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How to make perfect magic squares & cubes
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Dürer & Franklin transformation
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How to produce famous magic squares?
  
It is possible to use transformation to produce the famous magic square of Albrecht Dürer or the famous
magic square of Benjamin Franklin. Transformation means, that you swap systematically (in one or more
steps) the digits of a square with sequencing digits into a pure magic square.
 

 
Dürer transformation

The easiest transformation consists of only one step. Swap the digits in the corners and in the middle of a
4x4 square with sequencing digits, crosswise:
 
 
 
 
28
32
36
40
 
 
 
 
 
34
34
34
34
 
 
34
 
 
 
 
34
 
 
 
34
 
 
 
 
34
10
 
1
2
3
4
 
 
 
34
 
16
2
3
13
 
26
 
5
6
7
8
 
 
 
34
 
5
11
10
8
 
42
 
9
10
11
12
 
 
 
34
 
9
7
6
12
 
58
 
13
14
15
16
 
 
 
34
 
4
14
15
1
 
 
 
Use only one extra step to produce the famous magic square of Albrecht Dürer:
 
 
                                                                                                                                                                      Dürer’s magic square:
 
 
 
28
32
36
40
 
 
 
 
 
28
36
32
40
 
 
 
 
 
34
34
34
34
 
 
34
 
 
 
 
34
 
 
 
34
 
 
 
 
34
 
 
 
34
 
 
 
 
34
10
 
1
2
3
4
 
 
 
10
 
1
3
2
4
 
 
 
34
 
16
3
2
13
 
26
 
5
6
7
8
 
 
 
26
 
5
7
6
8
 
 
 
34
 
5
10
11
8
 
42
 
9
10
11
12
 
 
 
42
 
9
11
10
12
 
 
 
34
 
9
6
7
12
 
58
 
13
14
15
16
 
 
 
58
 
13
15
14
16
 
 
 
34
 
4
15
14
1
 
 
 
N.B.: If you substitute the digits 4 and 1 by the 4th respectively 1st character of the alphabet, you read the last
row as D1514A, that are the initials of the German renaissance artist/mathematician Albrecht Dürer (1471-1528)
plus the year, in which he ‘published’ the magic square by making the etching Melancholia. If you substitute the
characters of ALBRECHT DURER by their number in the alphabet and you summarize the digits, you get the sum
of 135. Add 1 (the symbol of God; the 1 is larger than the other digits in the etching) and  ou get 136. That is
exactly the sum of all digits 1 up to 16 in the magic square!!!
 
 
Use two extra steps to transform the magic square of Albrecht Dürer into a panmagic 4x4 square (= smallest
most perfect magic square):
 
 
                                                                                                                              Panmagic 4x4 square:
 
 
 
34
34
34
34
 
 
 
 
 
34
34
34
34
 
 
 
 
 
34
34
34
34
 
 
 
 
34
 
 
 
 
34
 
 
 
52
 
 
 
 
20
 
 
 
34
 
 
 
 
34
 
 
34
 
16
3
2
13
 
 
 
34
 
16
3
13
2
 
 
 
34
 
16
3
13
2
 
 
 
34
 
5
10
11
8
 
 
 
34
 
5
10
8
11
 
 
 
34
 
5
10
8
11
 
34
34
34
 
9
6
7
12
 
 
 
34
 
9
6
12
7
 
 
 
34
 
4
15
1
14
 
34
34
34
 
4
15
14
1
 
 
 
34
 
4
15
1
14
 
 
 
34
 
9
6
12
7
 
34
34
 
 
 
 
Franklin transformation

It is also possible to transform a 8x8 square with sequencing digits in four steps into (a version of) the famous
Franklin magic 8x8 square:
 
 
  Swap column 3&4 with 7&8                          Combine the rows (1-8, 2-7, 3-6, 4-5)            Swap colours and bold with not bold
1
16
17
32
33
48
49
64
 
 
1
16
49
64
33
48
17
32
 
 
1
16
49
64
33
48
17
32
2
15
18
31
34
47
50
63
 
 
2
15
50
63
34
47
18
31
 
 
8
9
56
57
40
41
24
25
3
14
19
30
35
46
51
62
 
 
3
14
51
62
35
46
19
30
 
 
2
15
50
63
34
47
18
31
4
13
20
29
36
45
52
61
 
 
4
13
52
61
36
45
20
29
 
 
7
10
55
58
39
42
23
26
5
12
21
28
37
44
53
60
 
 
5
12
53
60
37
44
21
28
 
 
3
14
51
62
35
46
19
30
6
11
22
27
38
43
54
59
 
 
6
11
54
59
38
43
22
27
 
 
6
11
54
59
38
43
22
27
7
10
23
26
39
42
55
58
 
 
7
10
55
58
39
42
23
26
 
 
4
13
52
61
36
45
20
29
8
9
24
25
40
41
56
57
 
 
8
9
56
57
40
41
24
25
 
 
5
12
53
60
37
44
21
28
 
 
  Swap row 2 with 3 and 6 with 7                        Franklin magic 8x8 square:
16
1
64
49
48
33
32
17
 
 
16
1
64
49
48
33
32
17
9
8
57
56
41
40
25
24
 
 
50
63
2
15
18
31
34
47
50
63
2
15
18
31
34
47
 
 
9
8
57
56
41
40
25
24
55
58
7
10
23
26
39
42
 
 
55
58
7
10
23
26
39
42
14
3
62
51
46
35
30
19
 
 
14
3
62
51
46
35
30
19
11
6
59
54
43
38
27
22
 
 
52
61
4
13
20
29
36
45
52
61
4
13
20
29
36
45
 
 
11
6
59
54
43
38
27
22
53
60
5
12
21
28
37
44
 
 
53
60
5
12
21
28
37
44
 
 
 
Use two extra steps to produce a Franklin panmagic 8x8 square (= also a most perfect magic 8x8 square):
 
 
Swap column 1 with 2 and swap column 5 with 6
 
 
 
 
130
130
130
130
130
130
130
130
 
 
 
 
130
130
130
130
130
130
130
130
 
 
 
146
 
 
 
 
 
 
 
 
114
130
130
 
16
1
64
49
48
33
32
17
 
130
130
 
50
63
2
15
18
31
34
47
 
130
130
 
9
8
57
56
41
40
25
24
 
130
130
 
55
58
7
10
23
26
39
42
 
130
130
 
14
3
62
51
46
35
30
19
 
130
130
 
52
61
4
13
20
29
36
45
 
130
130
 
11
6
59
54
43
38
27
22
 
130
130
 
53
60
5
12
21
28
37
44
 
 
 
114
 
 
 
 
 
 
 
 
146
 
 
 
130
130
130
130
130
130
130
 
 
 
 
 
130
130
130
130
130
130
130
 
 
 
 
 
130
130
130
130
130
130
130
 
 
 
 
 
130
130
130
130
130
130
130
 
 
 
 
 
130
130
130
130
130
130
130
 
 
 
 
 
130
130
130
130
130
130
130
 
 
 
 
 
130
130
130
130
130
130
130
 
 
 
 
Swap in each 4x4 sub-square the yellow marked digits with the red marked digits
 
 
 
 
1
16
64
49
33
48
32
17
 
 
 
63
50
2
15
31
18
34
47
 
 
 
8
9
57
56
40
41
25
24
 
 
 
58
55
7
10
26
23
39
42
 
 
 
3
14
62
51
35
46
30
19
 
 
 
61
52
4
13
29
20
36
45
 
 
 
6
11
59
54
38
43
27
22
 
 
 
60
53
5
12
28
21
37
44
 
 
 
Franklin panmagic (= also most perfect magic) 8x8 square:
 
 
 
 
130
130
130
130
130
130
130
130
 
 
 
 
 
 
130
130
130
130
130
130
130
130
 
 
 
 
 
130
 
 
 
 
 
 
 
 
130
 
 
130
130
 
8
9
64
49
40
41
32
17
 
 
 
130
130
 
58
55
2
15
26
23
34
47
 
260
260
130
130
 
1
16
57
56
33
48
25
24
 
260
260
130
130
 
63
50
7
10
31
18
39
42
 
260
260
130
130
 
6
11
62
51
38
43
30
19
 
260
260
130
130
 
60
53
4
13
28
21
36
45
 
260
260
130
130
 
3
14
59
54
35
46
27
22
 
260
260
130
130
 
61
52
5
12
29
20