how the ancient scholar form the magic square and what was their techniques
thinkiit pre foundation class viii mental
exactly as the title suggests people
thinkiit prefoundation
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A scripture about the abramelin squares.Descripción completa
Description : A scripture about the abramelin squares.
A scripture about the abramelin squares.Full description
bookFull description
Descripción: book
OAM OAM Shri Maha Ganapataye Namah
Construction of magic squares Karanam L. Ramakumar India
A magic square is a n x n grid (n is an odd number) in which natural numbers between and including 1 to n2 are filled such that the sum of the numbers in any horizontal, vertical or diagonal rows is same. For example, let us consider a 3 x 3 grid. We have 32 = 9 numbers (1 to 9). We fill the 3 x 3 grid as shown below: 8
1
6
3
5
7
4
9
2
The sum of any horizontal, vertical or diagonal row is 15. Almost 50 years ago, I came across an ingenious method of constructing any n x n grid (n is odd) and finding out the “magic sum” and also the middle number. I thought I would share with others. To construct a magic square, filling the cells in the grid has to be done in a methodical manner. Let us take an example of 5 x 5 grid (n = 5). We have to fill the numbers 1 to 25 in the cells of the grid. The sequence sequence of filling is shown below:
17
24
Start
1
8
15
5
7
14
16
6
13
20
22
12
19
21
3
18
25
2
9
23 4 10 11
End
1. Start from the first row middle cell. Write 1 there.
2. Then come to the last row and shift to the adjacent cell on right. Write 2 there. 3. Then proceed diagonally to right filling each cell with subsequent numbers until you come to the right edge. 4. Proceed to the first cell of the same row on the left edge and go up to the next higher cell in the same column. Write the next number. 5. Proceed diagonally to right filling each cell with subsequent numbers until you come to a filled cell. 6. Come down by one cell and write the next number. 7. Again proceed diagonally to right filling each cell with subsequent numbers until you come to a filled cell or the top edge. 8. If it is a filled cell repeat the steps 6 and 7. 9. If it is the cell in the first row AND NOT the last cell in the first row, come to the last row and shift to the adjacent cell on right. Write the next number there. Repeat the steps 3 onwards. 10.If it is the last cell of the first row, come down by one cell and write the next number. Go to the first cell of the same row on the left edge and go up to the next higher cell in the same column. Write the next number. 11.Repeat the steps 2 onwards until you exhaust all the grids. If you have filled all the cells correctly, you will notice that the last number will be in the same column as the first number (middle cell of the last row). The middle number will always be average of the first and last numbers of the row, column or diagonal containing the middle cell. In general the middle number is always In the present case, n=5 and the middle number is
52
+
n2
+
2
1
1
.
= 13 2 In the 5 x 5 grid as shown above, the first number 1 and the last number 25 are in the same column in the first and last cells of the column. The middle number is 13 which is the average of 1 and 25 (column), 4 and 22 (row), 17 and 9 or 15 and 11 (diagonals).
The magic sum (sum of the numbers in any row, column or diagonal) is given by:
n (n2 + 1) 5 (52 + 1) 5x(26) Sum = = 65 = = 2 2 2 There could be many other ways of constructing magic squares. Following the above logic let us construct the 7 x 7 and 9 x 9 magic squares
7 x 7 magic square (we have to fill numbers 1 to 49) 30
39
48
1
10
19
28
38
47
7
9
18
27
29
46
6
8
17
26
35
37
5
14
16
25
34
36
45
13
15
24
33
42
44
4
21
23
32
41
43
3
12
22
31
40
49
2
11
20
The middle number is 25. It is the average of 1 and 49 (column), 5 and 45 (row), 30 and 20 or 28 and 22 (diagonals). Now the magic sum is
7 (7 (72 + 1) 7x(50) = = 175. Let us verify. The sum of the numbers in the first row = 2 2 is 30+39+48+1+10+19+28 = 175. 9 x 9 magic square (we have to fill numbers 1 to 81) 47
58
69
80
1
12
23
34
45
57
68
79
9
11
22
33
44
46 46
67
78
8
10
21 21
32
43
54
56 56
77
7
18
20
31 31
42
53
55
66 66
6
17
19
30
41
52
63
65
76
16
27
29
40
51 51
62
64
75
5
26
28
39
50
61 61
72
74
4
15
36
38
49
60
71 71
73
3
14
25 25
37
48
59
70
81
2
13
24
35
2
Middle number is (n +1)/2 = 41. The magic sum is
=
Go ahead. Construct other magic squares and enjoy!
