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How to produce all possible most perfect (Franklin pan)magic 8x8 squares?
First question is: how many (and which) most perfect magic 8x8 squares are there? According to the information in
the book of Arno van den Essen as well as on a several well-established websites there are 368640 most perfect
magic 8x8 squares excluding rotating and/or mirroring. From Aale de Winkel I have got a download with 368640
most perfect magic 8x8 squares.
Obvious in the download is that the digits 1 up to 64 are not in a proportional quantity in the upper left corner. If you
watch the smallest most perfect magic square, the panmagic 4x4 square, than you see that the digits 1 up to 16 are
each 3 times (= proportinal quantity) in the upper left corner of the 48 panmagic magic 4x4 squares excluding rotation
and/or mirroring. Including rotation and/or mirroring the digits 1 up to 16 are each (3 x 8 =) 24 times (= proportional
quantity) in the upper left corner. The proportionality is due to the shift of the 3 basic 4x4 panmagic squares (see page
‘panmagic 4x4 square’). It is clear which 3 (basic magic squares) x 16 (shifts) = 48 panmagic 4x4 squares are ‘original’.
By rotation and/or mirroring you get the remainder of the (48 x 8 =) 384 panmagic 4x4 squares including rotation and/or
mirroring. But it is not clear which most perfect magic 8x8 squares are ‘original’ and which are not.
The download starts with magic suares which have the digit 63 in the upper left corner. In the download there are
relatively a lot of magic squares which have the digit 63 in the upper left corner. To be precise, in the download there
are 23040 magic squares which have the digit 63, in the upper left corner. Based on magic squares including rotation
and/or mirroring I have produced by rotation with a quarter to te right and (vertical) mirroring (see page ‘panmagic
4x4 square’) 23040 other solutions. In total I have produced from the download (23040 + 23040 =) 46080 most perfect
magic 8x8 squares including rotation and/or mirroring which have the digit 63 in the upper left corner. Notify that 368640
x 8 / 64 = 46080; so 46080 most perfect magic 8x8 squares which have the digit 63 in the upper left corner is a propor-
tional part (1/64 x 2949120) of all most perfect magic 8x8 squares including rotation and/or mirroring.
I have sort the (8 rows x 8 digits = 64 digits in a row of all) 46080 most perfect magic 8x8 squares which have the digit
63 in the upper left corner. There turn out to be each time 384 magic squares which have the same first 4 digits in the
upper row, each time 48 magic squares which have the same upper row (= 8 digits), each time 8 magic squares which
have the same upper half (= 32 digits) and each time 2 magic squares which have (due to the swap possibility of row
6&8) the same five upper rows (= 40 digits).
I have used the different methods to produce most perfect magic 8x8 squares which have the digit 63 in the upper left
corner and I have identified that these magic squares are in the download of 46080 most perfect magic squares which
have the digit 63 in the upper left corner. So I have no doubts that the number of 368640 (x 8 = 2949120 including
rotation and or mirroring) is true.
Second question is: How can I get all 36840 (x 8 = 2949120) possibilities by using existing (or new) methods to produce
most perfect magic (8x8) squares? Besides the quadrant method of Willem Barink, Sudoku method 3 and basic pattern
method 1 have a big potential of solutions. Willem Barink has already asked Aale de Winkel to examine the scope of the
quadrant method. Ik hope to present in the future the results of this examination on my website. I have examined the
scope of Sudoku method 3 and basic pattern method 1.
Sudoku method 3 and basic pattern method 1 have both the panmagic 4x4 square as basic pattern. Using Sudoku method
3 or basic pattern 1 you get different solutions. I have taken a sample of 120 (I have selected each time at random one
magic square out of 384 magic squares, which have the same first 4 digits in the upper row; notify that 120 x 384 = 46080)
out of the 46080 most perfect magic 8x8 squares which have the digit 63 in the upper left corner. I have determined how
these magic squares can be produced by using a 4x4 panmagic square as basic pattern.
In the sample of 120 most perfect (Franklin pan)magic squares (which I have corrected by swapping rows and/or columns)
I have discovered not only Sudoku methode 3 and basic pattern method 1, but also Sudoku method 2 and three other groups.
In total I have determined the following 6 groups:
From the 120 magic squares 5 magic squares have 4x the same panmagic 4x4 square as basic pattern.
See for example:
Most perfect magic 8x8 Basic pattern Sudoku pattern
|
63
|
17
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40
|
10
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47
|
1
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56
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26
|
|
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15
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1
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8
|
10
|
15
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1
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8
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10
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|
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3
|
1
|
2
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0
|
2
|
0
|
3
|
1
|
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6
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44
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29
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51
|
22
|
60
|
13
|
35
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|
|
6
|
12
|
13
|
3
|
6
|
12
|
13
|
3
|
|
|
0
|
2
|
1
|
3
|
1
|
3
|
0
|
2
|
|
25
|
55
|
2
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48
|
9
|
39
|
18
|
64
|
|
|
9
|
7
|
2
|
16
|
9
|
7
|
2
|
16
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|
|
1
|
3
|
0
|
2
|
0
|
2
|
1
|
3
|
|
36
|
14
|
59
|
21
|
52
|
30
|
43
|
5
|
|
|
4
|
14
|
11
|
5
|
4
|
14
|
11
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5
|
|
|
2
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0
|
3
|
1
|
3
|
1
|
2
|
0
|
|
31
|
49
|
8
|
42
|
15
|
33
|
24
|
58
|
|
|
15
|
1
|
8
|
10
|
15
|
1
|
8
|
10
|
|
|
1
|
3
|
0
|
2
|
0
|
2
|
1
|
3
|
|
38
|
12
|
61
|
19
|
54
|
28
|
45
|
3
|
|
|
6
|
12
|
13
|
3
|
6
|
12
|
13
|
3
|
|
|
2
|
0
|
3
|
1
|
3
|
1
|
2
|
0
|
|
57
|
23
|
34
|
16
|
41
|
7
|
50
|
32
|
|
|
9
|
7
|
2
|
16
|
9
|
7
|
2
|
16
|
|
|
3
|
1
|
2
|
0
|
2
|
0
|
3
|
1
|
|
4
|
46
|
27
|
53
|
20
|
62
|
11
|
37
|
|
|
4
|
14
|
11
|
5
|
4
|
14
|
11
|
5
|
|
|
0
|
2
|
1
|
3
|
1
|
3
|
0
|
2
|
Group II [2x2 the same panmagic 4x4 square as basic pattern]
From the 120 magic squares 21 magic squares have 2x2 the same panmagic 4x4 square as basic pattern.
See for example:
Most perfect magic 8x8 Basic pattern Sudoku pattern
|
63
|
33
|
24
|
10
|
31
|
1
|
56
|
42
|
|
|
15
|
1
|
8
|
10
|
15
|
1
|
8
|
10
|
|
|
3
|
2
|
1
|
0
|
1
|
0
|
3
|
2
|
|
6
|
28
|
45
|
51
|
38
|
60
|
13
|
19
|
|
|
6
|
12
|
13
|
3
|
6
|
12
|
13
|
3
|
|
|
0
|
1
|
2
|
3
|
2
|
3
|
0
|
1
|
|
41
|
55
|
2
|
32
|
9
|
23
|
34
|
64
|
|
|
9
|
7
|
2
|
16
|
9
|
7
|
2
|
16
|
|
|
2
|
3
|
0
|
1
|
0
|
1
|
2
|
3
|
|
20
|
14
|
59
|
37
|
52
|
46
|
27
|
5
|
|
|
4
|
14
|
11
|
5
|
4
|
14
|
11
|
5
|
|
|
1
|
0
|
3
|
2
|
3
|
2
|
1
|
0
|
|
43
|
53
|
4
|
30
|
11
|
21
|
36
|
62
|
|
|
11
|
5
|
4
|
14
|
11
|
5
|
4
|
14
|
|
|
2
|
3
|
0
|
1
|
0
|
1
|
2
|
3
|
|
18
|
16
|
57
|
39
|
50
|
48
|
25
|
7
|
|
|
2
|
16
|
9
|
7
|
2
|
16
|
9
|
7
|
|
|
1
|
0
|
3
|
2
|
3
|
2
|
1
|
0
|
|
61
|
35
|
22
|
12
|
29
|
3
|
54
|
44
|
|
|
13
|
3
|
6
|
12
|
13
|
3
|
6
|
12
|
|
|
3
|
2
|
1
|
0
|
1
|
0
|
3
|
2
|
|
8
|
26
|
47
|
49
|
40
|
58
|
15
|
17
|
|
|
8
|
10
|
15
|
1
|
8
|
10
|
15
|
1
|
|
|
0
|
1
|
2
|
3
|
2
|
3
|
0
|
1
|
Group III [4x different panmagic 4x4 square as basic pattern]
From the 120 magic squares 11 magic squares have 4x a different panmagic 4x4 square as basic pattern
(and 4x the same Sudoku pattern). See for example:
Most perfect magic 8x8 Basic pattern Sudoku pattern
|
63
|
33
|
28
|
6
|
64
|
34
|
27
|
5
|
|
|
15
|
1
|
12
|
6
|
16
|
2
|
11
|
5
|
|
|
3
|
2
|
1
|
0
|
3
|
2
|
1
|
0
|
|
26
|
8
|
61
|
35
|
25
|
7
|
62
|
36
|
|
|
10
|
8
|
13
|
3
|
9
|
7
|
14
|
4
|
|
|
1
|
0
|
3
|
2
|
1
|
0
|
3
|
2
|
|
37
|
59
|
2
|
32
|
38
|
60
|
1
|
31
|
|
|
5
|
11
|
2
|
16
|
6
|
12
|
1
|
15
|
|
|
2
|
3
|
0
|
1
|
2
|
3
|
0
|
1
|
|
4
|
30
|
39
|
57
|
3
|
29
|
40
|
58
|
|
|
4
|
14
|
7
|
9
|
3
|
13
|
8
|
10
|
|
|
0
|
1
|
2
|
3
|
0
|
1
|
2
|
3
|
|
55
|
41
|
20
|
14
|
56
|
42
|
19
|
13
|
|
|
7
|
9
|
4
|
14
|
8
|
10
|
3
|
13
|
|
|
3
|
2
|
1
|
0
|
3
|
2
|
1
|
0
|
|
18
|
16
|
53
|
43
|
17
|
15
|
54
|
44
|
|
|
2
|
16
|
5
|
11
|
1
|
15
|
6
|
12
|
|
|
1
|
0
|
3
|
2
|
1
|
0
|
3
|
2
|
|
45
|
51
|
10
|
24
|
46
|
52
|
9
|
23
|
|
|
13
|
3
|
10
|
8
|
14
|
4
|
9
|
7
|
|
|
2
|
3
|
0
|
1
|
2
|
3
|
0
|
1
|
|
12
|
22
|
47
|
49
|
11
|
21
|
48
|
50
|
|
|
12
|
6
|
15
|
1
|
11
|
5
|
16
|
2
|
|
|
0
|
1
|
2
|
3
|
0
|
1
|
2
|
3
|
From the 120 magic squares 65 magic squares have 1x a splitted up panmagic 4x4 square as basic pattern
(notify that all 6 different basic patterns are in the sample). See for example:
Most perfect magic 8x8 Basic pattern Sudoku pattern
|
63
|
3
|
54
|
10
|
61
|
1
|
56
|
12
|
|
|
15
|
3
|
6
|
10
|
13
|
1
|
8
|
12
|
|
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
|
50
|
14
|
59
|
7
|
52
|
16
|
57
|
5
|
|
|
2
|
14
|
11
|
7
|
4
|
16
|
9
|
5
|
|
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
|
11
|
55
|
2
|
62
|
9
|
53
|
4
|
64
|
|
|
11
|
7
|
2
|
14
|
9
|
5
|
4
|
16
|
|
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
|
6
|
58
|
15
|
51
|
8
|
60
|
13
|
49
|
|
|
6
|
10
|
15
|
3
|
8
|
12
|
13
|
1
|
|
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
|
31
|
35
|
22
|
42
|
29
|
33
|
24
|
44
|
|
|
15
|
3
|
6
|
10
|
13
|
12
|
8
|
1
|
|
|
1
|
2
|
1
|
2
|
1
|
2
|
1
|
2
|
|
18
|
46
|
27
|
39
|
20
|
48
|
25
|
37
|
|
|
2
|
14
|
11
|
7
|
4
|
5
|
9
|
16
|
|
|
1
|
2
|
1
|
2
|
1
|
2
|
1
|
2
|
|
43
|
23
|
34
|
30
|
41
|
21
|
36
|
32
|
|
|
11
|
7
|
2
|
14
|
9
|
16
|
4
|
5
|
|
|
2
|
1
|
2
|
1
|
2
|
1
|
2
|
1
|
|
38
|
26
|
47
|
19
|
40
|
28
|
45
|
17
|
|
|
6
|
10
|
15
|
3
|
8
|
1
|
13
|
12
|
|
|
2
|
1
|
2
|
1
|
2
|
1
|
2
|
1
|
Basic patterns 1, 2, 3 and 4 can have a different Sudoku pattern (see for example):
Most perfect magic 8x8 Basic pattern Sudoku pattern
|
63
|
8
|
58
|
1
|
59
|
4
|
62
|
5
|
|
|
15
|
8
|
10
|
1
|
11
|
4
|
14
|
5
|
|
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
|
18
|
41
|
23
|
48
|
22
|
45
|
19
|
44
|
|
|
2
|
9
|
7
|
16
|
6
|
13
|
3
|
12
|
|
|
1
|
2
|
1
|
2
|
1
|
2
|
1
|
2
|
|
7
|
64
|
2
|
57
|
3
|
60
|
6
|
61
|
|
|
7
|
16
|
2
|
9
|
3
|
12
|
6
|
13
|
|
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
|
42
|
17
|
47
|
24
|
46
|
21
|
43
|
20
|
|
|
10
|
1
|
15
|
8
|
14
|
5
|
11
|
4
|
|
|
2
|
1
|
2
|
1
|
2
|
1
|
2
|
1
|
|
15
|
56
|
10
|
49
|
11
|
52
|
14
|
53
|
|
|
15
|
8
|
10
|
1
|
11
|
4
|
14
|
5
|
|
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
|
34
|
25
|
39
|
32
|
38
|
29
|
35
|
28
|
|
|
2
|
9
|
7
|
16
|
6
|
13
|
3
|
12
|
|
|
2
|
1
|
2
|
1
|
2
|
1
|
2
|
1
|
|
55
|
16
|
50
|
9
|
51
|
12
|
54
|
13
|
|
|
7
|
16
|
2
|
9
|
3
|
12
|
6
|
13
|
|
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
|
26
|
33
|
31
|
40
|
30
|
37
|
27
|
36
|
|
|
10
|
1
|
15
|
8
|
14
|
5
|
11
|
4
|
|
|
1
|
2
|
1
|
2
|
1
|
2
|
1
|
2
|
Group V [2x splitted up panmagic 4x4 square as basic pattern]
From the 120 magic squares 8 magic squares have 2x a splitted up panmagic 4x4 square as basic pattern.
See for example:
Most perfect magic 8x8 Basic pattern Sudoku pattern
| 
