 139. Also, the sum of all the main diagonals is 139.

But now follow several other points of interest:

The sum of the four corner elements of the square is the same number (note the numbers in the cells coloured red), i.e., 22 + 87 + 11 + 19 = 139.

The sums of the numbers in the two sets of like-coloured cells are again the same number (139).

The sums of the numbers in the two sets of like coloured cells here are yet again the same number (139).

The sum of the numbers in the four central cells is 139.

The sums of the numbers in these like-coloured 2 x 2 blocks are all 139.

And so also in these two coloured 2 × 2 blocks.

Pros and Cons of Ramanujan Magic Square

The Pros and Cons of Ramanujan Magic Square are as follows:

Here are the advantages of Ramanujan Magic Square

• Compared to other approaches, it is easy to understand and more aesthetically pleasing.
• Additionally, it is more adaptable, promoting innovation in the design of the square.
• It can be utilized to make many kinds of magic squares.

Here are the disadvantages of Ramanujan Magic Square

• It is not a widely utilized procedure, so it is applied sparingly.
• It could be more challenging to describe to others because it is less well-known.
• It can be difficult to extend to higher dimensions because it depends on geometry.

Applications of Ramanujan Magic Square

Here are some practical applications of Ramanujan’s Magic Square.

• It could be used in a wide spectrum of fixed resource allocation over 2-D topographical entities.
• It is used in Birkhoff — von Neumann decomposition, Quantum permutation matrices etc.

Formula of Ramanujan Magic Square

Here’s the formula of Ramanujan Magic Square

The numbers in the first row should be as follows:

The number in the second row should be as follows:

The number in the second row should be as follows:

The number in the second row should be as follows:

Hence, Ramanujan’s Magic square is of the form:

Here, DD = Date of birth.

MM= Month of birth.

CC= Century of birth.

YY= Year of birth.

Important Points on Ramanujan Magic Square

Here are some of the key points of Ramanujan Magic Square

• The sum of each row or column or diagonal is the same.
• The sum of the four corner elements of the square is the same number.
• Since 2012, National Mathematics Day is celebrated on December 22 in memory of one of the finest and most legendary of mathematicians, Srinivasa Ramanujan.

Solved Examples on Ramanujan Magic Square

Here are some solved examples of the Ramanujan Magic Square for you to prepare for your exam.

Example 1: Rohan’s Birthday Is on 02 April 1995. Draw a Rohan’s Magic Square.

Solution: Since Rohan’s birthday is on 02 April 1995.

We have DD = 02

MM = 04

CC = 19

YY = 95

Using the concept of Ramanujan’s Magic square, we have,

The numbers in the first row should be:

02, 04, 19, 95

The numbers in the second row should be:

95 + 1 = 96, 19 – 1 = 18, 04 – 3 = 01, 02 + 3 = 05

The numbers in the third row should be:

04 – 2 = 02, 02 + 2 = 4, 95 + 2 = 97, 19 – 2 = 17

The numbers in the fourth row should be:

19 + 1 = 20, 95 – 1 = 94, 02 + 1 = 03, 04 – 1 = 03

Therefore, the magic square is:

Example 2: Complete the following Ramanujan Magic Square.

Solution: From the first row, it is observed that:

DD = 08

MM = 10

CC = 19

YY = 90

The numbers in the second row should be as follows:

YY + 1. CC – 1, MM – 3, DD + 3, that is,

91, 18, 07, 11. Therefore, the missing numbers in the second row are 91 and 11.

The numbers in the third row should be as follows:

MM – 2, DD + 2, YY + 2, CC – 2, that is,

08, 10, 92, 17. Therefore, the missing numbers in the third row are 08 and 92.

The numbers in the fourth row should be as follows:

CC + 1, YY – 1, DD + 1, MM – 1, that is

20, 89, 09, 09. Therefore, the missing numbers in the third row are 89 and 09.

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If you are checking Ramanujan Magic Square article, check related maths articles:
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