COMMON DIFFERENCE METHOD
The ‘COMMON DIFFERENCE METHOD" suggested by Christna Rachel and Chris Mary is the easiest method for forming Magic Squares of the order 3 X 3.This method is called ‘COMMON DIFFERENCE METHOD’ as the Arithmetic Progression used for filling is formed by adding and subtracting the first 4 multiples of the common difference to the mid term . All you will have to do is to fill the respective cells after finding the mid term, as shown in Fig. MS: 1.
Step:1 Find the mid term by dividing the Magic Sum by 3 and write it in the center cell of the grid. For example if
The Magic Sum = 75
The mid term = 75/ 3= 25.
(Care should be taken to see that the Magic Sum chosen is a multiple of 3 so that we get Natural numbers for filling. We can make Magic Squares with any real number, but normally Natural numbers are used.)
Step: 2 Fill the remaining cells in the order given in Fig. MS: 2. We can start form any corner cell and fill either in clockwise or anticlockwise direction. The number to be filled in the first cell is obtained by adding the first multiple of the common difference and the number for the second cell is obtained by adding the second multiple . At the same time in the diagonally opposite cell we shall start the reverse operation in the same direction.
For example; if we choose 5 as common difference then we start our filling with 30 (25 + 5 =30), then with 35. At the same time from the diagonally opposite cell we shall start filling with 20 (25 – 5 =20), then with 15.
Step: 3 The third filling is done with difference of the mid term and thrice the common difference. In this example it is 25 – (3 x 5 )= 25 -15 = 10. At the very same time 25 +15= 40 is filled in the opposite cell.
Step: 4 Fill the fourth cell with the sum of mid term and four times the common difference. That is 25 + ( 4 x 5 ) = 45; and in the opposite cell use the difference of four times the common difference from the mid term. (25 – (4x 5) = 25 – 20 = 5 is written.)
Fig. MS 1- Filling Pattern.
Here M is the mid term and D is the common difference. +1D, +2D.....and -1D,-2D etc denote the multiples of D to be added or ubtracted with mid term.
Fig. MS 2
See the order in which each multiple of 5 is added or subtracted.
Even though this method is explained in steps, we can finish the Magic Square in the first step itself.
Using Common Difference Method, make 3 x 3 Magic Square of sum 129, where the common difference is 4. Now make your own questions and enjoy your leisure time.