Magic squares complete view odd and even square matriX

 Magic squares

Magic squares are among the Mathematical amusements.

The sum of the first k natural numbers is 

$\frac{k(k+1)}{2}$

.

In a magic square of size n x n , the sum of each row, column, and diagonal is the same.

Substitute k=n² into the formula:  

A magic square is a square array of integers from 1 to n² is said to be of order n. 

The sum of the integers from  1 to n²  is 

So in a magic square the numbers in each row column and diagonal have the magic sum 

This is to be a magic square. Each row, column, and diagonal is to add to the same number. 

How can this magic square be completed?

 Solution:

Odd Square Matrix

finding 4 corners    Value    D     C    A    B

for matrix 3 x 3 

finding 4 corners Value	D	C	A	B
n x n	n + 1	D - 2	B + 2	x n
for matrix 3 x 3	1 x 3= 3 3 + 1 = 4	4 - 2 = 2	6+2=8	2 x 3 = 6
5 x 5	2 x 5 = 10 10+1 = 11	11 - 2 = 9	15+2=17	3 x 5 = 15
7 x 7	3 x 7 = 21 21+1 = 22	22 - 2 = 20	28+2=30	4 x 7 = 28
9 x 9	4x 9 = 36 36 +1 =37	37 - 2 = 35	45 + 2 = 47	5 x 9 = 45
11 x 11	5 x 11 = 55 55 + 1 = 56	56 - 2 = 54	66 + 2 = 68	6 x 11 = 66
13 x 13	6 x 13 = 78 78 + 1 = 79	79 - 2 = 77	91 + 2 = 93	7 x 13 = 91
15 x 15	7 x 15 = 105 105 + 1 = 106	106 - 2 = 104	120 + 2 =122	8 x 15 = 120
17 x 17	8 x 17 = 136 136 + 1 = 137	137 - 2 = 135	153 + 2 = 155	9 x 17 =153
19 x 19	9 x 19 = 171 171 + 1 = 172	172 - 2 = 170	190 + 2 = 192	10 x 19 = 190
21 x 21	10 x 21 = 210 210 + 1 = 211	211- 2 = 209	231 + 2 = 233	11 x 21 = 231

3 x 3 odd square matrix  

Sum is 15

Sum is 6  ( Number can be repeated and use only1,2 & 3) 

Sum is 18

5 x 5 odd square matrix  

Sum is 65

Numbers 1 to 25

Start 1 -- --- I First Row middle

Complete 25 -- --- Last Row middle

Number Entering Method

 L shape

diagonal   /  if ends with no columns then L shape

diagonal   /  if ends with filled column next to the Row ↓

we can use this method for odd number matrix like 3,5,7,9,11,13,.....

step by step images

Step 1

7 x 7 odd square matrix  

Sum is 175

9 x 9 odd square matrix  

Sum is 369

11 x 11 odd square matrix  

Sum is 671

13 x 13 odd square matrix  

Sum is 1105

15 x 15 odd square matrix  

Sum is 1695

17 x 17 odd square matrix  

Sum is 2465

19 x 19 odd square matrix  

Sum is 3439

21 x 21 odd square matrix  

Sum is 4641

10 x 10 EVEN square matrix  

Sum is 505

I TRIED UP TO 21 X 21 MATRIX 

TRY MORE LEVEL