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How to produce bigger panmagic squares
From a panmagic 4x4 square, for example the famous Khajuraho square, you can produce a bigger
panmagic square (8x8, 12x12, 16x16, 20x20, …).
Rewrite the Khajuraho magic square as follows:
Khajuraho square Basic square
|
7
|
12
|
1
|
14
|
|
|
7
|
h-4
|
1
|
h-2
|
|
2
|
13
|
8
|
11
|
|
|
2
|
h-3
|
8
|
h-5
|
|
16
|
3
|
10
|
5
|
|
|
h
|
3
|
h-6
|
5
|
|
9
|
6
|
15
|
4
|
|
|
h-7
|
6
|
h-1
|
4
|
If you want to produce an 8x8 panmagic square, you need four 4x4 squares (the basic square and
three additional squares):
|
7
|
h-4
|
1
|
h-2
|
+8
|
-8
|
+8
|
-8
|
|
2
|
h-3
|
8
|
h-5
|
+8
|
-8
|
+8
|
-8
|
|
h
|
3
|
h-6
|
5
|
-8
|
+8
|
-8
|
+8
|
|
h-7
|
6
|
h-1
|
4
|
-8
|
+8
|
-8
|
+8
|
|
+16
|
-16
|
+16
|
-16
|
+24
|
-24
|
+24
|
-24
|
|
+16
|
-16
|
+16
|
-16
|
+24
|
-24
|
+24
|
-24
|
|
-16
|
+16
|
-16
|
+16
|
-24
|
+24
|
-24
|
+24
|
|
-16
|
+16
|
-16
|
+16
|
-24
|
+24
|
-24
|
+24
|
The highest digit of the (pure) 8x8 square is 64, so h is 64. Calculate first the digits of the basic square.
Then you take a digit from a cell of the basic square and add a digit from the same cell of the first, the
second or the third additional square. Result:
Panmagic 8x8 square
|
7
|
60
|
1
|
62
|
15
|
52
|
9
|
54
|
|
2
|
61
|
8
|
59
|
10
|
53
|
16
|
51
|
|
64
|
3
|
58
|
5
|
56
|
11
|
50
|
13
|
|
57
|
6
|
63
|
4
|
49
|
14
|
55
|
12
|
|
23
|
44
|
17
|
46
|
31
|
36
|
25
|
38
|
|
18
|
45
|
24
|
43
|
26
|
37
|
32
|
35
|
|
48
|
19
|
42
|
21
|
40
|
27
|
34
|
29
|
|
41
|
22
|
47
|
20
|
33
|
30
|
39
|
28
|
This square is almost Franklin panmagic. Only the sum of four 2x2 subsquares in the middle two columns
is not half of the magic sum (1/2 x 260 = 130). If you swap the coloured digits in the rows, you get the
following most perfect (Franklin pan)magic 8x8 square:
Most perfect (Franklin pan)magic 8x8 square
|
15
|
60
|
1
|
54
|
7
|
52
|
9
|
62
|
|
2
|
53
|
16
|
59
|
10
|
61
|
8
|
51
|
|
64
|
11
|
50
|
5
|
56
|
3
|
58
|
13
|
|
49
|
6
|
63
|
12
|
57
|
14
|
55
|
4
|
|
31
|
44
|
17
|
38
|
23
|
36
|
25
|
46
|
|
18
|
37
|
32
|
43
|
26
|
45
|
24
|
35
|
|
48
|
27
|
34
|
21
|
40
|
19
|
42
|
29
|
|
33
|
22
|
47
|
28
|
41
|
30
|
39
|
20
|
This method is an alternative (or the underlying method) of the basic pattern method. With the Khajuraho
method it is possible to produce bigger most perfect (Franklin pan)magic squares (16x16, 32x32, …). Are
you looking for an easier way to produce bigger Franklin panmagic squares, then use the basic key method.
See more information on page: Khajuraho method, explanation
Information for whiz kids:
From the above mentioned 8x8 Franklin panmagic square you can produce a 16x16 Franklin
panmagic square. Use the above mentioned square as basic square. Rewrite the digits 33
up to 64 in h-31, h-30, …, h. You need three additional 8x8 squares. Use in the first, second,
respectively third additional square +/- 32, +/- 64 respectevely +/- 96. Use the above men-
tioned colour pattern to swap digits in the rows (use a 2x2 carpet of the colour pattern; for
example in the first row: don’t swap digits 1&5, but swap digits 1&9).
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