1. MAGIC
SQUARES
© Dhaval Bathia

2. What is a magic-square ?
A magic square is a box filled with digits in such a manner that the total of all rows, columns and diagonals is the same..
Let us have a look at such a magic square…. Below is a magic square formed with the numbers from 1 to 9
As we can see, the total of each row, column and diagonal is the same..i.e. 15 4 3 8 9 5 1 2 7 6
© Dhaval Bathia

3. 4 3 8 9 5 5 1 2 7 6
Let us check the totals of each rows, columns and diagonals
ROWS
COLUMNS
DIAGONALS
= 15
= 15
= 15
= 15
= 15
= 15
= 15
= 15
© Dhaval Bathia

4. Since there were 9 squares in the grid, we used numbers from 1 to 9
Such a square where each row, column and diagonal gives the same total is called a Magic Square
What we saw just now was a (3 x 3) magic square…because there are 3 rows and 3 columns on each side
Since there were 9 squares in the grid, we used numbers from 1 to 9
Next, we will see a (5 x 5) magic square
© Dhaval Bathia

5. 11 10 4 23 17 18 12 6 5 24 25 19 13 7 1 2 21 20 14 8 9 3 22 16 15
This is a 5 by 5 magic square……
It has numbers from 1 to 25
The total of each column, row and diagonal is 65
© Dhaval Bathia

6. We will now study how to form these magic squares…
Often questions on Magic Squares are asked in competitive exams.
Those who do not know the art of forming these squares use ‘trial and error method.’ However, it is virtually impossible for most people to form these squares using ‘trial and error’ method
In this presentation, I will teach you the correct method.
I will teach you which number to put where so that you can get the correct answer instantly..
Its actually very simple
© Dhaval Bathia

7. There are 4 rules that we will follow :
A) Put 1 in the centre-square of last column
B) Go to South-East and put the next number there
C) If south east is full; go to west and put the next number
D) After filling the last square of the grid, GO TO WEST
DIRECTIONS
NORTH
WEST
EAST
SOUTH-EAST
SOUTH
© Dhaval Bathia

8. Let us Begin ! A) Put 1 in the centre-square of last column
B) Go to South-East and put the next number there
C) If south east is full; go to west and put the next number
D) After filling the last square of the grid, GO TO WEST
Let us Begin !
© Dhaval Bathia

9. 1 Rule 1 says put 1 in centre square of last column So we put 1 there
© Dhaval Bathia

10. 1 2 Rule 2 says go to south-east
However, there is no square in south-east. So we assume an imaginary square.
But you cannot put a number in an imaginary square. Hence 2 will be written in a square farthest from it. Look at the figure.
© Dhaval Bathia

11. 1 2 3
From 2 we go to south-east and put 3
© Dhaval Bathia

12. 4 1 2 3 From 3 we go to south-east.
But there is no square, so again we form an imaginary square and put the next number in the square farthest from it.
This is how it will be done
© Dhaval Bathia

13. 4 5 1 2 3
From 4 we go to south-east and put 5
© Dhaval Bathia

14. 4 6 5 1 2 3
From 5 we go to south-east to put 6. But, south east is full. In this case, we refer the third rule which says ‘if south-east is full go to west’
So we go to the west of 5 and put 6
© Dhaval Bathia

15. 4 6 5 7 1 2 8 3
From 6 we go to south-east to put 7. From 7 we go to south-east and put 8
© Dhaval Bathia

16. 10 4 6 5 7 1 2 8 9 3
From 8 we go to south-east, use imaginary square and put 9
From 9 we go to south-east use imaginary square put 10
© Dhaval Bathia

17. 11 10 4 12 6 5 13 7 1 2 14 8 9 3 15
Next, south-east of 10 is full, so we go to west and write 11
From 11 south-east we write 12, from 12 south-east we write 13 and similarly write down 14 and 15
© Dhaval Bathia

18. 11 10 4 12 6 5 13 7 1 2 14 8 9 3 16 15
15 is the last square of the grid. We come to the fourth and last rule which says :
‘After filling the last square of the grid, GO TO WEST’
So, from 15 we go to west and write down 16
© Dhaval Bathia

19. 11 10 4 17 18 12 6 5 19 13 7 1 2 20 14 8 9 3 16 15
From 16 we go to 17
From 17 we go to 18
From 18 we go to 19
From 19 we go to 20
© Dhaval Bathia

20. 11 10 4 23 17 18 12 6 5 24 25 19 13 7 1 2 21 20 14 8 9 3 22 16 15
From 20 we go to 21
From 21 we go to 22
From 22 we go to 23
From 23 we go to 24
From 24 we go to 25
© Dhaval Bathia

21. 11 10 4 23 17 18 12 6 5 24 25 19 13 7 1 2 21 20 14 8 9 3 22 16 15
And Lo !
Our Magic Square is Formed !!
© Dhaval Bathia