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Magic squares (most perfect, [Franklin] panmagic & inlaid)
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Analysis Franklin panm. 8x8 (1)
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Analysis of the (basic patterns  of the) 8x8 Franklin panmagic square

It is possible to choose sub-squares from a 2x2 carpet (see page 4x4 panmagic and each sub-square
is (Franklin) panmagic. It is also possible to swap rows and/or columns 1&3, 2&4, 5&7 and/or 6&8 of
each 8x8 Franklin pan magic square without loosing any magic feature.

 

[1st] Analysis of an 8x8 panmagic square from the book of Arno van den
        Essen

In the book “Magische vierkanten: van Lo Shu tot Sudoku” from Arno van den Essen, 2nd print, you
can find on page 152 an 8x8 Franklin panmagic square. By  swapping some rows and columns (as men-
tioned above), you can simplify the pattern of the square:
 
 
Franklin panmagic square p. 152
 
 
 
 
Simplified pattern
 
 
 
 
1
32
38
59
5
28
34
63
 
 
 
 
1
32
38
59
34
63
5
28
46
51
9
24
42
55
13
20
 
 
 
 
40
57
3
30
7
26
36
61
27
6
64
33
31
2
60
37
 
 
 
 
27
6
64
33
60
37
31
2
56
41
19
14
52
45
23
10
 
 
 
 
62
35
25
8
29
4
58
39
11
22
48
49
15
18
44
53
 
 
 
 
17
16
54
43
50
47
21
12
40
57
3
30
36
61
7
26
 
 
 
 
56
41
19
14
23
10
52
45
17
16
54
43
21
12
50
47
 
 
 
 
11
22
48
49
44
53
15
18
62
35
25
8
58
39
29
4
 
 
 
 
46
51
9
24
13
20
42
55
 
 
The simplified pattern of the 8x8 square can be traced back to (the pattern of) a 4x4 panmagic square
as follows:
 
 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
 
 
1
16
6
11
2
15
5
12
8
9
3
14
7
10
4
13
11
6
16
1
12
5
15
2
14
3
9
8
13
4
10
7
1
16
6
11
2
15
5
12
8
9
3
14
7
10
4
13
11
6
16
1
12
5
15
2
14
3
9
8
13
4
10
7
 
 
The basic pattern is a 4x4 pan magic square, which has been split up and filled in as follows:
 
 
  panmagic 4x4                               split up                                                 fill in
1
15
6
12
 
 
1
 
6
 
 
15
 
12
 
 
1
16
6
11
2
15
5
12
8
10
3
13
 
 
8
 
3
 
 
10
 
13
 
 
8
9
3
14
7
10
4
13
11
5
16
2
 
 
11
 
16
 
 
5
 
2
 
 
11
6
16
1
12
5
15
2
14
4
9
7
 
 
14
 
9
 
 
4
 
7
 
 
14
3
9
8
13
4
10
7
 
 
Please note: an alternative basic pattern has been found!

 
 
[2nd] Analysis of an 8x8 panmagic square on the website of Miguel Angel
        Amela

On the website www.region.com.ar/amela/franklinsquares/ you can find the square below - an 8x8 Franklin
panmagic square - at structure I. This square can be traced back to a simplified pattern. First row 6&8 and
column 5&7 have been swapped and secondly the coloured digits have been swapped (alternatively).
 
 
 Example structure I from website          Swap row 6&8 and column 5&7
1
46
51
32
35
62
17
16
 
1
46
51
32
17
62
35
16
60
23
10
37
26
7
44
53
 
60
23
10
37
44
7
26
53
14
33
64
19
48
49
30
3
 
14
33
64
19
30
49
48
3
55
28
5
42
21
12
39
58
 
55
28
5
42
39
12
21
58
9
38
59
24
43
54
25
8
 
9
38
59
24
25
54
43
8
63
20
13
34
29
4
47
50
 
52
31
2
45
36
15
18
61
6
41
56
27
40
57
22
11
 
6
41
56
27
22
57
40
11
52
31
2
45
18
15
36
61
 
63
20
13
34
47
4
29
50
 
 
 Alternative swap (coloured digits)
33
14
19
64
17
62
35
16
28
55
42
5
44
7
26
53
46
1
32
51
30
49
48
3
23
60
37
10
39
12
21
58
41
6
27
56
25
54
43
8
20
63
34
13
36
15
18
61
38
9
24
59
22
57
40
11
31
52
45
2
47
4
29
50
 
 
The simplified pattern of the 8x8 square can be traced back to (the pattern of) a 4x4 panmagic square
as follows:
 
 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
 
 
1
14
3
16
1
14
3
16
12
7
10
5
12
7
10
5
14
1
16
3
14
1
16
3
7
12
5
10
7
12
5
10
9
6
11
8
9
6
11
8
4
15
2
13
4
15
2
13
6
9
8
11
6
9
8
11
15
4
13
2
15
4
13
2
 
 
The basic pattern is a 4x4 panmagic square, which has been split up and filled in as follows:
 
 
  panmagic 4x4                                  split up                                             fill in
9
6
3
16
 
 
 
 
3
16
 
 
1
14
3
16
4
15
10
5
 
 
 
 
10
5
 
 
12
7
10
5
14
1
8
11
 
 
14
1
 
 
 
 
14
1
16
3
7
12
13
2
 
 
7
12
 
 
 
 
7
12
5
10
 
 
 
 
 
 
9
6
 
 
 
 
9
6
11
8
 
 
 
 
 
 
4
15
 
 
 
 
4
15
2
13
 
 
 
 
 
 
 
 
8
11
 
 
6
9
8
11
 
 
 
 
 
 
 
 
13
2
 
 
15
4
13
2
 
 
It appears to be a new basic pattern but it is in fact the basic pattern of a sub-square on the 2x2 carpet
of the already known basic pattern (see below).
 
 
 
x
x
 
 
x
x
 
x
 
 
x
x
 
 
X
x
 
 
x
x
 
 
X
 
x
x
 
 
x
x
 
 
x
x
 
 
x
x
 
x
 
 
x
x
 
 
X
x
 
 
x
x
 
 
X
 
x
x
 
 
x
x
 
 
 
Please note: that (in addition to the already known [classical] row- and column swaps) alternative digit
swaps have been found.
 

 
[3rd] Analysis of an 8x8 panmagic square produced by the basic key method

The following 8x8 Franklin panmagic square has been produced according to the basic key method of con-
struction (see page Basic key method(1) ):
 
 
33
26
40
31
35
28
38
29
48
23
41
18
46
21
43
20
25
34
32
39
27
36
30
37
24
47
17
42
22
45
19
44
49
10
56
15
51
12
54
13
64
7
57
2
62
5
59
4
9
50
16
55
11
52
14
53
8
63
1
58
6
61
3
60
 
 
This square has the following pattern:
 
 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
 
 
1
10
8
15
3
12
6
13
16
7
9
2
14
5
11
4
9
2
16
7
11
4
14
5
8
15
1
10
6
13
3
12
1
10
8
15
3
12
6
13
16
7
9
2
14
5
11
4
9
2
16
7
11
4
14
5
8
15
1
10
6
13
3
12
 
 
This 8x8 Franklin panmagic square can be produced from a 4x4 panmagic square as follows:

 
                           panmagic 4x4       Swap (coloured digits)                       Split up
1
8
10
15
 
1
10
8
15
 
1
10
8
15
 
 
 
 
14
11
5
4
 
14
5
11
4
 
 
 
 
 
14
5
11
4
7
2
16
9
 
9
2
16
7
 
9
2
16
7
 
 
 
 
12
13
3
6
 
6
13
3
12
 
 
 
 
 
6
13
3
12
 
 
                                 Fill in
1
10
8
15
3
12
6
13
16
7
9
2
14
5
11
4
9
2
16
7
11
4
14
5
8<