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From each (pattern of a) 4x4 pan magic square you can produce an 8x8 most perfect (Franklin pan)magic
square. The 8x8 square contains the following digits from 1 to 64 (= 4 x 16 digits):
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1
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2
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3
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4
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5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
16
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|
17
|
18
|
19
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20
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21
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22
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23
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24
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25
|
26
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27
|
28
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29
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30
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31
|
32
|
|
33
|
34
|
35
|
36
|
37
|
38
|
39
|
40
|
41
|
42
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43
|
44
|
45
|
46
|
47
|
48
|
|
49
|
50
|
51
|
52
|
53
|
54
|
55
|
56
|
57
|
58
|
59
|
60
|
61
|
62
|
63
|
64
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|
16
|
3
|
10
|
5
|
|
|
16
|
|
|
5
|
|
|
|
3
|
10
|
|
|
2
|
13
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8
|
11
|
|
|
|
13
|
8
|
|
|
|
2
|
|
|
11
|
|
7
|
12
|
1
|
14
|
|
|
|
12
|
1
|
|
|
|
7
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|
|
14
|
|
9
|
6
|
15
|
4
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|
|
9
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|
|
4
|
|
|
|
6
|
15
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|
Enter digits from the same sub-square in the empty cells crosswise:
|
16
|
4
|
9
|
5
|
|
|
15
|
3
|
10
|
6
|
|
1
|
13
|
8
|
12
|
|
|
2
|
14
|
7
|
11
|
|
8
|
12
|
1
|
13
|
|
|
7
|
11
|
2
|
14
|
|
9
|
5
|
16
|
4
|
|
|
10
|
6
|
15
|
3
|
Combine the 2 sub-squares and add the same 2 sub-squares to the bottom:
|
16
|
4
|
9
|
5
|
15
|
3
|
10
|
6
|
|
1
|
13
|
8
|
12
|
2
|
14
|
7
|
11
|
|
8
|
12
|
1
|
13
|
7
|
11
|
2
|
14
|
|
9
|
5
|
16
|
4
|
10
|
6
|
15
|
3
|
|
16
|
4
|
9
|
5
|
15
|
3
|
10
|
6
|
|
1
|
13
|
8
|
12
|
2
|
14
|
7
|
11
|
|
8
|
12
|
1
|
13
|
7
|
11
|
2
|
14
|
|
9
|
5
|
16
|
4
|
10
|
6
|
15
|
3
|
Substitute the digits for the digits with the right colour from the above mentioned table:
|
16
|
52
|
9
|
53
|
15
|
51
|
10
|
54
|
|
1
|
61
|
8
|
60
|
2
|
62
|
7
|
59
|
|
56
|
12
|
49
|
13
|
55
|
11
|
50
|
14
|
|
57
|
5
|
64
|
4
|
58
|
6
|
63
|
3
|
|
32
|
36
|
25
|
37
|
31
|
35
|
26
|
38
|
|
17
|
45
|
24
|
44
|
18
|
46
|
23
|
43
|
|
40
|
28
|
33
|
29
|
39
|
27
|
34
|
30
|
|
41
|
21
|
48
|
20
|
42
|
22
|
47
|
19
|
There are 6 options to split up a 4x4 panmagic square and to produce a basic pattern.
1 split up panmagic 4x4 Produce basic pattern 8x8 most perfect (Franklin pan)magic square
|
16
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|
|
5
|
|
3
|
10
|
|
|
|
16
|
4
|
9
|
5
|
15
|
3
|
10
|
6
|
|
|
16
|
52
|
9
|
53
|
15
|
51
|
10
|
54
|
|
|
13
|
8
|
|
2
|
|
|
11
|
|
|
1
|
13
|
8
|
12
|
2
|
14
|
7
|
11
|
|
|
1
|
61
|
8
|
60
|
2
|
62
|
7
|
59
|
|
|
12
|
1
|
|
7
|
|
|
14
|
|
|
8
|
12
|
1
|
13
|
7
|
11
|
2
|
14
|
|
|
56
|
12
|
49
|
13
|
55
|
11
|
50
|
14
|
|
9
|
|
|
4
|
|
6
|
15
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|
|
|
9
|
5
|
16
|
4
|
10
|
6
|
15
|
3
|
|
|
57
|
5
|
64
|
4
|
58
|
6
|
63
|
3
|
|
|
|
|
|
|
|
|
|
|
|
16
|
4
|
9
|
5
|
15
|
3
|
10
|
6
|
|
|
32
|
36
|
25
|
37
|
31
|
35
|
26
|
38
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|
|
|
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1
|
13
|
8
|
12
|
2
|
14
|
7
|
11
|
|
|
17
|
45
|
24
|
44
|
18
|
46
|
23
|
43
|
|
|
|
|
|
|
|
|
|
|
|
8
|
12
|
1
|
13
|
7
|
11
|
2
|
14
|
|
|
40
|
28
|
33
|
29
|
39
|
27
|
34
|
30
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|
|
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|
9
|
5
|
16
|
4
|
10
|
6
|
15
|
3
|
|
|
41
|
21
|
48
|
20
|
42
|
22
|
47
|
19
|
2 split up panmagic 4x4 Produce basic pattern 8x8 most perfect (Franklin pan)magic square
|
16
|
3
|
|
|
|
|
10
|
5
|
|
|
16
|
3
|
9
|
6
|
15
|
4
|
10
|
5
|
|
|
16
|
51
|
9
|
54
|
15
|
52
|
10
|
53
|
|
|
|
8
|
11
|
2
|
13
|
|
|
|
|
1
|
14
|
8
|
11
|
2
|
13
|
7
|
12
|
|
|
1
|
62
|
8
|
59
|
2
|
61
|
7
|
60
|
|
|
|
1
|
14
|
7
|
12
|
|
|
|
|
8
|
11
|
1
|
14
|
7
|
12
|
2
|
13
|
|
|
56
|
11
|
49
|
14
|
55
|
12
|
50
|
13
|
|
9
|
6
|
|
|
|
|
15
|
4
|
|
|
9
|
6
|
16
|
3
|
10
|
5
|
15
|
4
|
|
|
57
|
6
|
64
|
3
|
58
|
5
|
63
|
4
|
|
|
|
|
|
|
|
|
|
|
|
16
|
3
|
9
|
6
|
15
|
4
|
10
|
5
|
|
|
32
|
35
|
25
|
38
|
31
|
36
|
26
|
37
|
|
|
|
|
|
|
|
|
|
|
|
1
|
14
|
8
|
11
|
2
|
13
|
7
|
12
|
|
|
17
|
46
|
24
|
43
|
18
|
45
|
23
|
44
|
|
|
|
|
|
|
|
|
|
|
|
8
|
11
|
1
|
14
|
7
|
12
|
2
|
13
|
|
|
40
|
27
|
33
|
30
|
39
|
28
|
34
|
29
|
|
|
|
|
|
|
|
|
|
|
|
9
|
6
|
16
|
3
|
10
|
5
|
15
|
4
|
|
|
41
|
22
|
48
|
19
|
42
|
21
|
47
|
20
|
3 split up panmagic 4x4 Produce basic pattern 8x8 most perfect (Franklin pan)magic square
|
16
|
|
|
5
|
|
3
|
10
|
|
|
|
16
|
11
|
2
|
5
|
8
|
3
|
10
|
13
|
|
|
16
|
59
|
2
|
53
|
8
|
51
|
10
|
61
|
|
2
|
|
|
11
|
|
13
|
8
|
|
|
|
2
|
5
|
16
|
11
|
10
|
13
|
8
|
3
|
|
|
50
|
5
|
64
|
11
|
58
|
13
|
56
|
3
|
|
|
12
|
1
|
|
7
|
|
|
14
|
|
|
15
|
12
|
1
|
6
|
7
|
4
|
9
|
14
|
|
|
63
|
12
|
49
|
6
|
55
|
4
|
57
|
14
|
|
|
6
|
15
|
|
9
|
|
|
4
|
|
|
1
|
6
|
15
|
12
|
9
|
14
|
7
|
4
|
|
|
1
|
54
|
15
|
60
|
9
|
62
|
7
|
52
|
|
|
|
|
|
|
|
|
|
|
|
16
|
11
|
2
|
5
|
8
|
3
|
10
|
13
|
|
|
32
|
43
|
18
|
37
|
24
|
35
|
26
|
45
|
|
|
|
|
|
|
|
|
|
|
|
2
|
5
|
16
|
11
|
10
|
13
|
8
|
3
|
|
|
34
|
21
|
48
|
27
|
42
|
29
|
40
|
19
|
|
|
|
|
|
|
|
|
|
|
|
15
|
12
|
1
|
6
|
7
|
4
|
9
|
14
|
|
|
47
|
28
|
33
|
22
|
39
|
20
|
41
|
30
|
|
|
|
|
|
|
|
|
|
|
|
1
|
6
|
15
|
12
|
9
|
14
|
7
|
4
|
|
|
17
|
38
|
31
|
44
|
25
|
46
|
23
|
36
|
4 split up panmagic 4x4 Produce basic pattern 8x8 most perfect (Franklin pan)magic square
|
16
|
3
|
|
|
|
|
10
|
5
|
|
|
16
|
3
|
2
|
13
|
8
|
11
|
10
|
5
|
|
|
16
|
51
|
2
|
61
|
8
|
59
|
10
|
53
|
|
2
|
13
|
|
|
|
|
8
|
11
|
|
|
2
|
13
|
16
|
3
|
10
|
5
|
8
|
11
|
|
|
50
|
13
|
64
|
3
|
58
|
5
|
56
|
11
|
|
|
|
1
|
14
|
7
|
12
|
|
|
|
|
15
|
4
|
1
|
14
|
7
|
12
|
9
|
6
|
|
|
63
|
4
|
49
|
14
|
55
|
12
|
57
|
6
|
|
|
|
15
|
4
|
9
|
6
|
|
|
|
|
1
|
14
|
15
|
4
|
9
|
6
|
7
|
12
|
|
|
1
|
62
|
15
|
52
|
9
|
54
|
7
|
60
|
|
|
|
|
|
|
|
|
|
|
|
16
|
3
|
2
|
13
|
8
|
11
|
10
|
5
|
|
|
32
|
35
|
18
|
45
|
24
|
43
|
26
|
37
|
|
|
|
|
|
|
|
|
|
|
|
2
|
13
|
16
|
3
|
10
|
5
|
8
|
11
|
|
|
34
|
29
|
48
|
19
|
42
|
21
|
40
|
27
|
|
|
|
|
|
|
|
|
|
|
|
15
|
4
|
1
|
14
|
7
|
12
|
9
|
6
|
|
|
47
|
20
|
33
|
30
|
39
|
28
|
41
|
22
|
|
|
|
|
|
|
|
|
|
|
|
1
|
14
|
15
|
4
|
9
|
6
|
7
|
12
|
|
|
17
|
46
|
31
|
36
|
25
|
38
|
23
|
44
|
5 splitup panmagic 4x4 Produce basic pattern 8x8 most perfect (Franklin pan)magic square
|
16
|
|
10
|
|
|
3
|
|
5
|
|
|
16
|
7
|
10
|
1
|
12
|
3
|
14
|
5
|
|
|
48
|
23
|
58
|
1
|
60
|
3
|
46
|
21
|
|
2
|
|
8
|
|
|
13
|
|
11
|
|
|
2
|
9
|
8
|
15
|
6
|
13
|
4
|
11
|
|
|
18
|
| |