LOGIC BEHIND MAGIC SQUARE

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.