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LOGIC BEHIND MAGIC SQUARE

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Assume a magic square of 3x3
A
B
C
D
E
F
G
H
I
The above figure shows a 3x3 magic square. In the magic square sum of colum, row, diagnol numbers should be same
A+B+C=D+E+F=G+H+I=A+D+G=B+E+H=C+F+I=A+E+I=C+E+G=sum
A+B+C
D+E+F
G+H+I
A+D+G
B+E+H
C+F+I
A+E+I
C+E+G
By  seeing the  above there is 8 type of sum is available. Row wise-3,  column wise-3, diagnol wise-2. Condition#1:
There are 8 different combinations are there with same sum in a magic square.
In the above 8 combinations A=2 times, B= 3 times, C= 2 times, D=3 times E=4 times, F=3 times, G=2 times, H=3 times, I= 2 times
Hence the above forms the condition#2. In simply condition#2, middle number E appears 4 times in 8 combinations.
Hence row, column, diagnol sum should be equal. In 3x3 magic square we are taking numbers from 1 to 9. From 9 numbers we need to take 3 numbers. From combination formula we can find out the the number of ways we can write
Combinations= 9c3  =84.
From 9 numbers by taking 3 numbers at a time we can form 84 combinations. All 84 combinations we are showing here.






1+2+3=6
2+3+4=9
3+4+5=12
4+5+6=15
5+6+7=18
6+7+8=21
7+8+9=24
1+2+4=7
2+3+5=10
3+4+6=13
4+5+7=16
5+6+8=19
6+7+9=22
(1)
1+2+5=8
2+3+6=11
3+4+7=14
4+5+8=17
5+6+9=20
6+8+9=23

1+2+6=9
2+3+7=12
3+4+8=15
4+5+9=18
5+7+8=20


1+2+7=10
2+3+8=13
3+4+9=16
4+6+7=17
5+7+9=21
(3)

1+2+8=11
2+3+9=14
3+5+6=14
4+6+8=18
5+8+9=22


1+2+9=12
2+4+5=11
3+5+7=15
4+6+9=19



1+3+4=8
2+4+6=12
3+5+8=16
4+7+8=19
(6)


1+3+5=9
2+4+7=13
3+5+9=17
4+7+9=20



1+3+6=10
2+4+8=14
3+6+7=16
4+8+9=21



1+3+7=11
2+4+9=15
3+6+8=17




1+3+8=12
2+5+6=13
3+6+9=18
(10)



1+3+9=13
2+5+7=14
3+7+8=18




1+4+5=10
2+5+8=15
3+7+9=19




1+4+6=11
2+5+9=16
3+8+9=20




1+4+7=12
2+6+7=15





1+4+8=13
2+6+8=16
(15)




1+4+9=14
2+6+9=17





1+5+6=12
2+7+8=17





1+5+7=13
2+7+9=18





1+5+8=14
2+8+9=19





1+5+9=15






1+6+7=14
(21)





1+6+8=15






1+6+9=16






1+7+8=16






1+7+9=17






1+8+9=18






(28)

Total 84 combinations

As you seen above table total combinations from 9 numbers by taking 3 numbers at a time is
28+21+15+10+6+3+1=84.
Now we will verify the sum of the combinations







Sum=6 combinations
1
Sum=7 combinations
1
Sum=8 combinations
2
Sum=9 combinations
3
Sum=10 combinations
4
Sum=11combinations
5
Sum=12 combinations
7
Sum=13combinations
7
Sum=14 combinations
8
Sum=15 combinations
8
Sum=16combinations
8
Sum=17 combinations
7
Sum=18 combinations
7
Sum=19 combinations
5
Sum=20 combinations
4
Sum=21 combinations
3
Sum=22 combinations
2
Sum=23 combinations
1
Sum=24 combinations
1
TOTAL
84 COMBINATION
By seeing the above table in SUM=14,SUM=15,SUM=16,  there are 8 combinations are available. So condition#1 is satisfied for sum=14, sum=15, sum=16. Since we will analyse the above 3 for condition#2. The following table analyses the condition#2.
Sum=14 combinaztions
Sum=15 combinaztions
Sum=16 combinations
1+4+9
1+5+9
4+5+7
1+5+8
1+6+9
3+4+9
1+6+7
2+4+9
3+5+8
2+3+9
2+5+8
3+6+7
2+4+8
2+6+7
2+5+9
2+5+7
3+4+8
2+6+8
3+4+7
3+5+7
1+6+9
3+5+6
4+5+6
1+7+8
In the above condition number#2 is satisfied for the sum=15. Since to prepare a magic square sum=15 combinations are suitable because 5 is appeared 4 times. Remaining combinations are not suitable. Now we can prepare the magic square with sum=15 combinations. The different combinations to prepare the magic square 1+5+9, 1+6+9, 2+4+9, 2+5+8, 2+6+7, 3+4+8, 3+5+7, 4+5+6. In this combinations 5 appeared 4 times. By using the above combinations the magic square is like follows.
4
9
2
 3
5
7
8
1
6
In the above magic square 3x3 we are used numbers from1 to 9 arranged in 9 boxes. In any direction the sum equal to 15.
Row wise  4+9+2=15, 3+5+7=15, 8+1+6=15.
Column wise 4+3+8=15, 9+5+1=15, 2+7+6=15.
Diagnol wise: 4+5+6=15, 2+5+8=15.
 



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