Khajuraho method
Use the famous Khajuraho 4x4 panmagic square to construct larger magic squares which are a multiple of 4 (= 8x8, 12x12, 16x16, 20x20, … magic square).
Rewrite the Khajuraho magic square as follows:
Khajuraho magic square Basic magic 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 |
To construct an 8x8 panmagic square, you need the basic square and 3 extending magic 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 number in the 8x8 square is 64. Fill in 64 for h and calculate all the numbers. You get the following 8x8 panmagic square.
Panmagic 8x8 magic 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 magic square is almost Franklin panmagic. Only four 2x2 sub-squares in the middle two columns give not 1/2 of the magic sum (1/2 x 260 = 130). If you swap the colours you get the following Franklin panmagic 8x8 square:
Franklin panmagic 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 |
Use the Khajuraho method to construct magic squares of order is multiple of 4 from 8x8 to infinity. See 8x8, 12x12, 16x16, 20x20, 24x24, 28x28 and 32x32
It is possible to use each 4x4 panmagic square to construct a 8x8 Franklin panmagic square.
See above how to construct the almost perfect 8x8 Franklin panmagic square (replace the numbers 9 up to 16 of the 4x4 panmagic square by 57 up to 64 to create the first 4x4 sub-square and add each time 8 to the eight low numbers and -/- 8 to the eight high numbers to create the three other 4x4 sub-squares).
You must swap half of the numbers to get a perfect 8x8 Franklin panmagic square. Which numbers you must swap and how to swap the numbers, depends on the place of the 1 and the 8 in the 4x4 panmagic square.
| 1 | 2 |
| 3 | 4 |
If the 1 and the 8 are in the same column, than you must swap half of the numbers of sub-square 1 with 2 and 3 with 4 (= horizontally).
If the 1 and the 8 are in the same row, than you must swap half of the numbers of sub-square 1 with 3 and 2 with 4 (= vertically).
Correction sheet 1 Correction sheet 2
If the 1 and the 8 are in position 1 & 2 or 3 & 4 of the row/column, than you must use correction sheet 1.
If the 1 and the 8 are in position 2 & 3 or 1 & 4 of the row/column, than you must use correction sheet 2.