Category Archives: Magic Square

Srinivasa Ramanujan – 100-by-100 Biography Magic Square

Ramanujan and Magic Squares

Srinivasa Ramanujan had a special affinity toward numbers. His taxi-cab number (1729) incident is popular. 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.

Incidentally, in the opening page of the first Ramanujan’s notebook, there begins by working out a 3×3 Magic Square!

Having worked on a variety of special Magic squares ourselves, we could not think of a greater tribute to Srinivasa Ramanujan than this!

Summary

This is one of the biggest number puzzles we have done so far! This Biography Magic Square summarizes the important events that happened in the life of Sri Srinivasa Ramanujan.

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, Ramanujan-style.

Srinivasa Ramanujan 100x100 biography MagicSquare

Srinivasa Ramanujan 100×100 biography MagicSquare

Construction

Important dates from Ramanujan’s life were collected and these were then arranged horizontally in a row, from left to right. This row would form the top row of this biography magic square. The rest of the magic square is constructed after assembling this row.

This magic square has the properties of a conventional magic square, namely the sum of the entries along each row/column/diagonal sum up to the same magic-sum 2183.
It has these additional properties:
- starting from left to right, or, from top to bottom, we have embedded magic squares of orders 4 x 4 , 8 x 8, 12 x 12, 16 x 16, 20 x 20, and then in increased orders of 25 x25, 30 x 30, 36 x 36, 42 x 42, 49 x 49, 56 x 56, 64 x 64, 72 x 72, 81 x 81, 90 x 90, and finally 100 x 100. This is thus a cascade of magic-squares-inside-a-magic-sqaure!

Thus the total 100 x 100 Ramanujan Biography Magic square will contain the following 184 smaller magic squares of sizes as listed below:


Size of Magic Square      Number of such Magic Squares                       Total Entries


4 x 4  Magic squares          25             25   ( 4 x 4 )           =       400   squares

5 x 5  Magic squares          20             20   ( 5 x 5 )           =       500   squares

6 x 6  Magic squares          24             24   ( 6 x 6 )           =       864   squares

7 x 7  Magic squares          28             28   ( 7 x 7 )           =     1372   squares

8 x 8  Magic squares          32             32   ( 8 x 8 )           =     2048   squares

9 x 9  Magic squares          36             36   ( 9 x 9 )           =     2916   squares

10 x 10  Magic squares     19             19 (10 x 10 )          =     1900   squares


Total                                         184  (Different sized squares)   10,000   Squares


Sidenote

We have constructed a smaller 16 x 16 version of this Biography Magic Square with fewer details, which you can find here.

This was earlier published in an article ”A Unique Novel Homage to the Great Indian Mathematician“ in the March 2013 (Volume 23, Pg 146-147) Mathematics Newsletter published by the Ramanujan Mathematics Society. (download free).

How to construct a Magic Square for a Given Date

Let us do magic square to the date of Birth of our Greatest Indian Mathematician Sri Srinivasa Ramanujan.

His date of Birth is 22nd Dec 1887. The following is a Date-Of-Birth Magic Square with a Magic Total 139.

[table id=magic-square-for-given-date__4x4 /]

The following is a table with variables, which we will use in the explanation

[table id=magic-square-for-given-date__4x4_using_alphabets /]

  1. Write the date in the top row first square, month in the second square and year in two parts in the third and fourth square.
  2. Add all the four numbers and write on the top of the square (139).
  3. Now we have to make a 4×4 magic square.
  4. Draw a empty 4×4 square and replace 22 by A, 12 by B, 19 by C and 87 by D.
  5. Now we know A,B,C and D. Rest of the values we have to find out.
  6. Count all numbers in the total 1+3+9=13. Then 1+3=4. (add all numbers and make it a single digit. Write this in the H square
  7. Now add B+C ( 12 + 18 =30). Divide 30 into two parts. i.e 14 and 16. Write 14 in W square and 16 in the Z square
  8. By using the properties of Magic square, i.e all vertical, Horizontal and both Diagonal totals are equal, we are going to solve this Magic Square.
  9. In the Fourth vertical column, we know the values of D,H,Z. we have to find Out the value of S. Hence S = 139 – (D+H+Z) = 139 — ( 87+4+16) = 139 – 107 = 32. This is the value of S. Write 32 in the S square
  10. Now in Diagonals, we know the value of A and Z . Hence the value of G+Q = 139 – (22 + 16) = 139-38= 101. Divide it into two parts 50 and 51 and write it in the F and R squares. F = 50 and R = 51.
  11. In another diagonal, we know the value of D and W . Hence the value of G+Q = 139 – (87 + 14) = 139-101= 38. Divide it into two parts 17 and 21 and write it in the F and R squares. G = 17 and Q = 21.
  12. In the second row, we know the value of E,G,H. Hence value of E=139-(E+G+H)= 139 –( 50+17+4)= 139 – 71 = 68. Write 68 in E.
  13. In the third row, we know the value of Q,R,S. Hence value of P=139-(Q+R+S)= 139 –(21+51+32)= 139 – 104 = 35. Write 35 in P.
  14. Now in the second vertical column, we know the values of B,F,Q. Hence value of X = 139 – (12+50+21) = 139- 83 = 56. Write 56 in X
  15. Now in the third vertical column, we know the values of C,G,R. Hence value of X = 139 – (18+17+51) = 139- 86 = 53. Write 53 in Y
  16. Now add the values of W,X,Y and Z. If you get 139 is the total,

Hurray!!! MAGIC SQUARE FOR 22-12-1887 IS READY

Enjoy Magic square for your date of Birth!!!

How to construct a 3×3 Magic Square for a Given Total

Let the given Number be 63. We will get to the following 3 X 3 Magic-Total 63.

[table id=construct-magic-square-for-given-total-63__3x3 /]

  1. Let us do 3 x 3 magic square with Magic Sum of 63.
  2. From 63, minus 15 ( the row total of 3×3 base magic square. (63-15=48)
  3. Divide 48 by 3(since we have decided to do 3×3 magic square (48/3=16).
  4. Add one to this (16+1=17)
  5. Now start constructing a 3 x 3 magic square with 17 as the beginning number and (Top row middle – Just in the place of 1).

Note:

  • For constructing 3×3 magic square -15, then divide by 3, then add 1.
  • For constructing 4×4 magic square -34, then divide by 4, then add 1. and
  • For constructing 5×5 magic square -65, then divide by 5, then add 1.

While dividing by 3 or 4 or 5, if any fraction comes, put that fraction in all the squares. You should not delete or make it decimal.

How to construct 6 x 6 Magic square

Here is a 6 x 6 Magic Square with Magic-Total 111

[table id=6x6-magic-square--total-111 /]

  1. Draw a 6 x 6 empty square.
  2. Draw a bold line after the third square, Horizontally and vertically.
  3. Now the 6×6 magic square will be divided into four 3 x 3 Magic squares.
  4. Start filling the 3 x 3 magic square on the top left with numbers 1 to 9. and top right from 19 to 27, bottom left with 28 to 36 and bottom right with 10 to 18.
  5. Now exchange the numbers 8,5,4 from the top left 3×3 square to the bottommost left 3×3 squares with the numbers 35, 32, 31 and vice versa.
  6. Your 6×6 magic square is ready. All vertical, Horizontal and both diagonal totals are equal ( 111) .

How to construct a 4 x 4 magic square

This is a 4 x 4 Magic Square with Magic-Total 34

[table id=4x4-magic-square /]

  1. Draw a 4×4 empty square.
  2. Draw two diagonals.
  3. Start counting all the squares from the top left towards right.
  4. Now start writing numbers 1 to 16, count all squares and write numbers ONLY IN THE BLANK SQUARES (2,3,5,8,9,12,14,15 are to be written)
  5. Now start writing number from the bottom most right square towards left, count all squares, BUT WRITE NUMBERS ONLY IN THE SQUARES HAVING DIAGONAL LINES ONLY (1,4,6,7,10,11,13 AND 16) ARE TO BE WRITTEN.
  6. Magic square 4×4 is ready.

Note:  This method is applicable only for 4×4 magic square.  For 6×6, 8×8 there are separate methods available.

 

How to construct an odd order magic square?

This is a 3 X 3 Magic square with Magic-Total 15

[table id=3x3-magic-square__total-15 /]

We will discuss the construction of this, step-by-step.

  1.  Put number one in the upper middle square.
  2. Always write the next number ONE SQUARE RIGHT AND ONE SQUARE ABOVE
  3. AND WHEN IT GOES:
    • Beyond boundary in the upper side, write it vertically down most square.
    • Beyond the boundary in the right side, write it in the horizontally leftmost Square.
    • When the square is already filled with a number, just step-down one square vertically below.
  4. And after the Top right square always write the next number one square vertically below.

With these simple steps you can make any biggest “ODD NUMBER MAGIC SQUARE IN THE WORLD”

These steps are useful ONLY FOR ODD NUMBER MAGIC SQUARE AND WILL NOT BE USEFUL FOR EVEN NUMBER MAGIC SQUARES.

Magic Squares – What are they?

Magic squares are a set of numbers filled in a 3×3, 4×4,… squares and after filling it, all the vertical, Horizontal and both diagonal totals are equal.

Properties of the Magic Square:

  1. Magic squares are made in 3×3, 4×4, 5×5….etc.
  2. Generally a 3×3 magic square is filled with numbers 1 to 9 and 4×4 magic square is filled with 1 to 16,…… In general a “nxn” magic square will be filled with 1 to n2 numbers.
  3. In general in a “nxn” magic square will have “n” rows and “n” columns and n2 Small squares.
  4. After filling all columns, ALL VERTICAL, ALL HORIZONTAL AND BOTH DIAGONAL THE TOTALS ARE EQUAL.
  5. A magic square done from 1 to 9, 1 to 16,… are called as Base Magic square.
  6. The total of a row/column of a Base Magic Square will be “First number plus the Last number added together, divided by two, and then multiplied by the Number of rows”. i.e. {( 1+n2 ) / 2} ……. {( 1+n2 ) divided 2}
  7. Generally natural numbers 1,2,3,… are used for forming the magic squares and we can use fractions, zero, negative numbers and squares, cubes etc etc…
  8. We can make 3×3, 4×4, 5×5… magic squares for a particular number, and for a given space interval at our choice.
  9. Also we can make magic squares for a given number, given year or given date of birth etc.

Srinivasa Ramanujan – 16×16 Biography Magic Square

This Biography Magic Square summarizes the important events that in the life of Sri Srinivasa Ramanujan.

How it was constructed:

Important dates in the life of Ramanujan were taken, two digits at a time, representing either the date or the month or the first or second part of the four-digit year. As an example, Ramanujan’s birth-day 22-12-1887 is taken in four separate entries as 22 12 18 87. These were then laid out in the top of the Magic Square, in the first column. Then, a complete Magic Square was built on top of these numbers, with the following additional feature : each  square indicated by a separate color (in this case, there are 4 such 4×4 sub-sqaures), which are magic squares themselves!

Srinivasa Ramanujan 16x16 biography Magic Square

Srinivasa Ramanujan 16×16 biography Magic Square

(Download Ramanujan_16x16_biography_Magic_Square in excel format).

This is a smaller version of the 100-by-100 and 125-by-125 biography magic squares that we have constructed.

This was earlier published in an article ”A Unique Novel Homage to the Great Indian Mathematician“ in the March 2013 (Volume 23, Pg 146-147) Mathematics Newsletter published by the Ramanujan Mathematics Society. (download free).

If you find this interesting, you could construct your own! If you want some help, drop a mail to me at contact[at]jollymaths[dot]com.