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 de-
vided by four, which gives a maximum remainder 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 addi-
tional 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 bottom right 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 most perfect (Franklin pan)magic 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 most perfect (Franklin pan)magic 8x8 square.
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