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Perfect magic squares
Analysis Franklin panm. 8x8
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Analysis of the (basic patterns of the) 8x8 Franklin panmagic square


Information for whiz kids:
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 mentioned above), you can
simplify the pattern of the square:
 
 
Franklin panmagic square page 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
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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
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31
32
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34
35
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40
41
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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 construction
(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
15
1
10
6
13
3
12
 
 
Please note that an alternative swap of digits is necessary to translate the basic key method of construction into
the basic pattern method of construction.
 
 
[4th] Analysis of 8x8 panmagic square(s) of Willem Barink (medjig method)
The following 8x8 panmagic square can be found on Willem Barink’s (medjig method) website
wba.novaloka.nl/magic-squares.html  
 
 
62
4
13
51
46
20
29
35
5
59
54
12
21
43
38
28
52
14
3
61
36