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Magic squares (most perfect, [Franklin] panmagic & inlaid)
Detailed explanation about the structure and construction of magic squares
Khajuraho method, explanation
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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 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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Magic squares|Contact / guestbook|Most magic square per order|3x3 magic square|3x3 magic square, explanation|Sudoku method (1)|Sudoku method (2)|Sudoku method (3)|Pan magic 4x4 square|Pan magic 4x4 square, explanation|Pan magic 4x4 square, binary|Dürer & Franklin transformation|Transformation method|Transformation method, analysis|[ultra] pan magic 5x5 square|Pan magic 5x5 square, explanation|6x6 magic square|Ultra (pan)magic 8x8 square|Most perfect magic squares, explanation|8x8 most perfect magic squares, binary|Khajuraho method|Khajuraho method, explanation|Basic pattern method (1a)|Basic pattern method (1b)|Basic pattern method (2)|Basic pattern method (3a)|Basic pattern method (3b)|Basic pattern method (3c)|Basic pattern method (4)|Basic pattern method (5)|Basic pattern method (6)|Basic pattern method (7a)|Basic pattern method (7b)|Analysis Franklin panm. 8x8 (1)|Analysis Franklin panm. 8x8 (2)|Basic key method (1)|Basic key method (2)|Quadrant method (Willem Barink)|Quadrant method group 1 up to 5|Quadrant method group 6 up to 10|Quadrant method group 11 up to 19|[ultra] pan magic 9x9 square (1)|pan magic 9x9 square (2)|pan magic 9x9 square (3)|3x extra magic 9x9 square|10x10 magic square|Composite 12x12 magic square|14x14 magic square|[Ultra] pan magic 15x15 square|3x extra magic 15x15 square|The perfect magic square|3x extra magic 18x18 square|Ultra pan magic 25x25 square|[ultra] pan magic 27x27 square|[ultra] pan magic 35x35 square|extra magic 35x35 square|Bordered squares|Inlaid square (1)|Inlaid square (2)|Each magic sum|Water retention challenge|Most magic 4x4x4 cube|Perfect (Nasik) magic 8x8x8 cube|[More than] perfect magic 9x9x9 cube|Trick with 8x8 bimagic square|Favorite Links