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What to know about the Khajuraho method
How I discovered the Khajuraho method (and the basic pattern method).
I think it will be interesting to tell how I have discovered the Khajuraho method of construction. Strange but true,
the story begins with the discovery of the method of construction to produce squares for each random chosen
magic sum (see page Each magic sum). The key of this method is an impure 4x4 magic square with (8 different)
positive and (8 different) negative digits and the magic sum of 0. The minimum difference between digits is four
because of the remainder. The random chosen magic sum must be devided by four, which gives a maximum remain-
der of 3 (for example 403 / 4 = 100 remainder 3). Because the minimum difference between the digits is four, after
correction of the maximum remainder of 3, there are still 16 different digits in the square.
What I have discovered is that it is possible to translate a panmagic 4x4 square, for example the Khajuraho square,
to the key of the magic sum of 0. Translate the digits 1 up to 16 in -30, -26, -22, -18, -14, -10, -6, -2, +2, +6, +10,
+14, +18, +22, +26 en +30; see below.
Khajuraho square translation in key of magic sum of 0
|
7
|
12
|
1
|
14
|
|
-6
|
14
|
-30
|
22
|
|
2
|
13
|
8
|
11
|
|
-26
|
18
|
-2
|
10
|
|
16
|
3
|
10
|
5
|
|
30
|
-22
|
6
|
-14
|
|
9
|
6
|
15
|
4
|
|
2
|
-10
|
26
|
-18
|
I had already discovered that it is possible to produce from the 4x4 key square of the magic sum of 0 a bigger, for
example a 8x8 key square of the magic sum of 0. You need three additional 4x4 squares. In the first additional
square you use the digits +/- 34, 38, 42, 46, 50, 54, 58 en 62 (put the lowest, second lowest, …, second highest
and highest digit in exactly the same place as in the 4x4 key square). In the second additional square you use the
digits +/- 66, 70, 74, 78, 82, 86, 90 en 94. In the third additional square you use the digits +/- 98, 102, 106, 110,
114, 118, 122 en 126. Produce the following square:
|
-6
|
14
|
-30
|
22
|
-38
|
46
|
-62
|
54
|
|
-26
|
18
|
-2
|
10
|
-58
|
50
|
-34
|
42
|
|
30
|
-22
|
6
|
-14
|
62
|
-54
|
38
|
-46
|
|
2
|
-10
|
26
|
-18
|
34
|
-42
|
58
|
-50
|
|
-70
|
78
|
-94
|
86
|
-102
|
110
|
-126
|
118
|
|
-90
|
82
|
-66
|
74
|
-122
|
114
|
-98
|
106
|
|
94
|
-86
|
70
|
-78
|
126
|
-118
|
102
|
-110
|
|
66
|
-74
|
90
|
-82
|
98
|
-106
|
122
|
-114
|
Translate this square (back) to a pure 8x8 square. Translate the lowest digit (= most negative) digit in 1 and the highest
(most positive) digit in 64; see below:
|
31
|
36
|
25
|
38
|
23
|
44
|
17
|
46
|
|
26
|
37
|
32
|
35
|
18
|
45
|
24
|
43
|
|
40
|
27
|
34
|
29
|
48
|
19
|
42
|
21
|
|
33
|
30
|
39
|
28
|
41
|
22
|
47
|
20
|
|
15
|
52
|
9
|
54
|
7
|
60
|
1
|
62
|
|
10
|
53
|
16
|
51
|
2
|
61
|
8
|
59
|
|
56
|
11
|
50
|
13
|
64
|
3
|
58
|
5
|
|
49
|
14
|
55
|
12
|
57
|
6
|
63
|
4
|
As key square of the Khajuraho method of construction (see page ‘Khajuraho method’) I took the third additional 4x4
square (see at the right of the bottom of the 8x8 square).
4x4 sub-square at the right of the bottom Key square
|
7
|
60
|
1
|
62
|
|
|
7
|
h-4
|
1
|
h-2
|
|
2
|
61
|
8
|
59
|
|
|
2
|
h-3
|
8
|
h-5
|
|
64
|
3
|
58
|
5
|
|
|
h
|
3
|
h-6
|
5
|
|
57
|
6
|
63
|
4
|
|
|
h-7
|
6
|
h-1
|
4
|
The square produced by the Khajuraho method of construction is almost Franklin panmagic. Only the sum of four 2x2
sub-squares in the middle two columns is not half of the magic sum (1/2 x 260 = 130). If you swap digits in the rows,
you get a (perfect) Franklin panmagic 8x8 square. In stead of the Khajuraho method of construction and swapping digits
in the rows you can use directly the basic pattern method of construction (see page ‘basic pattern method’) to produce
the same (perfect) Franklin panmagic 8x8 square.
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