|  |  | | <HOME> <<PREVIOUS] [NEXT>>
How to produce ultra panmagic 8x8 squares?
The Franklin panmagic (= most perfect magic) 8x8 square has the most magic features (click on [NEXT>>).
The most magic odd squares are ultra (pan)magic squares. But also (even, for example) 8x8 ultra (pan)
magic squares exist. Just like the the most perfect magic square, the ultra magic 8x8 square consists of four
4x4 squares with the same structure. These 4x4 squares have the following structure:
The sum per colour is the lowest plus the highest digit, that is (as we take a 8x8 square) 1 + 64 = 65.
First we produce (as basic pattern) a 4x4 square using four binary grids. N.b.: Note that the four binary
grids have the right (see above) structure!!! Secondly we use the right Sudoku grid to get all digits from
1 up to 64 in the ultra (pan)magic 8x8 square.
|
1x digit
|
|
|
|
2x digit
|
|
|
|
4x digit
|
|
|
|
8x digit +1
|
|
|
Magic 4x4 square
|
|
0
|
0
|
1
|
1
|
|
|
0
|
1
|
1
|
0
|
|
|
0
|
1
|
1
|
0
|
|
|
0
|
1
|
0
|
1
|
|
|
1
|
15
|
8
|
10
|
|
1
|
1
|
0
|
0
|
|
|
0
|
1
|
1
|
0
|
|
|
1
|
0
|
0
|
1
|
|
|
1
|
0
|
1
|
0
|
|
|
14
|
4
|
11
|
5
|
|
1
|
1
|
0
|
0
|
|
|
1
|
0
|
0
|
1
|
|
|
0
|
1
|
1
|
0
|
|
|
1
|
0
|
1
|
0
|
|
|
12
|
6
|
13
|
3
|
|
0
|
0
|
1
|
1
|
|
|
1
|
0
|
0
|
1
|
|
|
1
|
0
|
0
|
1
|
|
|
0
|
1
|
0
|
1
|
|
|
7
|
9
|
2
|
16
|
N.B.: The 4x4 square is semi panmagic (group III of the 880 possible 4x4 squares excluding rotating and/or
mirroring). You can put the binary grids in random order (and that gives 4x3x2x1 is 24 possibilities). It is
possible for each binary grid to swap digits 0 and 1 or not (and that gives 2x2x2x2 is 16 possibilities). In
total there are including rotating and/or mirroring (24 x 16 =) 384 different semi panmagic 4x4 squares of
group III.
The problem is that the 4x4 square is (semi panmagic, but) not panmagic. The solution is to use in the upper
half of the 8x8 grid two times the 4x4 square and to use in the lowest half of the 8x8 grid two times the 4x4
square up site down.
| 1x digit from 4x 4x4 magic square[up site down] |
|
| 1 |
15 |
8 |
10 |
1 |
15 |
8 |
10 |
| 14 |
4 |
11 |
5 |
14 |
4 |
11 |
5 |
| 12 |
6 |
13 |
3 |
12 |
6 |
13 |
3 |
| 7 |
9 |
2 |
16 |
7 |
9 |
2 |
16 |
| 16 |
2 |
9 |
7 |
16 |
2 |
9 |
7 |
| 3 |
13 |
6 |
12 |
3 |
13 |
6 |
12 |
| 5 |
11 |
4 |
14 |
5 |
11 |
4 |
14 |
| 10 |
8 |
15 |
1 |
10 |
8 |
15 |
1 |
| |
|
|
|
|
|
|
|
| |
|
|
|
|
|
|
|
| + 16x digit from Sudoku grid |
|
| 0 |
3 |
1 |
2 |
1 |
2 |
0 |
3 |
| 3 |
0 |
2 |
1 |
2 |
1 |
3 |
0 |
| 2 |
1 |
3 |
0 |
3 |
0 |
2 |
1 |
| 1 |
2 |
0 |
3 |
0 |
3 |
1 |
2 |
| 0 |
3 |
1 |
2 |
1 |
2 |
0 |
3 |
| 3 |
0 |
2 |
1 |
2 |
1 |
3 |
0 |
| 2 |
1 |
3 |
0 |
3 |
0 |
2 |
1 |
| 1 |
2 |
0 |
3 |
0 |
3 |
1 |
2 |
| |
|
|
|
|
|
|
|
| |
|
|
|
|
|
|
|
| = ultra (pan)magic 8x8 square |
| 1 |
63 |
24 |
42 |
17 |
47 |
8 |
58 |
| 62 |
4 |
43 |
21 |
46 |
20 |
59 |
5 |
| 44 |
22 |
61 |
3 |
60 |
6 |
45 |
19 |
| 23 |
41 |
2 |
64 |
7 |
57 |
18 |
48 |
| 16 |
50 |
25 |
39 |
32 |
34 |
9 |
55 |
| 51 |
13 |
38 |
28 |
35 |
29 |
54 |
12 |
| 37 |
27 |
52 |
14 |
53 |
11 |
36 |
30 |
| 26 |
40 |
15 |
49 |
10 |
56 |
31 |
33 |
Alternative
It is also possible to choose the following structure:
Now we produce a semi panmagic 4x4 square of group II.
|
1x digit
|
|
|
|
2x digit
|
|
|
|
4x digit
|
|
|
|
8x digit +1
|
|
|
Mag. 4x4 sq.
|
|
0
|
0
|
1
|
1
|
|
|
0
|
1
|
1
|
0
|
|
|
0
|
0
|
1
|
1
|
|
|
0
|
1
|
0
|
1
|
|
|
1
|
11
|
8
|
14
|
|
1
|
1
|
0
|
0
|
|
|
0
|
1
|
1
|
0
|
|
|
1
|
1
|
0
|
0
|
|
|
0
|
1
|
0
|
1
|
|
|
6
|
16
|
3
|
9
|
|
1
|
1
|
0
|
0
|
|
|
1
|
0
|
0
|
1
|
|
|
0
|
0
|
1
|
1
|
|
|
1
|
0
|
1
|
0
|
|
|
12
|
2
|
13
|
7
|
|
0
|
0
|
1
|
1
|
|
|
1
|
0
|
0
|
1
|
|
|
1
|
1
|
0
|
0
|
|
|
1
|
0
|
1
|
0
|
|
|
15
|
5
|
10
|
4
|
And now we have to puzzle to find the right Sudoku grid to produce another ultra (pan)magic 8x8 square:
| 1x digit from 4x4 [up site down] |
|
| 1 |
11 |
8 |
14 |
1 |
11 |
8 |
14 |
| 6 |
16 |
3 |
9 |
6 |
16 |
3 |
9 |
| 12 |
2 |
13 |
7 |
12 |
2 |
13 |
7 |
| 15 |
5 |
10 |
4 |
15 |
5 |
10 |
4 |
| 4 |
10 |
5 |
15 |
4 |
10 |
5 |
15 |
| 7 |
13 |
2 |
12 |
7 |
13 |
2 |
12 |
| 9 |
3 |
16 |
6 |
9 |
3 |
16 |
6 |
| 14 |
8 |
11 |
1 |
14 |
8 |
11 |
1 |
| |
|
|
|
|
|
|
|
| |
|
|
|
|
|
|
|
| + 16x digit from Sudoku grid |
|
| 0 |
1 |
2 |
3 |
2 |
3 |
0 |
1 |
| 2 |
3 |
0 |
1 |
0 |
1 |
2 |
3 |
| 0 |
1 |
2 |
3 |
2 |
3 |
0 |
1 |
| 2 |
3 |
0 |
1 |
0 |
1 |
2 |
3 |
| 0 |
1 |
2 |
3 |
2 |
3 |
0 |
1 |
| 2 |
3 |
0 |
1 |
0 |
1 |
2 |
3 |
| 0 |
1 |
2 |
3 |
2 |
3 |
0 |
1 |
| 2 |
3 |
0 |
1 |
0 |
1 |
2 |
3 |
| |
|
|
|
|
|
|
|
| |
|
|
|
|
|
|
|
| = ultra (pan)magic 8x8 square |
| 1 |
27 |
40 |
62 |
33 |
59 |
8 |
30 |
| 38 |
64 |
3 |
25 |
6 |
32 |
35 |
57 |
| 12 |
18 |
45 |
55 |
44 |
50 |
13 |
23 |
| 47 |
53 |
10 |
20 |
15 |
21 |
42 |
52 |
| 4 |
26 |
37 |
63 |
36 |
58 |
5 |
31 |
| 39 |
61 |
2 |
28 |
7 |
29 |
34 |
60 |
| 9 |
19 |
48 |
54 |
41 |
51 |
16 |
22 |
| 46 |
56 |
11 |
17 |
14 |
24 |
43 |
49 |
N.B.: The groups IV, V and VI of the 4x4 squares are semi panmagic as well. Puzzle yourself to produce
ultra (pan)magic 8x8 squares using the 4x4 magic squares of group IV, V and VI!!!
If you want to know how to produce ultra magic squares, which are symmetric, panmagic and compact
for each order that is a multiple of four, find the key on page 'Basic key method (2)'.
<HOME> <<PREVIOUS] [NEXT>> |
|
|
