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Magic Square: Step by Step Explanation


Magic Square is  square  which is filled with  numbers in such a way that total of all digits in a row, column or diagonal is always same. If we add the numbers in rows, Columns & diagonals the result is same number. That number is known as square constant.

We will easily find out that square constant by using following formula.


  Constant(Sum) = n[(n2 + 1)/2]

   Where, n is number of rows or columns in the square.

  For example, if n = 3,

          Constant = 3[(32 + 1)/2]

= 3 X 5 = 15

It means,  if we add the numbers of a row or column or diagonal, we get the answer 15.

Similarly, the magic constant for 5 x 5 is 5 x 13 = 65.


Many people have found these squares fascinating and so they started to call it as magic.


There are three different types of Magic squares:

1)Odd numbered Magic square

2)Singly even Magic square

3)Doubly even Magic square


1)Odd numbered Magic square: In odd number Magic square odd number of rows or columns are there e.g. 3×3, 5×5, 7×7, 9×9 …….

In general, odd number square is more known and in competitive exams also it is  asked often,  so we will solve it first.

Example: solve 3×3 square.


Step 1: Place the number 1 in center box of the Top Row.

When we solve for odd number magic square our first step is always this.

We can put the 1 in center box of bottom row or left most column or right most column also but here we start with Top row.

magic square


Step 2: Fill in the next number by using up one –  right one  pattern.

For number 2  move 1 row up and right side to the column. When we move like this, we didn’t get actual box there, initially we consider imaginary box.


magic square


Now in such condition move 2 to the bottom most box in the same column.


Step 3: Now for next number repeat the step 2, move one row up and one column right.


magic square 

Now in this condition move 3 in the farthest box of the same row. 


magic square


Step 4: Again, we move one row up & one column right , but now our box is already occupied as no. 1 is there. So, in such  condition we come back to our number & place the next number directly below it.


Step 5: We fill next number 5 & 6 by using same one row up one column right method.




Step 6: Again, when we move one row up & one column right, we will get the imaginary box as shown in fig. As there is no row or column in same alignment so we move back to the number 6 & write down next number directly below it.





Step 6: When we move one row up & one column right, we got number 8 as shown in fig. So we move it to the farthest place in same column.


Magic square   


Step 7: Again, follow the rule one row up & one column right, we got number 9 as shown in fig. So, we move it to the farthest place in same row.  

Magic square


By following some simple rules our magic square is filled completely. We can summarize these steps as follow:

If we start our magic square by filling one in the center of top row, then remaining box should be filled by one up row & one right column rule. But some exceptions are there:


i)If the movement takes you to the box which is above the top row then move this number to the bottom row of that column.


ii) If the movement takes you to the box which is right side of the magic square’s right most column then move that number to the farthest place of the same row.


iii) If the movements take you to the box that is already occupied then go back to the previous box and write down the next number directly below it.


There are many possible ways to make this magic square.

Here I show some possible ways of making 3 x 3 Magic Square.



So, these are 4 possible ways of making magic square. If we observe them carefully, we understand that if we tilt any one square from these in four directions, we will get other squares.


To study more find below links:

i. Find the square of specific number in 4-5 sec.

ii. Multiply any number by 9,99,999…in some sec.

iii. Calendar problem tricks 



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